Advances and New Developments in Fuzzy Logic and Technology: Selected Papers from IWIFSGN'2019 – The Eighteenth International Workshop on October 24-25, 2019 in Warsaw, Poland 3030777154, 9783030777159

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Advances in Intelligent Systems and Computing 1308

Krassimir T. Atanassov · Vassia Atanassova · Janusz Kacprzyk · Andrzej Kałuszko · Maciej Krawczak · Jan W. Owsiński · Sotir S. Sotirov · Evdokia Sotirova · Eulalia Szmidt · Sławomir Zadrożny   Editors

Advances and New Developments in Fuzzy Logic and Technology Selected Papers from IWIFSGN’2019 – The Eighteenth International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets held on  October 24–25, 2019 in Warsaw, Poland

Advances in Intelligent Systems and Computing Volume 1308

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by DBLP, EI Compendex, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST). All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/11156

Krassimir T. Atanassov Vassia Atanassova Janusz Kacprzyk Andrzej Kałuszko Maciej Krawczak Jan W. Owsiński Sotir S. Sotirov Evdokia Sotirova Eulalia Szmidt Sławomir Zadrożny •

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Advances and New Developments in Fuzzy Logic and Technology Selected Papers from IWIFSGN’2019 – The Eighteenth International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets held on October 24–25, 2019 in Warsaw, Poland

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Editors Krassimir T. Atanassov Department of Bioinformatics and Mathematical Modelling Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences Sofia, Bulgaria

Vassia Atanassova Department of Bioinformatics and Mathematical Modelling Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences Sofia, Bulgaria

Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences Warszawa, Poland

Andrzej Kałuszko Systems Research Institute Polish Academy of Sciences Warsaw, Poland

Maciej Krawczak WIT- Warsaw School of Information Technology Warsaw, Poland

Jan W. Owsiński Systems Research Institute Polish Academy of Sciences Warsaw, Poland

Sotir S. Sotirov Intelligent Systems Laboratory Faculty of Technical Sciences Prof. Assen Zlatarov University Burgas, Bulgaria

Evdokia Sotirova Intelligent Systems Laboratory Faculty of Technical Sciences Prof. Assen Zlatarov University Burgas, Bulgaria

Eulalia Szmidt Systems Research Institute Polish Academy of Sciences Warsaw, Poland

Sławomir Zadrożny Systems Research Institute Polish Academy of Sciences Warszawa, Poland

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-3-030-77715-9 ISBN 978-3-030-77716-6 (eBook) https://doi.org/10.1007/978-3-030-77716-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume is composed of selected papers presented at IWIFSGN’2019—The Eighteenth International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets—held on October 24–25, 2019, in Warsaw, Poland, which is one of main conferences on fuzzy logic, notably on extensions of the traditional fuzzy sets, in particular on the intuitionistic fuzzy sets. A considerable part of the conference sessions is also concerned with recent developments and challenges in the theory and applications of other topics exemplified by uncertainty, incompleteness and imprecision modeling, the generalized nets (GNs), a powerful extension of the traditional Petri net paradigm, and the InterCriteria Analysis, a new method for the feature selection and analyses in multicriteria and multiattribute decision-making problems. Some more general problems of computational and artificial intelligence, exemplified by evolutionary computations, machine learning, etc. are also dealt with. The papers included yield a good perspective on all of these important issues and problems and contain contributions by well-known researchers and scholars. The volume is composed of some parts that cover the main areas and challenges related to the above general vision and rationale. Part I, “Issues in the representation, processing and analyses of uncertain information”, contains papers which concern issues and problems of a more general interest, notably related to various aspects of uncertainty and imprecision. Krassimir Atanassov (“Intuitionistic fuzzy temporal-modal operators”) presents a new type of intuitionistic fuzzy operators that are extensions similar to the intuitionistic fuzzy modal operators, as well as to the temporal operators defined over the intuitionistic fuzzy sets (IFSs), and an analysis of their properties. Some novel definition of the some specific intuitionistic fuzzy implications is given, and possible future directions are outlined. Andrzej Piegat and Marek Landowski (“In direction of intuitionistic fuzzy arithmetic”) deal with an approach to improve the current intuitionistic fuzzy arithmetic by using a new alternative definition of the type 2 fuzzy set which is defined from the point of view of fuzzy arithmetic and not control and reasoning as in the case of the traditional definition.

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Gabriela E. Martínez, Patricia Melin and Oscar Castillo (“A new approach for an intuitionistic fuzzy Sugeno integral using morphological gradient edge detector”) propose an extension to the concept of the Sugeno integral using the intuitionistic fuzzy sets. An application to the morphological gradient edge detection is presented. The approach is used as an aggregation operator to combine the four gradients of the morphological gradient edge detector and makes it possible to calculate the Sugeno integral for combining multiple sources of information with a membership degree and non-membership degree. Emphasis is on the aggregation operators that use measures with the intuitionistic fuzzy sets, in particular, the Sugeno integral. The new method is compared with the traditional Sugeno integral and other aggregation operators on the Berkeley database (BSDS) and with synthetic images. Piotr Nowak and Olgierd Hryniewicz (“M-probabilistic versions of the strong law of large numbers”) present the M-probability theory introduced for the intuitionistic fuzzy events (IFEs), defined by Atanassov’s intuitionistic fuzzy sets (IFSs). Generalized versions of the strong law of large numbers (SLLN), i.e., the Brunk–Prokhorov SLLN, the Marcinkiewicz–Zygmund SLLN, the Korchevsky SLLN, are formulated within the M-probability theory and are illustrated on an example. Katarína Čunderlíková (“Intuitionistic fuzzy probability and almost everywhere convergence”) presents a new formulation of almost everywhere convergence using an intuitionistic fuzzy probability. Two concepts of the almost everywhere convergence are compared, and an analysis of some specific almost everywhere convergence of a sequence of intuitionistic fuzzy observables induced by a Borel measurable function is provided. Alžbeta Michalíková (“Classification of images by using distance functions defined on intuitionistic fuzzy sets”) is concerned with the classification of images, to be more specific the classification of automobile tire threads into some specific classes. Tools and techniques of the intuitionistic fuzzy sets are employed. Each class is characterized by its pattern. In the pre-processing, a transformation of an image into the numeric data, represented by a vector, is done. Then, the values of the membership function, non-membership function and hesitance margin for each coordinate of the vector are calculated. Finally, the values of a distance function between the image and class patterns are computed, and the classification is performed via a special function based on the distance function. Urszula Bentkowska and Barbara Pękala (“A study on local properties and local contrast in a fuzzy setting”) are concerned with the problem of local properties and local contrast of a fuzzy relation. The importance of these two concepts which measure, in a different way, the influence of neighboring elements on the element itself is studied. Modified versions of the local properties of a fuzzy relation are proposed. Moreover, dependencies between the local properties and local contrast are examined. Adam Bzowski, Paweł M. Wójcicki, Kinga M. Wójcicka and Michał K. Urbański (“Note on Zadeh’s extension principle based on fuzzy variable approach”) are concerned with Zadeh’s extension principle, a powerful tool for extending

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relationships, functions, etc. from non-fuzzy to fuzzy variables. Specifically, a new approach is presented in which the extension principle is derived from the assumption that the underlying possibility measure is maxitive and the fuzzy sets are defined using the fuzzy variable approach. Dawid Ewald, Jacek M. Czerniak and Marcin Paprzycki (“OFNBee method applied for the solution of problems with multiple extremes”) are concerned with an algorithm whose operation is based on the behavior of bees, more specifically the most intensively developed concept is the ABC algorithm. Its main advantage is the ease of customization and a fast operation. One of the older solutions is the MBO concept. However, due to its high complexity, the algorithm had limited application. In this work, a new algorithm is presented based on the bee swarm optimization using the ordered fuzzy numbers which can improve the efficiency. Krzysztof Piasecki and Anna Łyczkowska-Hanćkowiak (“Imprecision indexes of oriented fuzzy numbers”) are concerned with the oriented fuzzy number (OFN) which are a kind of imprecise (fuzzy) numbers. Some imprecision ratings dedicated to OFNs are proposed. Imprecision is meant as the ambiguity and indistinguishability. The OFN ambiguity is evaluated by an ambiguity index defined as a generalization of an energy measure determined for the fuzzy numbers (FNs). The OFN indistinctness is evaluated by an indistinctness index defined as a generalization of the Czogała–Gottwald–Pedrycz entropy measure for the fuzzy numbers. These measures are extended to the OFNs. Some basic properties of these measures are proven. An application for financial portfolio analysis is shown. Sebastian Porębski (“Detailed evaluation of fuzzy sets in rule conditions as a key for accurate and explainable rule-based systems”) is concerned with a search for a compromise between the accuracy and interpretability of decision support systems. The most accurate of these systems are characterized by high incomprehensibility. On the other hand, rule systems are interpretable but their weakness is accuracy. The search for balanced solutions is therefore necessary. The evaluation of fuzzy sets is assumed to be a basic element of rule sets. The evaluation of fuzzy set matching to training data makes it possible to choose the best components in the beginning of rule extraction. The approach proposed results in a high accuracy and a satisfactory interpretability. Leszek Klukowski (“Tests for estimates of the tolerance relation based on the pairwise comparisons in binary and multivalent form”) presents original tests for the verification of estimates of a tolerance relation obtained on the basis of multiple independent pairwise comparisons, in the binary and multivalent form, with random errors. The estimates are obtained with the use of the nearest adjoining order (NAO), i.e., by the minimization of differences between the relation form and pairwise comparisons. The tests are based on exact or limiting distributions of the proposed statistics and do not need simulations. Part II, “Applications in healthcare, medicine, and sports”, is concerned with the use of modern methods based on the analysis of uncertain and imprecise data for the broadly perceived health care and medicine which is considered to be one of major applications of modern technologies for the society and economy.

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Minko Minkov and Evdokia Sotirova (“Opportunity for obtaining an intuitionistic fuzzy estimation for health-related quality of life data”) present a method for obtaining a degree of happiness of the quality of life of the residents of the city of Burgas, Bulgaria, using real data of 1200 persons (men and women). The analysis of health status data is very important for the identification and diagnosis of diseases. For the determination of the degree of happiness, intuitionistic fuzzy assessments (optimistic, strong optimistic, pessimistic and strong pessimistic) on the output of the multilayer perceptron are obtained. Nikolay Andreev and Vassia Atanassova (“InterCriteria Analysis of the Blood Group Distribution of Patients of Saint Anna Hospital in 2015–2019”) present some results obtained in the application of the method of InterCriteria Analysis to a dataset of 33782 patients of the Saint Anna Hospital in Sofia, Bulgaria, whose blood was tested in the period of 2015–2019. The dataset, after a cleansing and anonymization, contains data of blood groups of the tested patients, their birth years and sex, and the research is aimed at the detection of dependences and changes over time of the distribution of different blood types over the Bulgarian population represented in this sample. The results of this study can potentially be useful for the blood transfusion specialists in the country who deal with the blood transfusion capacity of the population. Evdokia Sotirova, Greta Bozova, Hristo Bozov, Sotir Sotirov and Valentin Vasilev (“Application of the InterCriteria Analysis Method to a data of Malignant Melanoma Disease for the Burgas region for 2014–2018”) analyze statistical data for the registered patients with malignant melanoma in the Burgas region for the period 2014–2018. The InterCriteria Analysis approach is applied. Relations between gender, marital status and the type of malignant melanoma, as well as relations between the year of registration of a patient and the type of malignant melanoma are studied. The results obtained by using the InterCriteria Analysis method are confirmed by statistical analyses. Krassimir Atanassov, Valentin Vasilev, Velin Andonov and Evdokia Sotirova (“A Generalized Net Model of the Abdominal Aorta and Its Branches as a Part of the Vascular System”) present a further development of the use of the generalized net (GN) for the modeling and analysis of the functioning of different systems and organs in the human body. Specifically, in this paper a generalized net model of the abdominal aorta and its branches of the vascular system is proposed. Martin Lubich, Velin Andonov, Anthony Shannon, Chavdar Slavov, Tania Pencheva and Krassimir Atanassov (“A generalized net model of the human body excretory system”) are concerned with the use of the generalized nets (GNs) that have been proved to be a successful tool for the modeling of parallel processes, notably for the modeling of different human body systems. In this work, the generalized nets are applied for the first time for a detailed description of the human body excretory system. Sotir Sotirov, Greta Bozova, Valentin Vasilev and Maciej Krawczak (“Clustering of InterCriteria Analysis data using a malignant neoplasms of the digestive organs data”) propose the use of a specific type of the artificial neural networks for the clustering of data. The aim is to obtain an aggregation of such data.

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The example considered has nine neurons and nine clusters along with the centers of gravity of these clusters. To analyze the patients with oncological diseases, registered in Burgas, Bulgaria, from 2014 to 2018, the InterCriteria Analysis (ICA) approach is applied. Tomasz Pander and Tomasz Przybyła (“Fuzzy-based algorithm for QRS detection”) are concerned the ECG signal analysis which is an important task for the modern cardiological diagnostics. The accuracy and precision of the ECG signal measurements or classification depend on the quality of the method for determining the position of QRS complexes in the electrocardiogram. A simple method of R-wave detection of the QRS complex based on an exceeding value of the amplitude threshold by the samples of the so-called detection function waveform is proposed. The study is carried out for the MIT-BIH Arrhythmia Database, Noise Stress Test Database, Fantasia Database and QT Database. In the case of QRS detection, a high performance of the new method was obtained and shown to be competitive in comparison to the reference methods. Antonio Antonov, Dafina Zoteva and Olympia Roeva (“Influence of the Indoor Hockey “Push & Flick” Methodology on the Ball Speed During the Penalty Corner Shooting”) are concerned with indoor hockey that is an official non-Olympic discipline regulated by International Hockey Federation (FIH) and practiced in a hall with a handball size pitch. The main objective of the game is to win by scoring more goals than the opponent. The penalty corner is one of the most important game situations in hockey (both outdoor and indoor field hockey) with 40% of all goals resulting from this tactical situation. This number may reach 46% or even 68%. The purpose of this paper is to study the influence of the indoor hockey’s “Push & Flick” methodology on the ball speed improvement during the penalty corner shooting in potentially effective goal zones. Using tools and techniques of both the variation analysis and the InterCriteria Analysis, it was possible to find values, and possible relations and dependencies between indicators reflecting the ball speed of zone shooting. Four elite indoor hockey players from the team of the National Sports Academy in Bulgaria, participants in the European Indoor Hockey Clubs Challenge, were involved in the experiment. The results of using the InterCriteria Analysis are shown to be promising. Simeon Ribagin and Spas Stavrev (“InterCriteria Analysis of Data Obtained from University Students Practicing Sports Activities”) propose the application of InterCriteria analysis to the analysis of data on intellectual status parameters obtained from university students practicing sports activities. Part III, “Applications in industry, business and critical infrastructure”, is concerned with very important applications in issues and problems which are crucial in the complex present world that is full of dangers, risks and decision challenges, and yet all this is to be considered and solved under an extreme uncertainty and imprecision of data, information and knowledge. Good decisions may have a huge economic and social impact. Dafina Zoteva, Olympia Roeva and Hristo Tsakov (“Forest Fire Analysis Based on InterCriteria Analysis”) are concerned with forest fires which annually affect large areas all over the world. The present research extends and builds up an

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analysis of the forest fires risk assessment in Bulgaria over the last 20 years. Two methodologies, different in their essence, are used in the study: a common approach (Lubenov’s methodology) and the use of InterCriteria Analysis (ICrA). Lubenov’s methodology classifies the different regions of Bulgaria in groups according to the risk of forest fires. The ICrA, which seeks to find relations between some predefined criteria, is used as an additional approach to refine this classification. The result shown confirms the advantages of the approach presented. Velin Andonov, Stoyan Poryazov and Emiliya Saranova (“Generalized Net Model of the Overall Telecommunication System with Queuing”) propose the use of Atanassov’s generalized nets to model an overall telecommunication system with queueing. The model is based on the classic conceptual model of a telecommunication system which takes into account the users’ behavior, a finite number of users and terminals, losses due to abandoned and interrupted dialing, blocked and interrupted switching, blocked and abandoned ringing, abandoned communication and other undesirable aspects. A queuing system with finite capacities of the server and buffer and the FIFO (first-in first-out) discipline of service of requests is included in the switching stage. The proposed model can be used in the analytical modeling of overall telecommunication systems. Veselina Bureva (“Generalized Net Model of Information Security Activities in the Automated Information Systems”) presents a generalized net model of information security activities in automated information systems. The processes of information security are investigated, and particular steps of such activities are analyzed. The constructed generalized net model provides a model of procedures of the detection of security vulnerabilities, calculation of risk of an attack, implementation of threats and execution of a security policy. Hubert Zarzycki, Wojciech T. Dobrosielski, Jacek M. Czerniak and Dawid Ewald (“Use of OFN in the short-term prediction of exchange rates”) deal with the search for short-term trends based on the exchange rates of the leading currencies. A novel method of detecting patterns in trends represented by linguistic terms deals with the linguistic variables with values given as the ordered fuzzy numbers. The solution is based on the application of fuzzification of the source data, using the transposition of the parameters (change direction, max, min, opening and closing prices) of the daily exchange rates of EUR to USD. The method proposed is a basis for the rule-based forecasting methodology for financial applications. Łukasz Apiecionek, Wojciech T. Dobrosielski and Dawid Ewald (“Ordered Fuzzy Numbers for IoT Smart Home Solution) discuss issues related to the use of the Internet of Things (IoT), which consists in connecting all possible devices to the Internet in order to provide them with new functionalities and thus to improve the user’s life standard. One of such important solutions is that of Intelligent Homes called Smart Home for which there is a possibility of monitoring inner environment which provides a potential for, e.g., a better heating control. The authors propose to use the ordered fuzzy numbers for the development and analysis of some heating and cooling method. The proposed solution is tested in a special climate chamber.

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Krzysztof Szkatuła (“When Two-Constraint Binary Knapsack Problem is equivalent to Classical Knapsack Problem?”) is concerned with the famous Knapsack problem which is a general model for a multitude of problems in many areas of science and technology, notably in operations research, logistics, management science, etc. He considers when the two-constraint 0–1 Knapsack problem is equivalent to the single-constraint classical Knapsack problem. It is assumed that some of the problem coefficients are realizations of mutually independent random variables. For the considered asymptotical random model of the problem, the case when the corresponding Lagrange multiplier is equal to zero is identified which in turn means that the corresponding constraint is redundant. We strongly believe that the high-quality, interesting and inspiring contributions, included in this volume, will be of much interest and use for a wide research community. We wish to thank the contributors for their great works, and well as other participants of the IWIFSGN’2019 conferences, for their active participation in the event, for vivid discussions, eagerness to exchange and share new ideas, and a friendly atmosphere. Special thanks are due to anonymous referees whose deep and constructive remarks and suggestions have helped to greatly improve the quality and clarity of contributions. And last but not least, we wish to thank Dr. Tom Ditzinger, Dr. Leontina di Cecco and Mr. Holger Schaepe for their dedication and help to implement and finish this important publication project on time, while maintaining the highest publication standards. Krassimir T. Atanassov Janusz Kacprzyk Andrzej Kałuszko Maciek Krawczak Jan W. Owsiński Sotir Sotirov Evdokia Sotirova Eulalia Szmidt Sławomir Zadrożny

Contents

Issues in the Representation, Processing and Analyses of Uncertain Information Intuitionistic Fuzzy Temporal-Modal Operators . . . . . . . . . . . . . . . . . . . Krassimir Atanassov

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In Direction of Intuitionistic Fuzzy Arithmetic . . . . . . . . . . . . . . . . . . . . Andrzej Piegat and Marek Landowski

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A New Approach for an Intuitionistic Fuzzy Sugeno Integral Using Morphological Gradient Edge Detector . . . . . . . . . . . . . . . . . . . . . . . . . Gabriela E. Martínez, Patricia Melin, and Oscar Castillo M-Probabilistic Versions of the Strong Law of Large Numbers . . . . . . . Piotr Nowak and Olgierd Hryniewicz

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Intuitionistic Fuzzy Probability and Almost Everywhere Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katarína Čunderlíková

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Classification of Images by Using Distance Functions Defined on Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alžbeta Michalíková

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A Study on Local Properties and Local Contrast in Fuzzy Setting . . . . Urszula Bentkowska and Barbara Pȩkala Note on the Zadeh’s Extension Principle Based on Fuzzy Variable Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adam Bzowski, Michał K. Urbański, Kinga M. Wójcicka, and Paweł M. Wójcicki OFNBee Method Applied for Solution of Problems with Multiple Extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dawid Ewald, Jacek M. Czerniak, and Marcin Paprzycki

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Imprecision Indexes of Oriented Fuzzy Numbers . . . . . . . . . . . . . . . . . . 112 Krzysztof Piasecki and Anna Łyczkowska-Hanćkowiak Detailed Evaluation of Fuzzy Sets in Rule Conditions as a Key for Accurate and Explainable Rule-Based Systems . . . . . . . . . . . . . . . . 125 Sebastian Porębski Tests for Estimates of the Tolerance Relation Based on Pairwise Comparisons in Binary and Multivalent Form . . . . . . . . . . . . . . . . . . . . 136 Leszek Klukowski Applications in Healthcare, Medicine, and Sports Opportunity for Obtaining an Intuitionistic Fuzzy Estimation for Health-Related Quality of Life Data . . . . . . . . . . . . . . . . . . . . . . . . . 151 Minko Minkov and Evdokia Sotirova InterCriteria Analysis of the Blood Group Distribution of Patients of Saint Anna Hospital in 2015–2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Nikolay Andreev and Vassia Atanassova Application of the InterCriteria Analysis Method to a Data of Malignant Melanoma Disease for the Burgas Region for 2014–2018 . . . 166 Evdokia Sotirova, Greta Bozova, Hristo Bozov, Sotir Sotirov, and Valentin Vasilev A Generalized Net Model of the Abdominal Aorta and Its Branches as a Part of the Vascular System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Krassimir Atanassov, Valentin Vasilev, Velin Andonov, and Evdokia Sotirova A Generalized Net Model of the Human Body Excretory System . . . . . 186 Martin Lubich, Velin Andonov, Anthony Shannon, Chavdar Slavov, Tania Pencheva, and Krassimir Atanassov Clustering of InterCriteria Analysis Data Using a Malignant Neoplasms of the Digestive Organs Data . . . . . . . . . . . . . . . . . . . . . . . . 193 Sotir Sotirov, Greta Bozova, Valentin Vasilev, and Maciej Krawczak Fuzzy-Based Algorithm for QRS Detection . . . . . . . . . . . . . . . . . . . . . . 202 Tomasz Pander and Tomasz Przybyła Influence of the Indoor Hockey “Push & Flick” Methodology on the Ball Speed During the Penalty Corner Shooting . . . . . . . . . . . . . 216 Antonio Antonov, Dafina Zoteva, and Olympia Roeva InterCriteria Analysis of Data Obtained from University Students Practicing Sports Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Simeon Ribagin and Spas Stavrev

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Applications in Industry, Business and Critical Infrastructure Forest Fire Analysis Based on InterCriteria Analysis . . . . . . . . . . . . . . . 241 Dafina Zoteva, Olympia Roeva, and Hristo Tsakov Generalized Net Model of Overall Telecommunication System with Queuing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Velin Andonov, Stoyan Poryazov, and Emiliya Saranova Generalized Net Model of Information Security Activities in the Automated Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . 280 Veselina Bureva Use of OFN in the Short-Term Prediction of Exchange Rates . . . . . . . . 289 Hubert Zarzycki, Wojciech T. Dobrosielski, Jacek M. Czerniak, and Dawid Ewald Ordered Fuzzy Numbers for IoT Smart Home Solution . . . . . . . . . . . . 302 Łukasz Apiecionek, Wojciech T. Dobrosielski, and Dawid Ewald When Two-Constraint Binary Knapsack Problem is Equivalent to Classical Knapsack Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Krzysztof Szkatuła Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Issues in the Representation, Processing and Analyses of Uncertain Information

Intuitionistic Fuzzy Temporal-Modal Operators Krassimir Atanassov1,2(B) 1

Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bl. 105, 1113 Sofia, Bulgaria [email protected] 2 Intelligent Systems Laboratory, Prof. Asen Zlatarov University, 8010 Bourgas, Bulgaria

Abstract. New type of intuitionistic fuzzy operators that are extensions similar to the intuitionistic fuzzy modal operators, as well as to the temporal operators defined over intuitionistic fuzzy sets (IFSs), are introduced and some of their basic properties are discussed. For the first time, the definitions of the IFS-forms of intuitionistic fuzzy implications →139 , ..., →185 are given. Some ideas for future research are described. Keywords: Intuitionistic fuzzy implication · Intuitionistic fuzzy modal operator · Intuitionistic fuzzy set · Temporal operator

1

Introduction

The intuitionistic fuzzy modal operators were introduced in the first paper on intuitionistic fuzzy sets [1] together with the very concept of an Intuitionistic Fuzzy Set (IFS). In practice, their existence proved the fact that IFSs are essential extensions of the ordinary fuzzy sets of L. Zadeh [8], because they lose their sense in the case of fuzzy sets. In [2] some temporal operators over intuitionisitc fuzzy logic are described. Here, the intuitionistic fuzzy modal operators that are analogous of the modal operators “necessity” ( ) and “possibility” (♦), will be extended to the intuitionistic fuzzy temporal-modal operators and their basic properties will be studied. We will comment on the reason for the new operators to bear this particular name.

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 3–15, 2021. https://doi.org/10.1007/978-3-030-77716-6_1

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K. Atanassov

2

Preliminary Remarks

Let E be a universe, and T = [T0 , T1 ] be a non-empty set. In the particular case, T0 = −∞ and/or T1 = +∞. We call the elements of T “time-moments”. We define a Temporal IFS (TIFS) as the following: A(T ) = {x, μA (x, t), νA (x, t)|x ∈ E & t ∈ T }, where (a) A ⊂ E is a fixed set, (b) μA (x, t), νA (x, t), μA (x, t) + νA (x, t) ∈ [0, 1] for every x, t ∈ E × T , (c) μA (x, t) and νA (x, t) are the degrees of membership and non-membership, respectively, of the element x ∈ E at the time-moment t ∈ T . Suppose that we have two TIFSs: A(T  ) = {x, μA (x, t), νA (x, t)|x ∈ E & t ∈ T  }, and

B(T  ) = {x, μB (x, t), νB (x, t)|x ∈ E & t ∈ T  },

where T  and T  have finite number of distinct time-elements or they are timeintervals. Then we can define the intuitionistic fuzzy operations ∩, ∪, etc. and the intuitionistic fuzzy modal operators ( and ♦) as follows. A(T  ) ∪ B(T  ) = {x, μA(T  )∪B(T  ) (x, t), νA(T  )∪B(T  ) (x, t)|x, t ∈ E × (T  ∪ T  )},

where x, μA(T  )∪B(T  ) (x, t), νA(T  )∪B(T  ) (x, t) ⎧ x, μA (x, t ), νA (x, t ), if t = t ∈ T  − T  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x, μB (x, t ), νA (x, t ), if t = t ∈ T  − T  ⎪ ⎪ ⎨ =

, x, max(μA (x, t ), μB (x, t )), min(νA (x, t ), νA (x, t )), ⎪ ⎪ ⎪     ⎪ if t = t = t ∈ T ∩ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x, 0, 1, otherwise

A(T  ) ∩ B(T  ) = {x, μA(T  )∩B(T  ) (x, t), νA(T  )∩B(T  ) (x, t)|x, t ∈ E × (T  ∪ T  )},

where x, μA(T  )∩B(T  ) (x, t), νA(T  )∩B(T  ) (x, t) ⎧ x, μA (x, t ), νA (x, t ), if t = t ∈ T  − T  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x, μB (x, t ), νA (x, t ), if t = t ∈ T  − T  ⎪ ⎪ ⎨ =

, x, min(μA (x, t ), μB (x, t )), max(νA (x, t ), νA (x, t )), ⎪ ⎪ ⎪     ⎪ if t = t = t ∈ T ∩ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x, 0, 1, otherwise

Intuitionistic Fuzzy Temporal-Modal Operators

5

¬A(T ) = {x, νA (x, t), μA (x, t)|x ∈ E & t ∈ T }, A(T ) = {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ T }, ♦A(T ) = {x, 1 − νA (x, t), νA (x, t)|x ∈ E & t ∈ T }. Similarly to the respective definitions for IFSs (see, e.g., [3]), the TIFS A(T ) is a tautological set if and only it (iff) A(T ) = E ∗ ≡ {x, 1, 0|x ∈ E & t ∈ T } and A(T ) is an intuitionistic fuzzy tautological set iff for each x ∈ E and t ∈ T : μA (x, t) ≥ νA (x, t). For each IFS A Nα,β (A) = {x, μA (x), νA (x)|x ∈ E & μA (x) ≥ α & νA (x) ≤ β}. This definition is valid for the TIFSs, too.

3

Definitions of the Intuitionistic Fuzzy Implications →139 , ..., →185 over Intuitionistic Fuzzy Sets

In [3], the definitions of the first 138 intuitionistic fuzzy implications over IFSs are given, while in [4], there are other 47 definitions of intuitionistic fuzzy implications, but in intuitionistic fuzzy logical form. Below, for the first time, we give their IFS-forms (Table 1). We use the following two functions: ⎧ ⎨ 1 if x > 0 sg(x) = , ⎩ 0 if x ≤ 0 ⎧ ⎨ 0 if x > 0 sg(x) =

⎩

1 if x ≤ 0

,

6

K. Atanassov

Table 1. Intuitionistic fuzzy implications →139 , ..., →185 νA (x)+μB (x) μA (x)+νB (x) , |x ∈ E} 2 2 νA (x)+μB (x)+min(νA (x),μB (x)) μA (x)+νB (x)+max(μA (x),νB (x)) ,  3 3

→139

{x,

→140

{x, |x ∈ E}

→141

A (x),μB (x)) {x, νA (x)+μB (x)+max(ν , 3 |x ∈ E}

→142

A (x),νB (x)) {x, 3−μA (x)−νB (x)−max(μ , 3 |x ∈ E}

→143

{x,

→144 →145 →146 →147 →148 →149

μA (x)+νB (x)+min(μA (x),νB (x))  3 μA (x)+νB (x)+max(μA (x),νB (x))  3

1−μA (x)+μB (x)+min(1−μA (x),μB (x)) , 3 2+μA (x)−μB (x)−min(1−μA (x),μB (x)) |x ∈ E} 3 1+νA (x)−νB (x)+min(νA (x),1−νB (x)) {x, , 3 2−νA (x)+νB (x)−min(νA (x),1−νB (x)) |x ∈ E} 3 νA (x)+μB (x)+min(νA (x),μB (x)) {x, , 3 3−νA (x)−μB (x)−min(νA (x),μB (x)) |x ∈ E} 3 A (x),νB (x)) {x, 3−μA (x)−νB (x)−min(μ , 3 μA (x)+νB (x)+min(μA (x),νB (x)) |x ∈ E} 3 A (x),μB (x)) {x, 1−μA (x)+μB (x)+max(1−μ , 3 2+μA (x)−μB (x)−max(1−μA (x),μB (x)) |x ∈ E} 3 A (x),1−νB (x)) {x, 1+νA (x)−νB (x)+max(ν , 3 2−νA (x)+νB (x)−max(νA (x),1−νB (x)) |x ∈ E} 3 νA (x)+μB (x)+max(νA (x),μB (x)) 3−νA (x)−μB (x)−max(νA (x),μB (x)) {x, ,  3 3

|x ∈ E} →150,λ

{x,

→151,γ

{x,

→152,α,β {x,

νA (x)+μB (x)+λ−1 μA (x)+νB (x)+λ−1 , |x ∈ E}, where λ ≥ 1 2λ 2λ νA (x)+μB (x)+γ μA (x)+νB (x)+γ−1 , |x ∈ E}, where γ ≥ 1 2γ+1 2γ+1 νA (x)+μB (x)+α−1 μA (x)+νB (x)+β−1 , |x ∈ E} where α ≥ 1, α+β α+β

β ∈ [1, α]

→153,ε,η {x, min(1, max(μB (x), νA (x) + ε)), max(0, min(νB (x), μA (x) − η)) |x ∈ E}, where ε, η ∈ [0, 1] and ε ≤ η < 1 →154,λ

{x,

→155,λ

{x,

→156,λ

{x,

→157,λ

{x,

→158,γ

{x,

→159,γ

{x,

→160,γ

{x,

→161,γ

{x,

→162,α,β {x,

−μA (x)+μB (x)+λ μA (x)−μB (x)+λ , |x ∈ E}, where λ ≥ 1 2λ 2λ 1−μA (x)−νB (x)+λ μA (x)+νB (x)+λ−1 , |x ∈ E}, where λ ≥ 1 2λ 2λ νA (x)+μB (x)+λ−1 1−νA (x)−μB (x)+λ , |x ∈ E}, where λ ≥ 1 2λ 2λ νA (x)−νB (x)+λ −νA (x)+νB (x)+λ , |x ∈ E}, where λ ≥ 1 2λ 2λ 1−μA (x)+μB (x)+γ μA (x)−μB (x)+γ , |x ∈ E}, where γ ≥ 1 2γ+1 2γ+1 2−μA (x)−νB (x)+γ μA (x)+νB (x)+γ−1 , |x ∈ E}, where γ ≥ 1 2γ+1 2γ+1 νA (x)−νB (x)+γ+1 −νA (x)+νB (x)+γ , |x ∈ E}, where γ ≥ 1 2γ+1 2γ+1 νA (x)+μB (x)+γ 1−νA (x)−μB (x)+γ , |x ∈ E}, where γ ≥ 1 2γ+1 2γ+1 −μA (x)+μB (x)+α μA (x)−μB (x)+β , |x ∈ E}, where α ≥ 1, β α+β α+β

∈ [1, α] (continued)

Intuitionistic Fuzzy Temporal-Modal Operators Table 1. (continued) →163,α,β {x,

1−μA (x)−νB (x)+α μA (x)+νB (x)+β−1 , |x α+β α+β

∈ E}, where α ≥ 1,

β ∈ [1, α] →164,α,β {x,

νA (x)−νB (x)+α −νA (x)+νB (x)+β , |x α+β α+β

∈ E}, where α ≥ 1,

β ∈ [1, α] →165,α,β {x,

νA (x)+μB (x)+α−1 1−νA (x)−μB (x)+β , |x α+β α+β

∈ E}, where α ≥ 1,

β ∈ [1, α] →166

{x, max(νA (x), min(μA (x), μB (x))), min(μA (x), max(νA (x), νB (x))) |x ∈ E}

→167

{x, max(1 − μA (x), min(μA (x), μB (x))), min(μA (x), 1 − min(μA (x), μB (x)))|x ∈ E}

→168

{x, max(1 − μA (x), min(μA (x), 1 − νB (x))), 1 − max(1 − μA (x), min(μA (x), 1 − νB (x)))|x ∈ E}

→169

{x, max(νA (x), min(1 − νA (x), μB (x))), 1 − max(νA (x), min(1 − νA (x), μB (x)))|x ∈ E}

→170

{x, max(νA (x), min(1 − νA (x), 1 − νB (x))), 1 − max(νA (x), min(1 − νA (x), 1 − νB (x)))|x ∈ E}

→171

{x, sg(max(μA (x), νB (x)) − max(νA (x), μB (x))), sg(max(μA (x), νB (x)) − max(νA (x), μB (x)))|x ∈ E}

→172

{x, sg(μA (x) − μB (x)), sg(μA (x) − μB (x))|x ∈ E}

→173

{x, sg(μA (x) + νB (x) − 1), sg(μA (x) + νB (x) − 1)|x ∈ E}

→174

{x, sg(1 − νA (x) − μB (x)), sg(1 − νA (x) − μB (x))|x ∈ E}

→175

{x, sg(νB (x) − νA (x)), sg(νB (x) − νA (x))|x ∈ E}

→176

{x, sg(μA (x) − μB (x)) + sg(μA (x) − μB (x)) max(νA (x), μB (x)), sg(μA (x) − μB (x)) min(μA (x), νB (x))|x ∈ E}

→177

{x, sg(μA (x) − μB (x)) + sg(μA (x) − μB (x)) max(1 − μA (x), μB (x)), sg(μA (x) − μB (x)) min(μA (x), 1 − μB (x))|x ∈ E}

→178

{x, sg(μA (x) − 1 + νB (x)) + sg(μA (x) − 1 + νB (x)) .(1 − min(μA (x), νB (x))), sg(μA (x) − 1 + νB (x)) min(μA (x), νB (x))|x ∈ E}

→179

{x, sg(1 − νA (x) − μB (x)) + sg(1 − νA (x) − μB (x)) max(νA (x), μB (x)), sg(1 − νA (x) − μB (x))(1 − max(νA (x), μB (x)))|x ∈ E}

→180

{x, sg(νB (x) − νA (x)) + sg(νB (x) − νA (x)) max(νA (x), 1 − νB (x)), sg(νB (x) − νA (x)) min(1 − νA (x), νB (x))|x ∈ E}

→181

{x, 1 − sg(μA (x)).(1 − μB (x)), νB (x).sg(μA (x))|x ∈ E}

→182

{x, 1 − sg(μA (x)).(1 − μB (x)), (1 − μB (x)).sg(μA (x))|x ∈ E}

→183

{x, 1 − sg(μA (x)).νB (x), νB (x).sg(μA (x))|x ∈ E}

→184

{x, 1 − sg(1 − νA (x)).νB (x), νB (x).sg(1 − νA (x))|x ∈ E}

→185

{x, 1 − sg(1 − νA (x)).(1 − μB (x)), (1 − μB (x)).sg(1 − νA (x)) |x ∈ E}

7

8

4

K. Atanassov

Intuitionistic Fuzzy Temporal-Modal Operators “Necessity” and “Possibility”

Let T = [T0 , T1 ] be a well-ordered set of time-moments, where T0 < T1 . Let us define Tτ = {t|t ∈ T & t ≥ τ }, T τ = {t|t ∈ T & t ≤ τ }. Therefore,

Tτ ∪∗ T τ = T, Tτ ∩∗ T τ = {τ },

where ∩∗ and ∪∗ are the set-theoretical operations “union” and “intersection”. Let A(T ) be a TIFS and let τ ∈ T be fixed. We define: τ A(T ) τ

= {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ },

A(T ) = {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ T τ },

♦τ A(T ) = {x, 1 − νA (x, t), νA (x, t)|x ∈ E & t ∈ Tτ }, ♦τ A(T ) = {x, 1 − νA (x, t), νA (x, t)|x ∈ E & t ∈ T τ }. Obviously, for each TIFS A(T ) and for τ ∈ T : τ A(T ) τ

=

A(Tτ ),

A(T ) =

A(T τ ),

♦τ A(T ) = ♦A(Tτ ), ♦τ A(T ) = ♦A(T τ ). Theorem 1. Let A(T ) and B(T ) be two TIFSs and let τ ∈ T be fixed. Then (a) (b) (c) (d) (e) (f )

τ (A(T ) τ

(A(T ) ∩ B(T )) =

τ (A(T ) τ

∩ B(T )) =

∪ B(T )) =

(A(T ) ∪ B(T )) =

τ (A(T )@B(T )) τ

=

(A(T )@B(T )) =

τ A(T ) τ

τ B(T ), τ

A(T ) ∩

τ A(T ) τ

∩

∪

B(T ),

τ B(T ), τ

A(T ) ∪

B(T ),

τ A(T )@

τ B(T ),

τ

τ

A(T )@

B(T ),

(g) ♦τ (A(T ) ∩ B(T )) = ♦τ A(T ) ∩ ♦τ B(T ),

Intuitionistic Fuzzy Temporal-Modal Operators

9

(h) ♦τ (A(T ) ∩ B(T )) = ♦τ A(T ) ∩ ♦τ B(T ), (i) ♦τ (A(T ) ∪ B(T )) = ♦τ A(T ) ∪ ♦τ B(T ), (j) ♦τ (A(T ) ∪ B(T )) = ♦τ A(T ) ∪ ♦τ B(T ), (k) ♦τ (A(T )@B(T )) = ♦τ A(T )@♦τ B(T ), (l) ♦τ (A(T )@B(T )) = ♦τ A(T )@♦τ B(T ). Proof. We will only prove (a) since the other equalities are proved in the same manner. τ (A(T )

∩ B(T ))

τ ({x, μA (x, t), νA (x, t)|x

=

∈ E & t ∈ Tτ }

∩{x, μB (x, t), νB (x, t)|x ∈ E & t ∈ Tτ }) = τ {x, min(μA (x, t), μB (x, t)), max(νA (x, t), νB (x, t))|x ∈ E & t ∈ Tτ } = {x, min(μA (x, t), μB (x, t)), 1 − min(μA (x, t), μB (x, t))|x ∈ E & t ∈ Tτ } = {x, min(μA (x, t), μB (x, t)), max(1 − μA (x, t), 1 − μB (x, t))|x ∈ E & t ∈ Tτ } = {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ } ∩{x, μB (x, t), 1 − μB (x, t)|x ∈ E & t ∈ Tτ } τ A(T )

=

∩

τ B(T ).



Lemma. Let τ ∈ T be a fixed time-moment. Then (a) If t0 ≥ τ . Then τ A({t0 })

(b) If t0 ≤ τ . Then

τ

=

A({t0 }),

A({t0 }) =

A({t0 }),

Proof. For (a) we obtain: τ A({t0 })

=

τ {x, μA (x, t), νA (x, t)|x

∈ E & t ∈ T τ & t = t0 }

= {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ & t = t0 } = {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t = t0 } = {x, μA (x, t0 ), 1 − μA (x, t0 )|x ∈ E} =

A({t0 }). 

(b) is proved analogously. Corollary. Let τ ∈ T be a fixed time-moment. Then τ A({τ })

=

A({τ }) =

τ

A({τ }).

10

K. Atanassov

Theorem 2. Let A(T ) be a TIFS and let τ ∈ T be fixed. Then (a)

τ

τ A(T ) τ

=

A(T ) =

τ A(T ),

A({τ }),

(b)

τ

(c)

τ ♦τ A(T )

= ♦τ A(T ),

(d)

τ τ ♦ A(T )

= ♦A({τ }),

(e) (f ) (g) (h)

τ τ τ τ

(i) ♦τ (j) ♦τ

τ A(T ) τ

=

A(T ) =

A({τ }), τ

A(T ),

♦τ A(T ) = ♦A({τ }), ♦τ A(T ) = ♦τ A(T ), τ A(T ) τ

=

A(T ) =

τ A(T ),

A({τ }),

(k) ♦τ ♦τ A(T ) = ♦τ A(T ), (l) ♦τ ♦τ A(T ) = ♦A({τ }), (m) ♦τ (n) ♦τ

τ A(T ) τ

=

A(T ) =

A({τ }), τ

A(T ),

(o) ♦τ ♦τ A(T ) = ♦A({τ }), (p) ♦τ ♦τ A(T ) = ♦τ A(T ). Proof. For the first equality, we obtain: τ

τ A(T )

= τ {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ } = {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ } =

τ A(T ).

The rest of the equalities are proved analogously. Theorem 3. Let A(T ) be a TIFS and let τ ∈ T be fixed. Then, τ A(T ) τ

⊆ A(Tτ ) ⊆ ♦τ A(T ),

A(T ) ⊆ A(T τ ) ⊆ ♦τ A(T ).



Intuitionistic Fuzzy Temporal-Modal Operators

11

Theorem 4. Let A(T ) be a TIFS and let τ ∈ T be fixed. Then, (a) ¬

τ (¬A(Tτ ))

(b) ¬

τ (¬A(T τ

(c) ¬ (d) ¬

τ

= ♦τ A(Tτ ),

)) = ♦A({τ }),

(¬A(Tτ )) = ♦A({τ }),

τ

(¬A(T τ )) = ♦τ A(T τ ),

(e) ¬♦τ (¬A(Tτ )) =

τ A(Tτ ),

(f ) ¬♦τ (¬A(T τ )) =

A({τ }),

(g) ¬♦τ (¬A(Tτ )) =

A({τ }), τ

(h) ¬♦τ (¬A(T τ )) =

A(T τ ),

Proof. For the first equality we obtain: ¬

τ (¬A(Tτ ))

=¬

=¬

τ (¬{x, μA (x, t), νA (x, t)|x

τ {x, νA (x, t), μA (x, t)|x

∈ E & t ∈ Tτ })

∈ E & t ∈ T & t ≥ τ}

= ¬{x, νA (x, t), 1 − νA (x, t)|x ∈ E & t ∈ T & t ≥ τ } = ¬{x, νA (x, t), 1 − νA (x, t)|x ∈ E & t ∈ Tτ } = {x, 1 − νA (x, t), νA (x, t)|x ∈ E & t ∈ Tτ } = ♦τ A(Tτ ). For the second equality we obtain: ¬

τ (¬A(T

τ

)) = ¬

τ (¬{x, μA (x, t), νA (x, t)|x

∈ E & t ∈ T τ })

= ¬ τ {x, νA (x, t), μA (x, t)|x ∈ E & t ∈ T & t ≤ τ } = ¬{x, νA (x, t), 1 − νA (x, t)|x ∈ E & t ∈ T & t ≤ τ & t ≥ τ } = ¬{x, νA (x, t), 1 − νA (x, t)|x ∈ E & t ∈ T & t = τ }) = {x, 1 − νA (x, t), νA (x, t)|x ∈ E & t ∈ T & t = τ }) = {x, 1 − νA (x, τ ), νA (x, τ )|x ∈ E}) = ♦A({τ }). 

The rest of the equalities are proved analogously. Theorem 5. Formulas τ (A(T )

and

τ

→i B(T )) →i (

(A(T ) →i B(T )) →i (

τ A(T ) τ

→i

A(T ) →i

τ B(T )) τ

B(T ))

are tautological sets for i = 2, 3, 5, 8, 11, 14, 20, 24, 25, 27, 29, 47, 48, 49, 52, 55, 57, 58, 60, 69, 77, 79, 81, 92, 93, 97, 99, 177, 179, 181, 182, 184.

12

K. Atanassov

Proof. Let A(T ) and B(T ) be TIFSs and τ ∈ T . Then, for example, for the intuitionistic fuzzy implication →2 , that is defined by A →2 B = {x, sg(μA (x) − μB (x)), νB (x).sg(μA (x) − μB (x))|x ∈ E} holds τ (A(T )

→2 B(T )) →i (

τ A(T )

τ ({x, μA (x, t), νA (x, t)|x

=

→i

τ B(T ))

∈ E & t ∈ T}

→2 {x, μB (x, t), νB (x, t)|x ∈ E & t ∈ T }) τ {x, μA (x, t), νA (x, t)|x

→2 (

τ {x, μA (x, t), νA (x, t)|x

→2 =

τ {x, sg(μA (x, t)

∈ E & t ∈ T}

∈ E & t ∈ T })

− μB (x, t)), νB (x, t).sg(μA (x, t) − μB (x, t))|x ∈ E & t ∈ T }

→2 ({x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ } →2 {x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ Tτ }) = {x, sg(μA (x, t) − μB (x, t)), 1 − sg(μA (x, t) − μB (x, t))|x ∈ E & t ∈ Tτ } →2 {x, sg(μA (x, t) − μB (x, t)), (1 − μA (x, t)).sg(μA (x, t) − μB (x, tt)) |x ∈ E & t ∈ Tτ } = {x, sg(sg(μA (x, t) − μB (x, t)) − sg(μA (x, t) − μB (x, t))), (1 − μA (x, t)).sg(μA (x, t) − μB (x, t)).sg(sg(μA (x, t) − μB (x, t)) −sg(μA (x, t) − μB (x, t)))|x ∈ E & t ∈ Tτ } = {x, sg(0), (1 − μA (x, t)).sg(μA (x, t) − μB (x, t)).sg(0)|x ∈ E & t ∈ Tτ } = {x, 1, 0|x ∈ E & t ∈ Tτ }

that is a tautological set. For the other implications, the checks are similar.



Theorem 6. Formulas τ (A(T )

and

τ

→i B(T )) →i (

(A(T ) →i B(T )) →i (

τ A(T ) τ

→i

A(T ) →i

τ B(T )) τ

B(T ))

are IFTSs for i = 1, ..., 9, 11, ..., 14, 17, 18, 20, 21, 24, 25, 27, ..., 29, 46, ..., 53, 55, 57, ..., 61, 64, 66, 69, 71, 72, 75, ..., 77, 79, ..., 81, 91, ..., 94, 97, ..., 102, 108, ..., 113, 118, ..., 120, 128, 134, ..., 137, 139, 141, 147, 149, ..., 154, 156, 158, ..., 162, 165, ..., 167, 169, 177, 179, 181, 182, 184. Proof. Let A(T ) and B(T ) be TIFSs and τ ∈ T . Then, for example, for the intuitionistic fuzzy implication →4 , that is defined by A →4 B = {x, max(νA (x), μB (x)), min(μA (x), νB (x))|x ∈ E} holds X≡

τ

(A(T ) →4 B(T )) →4 (

τ

A(T ) →4

τ

B(T ))

Intuitionistic Fuzzy Temporal-Modal Operators τ

=

13

({x, μA (x, t), νA (x, t)|x ∈ E & t ∈ T }

→4 {x, μB (x, t), νB (x, t)|x ∈ E & t ∈ T }) τ

→4 ( →4 τ

=

τ

{x, μA (x, t), νA (x, t)|x ∈ E & t ∈ T }

{x, μB (x, t), νB (x, t)|x ∈ E & t ∈ T })

{x, max(νA (x, t), μB (x, t)), min(μA (x, t), νB (x, t))|x ∈ E & t ∈ T } →4 ({x, μA (x, t), 1 − μA (x, t)|x ∈ E & t ∈ T τ } →4 {x, μB (x, t), 1 − μB (x, t)|x ∈ E & t ∈ T τ })

= {x, max(νA (x, t), μB (x, t)), 1 − max(νA (x, t), μB (x, t))|x ∈ E & t ∈ T τ } →4 {x, max(1 − μA (x, t), μB (x, t)), min(μA (x, t), 1 − μB (x, t))|x ∈ E & t ∈ T τ } = {x, max(1 − max(νA (x, t), μB (x, t)), 1 − μA (x, t), μB (x, t)), min(max(νA (x, t), μB (x, t)), μA (x, t), 1 − μB (x, t))|x ∈ E & t ∈ T τ }.

If νA (x, t) ≥ μB (x, t). Then max(1 − max(νA (x, t), μB (x, t)), 1 − μA (x, t), μB (x, t)) − min(max(νA (x, t), μB (x, t)), μA (x, t), 1 − μB (x, t)) = max(1 − νA (x, t), 1 − μA (x, t), μB (x, t)) − min(νA (x, t), μA (x, t), 1 − μB (x, t)) ≥ 1 − νA (x, t) − νA (x, t) ≥ 0. If νA (x, t) ≤ μB (x, t). Then max(1 − max(νA (x, t), μB (x, t)), 1 − μA (x, t), μB (x, t)) − min(max(νA (x, t), μB (x, t)), μA (x, t), 1 − μB (x, t)) = max(1 − μB (x, t), 1 − μA (x, t), μB (x, t)) − min(μB (x, t), μA (x, t), 1 − μB (x, t)) ≥ μB (x, t) − μB (x, t) ≥ 0. Therefore, in both cases X is an IFTS. The proofs for the other implications, as well as the proofs of the next two theorems are similar. 

Theorem 7. Formulas and

τ A(T ) τ

→i A(T )

A(T ) →i A(T )

are tautological sets for i = 2, 3, 5, 8, 11, 14, 15, 20, 23, 24, 27, 31, 32, 34, 37, 40, 42, 47, ..., 49, 52, 55, 57, 62, 63, 65, 68, 69, 74, 77, 79, 83, 84, 88, 92, 93, 97, 176, ..., 185. Theorem 8. Formulas and

τ A(T ) τ

→i A(T )

A(T ) →i A(T )

are IFTSs for i = 1, ..., 9, 11, ..., 15, 17, 18, 20, ..., 24, 27, ..., 38, 40, 42, 44, ..., 53, 55, 57, 59, 66, 68, 69, 71, 72, 74, ..., 77, 79, ..., 85, 88, ..., 94, 97, ..., 139, 141, 146, ..., 170, 176, ..., 185.

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5

Conclusion

Now, we must answer the question “why the new operators were called intuitionistic fuzzy temporal-modal operators”. First, of course, they are intuitionistic fuzzy operators. Second, they are modal operators, because they satisfy Theorems 2, 3, ..., 8, that are analogous to the modal axioms (see, e.g., [5,7]). Let A(T )(x) = {x, μA (x, t), νA (x, t)| t ∈ T }. Therefore, A(T ) =

 x∈E

A(T )(x).

Let |X| be the cardinality of set X. Third, the operators are temporal ones by the following reason. Let τ ∈ T be the present time-moment. Let us assume that the object x is valid in time-moment t if μA (x, t) ≥ α and νA (x, t) ≤ β, where α, β, α+β ∈ [0, 1] are fixed constants. In [6], the following temporal operators are discussed that can be interpreted by the new operators. never has been x iff |Nα,β (A(T τ )(x))| = 0, sometimes has been x iff 0 < |Nα,β (A(T τ )(x))| < |A(T τ )(x)|, always has been x iff 0 < |Nα,β (A(T τ )(x))| = |A(T τ )(x)|, never will be x iff |Nα,β (A(Tτ )(x))| = 0, sometimes will be x iff 0 < |Nα,β (A(Tτ )(x))| < |A(Tτ )(x)|, always will be x iff 0 < |Nα,β (A(Tτ )(x))| = |A(Tτ )(x)|. In future, the two intuitionistic fuzzy temporal-modal operators will be extended by analogy to the extensions of the intuitionistic fuzzy modal operators. Acknowledgements. The research was supported by the National Scientific Fund of Bulgaria under the Grants DN02/10 “New Instruments for Knowledge Discovery from Data, and their Modelling”.

Intuitionistic Fuzzy Temporal-Modal Operators

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References 1. Atanassov, K.: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci. - Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulg.) 2. Atanassov K.: Remark on a temporal intuitionistic fuzzy logic. In: Second Scientific Session of the “Mathematical Foundation Artificial Intelligence” Seminar, Sofia, 30 March 1990, Preprint IM-MFAIS-1-90, Sofia pp. 1–5. (1990), Reprinted: Int. J. Bioautomation, 20(S1), S63–S68 (2016) 3. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012). https:// doi.org/10.1007/978-3-642-29127-2 4. Atanassov, K.: Intuitionistic Fuzzy Logics. Springer, Cham (2017). https://doi.org/ 10.1007/978-3-642-29127-2 5. Feys, R.: Modal logics. Gauthier-Villars, Paris (1965) 6. Karavaev, E.: Foundations of Temporal Logic. Leningrad University Press, Leningrad (1983). (in Russian) 7. Lewis, C.: A Survey of Symbolic Logic. University of California Press, Berkeley (1918) 8. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

In Direction of Intuitionistic Fuzzy Arithmetic Andrzej Piegat1 and Marek Landowski2(B) 1

Faculty of Computer Science, West Pomeranian University of Technology, Zolnierska 49, 71-210 Szczecin, Poland [email protected] 2 Faculty of Computer Science and Telecommunications, Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500 Szczecin, Poland [email protected] Abstract. The article presents some opportunity to improve the current intuitionistic fuzzy arithmetic. This possibility consists in using the new alternative definition of the fuzzy set type 2. The current definition of this set is rather suitable for control and fuzzy reasoning. However, the new definition was developed mainly for fuzzy arithmetic. The article presents this definition and shows how to use it in the arithmetic of intuitionistic fuzzy numbers.

1

Introduction

The task of intuitionistic fuzzy arithmetic (IntF-arithmetic) is to implement basic arithmetic operations such as addition, subtraction, multiplication, division of intuitionistic fuzzy numbers (IntFNs). However, this arithmetic also is used to solve various types of mathematical problems of a more complex nature, e.g. solving simple equations AI NT + X = B I NT (AI NT , B I NT are IntFNs, X-solution of the equation), solving systems of linear equations as A1I NT x1 + B1I NT x2 = C1I NT , A2I NT x1 + B2I NT x2 = C2I NT , and other difficult problems. There are currently several versions of IntF-arithmetic. Examples are the versions presented in [4,5]. According to the authors’ opinion, the current IntF-arithmetic is not perfect. It solves some problems, usually easier ones, exactly. Some problems are solved more or less accurately (approximately) and some problems cannot be solved at all. This means that IntF-arithmetic requires improvement. How can this arithmetic be improved? In the authors’ opinion, one way to improve it is to introduce a new alternative definition of type 2 fuzzy set (T2FS). This statement may seem surprising, therefore it requires clarification. Intuitionistic fuzzy sets can be interpreted as type 2 fuzzy sets. This statement can be found both in the books of Professor Atanassov, the creator of the concept of IntFSs [1], and in the works of other scientists [3,8].

2

Intuitionistic Fuzzy Set and Type 2 Fuzzy Set

The possibility of transforming IntFS into T2FS enables the implementation of IntF-arithmetic operations using T2F-arithmetic. Here, however, we face similar c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 16–25, 2021. https://doi.org/10.1007/978-3-030-77716-6_2

In Direction of Intuitionistic Fuzzy Arithmetic

17

difficulties as in the case of IntF-arithmetic, because the current T2F-arithmetic is also not perfect. Figure 1 illustrates the transformation of IntFs AI N into ˜ T2FS A.

Fig. 1. Visualization of intuitionistic FS AI N (a) and the corresponding T2FS A˜ represented in the form of a footprint (FOU), where μ A - membership function, ω A confidence degree in expert opinion regarding membership, ν A - non-membership function, (1− δ A) - confidence degree in the expert opinion on non-membership, μ A + ν A ≤ 1, μ L A˜ = μ A (lower T2MF), μU A˜ = (1 − ν A) (upper T2MF).

In practice, the standard, interval type 2 fuzzy arithmetic (IT2F-arithmetic) is most commonly used [2]. This arithmetic also only solves some problems thoroughly, others approximately, and some problems cannot be solved at all. So, the current T2F-arithmetic also requires improvement. Improvement of T2Farithmetic will automatically trigger improvement of IntF-arithmetic. But how can you improve T2F-arithmetic? This can be done by introducing into T2Farithmetic a new definition of T2FS, a definition that would be more appropriate for this arithmetic than the current definition. In the authors’ opinion, the reason for the imperfection of the present T2F-arithmetic is that this arithmetic makes calculations only with borders of T2 fuzzy numbers (T2FNs) and not with their insides (with full bodies). This is because the arithmetic is based on the present “border” definition of the T2Fs. Let us recall this present definition, formula (1), cited from [7], (1) A˜ = {((x, u), μ A˜(x, u))|x ∈ X, u ∈ [0, 1]} where x is primary variable and u-secondary variable (uncertain primary membership). The very concept of T2Fs was introduced by the Father of FSs Lotfi Zadeh in [14]. An important concept of the present definition of T2FS is the ˜ allowing 2D-visualization of this set, (2). footprint of T2FS A˜ (FOU A) ˜ = {(x, u)|x ∈ X and u ∈ [μ (x), μ ˜(x)]} FOU( A) A ˜ A

(2)

˜ In formula (2) μ ˜(x) means the lower and μ A˜(x) upper MF of the set A. A Figure 2a shows 3D-visualization of the triangular T2FS and Fig. 2b footprint of the T2FS shown in Fig. 2a.

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Fig. 2. A T2FS A˜ on 3D space according to the border definition (1), (a). Footprint of ˜ (b). the triangular T2FS A,

In Fig. 2 it is easy to see that some pairs (x, u) do not belong to the T2FS at all because they lie inside of this set. Thus, definition (1) defines only the borders of a T2FS. This is a direct cause of its imperfections. This does not mean, however, that the current definition (1) of T2FS is not useful. Indeed, its usefulness has been demonstrated for fuzzy control, reasoning and modeling [7]. However, its usefulness for T2F-arithmetic is limited. The motive for developing the new, non-border definition of a T2FS was the observation that the current border definition of T2FS is an analogy to the first definition of T1FS of Zadeh and that he developed and published not one but two definitions of T1FS. The first and best known definition of T1FS was published by Zadeh in [13]. This definition, which can be called “border” definition, is given by (3). A = {(x, μ A(x))|x ∈ X }, 0 ≤ μ A(x) ≤ 1

(3)

Figure 3 shows a visualization of an exemplary triangular T1FS.

Fig. 3. Triangular type 1 fuzzy set, μ L (x), μ R (x) - left and right borders of the membership function.

Formula (4) defines the left and right borders of the triangular T1FS.  μ L (x) = (x − a)/(b − a) for x ∈ [a, b] μ(x) = μR (x) = (c − x)/(c − b) for x ∈ (b, c]

(4)

In Direction of Intuitionistic Fuzzy Arithmetic

19

Formula (4) defines only the borders of T1Fs. It does not define the inside, the body of this set. In 1975, L. Zadeh introduced the second definition of T1FS in [14]. According to this definition, T1FS A is the sum (set) of Aα cuts made at different levels μ = α. A single α-cut Aα is given by formula (5). Aα = {x| μ A(x) ≥ α}

(5)

The set T1FS A can be approximately defined in the discrete (6) version or exactly (continuous definition) by the formula (7).  αAα (6) A= α

∫1 A=

αAα

(7)

0

Both versions of the compositional definition of T1FS are shown in Fig. 4.

Fig. 4. Discrete, approximate (a) and continuous, accurate (b) composition of T1FS A from horizontal α-cuts Aα : x ∈ [x Lα, x Rα ].

Figure 5 shows a comparison of Zadeh’s border and body definition of T1FS.

Fig. 5. Illustration of the sense of borders and the corpus of a T1FS.

20

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A. Piegat and M. Landowski

Alternative Definition of the Fuzzy Set Type 2

To specify a single x value included in the Aα set, one needs to use the term Relative-Distance-Measure (RDM), [10], which here will be marked with the symbol γ, γ ∈ [0, 1]. On the basis of RDM-notion a multi-dimensional RDMT1F-arithmetic [10–12] has been developed. It has been described in over 40 papers. Examples of its applications are given in [6,9]. As its value increases, the left border x Lα is transited to the right border xRα , Fig. 5. The model specifying any x value is given by the formula (8). It is also a mathematical model of an horizontal α-cut of the T1FS-body. Aα : x = xα + γ(xRα − x Lα ), γ ∈ [0, 1]

(8)

The model of the whole body of a T1FS A is expressed by the formula (9). A : x(μ, γ) = x L (μ) + γ[xR (μ) − x L (μ)], μ, γ ∈ [0, 1]

(9)

In (9) x L (μ) and xR (μ) are inverse mathematical models of the left and right borders of MFs μ L (x) and μR (x), Fig. 5. For the triangular T1MF case shown in Fig. 5, the model of its body is given by formula (10). A : x(μ, γ) = [a + μ(b − a)] + γ{[c − μ(c − b)] − [a − μ(b − a)]}

(10)

For a = 2, b = 3, c = 5, Fig. 5, formula (10) takes a less complex form (11). A : x(μ, γ) = (4 + μ) + γ(2 − μ), μ, γ ∈ [0, 1]}

(11)

Understanding the mathematical model of the T1FS-body (9) and (10) allows for easier understanding the new definition of the fuzzy set type 2 (T2FS). ˜ Figure 6 shows the footprint of a T2FS A.

˜ Fig. 6. Footprint FOU A˜ of a T2FS A.

The fuzzy set type 2 is defined by Zadeh in 3D space. The lower MF, Fig. 6, ˜ i.e. as the MF with the greatest confidence. can be interpreted as core-MF( A), ˜ ˜ in analogy In contrast, the upper MF( A) can be interpreted as support-MF( A) to Zadeh’s names of α-cuts of T1MF, as shown in Fig. 7.

In Direction of Intuitionistic Fuzzy Arithmetic

21

˜ in the case of T1FS. Fig. 7. Core and support of MF( A)

In T1FS, support of A is a set of all values of the variable x satisfying the μ A > 0 condition, i.e. of those values of x that have, even to a minimal degree, ˜ support of MF( A) ˜ the characteristic feature of the set A. In the case of T2FS A, contains all possible (even minimally possible) membership functions T1MFs of A. In the case of T1FS A, the core of MF(A) contains all the values of the x ˜ the variable that fully have the feature of the set A. In the case of T2FS A, ˜ can be interpreted as the one of all possible MFs of A˜ that has the core-MF( A) ˜ and corehighest confidence of the problem experts. Between support-MF( A) ˜ there is an infinite number of possible T1MFs with increasing confidence MF( A) ˜ can be called transient MFs ( A), ˜ because c, Fig. 7. These intermediate T1MFs ( A) ˜ in core MF ( A), ˜ as shown in the variable c ∈ [0, 1] transitions support MF( A) Fig. 8.

˜ μ ˜(x, c) of a T2FS A˜ according to the new Fig. 8. Example of a 3-dimensional MF( A) A ∫1 definition of the T2FS-body ( A˜ = c A˜αc ). 0

˜ ˜ Transition of the MF( A)-support in MF( A)-core can be linear or nonlinear. What transition should be used in the specific case under study can be identified based on the distribution of expert evaluations. However, this is a problem for a separate article and will not be considered here. Linear transition (linear trans˜ ˜ formation of the MF ( A)-support into MF( A)-core) can be used as a basic and

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A. Piegat and M. Landowski

practical type of transitions. Similarly to T1FSs, Fig. 6, where most often linear transitions of the support in the core are performed. It will still be explained how you can develop the mathematical model x A˜(c, γ, μ) of the horizontal T2FS A˜ based on the new definition of T2FS presented earlier. Based on this defini˜ μ ˜(x, c, γ) shown in Fig. 8 can be developed. Vertical tion, also a vertical MF( A) A ˜ MF( A) is useful in fuzzy reasoning and control. Mathematically, it describes only ˜ The horizontal MF( A) ˜ defines the whole body of A, ˜ Fig. 9. the outer shell of A. It is very useful in fuzzy arithmetic, where calculations are made primarily on the values of variables x, y, z, . . . etc.

˜ Fig. 9. Illustration of the concept of outer shell of A˜ associated with vertical MF( A) ˜ and body of A˜ associated with horizontal MF ( A).

˜ x ˜(μ, c, γ), it is necessary In order to derive the formula for horizontal MF( A) A to enter the denotations shown in Fig. 10.

˜ achieved by transition of the support-MF( A) ˜ (sector Fig. 10. Horizontal T2MF( A) ˜ (sector (P5 P6 P7 P8 )) with use of RDM-variable c. (P1 P2 P3 P4 )) into the core MF( A),

˜ The horizontal mathematical model of the MF( A)-support (sector (P1 P2 P3 P4 )) is given by the formula (12).

In Direction of Intuitionistic Fuzzy Arithmetic

x A˜(c = 0, γ, μ) = x P 1 P 2 P 3 P 4 = x P 1 P 2 + γ(x P 3 P 4 − x P 1 P 2 ) = [a1 + μ(a2 − a1 )] + γ{[a4 − μ(a4 − a3 )] − [a1 + μ(a2 − a1 )]} = (1 + μ) + γ[(9 − 2μ) − (1 + μ)], μ, γ ∈ [0, 1]

23

(12)

˜ concerning the sector The horizontal mathematical model of the core-MF( A) (P5 P6 P7 P8 ) is given by the formula (13). x A˜(c = 1, γ, μ) = x P 5 P 6 P 7 P 8 = x P 5 P 6 + γ(x P 7 P 8 − x P 5 P 6 ) = [b1 + μ(b2 − b1 )] + γ{[b3 + μ(b4 − b3 )] − [b1 + μ(b2 − b1 )]} = (3 + μ) + γ[(8 − 2μ) − (3 + μ), μ, γ ∈ [0, 1]

(13)

˜ is obtained by transiting of the supportThe complete horizontal T2MF( A) ˜ ˜ MF( A) into core-MF( A) using the transition variable c ∈ [0, 1], formula (14). x A˜(c, γ, μ) = x P 1 P 2 P 3 P 4 + c(x P 5 P 6 P 7 P 8 − x P 1 P 2 P 3 P 4 ) = x A˜(c = 0, γ, μ) + c[x A˜(c = 1, γ, μ) − x A˜(c = 0, γ, μ)], c, μ, γ ∈ [0, 1]

(14)

In this case, formulas (12), (13), (14) define T2FN with trapezoidal supportand core-MFs. These formulas can also be used for triangular or rectangular MFs, which are special cases of trapezoidal MFs. How can the new definition of T2FS be applied to intuitionistic fuzzy numbers? The description of this arithmetic is a bit complicated and requires a separate article. Such an article will soon be prepared by the authors. However, in this article general guidelines and description of calculation steps will be presented on the example of solving the equation AI N + X = BI N , where AI N , BI N are intuitionistic FNs and X is the solution sought for the equation, Fig. 11.

Fig. 11. Input data in form of intuitionistic FNs AI N , BI N in equation AI N + X = BI N .

Step 1. Transform the intuitionistic FNs in footprints (FOUs) of T2FNs, Fig. 12. ˜ in 3D T2FNs Step 2. Transformation of footprints (FOU A˜ and FOU B) according to the new definition of the T2FS-body, Fig. 13. Step 3. Solving the equation A˜ + X = B˜ according to the rules of multidimensional RDM-fuzzy arithmetic. Important note! The algebraic and universal solution of the analyzed equa˜ B. ˜ It is the fuzzy tion A˜ + X = B˜ is not the same 3D-fuzzy number as A, number x(c, μ, γ A, γB ) existing in the dimension higher than 3D. Because this algebraic solution cannot be visualized, you need to provide some simplified,

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Fig. 12. FOUs of A˜ and B˜ achieved in the result of transformation of intuitionistic FNs in FOUs according to Step 1.

Fig. 13. Bodies of T2FNs A˜ and B˜ obtained using the new definition of T2FS in the ˜ equation A˜ + X = B.

low-dimensional information that would be understandable to people. For this purpose, you can specify 3D-span sx (c, μ, γx ) of this solution. This span has a geometric form similar to the 3D FNs shown in Fig. 13. And has support-MF and core-MF. The span sx is not, however, an algebraic, universal solution X of the equation under study. Step 4. Determine the footprint FOU(X) based on the sx of multidimensional result X. Step 5. Based on FOU(X), determine the intuitionistic FN XI N representing the obtained, universal, algebraic result X of the solved equation.

4

Conclusions

The article presents a new, alternative definition of T2FS that defines the whole body of this set while the current definition only defines its borders. It also shows on the example of the trapezoidal set how to develop an operational, mathematical definition of this set in the form of a membership function, which is particularly suitable for fuzzy arithmetic. Since intuitionistic FS is equivalent to the type 2 set, the new definition can be used not only in T2F-arithmetic but also in intuitionistic F-arithmetic. In this limited-volume article, the use of the new definition in intuitionistic arithmetic is described only generally. A more detailed description of this arithmetic along with examples of applications will be presented in the next articles of the authors.

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References 1. Atanassow, K.T.: Intuitionistic Fuzzy Sets. Physica, Heidelberg (1999) 2. Chen, S.M., Lee, L.W.: Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. Expert Syst. Appl. 37(1), 824–833 (2010) 3. Deschrijver, G., Kerre, E.E.: On the position of intuitionistic fuzzy set theory in the framework of theories modeling imprecision. Inf. Sci. 177(8), 1860–1866 (2007) 4. Dutta, P., Talakdar, P.: A novel arithmetic technique for generalized interval-valued triangular intuitionistic fuzzy numbers and its application in decision making. Open Cybern. Syst. J. 12, 72–120 (2018) 5. Mahapatra, G.S., Roy, T.K.: Intuitionistic fuzzy number and its arithmetic operation with application on system failure. J. Uncertain Syst. 7(2), 92–107 (2013) 6. Mazandarani, M., Zhao, Y.: Fuzzy bang-bang control problem under granular differentiability. J. Frankl. Inst. Eng. Appl. Math. 355(12), 4931–4951 (2018) 7. Mendel, J.M.: Uncertain Rule-Based Fuzzy Systems. Introduction and New Directions, 2nd edn., Springer, Cham (2017) 8. Naim, S., Hagras, H.: A type 2-hesitation fuzzy logic based multi-criteria group decision making system for intelligent shared environments. Soft Comput. 18(7), 1305–1319 (2014) 9. Najarijan, M., Zhao, Y.: Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives. IEEE Trans. Fuzzy Syst. 26(4), 2273–2288 (2018) 10. Piegat, A., Landowski, M.: Horizontal membership function and examples of its application. Int. J. Fuzzy Syst. 17(1), 22–30 (2015) 11. Piegat, A., Plucinski, M.: Fuzzy number division and the multi-granularity phenomenon. Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 497–511 (2017) 12. Piegat, A., Landowski, M.: Fuzzy arithmetic type 1 with horizontal membership functions. In: Kreinovich, V. (ed.) Uncertainty Modeling. Studies in Computational Intelligence, pp. 233–250. Springer, Cham (2017). https://doi.org/10.1007/978-3319-51052-1 14 13. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 14. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8(3), 199–249 (1975)

A New Approach for an Intuitionistic Fuzzy Sugeno Integral Using Morphological Gradient Edge Detector Gabriela E. Martínez, Patricia Melin(B) , and Oscar Castillo Tijuana Institute of Technology, Tijuana, Mexico [email protected]

Abstract. In this paper, an extension to the Sugeno integral using intuitionistic fuzzy sets for morphological gradient edge detection is presented. The proposed method is used as an aggregation operator to combine the four gradients of the morphological gradient edge detector and enables the calculation of the Sugeno integral for combining multiple sources of information with a membership degree and non-membership using intuitionistic fuzzy sets. In this paper, the focus is on aggregation operators that use measures with intuitionistic fuzzy sets, in particular, the Sugeno integral. The performance of the proposed method is compared with the traditional Sugeno integral and other aggregation operators, using images of the Berkeley database (BSDS) and with synthetic images. Keywords: Intuitionistic fuzzy sets · Sugeno integral · Aggregation operators · Morphological gradient edge detection

1 Introduction In the literature, we can find applications that handle inaccurate or imprecise data, so problems can be found due to the uncertainty they generate. In this sense, Zadeh proposed a solution in 1965 by defining fuzzy sets [1]. However, Michio Sugeno completed that definition with the introduction of the fuzzy integral and fuzzy measure terms [2] as a more appropriate way to measure parameters that depend on human subjectivity with a certain degree of uncertainty. The concept of intuitionistic fuzzy sets was introduced by Atanassov in 1983 [3], which allow having degrees of membership and degrees of non-membership in defining a fuzzy set. An aggregation operator can combine and reduce numerical information into a single representative value. There exist several aggregation operators that use measures, such as the arithmetic mean, weighted average, harmonic mean, geometric mean, ordered weighted averaging (OWA), OWA weighting, Choquet integral [2] and the Sugeno integral [4]. There are several applications in which the Sugeno integral has been successfully implemented. For example, in [5] the Sugeno integral is used for building the Sugeno based mean value for some specific fuzzy quantities. A comparison between Choquet © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 26–45, 2021. https://doi.org/10.1007/978-3-030-77716-6_3

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and Sugeno integrals as aggregation operators for pattern recognition is presented in [6], and in [7] interval type-2 fuzzy logic is used for module relevance estimation in Sugeno integration of modular neural networks. On the other hand, the intuitionistic fuzzy sets have had a great boom in recent years [8–10]; in [11] the Choquet and Sugeno integrals and intuitionistic fuzzy integrals as aggregation operators are presented. In [12] the theories of intuitionistic fuzzy sets and artificial neural networks are combined for intrusion detection. The Choquet and Sugeno integrals and intuitionistic fuzzy integrals as aggregation operators are presented in [11], and the concepts of interval fuzzy-valued, intuitionistic fuzzy-valued and interval intuitionistic fuzzy-valued Sugeno integrals were introduced in [13]. A good alternative to enhance the accuracy in the edge detection process are Fuzzy techniques; in the literature, we can find some important contributions using type-1, interval type-2, interval-valued and general type-2 fuzzy sets. In [14] fuzzy edge detection was applied to noisy images obtaining satisfactory results. In [15] presented a method to generate fuzzy image edge detection where the image gradients are transformed into membership degrees implementing parametric membership functions. In [16, 17] proposed some image edge detectors methodologies implementing interval type-2 Fuzzy sets, and these achieved better results than the type-1 fuzzy edge detectors. In [18] presented a competitive fuzzy edge detection method. In [19, 20] presented interesting fuzzy edge detection approaches using GT2 FS theory, and these improved the edges compared to existing type-1 and interval type-2 fuzzy edge detection methods. In [21], A general type-2 fuzzy edge detection method is presented. In [22] was proposed an approach using the Interval type-2 Sugeno integral to aggregate the outputs of a face recognition system based on modular neural networks. Besides, in [23] introduced a method to integrate modular neural networks using the Sugeno integral; this was applied for time series forecasting. In both proposals, the authors present satisfactory results when compared to other aggregation methods. In [24], a hybrid algorithm has been developed using the Otsu method to calculate a threshold value depending on the images for the Intuitionistic fuzzy edge detection algorithm. In general, information aggregation is an important and necessary process to perform any decision-making task; nevertheless, this can be applied to any problem when we need to aggregate different information sources or different criteria. According to [13], the interval fuzzy-valued Sugeno integrals are mathematically equivalent to the intuitionistic fuzzy-valued Sugeno integrals, so it is of interest to implement the Sugeno integral with the operators of the intuitionist fuzzy logic as a method of integration of the four gradients of the morphological gradient edge detector. This method was implemented in synthetics images and Images of the Berkeley database (BSDS) [25]. The rest of the paper is structured as follows: the basic concepts of the Sugeno measures and fuzzy integrals is described in Sect. 2, in Sect. 3 Sugeno integral with intuitionistic fuzzy sets is presented. Section 4 shows the case study using the morphological gradient edge detection. We explain the advantages and the simulation results of the proposed technique with benchmark face databases in Sect. 5, and finally Sect. 6 offers some conclusions.

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2 Sugeno Measures and Fuzzy Integrals Some operators allow mathematical operations in a collection of information elements, and generally, the general result is a numerical value. Fusion consists of combining information, the integration uses information from various sources for a particular purpose. Aggregation operators combine information when mathematically formalized, for example, arithmetic mean. In recent years, fuzzy integrals and fuzzy measures have become popular in various areas of research on aggregation operators, so it is interesting to apply these operators in different study cases to analyze their behavior. In general, when an aggregation process is used, computer systems that need to combine or merge information require preliminary data processing to eliminate non-relevant information. In particular, for the case of the morphological gradient edge detector, the information sources are the gradients calculated in the four directions. In general, it is possible to distinguish among the process of aggregation, information integration, and information fusion. For a better understanding, this Section defines the measures of Sugeno and Sugeno Integral. 2.1 Monotonic Measures A measure is considered in the area of mathematics as one of the most important concepts. We can often find in the literature that monotonous measures refer to fuzzy measures. This is somewhat confusing, because fuzzy sets are not related to the definition of these measures, due to this, in families of fuzzy sets the term “fuzzy measures” must be used for the measures (not additive or additive). If we have a universe of discourse X and a nonempty family C of subsets of X, a monotone measure μ on X, C is a function of the form μ: C → [0, ∞]. It is assumed that the universal set X is finite and that C = P(X). That is, normally assumed that the monotonic measures are sets of functions μ: P(X) → [0, 1]. Definition 1: On space X, a monotonic set measure μ is mapping μ: P(X) → [0, 1] such that the following properties hold [26, 27]: 1) μ (φ) = 0 2) μ (X) = 1 3) For all A, B ∈ P(X), if A ⊆ B, then μ (A) ≤ μ (B) 2.2 Sugeno Measure Special types of monotonic measures are the Sugeno λ-measures [28, 29]. Definition 2. If we have a finite set X = {x 1 , …, x n }. A discrete fuzzy measure on X is a function μ:2X → [0, 1] with the following properties: 1) μ(φ) = 0 and μ(X) = 1; 2) Given A, B ∈ 2X if A ⊂ B then μ (A) ≤ μ (B) (monotonicity property).

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The identifiers of sources of information are considered in the set X. For a subset A ⊆ X, μ(A) is considered to be the relevance degree of this subset of information. Definition 3. Let X = {x1 , …, xn } be any finite set and let λ ∈ (−1, +∞). A Sugeno λ-measure is a function μ from 2X to [0, 1] with the following properties: 1) μ(X) = 1; 2) if A, B ⊆ X with A ∩ B = φ, then μ(A ∪ B) = μ(A) + μ(B) + λ μ(A)μ(B)

(1)

where λ > − 1, usually the Eq. (1) is called the λ-rule. The densities are interpreted as the importance of the individual information sources. If X is a finite set, the fuzzy densities represented by μ({x}) are given for each x ∈ X, the measure of a set A of information sources is interpreted as the importance of that subset of sources [28]. For each A ⊂ P(X), the value of μ(A) can be calculate by the recurrent application of the λ-rule, and can be represented in the following form.   ΠxεA (1 + λμ({x})) (2) μ(A) = λ For each x ∈ X, given the values of the fuzzy densities μ({x}), the value of λ can be determined by using the constraint μ({x}) = 1, which applied in (2) results in (3): n λ+1= (3) (1 + λμ({xi })) i=1

This type of measures uses the parameter λ and once the densities are known, these can be calculated applying (3); Sugeno proved that this polynomial has a real root greater than −1. The property (1), specifying the n different densities, thereby reducing the number of free parameters from 2n − 2 to n [29], using the Theorem 1, is possible determined the value of the λ parameter [30]. Theorem 1: For each x ∈ X, let μ({x}) < 1 and μ({x}) > 0 for at least two elements of X. Then, it is possible to determine a unique parameter λ using (3) as follows: • If Σ x∈X μ({x}) < 1, then λ is a unique root in the interval (0, ∞). • If Σ x∈X μ({x}) = 0, then λ = 0; that is the unique root of the equation. • If Σ x∈X μ({x}) > 1, then λ is a unique root in the interval (−1, 0). Considering the Theorem 1, three situations should be distinguished: • If Σ x∈X μ({x}) < 1, then μ qualifies as a lower probability, λ > 0. • If Σ x∈X μ({x}) = 1, then μ is a classical probability, λ = 0. • If Σ x∈X μ({x}) > 1, then μ qualifies as an upper probability, λ < 0.

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When μ is a λ-fuzzy measure, the values of μ(Ai ) can be computed by means of (2), or recursively, reordering of the sets X and μ({x}), with respect to the values of the elements of set X [31].

2.3 Sugeno Integral Sugeno develops the concept of fuzzy integral as nonlinear functions defined with respect to fuzzy measures as λ-fuzzy measure using the concept of fuzzy measures; we can interpret the fuzzy integral as finding the maximum degree of similarity between the target and the expected value. The Sugeno integral generalizes “max-min” operators. Sugenoμ (x1 , x2 , . . . , xn ) = max (min(f (xσ (i)), μ(Aσ (i)))) i=1,n

(4)

with A0 = 0, where xσ (i) indicates the  indices that must be permuted as shown in (5), and where Aσ (i) = Aσ (i) , . . . , Aσ (n) . 0 ≤ f (xσ (1)) ≤ f (xσ (2)) ≤ . . . ≤ f (xσ (n)) ≤ 1

(5)

The Sugeno integral can be applied to solve several problems, which consider a finite set of n elements X = {x1 ,…,xn }. In Algorithm 1, the steps to evaluate the Sugeno integral (SI) are presented.

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3 Sugeno Integral with Intuitionistic Fuzzy Sets The basic concepts of the Intuitionistic fuzzy set used for the combination of the Sugeno integral with the morphological gradient are presented in this section. 3.1 Intuitionistic Fuzzy Set Let a set E be fixed and let A ⊂ E be a fixed sets. Intuitionistic fuzzy sets A* is defined as (6) A∗ = {x, μA (x), VA (x)|x ∈ E}

(6)

where function μA :E →[0,1] and VA :E →[0,1] define the degree of membership and the degree of non-membership of the element x  E to the set A, respectively, and for every x  E 0 ≤ μA (x) + VA (x) ≤ 1

(7)

Obviously, every ordinary fuzzy set has the form {x, μA (x), 1 − μA (x)|x ∈ E}

(8)

πA (x) = 1 − μA (x) − VA (x)

(9)

if

then πA (x) is the degree of non-determinacy (uncertainty/hesitancy) of the membership of element x ∈ E to the set A. in the case of ordinary fuzzy sets, πA (x) = 0 for every x ∈ E [32]. 3.2 Sugeno Integral with Intuitionistic Fuzzy Set Is possible to extend the Sugeno integral (4) by the use of degrees of membership and degrees of non-membership, using the concept of Intuitionistic fuzzy sets, obtaining (10) IFSIu(μA ,VA )) (x1 , x2 , . . . , xn ) 

        = max min f xσ (i) , μA Aσ (i) , max min f xσ (i) , VA Aσ (i) i=1..n

i=1..n

(10)

The Sugeno integral and the intuitionistic fuzzy Sugeno integral can be applying in problems where there is a finite set of n elements X = {x1 ,…,xn }. The intuitionistic fuzzy Sugeno integral is summarized in Algorithm 2.

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3.3 Intuitionistic Fuzzy Sugeno Integral Using π A = 0.4 If we have that x = {0.9, 0.6, 0.3} represents the information sources and associated to each entry a fuzzy density or membership value as μi = {0.3, 0.4, 0.1}, we calculate λ = 1.  The calculated fuzzy measures are μA Aσ (i) = {0.3, 0.82, 1}. Using the intuitionistic fuzzy densities or not membership values as VA (μi ) = { 0.3, 0.2, 0.5}, it was calculated a λ = 0 and the intuitionistic fuzzy measures calculated are VA (Aσ (i)) = {0.3, 0.5, 1}. After that, using (10) we calculate the intuitionistic Sugeno integral IFSI (μA , VA ) = (max(min((0.9, 0.3), (0.6, 0.82), (0.3, 1))), max(min((0.9, 0.3), (0.6, 0.5), (0.3, 1))) once the minimums have been calculated, the maxima are calculated IFSI (μA , VA ) = (max(0.3, 0.6, 0.3), max(0.3, 0.5, 0.3)) we obtain the maxima of μA , VA IFSI (μA , VA ) = (0.6, 0.5) after that, the mean of the obtained interval is calculated, however, is possible defined other way for extract a representative value of the interval obtained IFSI (μA , VA ) = 0.55 The information from different sources can be integrated with the proposed operator.

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4 Edge Detection In computer vision systems and image processing an essential phase is edge detection; mostly in applications for detection/identification of significant elements found in the image or feature extraction. This procedure can significantly reduce the amount of information that should be processed. A filter can be applied to preserve the most important characteristics of an image. Since it is easy for a person to recognize objects having little information, such as their characteristics or edges, it is expected that image recognition systems can learn with more little information and achieve a good percentage of recognition. In the literature, we can find several ways to carry out the process of edge detection, however, the most used techniques are the focus on calculating the gradient by the first derivative of the image. Exist several methods based on this approach; the best known are the Sobel, Prewitt, and Roberts operators [33, 34]; these apply a convolution process. In this paper, we are proposing to use the morphological gradient (MG) approach that consists on calculating the first derivative, in the four directions of the image by the variables D1 , D2 , D3 , and D4 respectively, as is shown in Fig. 1 and are calculated as follows. The variable Gi represents the possible directions of the gradients that are calculated using Eq. (11) by applying a 3 × 3 matrix, where zi represents the coefficients of matrix positions and that are obtained with Eq. (12); in this case, f represents the image, x-axis the columns and y-axis the rows. Finally, the edge value G is integrated by using Eq. (13) [17, 35, 36].

Fig. 1. Gradients direction Gi .

z1 = f (x − 1, y − 1)z2 = f (x, y − 1) z3 = f (x + 1, y − 1)z4 = f (x − 1, y) z5 = f (x, y)z6 = f (x + 1, y) z7 = f (x − 1, y + 1)z8 = f (x, y + 1) z9 = f (x + 1, y + 1)  D1 =

(z5 − z2)2 + (z5 − z8)2  D2 = (z5 − z4)2 + (z5 − z6)2

(11)

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 D3 = D4 =



(z5 − z1)2 + (z5 − z9)2 (z5 − z7)2 + (z5 − z3)2

G = D1 + D2 + D3 + D4

(12) (13)

5 Simulation Results The edge detection process is used in computer vision and pattern recognition, however, the process of identifying the edges is not easy, since it is necessary to determine what is and is not an edge of the image and the existing methods they can confuse edges that in many cases are hidden or partially distorted.

Fig. 2. Integration of the image gradients using ISI + MG.

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The main objective of this work is to carry out the aggregation process by combining the image gradients obtained by the GM edge detector that was explained in Sect. 4 with the Intuitionistic Sugeno integral. For the intuitionistic fuzzy aggregation, the gradients of the image are calculated and after that, the Sugeno integral combined with the intuitionistic fuzzy sets operators is used (see Sect. 3) to integrate the gradients using IFSI + MG (10). The fuzzy densities of the Sugeno integral determine the degree of membership and the degree of non-membership associated with each of the gradients, so it is expected that by applying the proposed method, the edge detection should be more robust than applying any of the traditional methods. A general diagram about this methodology is in Fig. 2. The steps for integrating the gradients based on IFSI + GM are in Fig. 3. The first step consists in calculating the four gradients Gi of the image; after that, the assignment of fuzzy densities M({xi })V({xi }) and πA (x) for each Gi are performed. Next, the calculation of the lambdas M and V is carried out,  and once obtained, the next step is to Calculate the fuzzy measures μA ({xi })and VA xσ (i) for finally, perform the calculation of the proposed method IFSI + GM.

Fig. 3. Integration of the image gradients by using IFSI + MG.

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If we have the numerical representation of an image like the one in Fig. 4, and we select coefficients of the matrix positions 3 × 3 (as in Fig. 3) to calculate the morphological gradient method by applying Eq. (11–13).

Fig. 4. Numerical representation of an image.

In Fig. 5, we can appreciate the numerical representation of an image with the traditional MG edge detected.

Fig. 5. Numerical representation of an image using MG edge detector.

Taking as reference the numerical representation shown in Fig. 4, we calculate the morphological gradient method using (11–12); however, Eq. (13) that represents the integration of the gradients is replaced by the Sugeno integral (SI) (Eq. (4)), to obtain the numerical representation of the Sugeno integral combined with the morphological gradient edge detector (SI + MG) which can be seen in Fig. 6. The same idea is used to calculate the proposed edge detector (IFSI + MG). Equation (13) is replaced by the intuitionistic Sugeno integral (Eq. (10)), to obtain the numerical representation of Fig. 7. The parameters used for this test was πA (x) = 0.2, μA (x) = 0.4 and VA (x) = 0.4; these parameters were arbitrarily designated, so it is necessary to perform tests using different values to verify the effectiveness of the method. As can be seen, when applying the proposed method, for each intensity level or pixel an interval is obtained; one of the main draw-backs that we have when obtaining an interval, is to determine which of the values represents the best solution, so it is necessary to determine the optimal value that represents whether or not that pixel corresponds to an edge.

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Fig. 6. Numerical representation of an image using SI + MG edge detector.

Fig. 7. Numerical representation of the interval of an image using IFSI + MG edge detector.

For this work, one of the approximations that we use is to calculate the average to represent the solution. The result obtained can be seen in Fig. 8, however, it does not mean that this is the only or the best way to do it. For this particular edge detection application, the main goal was to determine whether the pixel corresponds to an edge.

Fig. 8. Numerical representation of the IFSI + MG edge detector.

Using the proposed method, the central pixel of the selected sub-matrix (see Fig. 4) is replaced by the obtained value. We can notice that the final value obtained by the IFSI + MG was 14, this differs from the value calculated by the traditional MG that was of 33; therefore, the proposed method could be useful to determine more accurately if the

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gradient represents an edge of the image. The latter is due to the property that the IFSI + MG method has in the handling of uncertainty. The results of the proposed intuitionistic fuzzy edge detection method are presented below, the implementation of which consists in the integration of the image gradients using the integral Sugeno combined with the intuitionistic fuzzy systems operators. In this section, two tests are presented; in the first test, a database of synthetic images was used, which can be seen in Table 1. In the second test, the proposed edge detector was implemented in a benchmark database of image segmentation and boundary detection (BSDS) that we can found in Table 3. Table 1. Synthetic images

Image number

1

2

3

Synthetic Image

Reference Image

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Table 2. Edge detectors in synthetic images.

Image number

Morphological gradient

Sugeno integral

Intuitionistic Sugeno integral

1

2

3

In both tables, we can observe the original images and the reference images. Also, to test the effectiveness of the method proposed, several edge detectors were analyzed as the Morphological gradient method, the MG using the Sugeno integral (SI + MG) and MG combined with Sugeno integral and the operators of intuitionistic fuzzy systems (IFSI + MG). For the first test, the traditional morphological gradient edge detection, Sugeno integral and intuitionistic Sugeno integral methods was applied in both image databases (Tables 1 and 3); the visual results of the edge detection are shown in Table 2 for the case of the synthetic images and in Table 4 for the BSDS database, respectively. In both tables, the image number is in the first column. The images obtained using the aggregation operator based on the morphological gradient are presented in the second column. The images produced after applying the traditional Sugeno integral are shown in the third column, and finally, we can appreciate the images in which the proposed method IFSI + GM was implemented in the fourth column.

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Image number

1

2

3

4

5

Original image

Reference image

A New Approach for an Intuitionistic Fuzzy Sugeno Integral Table 4. Edge detectors in the BSDS images of the Berkeley database.

Image number

1

2

3

4

5

Morphological gradient

Sugeno integral

Intuitionistic Sugeno integral

41

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When working with synthetic images, you have the advantage of having the real image and the reference image, which allows you to use some metric that allows you to evaluate the efficiency of the proposed method. In this case, the databases used for the experiments have reference images, so it is possible to evaluate the edges detected with IFSI + GM and compare them with other edge detection methods. To evaluate the efficiency of the proposed edge detection method, the metric based on the “Figure of merit” of Pratt’s (FOM) [37–39] was used. In Table 5, is possible to appreciate the FOM values obtained with the synthetic images, using the detectors MG, SI + MG, IFSI + MG, respectively. Comparing the obtained results, we can conclude that for this data-base, the IFSI + MG and SI + MG provides best results, in most of the images than with the traditional MG edge detector. Table 5. FOM of the synthetic images. Image MG number

SI + MG IFSI + MG

1

0.8744 0.9477

0.9478

2

0.8199 0.9408

0.9408

3

0.8621 0.8812

0.8710

In the results obtained in Table 6, it can be observed that due to the nature of the information present in the real images of the BSDS Berkeley database, the edge detection process turns out to be more complicated than in the synthetic image database. Table 6. FOM of the BSDS Berkeley database. Image MG number

SI + MG IFSI + MG

1

0.1809 0.1932

0.1937

2

0.5492 0.5397

0.5402

3

0.2133 0.2438

0.2432

4

0.1053 0.0968

0.0971

5

0.1406 0.1476

0.148

The BSDS database used for these tests differs from the previous one, since it contains real images of landscapes, animals or other things. However, this database also has reference images. For this database, in most cases, the values calculated with the metric are very low for all the edges detectors, however, the parameters obtained to serve as a reference point to evaluate the proposed method IFSI + GM, and in turn to compare those results with those calculated with other edge detection techniques existing in the literature; in this case, GM and SI + GM.

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6 Conclusion In summary, this paper presents a fuzzy approach for edge detection applying the intuitionistic fuzzy Sugeno integral as a method to aggregate the image gradients; this provides a more robust system because the images are viewed as fuzzy information and this gives the advantage of modeling the uncertainty contained in the data and the results values demonstrate that the proposed edge detection method outperforms or comparable with other existing algorithms. Also, we presented a comparative analysis of the different used methods to aggregate the image gradients, such as the traditional Morphological gradient, Sugeno integral, and IFSI + MG. The results showed that the proposed method is suitable to combine this type of information; above all to improve edge detection tasks based on gradients. According to the results shown in Tables 5 and 6, we can notice that the edges of the image were good after applying the proposed operator or working with fuzzy measures. However, it is necessary to perform more tests varying the fuzzy densities designated for each information source to verify if the results can be improved even more. Operators that use fuzzy measures manage degrees of memberships to each of the information sources, and when applying intuitionistic fuzzy operators, it is allowed managing the various degree of membership and degrees of non-memberships. Therefore, when using this type of aggregation operators, different alternatives can be explored when solving problems, so it is interesting to apply fuzzy measurements to several applications to explore a broader field of action, as in this case in the edges detectors. In this research, we present an approximation of the combination of the intuitionistic fuzzy sets with the Sugeno integral applied in the morphological gradient edge detector. The experimental evaluations are performed on synthetic and real images, and the accuracy is quantified using Pratt’s Figure of Merit. The results values demonstrate that the proposed edge detection method outperforms or is comparable with other existing algorithms, however, it is necessary to consider whether there are other ways to combine the proposed methods. The IFSI + GM was successfully implemented as an aggregation operator in an edge detection problem, however, the proposed method can be used in any application where the numerical aggregation is necessary. As future work, it is possible to implement optimization techniques to find the optimal values for the πA value and fuzzy densities to the gradients as well as perform tests with other databases to improve the proposed system and to take the most advantages of the aggregation operator. Besides, when applying the proposed IFSI + GM method and obtaining an interval as a result, some other technique other than the average could be considered for obtaining a particular value that determines whether or not the pixel studied corresponds to an edge in the image.

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24. Yalcin, E., Badem, H., Gunes, M.: CUDA-based hybrid intuitionistic fuzzy edge detection algorithm, pp. 1–6 (2015). https://doi.org/10.1109/fuzz-ieee.2015.7338008 25. Arbelaez, P., Maire, M., Fowlkes, C., Malik, J.: Contour detection and hierarchical image segmentation. IEEE TPAMI 33(5), 898–916 (2011) 26. Yager, R.: A knowledge-based approach to adversarial decision-making Int. J. Intell. Syst. 23(1), 1–21 (2008) 27. Klir, G.: Uncertainty and Information. Wiley, Hoboken, NJ (2005) 28. Mendez-Vazquez, A., Gader, P., Keller, J.M., Chamberlin, K.: Minimum classification error training for Choquet integrals with applications to landmine detection. IEEE Trans. Fuzzy Syst. 16(1), 225–238 (2008) 29. Bezdek, J.C., Keller, J., Krisnapuram, R., Pal, N.R.: Fuzzy models and algorithms for pattern recognition and image processing. THFSS, vol. 4. Springer, Boston, MA (1999). https://doi. org/10.1007/b106267 30. Torra, V., Narukawa, Y.: Modeling Decisions, Information Fusion and Aggregation Operators. Springer-Verlag, Heidelberg, Germany (2007). https://doi.org/10.1007/978-3-540-68791-7 31. Verikas, A., Lipnickas, A., Malmqvist, K., Bacauskiene, M., Gelzinis, A.: Soft combination of neural classifiers: a comparative study. Pattern Recogn. Lett. 20(4), 429–444 (1999) 32. Atanassov, K., Vassilev, P., Tsvetkov, R.: Intuitionistic fuzzy sets measures and integrals. Bulgarian Academic Monographs (2013) 33. Sobel, I.: Camera Models and Perception. Ph.D. thesis, Stanford University, Stanford, CA (1970) 34. Prewitt, J.M.S.: Object enhancement and extraction. Picture Analysis and Psychopictorics, pp. 75–149. Academic Press, New York (1970) 35. Evans, A.N., Liu, X.U.: A morphological gradient approach to color edge detection. IEEE Trans. Image Process. 15(6), 1454–1463 (2006) 36. Becerikli, Y., Karan, T.M.: A new fuzzy approach for edge detection. In: Cabestany, J., Prieto, A., Sandoval, F. (eds.) IWANN 2005. LNCS, vol. 3512, pp. 943–951. Springer, Heidelberg (2005). https://doi.org/10.1007/11494669_116 37. Pratt, W.: Digital Image Processing. Wiley, New York (1978) 38. Abdou, I., Pratt, W.: Quantitative design and evaluation of enhancement/thresholding edge detectors. Proc. IEEE 67(5), 753–763 (1979) 39. Perez-Ornelas, F., Mendoza, O., Melin, P., Castro, J.R., Rodriguez-Diaz, A., Castillo, O.: Fuzzy index to evaluate edge detection in digital images. PLoS ONE 10(6), 1–19 (2015)

M-Probabilistic Versions of the Strong Law of Large Numbers Piotr Nowak(B) and Olgierd Hryniewicz Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01–447 Warszawa, Poland {Piotr.Nowak,Olgierd.Hryniewicz}@ibspan.waw.pl

Abstract. The M-probability theory was introduced for intuitionistic fuzzy events (IFEs), defined by Atanassov’s intuitionistic fuzzy sets (IFSs). In this paper, we formulate and prove generalized versions of the strong law of large numbers (SLLN for short), i.e. the Brunk–Prokhorov SLLN, Marcinkiewicz–Zygmund SLLN, Korchevsky SLLN within the M-probability theory, and we illustrate our results with an example. Keywords: M-probability · Brunk–Prokhorov SLLN Marcinkiewicz–Zygmund SLLN · Korchevsky SLLN

1

·

Introduction

The Kolmogorov probability theory is a commonly used model of randomness. However, other probability theories are also used, including the boolean algebraic probability theory, proposed for quantum systems by von Neumann and Carath´eodory, who considered states and observables in place of probability measures and random variables (see [17]). In many practical situations, two sources of uncertainty, i.e. randomness and imprecision, often modelled by Zadeh’s fuzzy sets, should be taken into account. A very known generalization of fuzzy sets are IFSs (see [1] and references therein). They are defined by two functions on a universe of discourse, describing the degree of membership and nonmembership. The paper [16] is devoted to IF-probability theory and M-probability theory, involving the L  ukasiewicz and Zadeh connectives between IFSs. Mazurekov´a in [6] proved the M-probabilistic counterpart of the Kolmogorov SLLN. Results of probability theories concerning IFEs and other similar theories can be also found e.g. in [5,7,9–15,17]. From the applications’ viewpoint, a well-developed theory of statistical inference for imprecisely-perceived random observations data is necessary. A similar theory is required for practical problems, where more complex models of imprecision, such as IFSs, are considered. As in the case of classical statistical inference, generalized versions of the SLLN within the M-probability theory seem to be essential for the development of the appropriate statistical techniques for IFEs c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 46–53, 2021. https://doi.org/10.1007/978-3-030-77716-6_4

M-Probabilistic Versions of the SLLN

47

as well as of the M-probabilistic theory of stochastic processes, which are in the authors’ plans. In this paper, we consider the limit behaviour of the scaled sums of Mobservables. We formulate and prove generalized versions of the SLLN, i.e. the Marcinkiewicz–Zygmund SLLN, Brunk–Prokhorov SLLN for independent Mobservables and the Korchevsky SLLN, where the independence of M-observables is not assumed. To illustrate our theoretical results, we use the M-probabilistic version of the Brunk–Prokhorov SLLN to a sequence of M-observables, for which the SLLN proved in [6] can not be applied, assuming that M-probability has the form of the probability of intuitionistic fuzzy events, introduced in [18] and further developed. Generalized versions of the SLLN were also proved in [12,13] in MV-algebraic and IVM-probabilistic settings. The paper is organized as follows. Section 2 is devoted to the basic elements of the M-probability theory. The generalized versions of the SLLN are formulated and proved in Sect. 3. The last section contains the illustrative example.

2

Selected Elements of M-Probability Theory

In this section, we recall basic elements of M-probability theory from [15,16], which will be used in the following part of the paper. For a measurable space (Ω, S), we denote by F = F (Ω, S) the space of all IFEs, i.e. all pairs A = (μA , νA ) of S-measurable functions on Ω, for which μA ≥ 0, νA ≥ 0 and μA +νA ≤ 1. Functions μA and νA are called, respectively, the membership function and nonmembership function. M-probability theory concerns the space F with the Zadeh connectives: A ∨ B = (μA ∨ μB , νA ∧ νB ) , A ∧ B = (μA ∧ μB , νA ∨ νB ) for each A, B ∈ F. ∞ ∞ ∞ Let X be a set, {An }n=1 ⊂ 2X , A ⊂ X, {xn }n=1 ⊆R, x ∈ R, {Cn }n=1 ⊂ F, ∞ C ∈ F. We write: An A iff A1 ⊆ A2 ⊆ ... and n An = A, xn x iff x1 ≤ x2 ≤ ... and x = supn xn , Cn C ⇔ μCn μC , νCn  νC . The notations An  A, xn  x, Cn  C are defined analogously. Definition 1. An M-state on F is a function s : F → [0, 1] such that: s (1Ω ) = 1, s (0Ω ) = 0, where 0Ω = (0Ω , 1Ω ) , 1Ω = (1Ω , 0Ω ), 1Ω ≡ 1, 0Ω ≡ 0; (ii) s (A) + s (B) = s (A ∨ B) + s (A ∧ B) ; (iii) if An A, Bn  B, then s (An ) s (A) , s (Bn )  s (B) (i)

∞

∞

for all A, B ∈ F and {An }n=1 , {Bn }n=1 ⊂ F.

  Definition 2. An M-probability is a function P = P  ,P  : F → J , where J is the family of compact intervals on R, fulfilling the following conditions: (i) P (1Ω ) = [1, 1] , P (0Ω ) = [0, 0] ; (ii) P (A) + P (B) = P (A ∨ B) + P (A ∧ B) ; (iii) if An A, Bn  B, then P (An ) P (A), P (Bn )  P (B) for all ∞ ∞ A, B ∈ F and {An }n=1 , {Bn }n=1 ⊂ F.

48

P. Nowak and O. Hryniewicz

An M-probability space is a pair (F, P ), where P is an M-probability on F. We introduce the notation P (A) = P  (A) ,P  (A) for each A ∈ F. Definition 3. An n-dimensional M-observable, n ∈ N, is a map x : Bn → F, where Bn is the Borel σ-field on Rn , satisfying the following conditions for A, B ∈ ∞ Bn and {An }∞ n=1 , {Bn }n=1 ⊂ Bn : (i) x (Rn ) = 1Ω , x (∅) = 0Ω ; (ii) x (A ∪ B) = x (A) ∨ x (B) and x (A ∩ B) = x (A) ∧ x (B) ; (iii) if An A and Bn  B, then x (An ) x (A) and x (Bn )  x (B) . We have the following easy generalization of Proposition 6 from [16] for d > 1. Proposition 1. For any d ∈ N and M-observable x : Bd → F and M-state s : F → [0, 1] , sx = s ◦ x : Bd → [0, 1] is a probability measure. Definition 4. Let x1 , x2 , ..., xn : B1 → F be M-observables. Then their joint Mobservable is a n-dimensional M-observable h : Bn → F such that h (ni=1 Ai ) = n i=1 xi (Ai ) for each A1 , A2 , . . . , An ∈ B1 . For M-observables, the following theorem and remark hold (see [16] for proof of this theorem). Theorem 1. To any M-observables x1 , x2 , ..., xn : B1 → F, there exists their joint M-observable. Remark 1. Let x1 , x2 , ..., xn : B1 → F be M-observables, g : Rn → R be a Borel measurable function, and h : Bn → F be the joint M-observable of x1 , x2 , ..., xn . Then g (x1 , x2 , ..., xn ) = h ◦ g −1 is an M-observable. Definition 5. M-observables x1 , x2 , ..., xn : B1 → F are independent (with respect to P ) if there exists an M-observable h : Bn → F such that for each {Ci }ni=1 ⊂ B1 P  (h (ni=1 Ci )) =

n 

P xi (Ci ) , P  (h (ni=1 Ci )) =

i=1

n 

P xi (Ci ) .

i=1

Definition 6. Let P : F → J be an M-probability. An M-observable x : B1 → F is integrable if expectations   tP x (dt) , E (x) = tP x (dt) E (x) = R





R

exist, and is square-integrable if R t2 P x (dt) + R t2 P x (dt) < ∞. If x is squareintegrable, variances of x also exist and are described by the equalities  

2  2  2,  2, t − E (x) P x (dt) , D (x) = t − E (x) P x (dt) . D (x) = R

We write x ∈

LpP

for p ≥ 0 if

 R

|t|p P x (dt) +

R

 R

|t|p P x (dt) < ∞.

M-Probabilistic Versions of the SLLN

49

The following generalization of Lemma 48 from [10] of Propo  is a consequence sition 1 and Theorem 16.12 from [2] for (X, X ) = Rd , Bd , (X  , X  ) = (R, B1 ), T = ϕ and μ = mh . Lemma 1. Let P : F → J be an M-probability and d ∈ N. Then, for any Borel function ϕ : Rd → R and M-observables h : Bd → F, y = ϕ (h) : B1 → F,  the expectation E (y) exists if and only if Rd |ϕ (t) |dP h (t) < ∞. Moreover, if  this condition holds, then E (y) = Rd ϕ (t) dP h (t). Furthermore, the analogous assertion holds for E (y) and P h . Finally, we recall the definition of convergence of M-observables to zero p-a.e. and a shortened version of Theorem 3.3 from [15]. ∞

Definition 7. Let {zn }n=1 be a sequence of M-observables in F and p be an ∞ M-state on F. {zn }n=1 converges to zero p-almost everywhere (limn→∞ zn = 0

k+i

  1 1 p-a.e.) if limq→∞ limk→∞ limi→∞ p zn − q , q = 1. n=k

∞ Theorem 2. Let {xn }n=1 state on F, hn : Bn → F,

be a sequence of M-observables in F, p be an Mn n ∈ N, joint observables of {xi }i=1 , gn : Rn → R arbitrary Borel functions, and yn = gn (x1 , x2 , · · · , xn ), n ∈ N. Then there exists a probability space (X , G, P ) and a sequence of random variables, ξn : X → R, n ∈ N, such that if ηn = gn (ξ1 , ξ2 , . . . , ξn ), n ∈ N, converges to zero P -a.s., then ∞ {yn }n=1 converges to zero p-a.e. Remark 2. From the proof of the above theorem it follows that: X = RN , G = σ(C), where C is the family of cylinders in RN (G coincides with the Borel σ-field B on RN ), and P : G → [0, 1] is the probability measure such that for each n ∈ N P ◦¯ın−1 = p ◦ hn , where ¯ın : RN → Rn are the projections. Moreover, the random variables {ξn }n∈N are given by the equalities ξn ((ui )∞ i=1 ) = un .

3

M-Probabilistic Versions of the Strong Law of Large Numbers

In this section, we will proof generalized versions of the SLLN for M-observables. The M-probabilistic version of the Marcinkiewicz–Zygmund SLLN assumes independence and identical distributions of M-observables. ∞

Theorem 3. Let (F, P ) be an M-probability space and {xn }n=1 be an independent sequence of M-observables in (F, P ) having the same distributions P x1 , P x1 . Let p ∈ (0, 2), x1 ∈ LpP , c = c = 0 for 0 < p < 1 and c = E  (x1 ) , c = E  (x1 ) for 1 ≤ p < 2. Then: n n    i=1 xi − nc i=1 xi − nc lim = 0 P -a.e., lim = 0 P  -a.e. (1) n→∞ n→∞ n1/p n1/p

50

P. Nowak and O. Hryniewicz

Proof. Let n ∈ N. By Theorem 1, there exists a joint M-observable hn of Mn n i=1 ti −nc and observables {xi }i=1 . We consider the function gn (t1 , t2 , ..., tn ) = n1/p the M-observable yn = hn ◦ gn−1 . Let (X , G, P ) be the probability space and {ξn }n∈N the random variables, described in Remark 2, which existence follows from Theorem 2 for p = P  . We denote by E the expectation with respect to P . ∞ The sequence {ξn }n=1 is independent and identically distributed. Indeed, P (ξ1 ∈ A1 , ξ2 ∈ A2 , . . . ξn ∈ An ) = P ◦ ¯ın−1 (ni=1 Ai ) = P  ◦ hn (ni=1 Ai ) n n      = P (xi (Ai )) = P (h1 (A1 )) P  hi (Ri−1 × Ai ) i=1

i=2

 = P A1 × RN

n 

n   P Ri−1 × Ai × RN = P (ξi ∈ Ai ) , {Ai }ni=1 ⊂ B1 , (2)

i=2

i=1

  P (ξn ∈ A) = P Rn−1 × A × RN = P  hn (Rn−1 × A) = P xn (A) = P x1 (A)

(3)

for n > 1 and A ∈ B1 . Thus, applying Lemma 1, we obtain: Eξn = E  (xn ) = c for 1 ≤ p < 2 and E|ξn |p = E  (|xn |p ) < ∞, n ∈ N. Then, by the classical version of the Marcinkiewicz–Zygmund SLLN (see Theo∞ rem 2 from [13]), the sequence {ηn }n=1 converges to zero P -a.s. Thus, by Theorem 2, the first assertion of (1) holds. Repeating the above steps for P  in place of P  , we obtain the second assertion of (1).   The Brunk–Prokhorov SLLN, in the M-probabilistic version formulated and proved below, does not assume the same distribution of M-observables. ∞

Theorem 4. Let p ≥ 2. Let (F, P ) be an M-probability space and {xn }n=1 be an independent sequence of M-observables in (F, P ) such that E  xn = 0, E  xn = 0, xn ∈ LpP for each n ∈ N. Let: ∞  E  |xn |p + E  |xn |p < ∞. np/2+1 n=1

Then: lim

n→∞

(4)

x1 + x2 + ... + xn = 0 P  -a.e. and P  -a.e. n ∞

(5)

Proof. From Theorem 1, 2, applied to the sequence {xn }n=1 and functions n i=1 ti gn (t1 , t2 , ..., tn ) = n , n ∈ N, with the remaining notation as in the proof of the previous theorem, it follows that there exists a probability space (X , G, P ) and random variables {ξn }n∈N , described in Remark 2. Since the equality (2) holds, the random variables are independent. By Lemma 1, Eξn = E  (xn ) = 0 and E|ξn |p = E  |xn |p < ∞, n ∈ N.

M-Probabilistic Versions of the SLLN

51

Then, by (4) nand the classical Brunk–Prokhorov SLLN (see Theorem 3 from ξi = 0 P -a.s. Thus, applying Theorem 2 and repeating the [13]), lim i=1 n n→∞

above steps for P  in place of P  , we obtain (5).

 

Let Ψc denote the family of positive functions non-decreasing in the ∞ ψ(x), 1 < ∞. interval {x > x0 } for some x0 ∈ R, such that n=1 nψ(n) The following M-probabilistic version of the Korchevsky SLLN does not require the independence of M-observables. ∞

Theorem 5. Let (F, P ) be an M-probability space, {xn }n=1 ⊂ LpP for some ∞ p ≥ 1, and be non-negative, i.e. m (xn ([0, ∞)) = 1, n ∈ N. Let {an }n=1 be a non-decreasing unbounded sequence of positive numbers. If E  sn = O (an ), E  sn = O (an ), where sn = x1 + x2 + ... + xn , n ∈ N, and for

some ψ ∈ ap ap     p p n n Ψc E |sn − E (sn ) | = O ψ(an ) , E |sn − E (sn ) | = O ψ(an ) , then n

n xi − i=1 E  xi = 0 P  -a.e., n→∞ an n n  i=1 xi − i=1 E xi = 0 P  -a.e. lim n→∞ an i=1

lim

(1)

(6)

(2)

Proof. Let for each n ∈ N ϕn , ϕn , gn : Rn → R be functions of the form n n n (1) (2) ϕn (t1 , t2 , ..., tn ) = i=1 ti , ϕn (t1 , t2 , ..., tn ) = | i=1 ti − i=1 E  xi |p , and n

t −

n

E x

i gn (t1 , t2 , ..., tn ) = i=1 i an i=1 . We apply Theorem 1, 2 to the sequence ∞ {xn }n=1 and functions gn , n ∈ N, with the remaining notation as in the proof of the previous theorem. Therefore, there exists a probability space (X , G, P ) and random variables {ξn }n∈N , as it was described in Remark 2. We consider random variables η1,n = ϕ1,n (¯ın ) , η2,n = ϕ2,n (¯ın ) on (X , G, P ) and M-observables sn = ϕ1,n (x1 , x2 , ..., xn ) : B (R) → F, y2,n = ϕ2,n (x1 , x2 , ..., xn ) : B (R) → F. For n ∈ N we have:  P ◦ ξn−1 = P xn , P ◦ ¯ı−1 n = P hn

and by Lemma 1, Eξn = E  (xn ) , E|ξn |p = E  |xn |p < ∞,   Eη1,n = ϕ1,n (¯ın ) dP = ϕ1,n (t) dP hn (t) = E  sn = O (an ) , N n

p  R R an Eη2,n = ϕ2,n (¯ın ) dP = ϕ2,n (t) dP hn (t) = E  y2,n = O . ψ (an ) N n R R Clearly, P (ξn ∈ [0, ∞)) = P xn ([0, ∞)) = 1, n ∈ N. Therefore, application of ∞ Theorem 1 from [4] to random variables {ξn }n=1 and Theorem 2 gives the first assertion of (6). Repeating the above consideration for P  in place of P  , we obtain the second assertion of (6).  

52

P. Nowak and O. Hryniewicz

4

An Illustrative Example

We assume that M-observables take values in F = F (Ω, S) , where Ω = {ω1 , ω2 } , S = 2Ω . In particular, we consider the following elements of F: 0Ω , q1 = (μ1 , ν1 ), q2 = (μ2 , ν2 ), 1Ω , where for i, j ∈ {1, 2} μi (ωj ) = 1 − νi (ωj ) = 1 if i = j and 0 otherwise. S) such that: P0 ({ω1 }) = P0 ({ω2 }) = 12 . MLet P0 be the probability on (Ω, probability has the form: P (A) = Ω μA dP0 , 1 − Ω νA dP0 , A ∈ F, proposed in [3] and generalized in [8]. We assume that M-observables xn : B1 → F, given for each A ∈ B1 by ⎧ 0Ω if A ∩ {an1 , an2 } = ∅; ⎪ ⎪ ⎨ q1 if A ∩ {an1 , an2 } = {an1 } ; xn (A) = q2 if A ∩ {an1 , an2 } = {an2 } ; ⎪ ⎪ ⎩ 1Ω if {an1 , an2 } ⊂ A, 1

1

where −an1 = an2 = n 2 /[log2 (2n)] 4 , n ∈ N, are independent. For each n ∈ N p p E (xn ) = E (xn ) = 0, E (|xn |p ) = E (|xn |p ) = n 2 /[log2 (2n)] 4 , p ≥ 2. ∞ ∞   E |xn |2 1 = Since 1 = ∞, the Kolmogorov condition does not n2 n=1

hold. However,

∞  n=1

n=1 n[log2 (2n)] 2 ∞ 

E |xn |p np/2+1

=

1 p n=1 n[log2 (2n)] 4

< ∞ for p > 4 and therefore, the

assumptions of Theorem 4 are satisfied and this implies that (5) holds.

5

Conclusions

In this paper, we develop M-probability theory. We formulate and prove generalized versions of the strong law of large numbers, i.e. the M-probabilistic versions of the Marcinkiewicz–Zygmund SLLN, Brunk–Prokhorov SLLN, and Korchevsky SLLN. They describe the limit behaviour of the scaled sums of M-observables. In the proofs of the aforementioned theorems, the Kolmogorov theory of probability and measure theory are used. To illustrate our results, we prove the convergence of the scaled sums of M-observables, for which the Mprobabilistic version of the Kolmogorov SLLN can not be used. The generalized versions of the SLLN, proved in the paper, can be useful for the development of the appropriate statistical techniques for IFEs as well as of the M-probabilistic theory of stochastic processes.

References 1. Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica-Verlag, Heidelberg (1999) 2. Billingsley, P.: Probability and Measure, 2nd edn. Wiley, New York (1986) 3. Grzegorzewski, P., Mr´ owka, E.: Probability of intuitionistic fuzzy events. In: Grze´ (eds.) Soft Methods in Probability, Statisgorzewski, P., Hryniewicz, O., Gil, M.A. tics and Data Analysis, pp. 105–115. Physica-Verlag, Heidelberg (2002)

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4. Korchevsky, V.: A generalization of the Petrov strong law of large numbers. Stat. Probab. Lett. 104, 102–108 (2015) 5. Lendelov´ a, K., Petroviˇcov´ a, J.: Representation of IF-probability for MV-algebras. Soft Comput. 10(7), 564–566 (2006) 6. Mazurekov´ a, P.: Laws of large numbers for M-observables. Notes on Intuitionistic Fuzzy Sets 13(2), 30–35 (2007) 7. Michal´ıkov´ a, A., Rieˇcan, B.: On invariant IF-state. Soft Comput. 22, 5043–5049 (2018) 8. Nowak, P.: Monotone measures of intuitionistic fuzzy sets. In: Wagenknecht, M., Hampel, R. (eds.) Proceedings of the 3rd Conference of the European Society for Fuzzy Logic and Technology, pp. 172–176. University of Applied Sciences at Zittau/G¨ orlitz, Zittau (2003) 9. Nowak, P., Hryniewicz, O.: Generalized versions of MV-algebraic central limit theorems. Kybernetika 51(5), 765–783 (2015) 10. Nowak, P., Hryniewicz, O.: On generalized versions of central limit theorems for IF-events. Inf. Sci. 355, 299–313 (2016) 11. Nowak, P., Hryniewicz, O.: On central limit theorems for IV-events. Soft Comput. 22, 2471–2483 (2018) 12. Nowak, P., Hryniewicz, O.: On MV-algebraic versions of the strong law of large numbers. Entropy 21(7), 710 (2019) 13. Nowak, P., Hryniewicz, O.: Strong laws of large numbers for IVM-events. IEEE Trans. Fuzzy Syst. 27(12), 2293–2301 (2019) 14. Renˇcov´ a, M.: A generalization of probability theory on MV-algebras to IF-events. Fuzzy Sets Syst. 161(12), 1726–1739 (2010) ˇ epniˇcka, M., Nov´ 15. Rieˇcan, B.: M-probability theory on IF events. In: Stˇ ak, V., Bodenhofer, U. (eds.) New Dimensions in Fuzzy Logic and Related Technologies. In: Proceeding of the 5th EUSFLAT, vol. I, pp. 227–230. Universitas Ostraviensis, Ostrava (2007) 16. Rieˇcan, B.: Probability theory on IF events. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds.) Algebraic and Proof-theoretic Aspects of Non-classical Logics. Lecture Notes in Computer Science, vol. 4460, pp. 290–308. Springer, Heidelberg (2007) 17. Rieˇcan, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 869–909. Elsevier, Amsterdam (2002) 18. Szmidt, E., Kacprzyk, J.: A concept of a probability of an intuitionistic fuzzy event. In: FUZZ-IEEE 1999. 1999 IEEE International Fuzzy Systems. Conference Proceedings, pp. 1346–1349. IEEE, Seoul (1999)

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence ˇ Katar´ına Cunderl´ ıkov´ a(B) Mathematical Institute, Slovak Academy of Sciences, ˇ anikova 49, 814 73 Bratislava, Slovakia Stef´ [email protected]

Abstract. The aim of this paper is formulate an almost everywhere convergence using an intuitionistic fuzzy probability. We compare two concepts of almost everywhere convergence and we study P-almost everywhere convergence of sequence of intuitionistic fuzzy observables induced by Borel measurable function.

1

Introduction

In [1,2] K.T. Atanassov introduced the notion of intuitionistic fuzzy sets. Then in [5] Grzegorzewski and Mr´ owka defined the probability on the family of intuitionistic fuzzy events N = {(μA , νA ) ; μA + νA ≤ 1}, where μA , νA are S-measurable, as a mapping P from the family N to the set of all compact intervals in R by the formula    μA dP , 1 − νA dP , P((μA , νA )) = Ω

Ω

where (Ω, S, P ) is probability space. This IF -probability was axiomatically characterized by B. Rieˇcan (see [8]). Since the intuitionistic fuzzy probability P can be decomposed to two intuitionistic fuzzy states m (see [9,12]), then we can use the results from papers [3,4] and we can formulate a P-almost everywhere convergence. In this paper we compare two concepts of almost everywhere convergence and we study P-almost everywhere convergence of sequence of intuitionistic fuzzy observables induced by Borel measurable function. Remark that in a whole text we use a notation “IF ” for short a phrase “intuitionistic fuzzy”.

2

IF-Event, IF-Probability, IF-State and IF-Observable

In this section we introduce the basic notions from IF -probability theory, see [1,2,11–13]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 54–65, 2021. https://doi.org/10.1007/978-3-030-77716-6_5

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence

55

Definition 1. Let Ω be a nonempty set. An IF -set A on Ω is a pair (μA , νA ) of mappings μA , νA : Ω → [0, 1] such that μA + νA ≤ 1Ω . Definition 2. Start with a measurable space (Ω, S). Hence S is a σ-algebra of subsets of Ω. An IF -event is called an IF -set A = (μA , νA ) such that μA , νA : Ω → [0, 1] are S-measurable. The family of all IF -events on (Ω, S) will be denoted by F , μA : Ω −→ [0, 1] be called the nonmembership function, νA : Ω −→ [0, 1] be called the nonmembership function. If A = (μA , νA ) ∈ F , B = (μB , νB ) ∈ F , then we define the Lukasiewicz binary operations ⊕,  on F by A ⊕ B = ((μA + μB ) ∧ 1Ω , (νA + νB − 1) ∨ 0Ω )), A  B = ((μA + μB − 1) ∨ 0Ω , (νA + νB ) ∧ 1Ω )) and the partial ordering is given by A ≤ B ⇐⇒ μA ≤ μB , νA ≥ νB . In paper we use max-min connectives defined by A ∨ B = (μA ∨ μB , νA ∧ νB ), A ∧ B = (μA ∧ μB , νA ∨ νB ) and the de Morgan rules (a ∨ b)∗ = a∗ ∧ b∗ , (a ∧ b)∗ = a∗ ∨ b∗ , where a∗ = 1 − a. Example 1. Fuzzy set f : Ω −→ [0, 1] can be regarded as IF -set, if we put A = (f, 1Ω − f ). If f = χA , then the corresponding IF -set has the form A = (χA , 1Ω − χA ) = (χA , χA ). In this case A ⊕ B corresponds to the union of sets, A  B to the intersection of sets and ≤ to the set inclusion. Consider a probability space (Ω, S, P ). Then in [5] the IF -probability P(A) of an IF -event A = (μA , νA ) ∈ F has been defined as a compact interval by the equality    P(A) =

μA dP , 1 − Ω

νA dP . Ω

Let J be the family of all compact intervals. Then the mapping P : F → J can be defined axiomatically similarly as in [8].

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Definition 3. Let F be the family of all IF -events in Ω. A mapping P : F → J is called an IF-probability, if the following conditions hold: (i) P((1Ω , 0Ω )) = [1, 1], P((0Ω , 1Ω )) = [0, 0]; (ii) if A  B = (0Ω , 1Ω ), then P(A ⊕ B) = P(A) + P(B); (iii) if An A, then P(An ) P(A). (Recall that [αn , βn ] [α, β] means that αn α, βn β, but An = (μAn , νAn ) A = (μA , νA ) means μAn μA , νAn νA .) IF -probability P is called separating, if   P (μA , νA ) = [P (μA ), 1 − P (νA )], where the functions P , P : T → [0, 1] are probabilities. Of course, each P(A) is an interval, denote it by P(A) = [P (A), P (A)]. By this way we obtain two functions P : F → [0, 1], P : F → [0, 1] and some properties of P can be characterized by some properties of P , P , see [9]. Theorem 1. Let P : F → J and P(A) = [P (A), P (A)] for each A ∈ F . Then P is an IF-probability if and only if P and P are IF-states. Proof. In [9] Theorem 2.3.   Recall that by an intuitionistic fuzzy state (IF -state) m we understand each mapping m : F → [0, 1] which satisfies the following conditions (see [10]): (i) m((1Ω , 0Ω )) = 1, m((0Ω , 1Ω )) = 0; (ii) if A  B = (0Ω , 1Ω ) and A, B ∈ F , then m(A ⊕ B) = m(A) + m(B); (iii) if An A (i.e. μAn μA , νAn νA ), then m(An ) m(A). Now we explain the notion of an observable. Let J be the family of all intervals in R of the form [a, b) = {x ∈ R : a ≤ x < b}. Then the σ-algebra σ(J) is denoted by B(R) and it is called the σ-algebra of Borel sets, its elements are called Borel sets (see [14]). Definition 4. By an IF -observable on F we understand each mapping x : B(R) → F satisfying the following conditions: (i) x(R) = (1Ω , 0Ω ), x(∅) = (0Ω , 1Ω ); (ii) if A ∩ B = ∅ and A, B ∈ B(R), then x(A)  x(B) = (0Ω , 1Ω ) and x(A ∪ B) = x(A) ⊕ x(B); (iii) if An A and An , A ∈ B(R), n ∈ N , then x(An ) x(A).

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence

57

Similarly we can define the notion of n-dimensional IF -observable. Definition 5. By an n-dimensional IF -observable on F we understand each mapping x : B(Rn ) → F satisfying the following conditions: (i) x(Rn ) = (1Ω , 0Ω ), x(∅) = (0Ω , 1Ω ); (ii) if A∩B = ∅ and A, B ∈ B(Rn ), then x(A)x(B) = (0Ω , 1Ω ) and x(A∪B) = x(A) ⊕ x(B); (iii) if An A and An , A ∈ B(Rn ), n ∈ N , then x(An ) x(A).

3

Product Operation, Joint IF-Observable and Function of Several IF-Observables

In [6] we introduced the notion of product operation on the family of IF -events F and showed an example of this operation. Definition 6. We say that a binary operation · on F is product if it satisfying the following conditions: (i) (1Ω , 0Ω ) · (a1 , a2 ) = (a1 , a2 ) for each (a1 , a2 ) ∈ F ; (ii) the operation · is commutative and associative; (iii) if (a1 , a2 )  (b1 , b2 ) = (0Ω , 1Ω ) and (a1 , a2 ), (b1 , b2 ) ∈ F , then       (c1 , c2 ) · (a1 , a2 ) ⊕ (b1 , b2 ) = (c1 , c2 ) · (a1 , a2 ) ⊕ (c1 , c2 ) · (b1 , b2 ) and



   (c1 , c2 ) · (a1 , a2 )  (c1 , c2 ) · (b1 , b2 ) = (0Ω , 1Ω )

for each (c1 , c2 ) ∈ F ; (iv) if (a1n , a2n ) (0Ω , 1Ω ), (b1n , b2n ) (0Ω , 1Ω ) and (a1n , a2n ), (b1n , b2n ) ∈ F , then (a1n , a2n ) · (b1n , b2n ) (0Ω , 1Ω ). In the following theorem is the example of product operation for IF -events. Theorem 2. The operation · defined by (x1 , y1 ) · (x2 , y2 ) = (x1 · x2 , y1 + y2 − y1 · y2 ) for each (x1 , y1 ), (x2 , y2 ) ∈ F is product operation on F . Proof. In [6] Theorem 1.   In [11] B. Rieˇcan defined the notion of a joint IF -observable and he proved its existence.

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Definition 7. Let x, y : B(R) → F be two IF-observables. The joint IFobservable of the IF-observables x, y is a mapping h : B(R2 ) → F satisfying the following conditions: (i) h(R2 ) = (1Ω , 0Ω ), h(∅) = (0Ω , 1Ω ); (ii) if A, B ∈ B(R2 ) and A ∩ B = ∅, then h(A ∪ B) = h(A) ⊕ h(B) and h(A)  h(B) = (0Ω , 1Ω ); (iii) if A, A1 , . . . ∈ B(R2 ) and An A, then h(An ) h(A); (iv) h(C × D) = x(C) · y(D) for each C, D ∈ B(R). Theorem 3. For each two IF-observables x, y : B(R) → F there exists their joint IF-observable. Proof. In [11] Theorem 3.3.

 

Remark 1. The joint IF -observable of IF -observables x, y from Definition 7 is two-dimensional IF -observable. If we have several IF -observables and a Borel measurable function, we can define the IF -observable, which is the function of several IF -observables. About this says the following definition. Definition 8. Let x1 , . . . , xn : B(R) → F be IF-observables, hn their joint IFobservable and gn : Rn → R a Borel measurable function. Then we define the IF-observable gn (x1 , . . . , xn ) : B(R) → F by the formula   gn (x1 , . . . , xn )(A) = hn gn−1 (A) . for each A ∈ B(R).

4

Lower and Upper Limits, m-Almost Everywhere Convergence

In [3] we defined the notions of lower and upper limits for a sequence of IF observables and showed the connection between two kinds of m-almost everywhere convergence. Definition 9. We shall say that a sequence (xn )n of IF -observables has lim sup, n→∞

if there exists an IF -observable x : B(R) → F such that x((−∞, t)) =

∞  ∞  ∞  p=1 k=1 n=k

xn

− ∞, t −

1

p

for every t ∈ R. We write x = lim sup xn . n→∞

Note that if another IF -observable y satisfies the above condition, then m ◦ y = m ◦ x.

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence

59

Definition 10. A sequence (xn )n of IF -observables has lim inf , if there exists n→∞ an IF -observable x such that x((−∞, t)) =

∞  ∞  ∞ 

xn

− ∞, t −

p=1 k=1 n=k

1

p

for all t ∈ R. Notation: x = lim inf xn . n→∞

Definition 11. Let (xn )n be a sequence of IF -observables on an IF -space (F , m). We say that (xn )n converges m-almost everywhere to 0, if   k+i     ∞  ∞  ∞  1 1 1 1 m xn xn = lim lim lim m = 1. − , − , p→∞ k→∞ i→∞ p p p p p=1 k=1 n=k

n=k

Remark 2. The defining formula is equivalent to the following equality    ∞  ∞  ∞ 1 1 m xn R\ − , = 0. p p p=1 k=1 n=k

Definition 12. By the zero IF -observable 0F we shall denote the IF -observable defined by the following formula ⎧ ⎨ (1Ω , 0Ω ), if t > 0 0F ((−∞, t)) = ⎩ (0Ω , 1Ω ), if t ≤ 0 for each t ∈ R. Remark 3. The zero IF -observable 0F can be rewrite in the following form ⎧ ⎨ (1Ω , 0Ω ), if 0 ∈ A 0F (A) = ⎩ /A (0Ω , 1Ω ), if 0 ∈ for each A = (−∞, t) and t ∈ R. Proposition 1. A sequence (xn )n of an IF-observables converges m-almost everywhere to 0 if and only if     ∞  ∞  ∞  ∞  ∞ ∞ 1 1 m xn xn − ∞, t − − ∞, t − =m p p p=1 k=1 n=k p=1 k=1 n=k   = m 0F ((−∞, t)) , for every t ∈ R. Proof. In [3] Proposition 4.1.   In accordance to Proposition 1 we can extend the notion of m-almost everywhere convergence by the following way.

60

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Definition 13. A sequence (xn )n of an IF -observables converges m-almost everywhere to an IF -observable x, if     ∞  ∞  ∞  ∞  ∞ ∞ 1 1 m xn xn − ∞, t − − ∞, t − =m p p p=1 k=1 n=k p=1 k=1 n=k   = m x((−∞, t)) , for every t ∈ R. Recall, that the corresponding probability space is (RN , σ(C), P ), where C is the family of all sets of the form {(ti )∞ i=1 : t1 ∈ A1 , . . . , tn ∈ An }, and P is the probability measure determined by the equality     P {(ti )∞ i=1 : t1 ∈ A1 , . . . , tn ∈ An } = m x1 (A1 ) · . . . · xn (An ) . The corresponding projections ξn : RN → R are defined by the equality   ξn (ti )∞ i=1 = tn . Theorem 4. Let (xn )n be a sequence of IF-observables, (ξn )n be the sequence of Borel measurable functions of corresponding projections, (g  n )n be a sequence  gn : Rn → R. If the sequence gn (ξ1 , . . . , ξn ) n converges P -almost everywhere,   then the sequence gn (x1 , . . . , xn ) n converges m-almost everywhere and  

 

m lim sup gn (x1 , . . . , xn ) (−∞, t) = m lim inf gn (x1 , . . . , xn ) (−∞, t) n→∞

n→∞

for each t ∈ R. Moreover

       P {u ∈ RN : lim sup gn ξ1 (u), . . . , ξn (u) < t} = m lim sup gn (x1 , . . . , xn ) (−∞, t) n→∞

n→∞

for each t ∈ R. Proof. In [4] Theorem 5.1.

5

 

P-Almost Everywhere Convergence

In paper [7] we study P-almost everywhere convergence only for a separating IF -probability. Such the IF -probability can be decomposed to two IF -states, we try apply the results from [3,4] for an IF -probability. Definition 14. Let (xn )n be a sequence of IF -observables on an IF -space (F , P). We say that (xn )n converges P-almost everywhere to 0, if   k+i   ∞  ∞  ∞  1 1 1 1 P xn xn − , − , = lim lim lim P p→∞ k→∞ i→∞ p p p p p=1 k=1 n=k

n=k

= [1, 1] = 1.

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence

61

Remark 4. The defining formula is equivalent to the following equality P

 ∞  ∞  ∞ p=1 k=1 n=k

  1 1 xn R\ − , = [0, 0] = 0. p p

Theorem 5. A sequence (xn )n of an IF-observables converges P-almost everywhere to 0 if and only if it converges P -almost everywhere and P -almost everywhere to 0. Proof. “⇒” Let P be an IF -probability and let (xn )n converge P-almost everywhere. Then by Definition 14 we have P

 ∞  ∞  ∞

 xn

p=1 k=1 n=k

1 1 − , p p

= lim lim lim P p→∞ k→∞ i→∞

 k+i 

 xn

n=k

1 1 − , p p



= [1, 1] = 1. Using Theorem 1 we obtain that  ∞  ∞  ∞



1 1 P xn − , p p p=1 k=1 n=k       ∞  ∞  ∞  ∞  ∞ ∞ 1 1 1 1 xn xn − , − , = P ,P p p p p p=1 p=1 k=1 n=k

k=1 n=k

and  k+i 



1 1 − , p p n=k   k+i    k+i    1 1 1 1 xn xn − , − , = P ,P , p p p p P

xn

n=k

n=k

where P , P are IF -states. Therefore P



 ∞  ∞  ∞

 xn

p=1 k=1 n=k

P

 ∞  ∞  ∞

p=1 k=1 n=k

1 1 − , p p

 xn

−

1 1 , p p





= lim lim lim P p→∞ k→∞ i→∞



= lim lim lim P p→∞ k→∞ i→∞

 k+i 

 xn

n=k

 k+i 

n=k

1 1 − , p p

 xn

−

1 1 , p p

 = 1,  =1

i.e. a sequence (xn )n of IF -observables converges P -almost everywhere and P almost everywhere to 0. “⇐” The opposite direction can be proved by similar way.  

ˇ K. Cunderl´ ıkov´ a

62

Proposition 2. A sequence (xn )n of an IF-observables converges P-almost everywhere to 0 if and only if P

 ∞  ∞  ∞

 xn

p=1 k=1 n=k

1 − ∞, t − p

=P

 ∞  ∞  ∞

 xn

p=1 k=1 n=k

1 − ∞, t − p



  = P 0F ((−∞, t)) ,

for every t ∈ R. Proof. “⇒” Let a sequence (xn )n of an IF -observables converges P-almost everywhere to 0. Then by Theorem 5 the sequence (xn )n of an IF -observables converges P -almost everywhere and P -almost everywhere to 0. Hence by Proposition 1 P



 ∞  ∞  ∞ p=1 k=1 n=k

 xn

1 − ∞, t − p

=P



 ∞  ∞  ∞ p=1 k=1 n=k

 xn

1 − ∞, t − p



  = P 0F ((−∞, t)) ,

and     ∞  ∞  ∞  ∞  ∞ ∞ 1 1 P xn xn − ∞, t − − ∞, t − = P p p p=1 k=1 n=k p=1 k=1 n=k   = P 0F ((−∞, t)) , for every t ∈ R. Since by Theorem 1 we have    ∞  ∞  ∞ 1 P xn − ∞, t − p p=1 k=1 n=k        ∞  ∞  ∞  ∞  ∞ ∞ 1 1 xn xn = P ,P , − ∞, t − − ∞, t − p p p=1 k=1 n=k p=1 k=1 n=k    ∞  ∞  ∞ 1 xn P − ∞, t − p p=1 k=1 n=k        ∞  ∞  ∞ ∞ ∞  ∞  1 1 xn xn = P , P , − ∞, t − − ∞, t − p p p=1 k=1 n=k p=1 k=1 n=k     P 0F ((−∞, t)) = P 0F ((−∞, t)) , P 0F ((−∞, t)) ,

then

Intuitionistic Fuzzy Probability and Almost Everywhere Convergence

P

 ∞  ∞  ∞

 xn

p=1 k=1 n=k

− ∞, t −

1 p

=P

 ∞  ∞  ∞

 xn

− ∞, t −

p=1 k=1 n=k

63

1 p



 = P 0F ((−∞, t)) , 

for every t ∈ R. “⇐” The opposite direction can be proved by similar way.   In accordance to Proposition 2 we can extend the notion of P-almost everywhere convergence by the following way. Definition 15. A sequence (xn )n of an IF -observables converges P-almost everywhere to an IF -observable x, if     ∞  ∞  ∞  ∞  ∞ ∞ 1 1 xn xn − ∞, t − − ∞, t − P =P p p p=1 k=1 n=k p=1 k=1 n=k   = P x((−∞, t)) , for every t ∈ R. Sometimes we need to work with sequence of IF -observables induced by a Borel measurable function. Recall, that the corresponding probability spaces are (RN , σ(C), P  ) and N (R , σ(C), P  ), where C is the family of all sets of the form {(ti )∞ i=1 : t1 ∈ A1 , . . . , tn ∈ An }, and P  , P  are the probability measures determined by the equalities      P  {(ti )∞ i=1 : t1 ∈ A1 , . . . , tn ∈ An } = P x1 (A1 ) · . . . · xn (An ) ,      P  {(ti )∞ i=1 : t1 ∈ A1 , . . . , tn ∈ An } = P x1 (A1 ) · . . . · xn (An ) . The corresponding projections ξn : RN → R are defined by the equality   ξn (ti )∞ i=1 = tn . Theorem 6. Let (xn )n be a sequence of IF-observables, (ξn )n be the sequence of Borel measurable functions of corresponding projections, (g  n )n be a sequence  gn : Rn → R. If the sequence gn (ξ1 , . . . , ξn ) n converges P  -almost everywhere   and P  -almost everywhere, then the sequence gn (x1 , . . . , xn ) n converges Palmost everywhere and  

 

P lim sup gn (x1 , . . . , xn ) (−∞, t) = P lim inf gn (x1 , . . . , xn ) (−∞, t) n→∞

n→∞

for each t ∈ R. Moreover        P lim sup gn (x1 , . . . , xn ) (−∞, t) = P  E , P  E n→∞

  for each t ∈ R, where E = {u ∈ RN : lim supn→∞ gn ξ1 (u), . . . , ξn (u) < t}.

64

ˇ K. Cunderl´ ıkov´ a

  Proof. “⇒” Let the sequence gn (ξ1 , . . . , ξn ) n converges P  -almost everywhere   and P  -almost everywhere. Then by Theorem 4 the sequence gn (x1 , . . . , xn ) n converges P -almost everywhere and P -almost everywhere and       P lim sup gn (x1 , . . . , xn ) (−∞, t) = P lim inf gn (x1 , . . . , xn ) (−∞, t) n→∞ n→∞       P lim sup gn (x1 , . . . , xn ) (−∞, t) = P lim inf gn (x1 , . . . , xn ) (−∞, t) n→∞

n→∞

for each t ∈ R. Moreover    

P  E = P lim sup gn (x1 , . . . , xn ) (−∞, t) , n→∞    

  P E = P lim sup gn (x1 , . . . , xn ) (−∞, t) , n→∞

(1) (2)

(3) (4)

  for each t ∈ R, where E = {u ∈ RN : lim supn→∞ gn ξ1 (u), . . . , ξn (u) < t}. Let P be an IF -probability. By Theorem 1 it can be decomposed to two IF = [P (A),P (A)] for each A ∈ F . Therefore by states P , P such that P(A)  Theorem 5 the sequence gn (x1 , . . . , xn ) n converges P-almost everywhere and by (1), (2) we have  

 

P lim sup gn (x1 , . . . , xn ) (−∞, t) = P lim inf gn (x1 , . . . , xn ) (−∞, t) n→∞

n→∞

for each t ∈ R. Moreover by (3), (4) we obtain        P lim sup gn (x1 , . . . , xn ) (−∞, t) = P  E , P  E n→∞

  for each t ∈ R, where E = {u ∈ RN : lim supn→∞ gn ξ1 (u), . . . , ξn (u) < t}. “⇐” The opposite direction can be proved by similar way.

6

 

Conclusion

We proved a modification of P-almost everywhere convergence with help of lim sup and lim inf in an intuitionistic fuzzy case. We compare two kinds of definition of the P-almost everywhere convergence for the intuitionistic fuzzy events and we show that they are equivalent. The Theorem 6 is important for the proof the Individual ergodic theorem based on IF -probability in intuitionistic fuzzy case.

References 1. Atanassov, K.T.: Intuitionistic Fuzzy Sets: Theory and Applications. Studies in Fuzziness and Soft Computing. Physica-Verlag, Heidelberg (1999) 2. Atanassov, K.T.: On Intuitionistic Fuzzy Sets. Springer, Berlin (2012)

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ˇ 3. Cunderl´ ıkov´ a, K.: Upper and lower limits and m-almost everywhere convergence of intuitionistic fuzzy observables. In: Notes on Intuitionistic Fuzzy Sets, vol. 24, no. 4, pp. 40–49 (2018) ˇ 4. Cunderl´ ıkov´ a, K.: m-almost everywhere convergence of intuitionistic fuzzy observables induced by Borel measurable function. In: Notes on Intuitionistic Fuzzy Sets, vol. 25, no. 2, pp. 29–40 (2019) 5. Grzegorzewski, P., Mr´ owka, E.: Probability of intuistionistic fuzzy events. In: Grzegorzewski, P., et al. (eds.) Soft Methods in Probability, Statistics and Data Analysis, pp. 105–115. Physica Verlag, New York (2002) 6. Lendelov´ a, K.: Conditional IF-probability. In: Soft Methods for Integrated Uncertainty Modelling. Advances in Soft Computing, pp. 275–283 (2006) 7. Lendelov´ a, K.: Almost everywhere convergence in family of IF-events with product. In: New Dimensions in Fuzzy Logic and Related Technologies: Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, 11–14 September 2007, pp. 231–236 (2007) 8. Rieˇcan, B.: A descriptive definition of the probability on intuitionistic fuzzy sets. In: Wagenecht, M., Hampet, R. (eds.) EUSFLAT 2003, pp. 263–266. Zittau-Goerlitz University of Applied Sciences (2003) 9. Rieˇcan, B.: On the probability on IF-sets and MV-algebras. In: Notes on IFS, vol. 11, no. 6, pp. 21–25 (2005) 10. Rieˇcan, B.: On a problem of Radko Mesiar: general form of IF-probabilities. Fuzzy Sets Syst. 152, 1485–1490 (2006) 11. Rieˇcan, B.: On the probability and random variables on IF events. In: Ruan, D., et al. (eds.) Applied Artifical Intelligence. Proceedings of the 7th FLINS Conference, Genova, pp. 138–145 (2006) 12. Rieˇcan, B.: Probability theory on intuitionistic fuzzy events, algebraic and prooftheoretic aspects of non-classical logics. Papers in Honour of Daniele Mundici’s 60th Birthday. Lecture Notes in Computer Science, vol. 4460 (2007) 13. Rieˇcan, B.: Analysis of fuzzy logic models. In: Koleshko, V. (ed.) Intelligent Systems, pp. 219–244. INTECH (2012) 14. Rieˇcan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava (1997)

Classification of Images by Using Distance Functions Defined on Intuitionistic Fuzzy Sets Alˇzbeta Michal´ıkov´ a1,2(B) 1

Faculty of Natural Science, Matej Bel University, Bansk´ a Bystrica, Slovakia [email protected] 2 Mathematical Institut, Slovak Academy of Sciences, Bratislava, Slovakia

Abstract. Intuitionistic fuzzy sets are mathematical structures which could be, together with operations, relations and functions defined on them, applied in many different spheres. In this paper, the distance functions defined on intuitionistic fuzzy sets are used to solve the problem of classification of images. Our specific problem is that we need to classify tire tread images into selected classes. Each class is characterized by its pattern. In the first step, the pre-processing of the image into the numeric data is done. The numerical data are represented by the specific vector. Then the values of membership function, non-membership function and hesitance margin for each coordinate of vector are calculated. As the last step of algorithm, the values of distance function between image and class patterns are computed. Classification is performed with the use of special function Sim whose values are dependent on distance function.

1

Introduction

Intuitionistic fuzzy sets (shortly IFSs) were introduced by Krassimir Atanassov in 1983 [1]. Since that time many new operations, relations and functions on this mathematical structure have been defined. In this paper, we use distance functions defined on IFSs for tire tread images classification. The research presented in this paper is motivated by several consultations with active crime scene investigators. During these consultations, it became apparent, that criminology department is in critical need of advanced software for tire tread print identification. The current software used by crime scene investigators is outdated, time demanding and hard to work with. This led to development of new software tool for tire print identification. Prototype of this application is still being developed by department of Computer Science of Matej Bel University. The problem can be formalized as follows: While on the crime scene, criminologists often find various types of prints, such as foot prints, finger prints and one of most common types of prints - tire tread prints. If there is a possibility to recognize the brand or manufacturer of tires present on the crime c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 66–74, 2021. https://doi.org/10.1007/978-3-030-77716-6_6

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scene, they could be one of the important evidence which could contribute to the conviction of offender. To recognize the tire tread brand or manufacturer, the database of tire tread prints needs to be build [2–4]. The data for this database could be obtained by making the photos of tires for example at the concrete seller. By using this approach, one could make photos of high quality with the proper position of the tires, the proper lighting, etc. Of course this possibility is time consuming and resource demanding. On the other hand, the images of the tire treads could be obtained from the web pages. Not all images are in the required quality and in the proper position, but the access to these images is simpler. In our previous work, presented in [5], we described process of creating database of download tire tread images. We developed the web crawling application to extract the relevant images together with important information from the predefined web pages. Not all images are suitable for the additional processing and for comparing with the tire tread prints obtained at the crime scene. In [6] authors presented seven basic types/classes of the tire tread images and they described the advantages/disadvantages of each class from the point of comparison mentioned in the paper [6]. This paper presents one of the first steps of tire tread print identification tire tread position identification. The images obtained from the web crawling application [5] are classified also into the seven classes. The classes of tire tread images used in this article are displayed on Fig. 1.

Fig. 1. Used classes of tire tread images

The next step of processing of the images obtained from these classes is the specification of the tire tread prints position. The position of tire tread print is specific for each class. As an example, the position of tire tread print of the Class1 is presented on the Fig. 2b. After some pre-processing one could get the tire treat print as shown on Fig. 2c. This type of image could be added to the database of tire tread prints and used for matching with tire tread prints which are found at crime scene. The paper is structured as follows: In the Sect. 2 we give a brief introduction into the theory of intuitionistic fuzzy sets and we define the structures which are used in this paper. In the Sect. 3 we discuss the way how we could prepare the data for classification. In Sect. 4 the obtained results are summarized and finally in the Sect. 5 the conclusions and some ideas for future work are mentioned.

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Fig. 2. Visualization of position of tire tread print

2

Intuitionistic Fuzzy Sets

Definition 1. Let X be a universe. An intuitionistic fuzzy set A is a set A = {x, μ A(x), ν A(x)|x ∈ X }

(1)

of the functions μ A : X → [0, 1], ν A : X → [0, 1] such that 0 ≤ μ A(x) + ν A(x) ≤ 1.

(2)

Function μ A is called the membership function and function ν A is called the non-membership function. In this paper the short notation IFSs is used for intuitionistic fuzzy sets. The family of all IFSs is denoted by F . There is another function defined on F , function π A which is defined as π A(x) = 1 − μ A(x) − ν A(x).

(3)

This function is called hesitation margin. Let us have two IFSs A = (μ A(x), ν A(x)), B = (μB (x), νB (x)). Then it holds A = B ⇐⇒ [μ A(x) = μB (x) & ν A(x) = νB (x)]

(4)

A ⊆ B ⇐⇒ [μ A(x) ≤ μB (x) & ν A(x) ≥ νB (x)]

(5)

Definition 2. Let S be a real-value function such that S : F × F → [0, 1]. S is called similarity measure defined on IFSs if for every A, B, C ∈ F it holds • S(A, B) = S(B, A), • S(A, B) = 1 iff A = B, • If A ⊆ B ⊆ C then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). Definition 3. Let d be a real-value function such that d : F × F → [0, ∞]. Function d is called distance function defined on IFSs if for every A, B, C ∈ F it holds

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• d(A, B) = d(B, A), • d(A, B) = 0 iff A = B, • If A ⊆ B ⊆ C then d(A, C) ≥ d(A, B) and d(A, C) ≥ d(B, C). In the paper [7] authors considered the discrete universe X = {x1, x2, . . . , xh } and for classification of the tire tread images they used the cosine-base similarity measure defined on IFSs as follows SC (A, B) =

h μ A(xi )μB (xi ) + ν A(xi )νB (xi ) 1 .   h i=1 μ2A(xi ) + ν 2A(xi ) μ2B (xi ) + νB2 (xi )

(6)

This IFSs similarity measure has been designed by Ye [8] and reader could see that the third function, hesitance margin π, is not used in the notation of this measure. As presented in the paper [7] by the use of this similarity measure in specific experiment with classification of tire tread prints, 90.8% of 326 images were classified correctly. In the past, there were designed various types of similarity measures defined on IFSs. We would like to improve the results of classification and therefore we decided to use another similarity measure. During the study of different articles we found some interesting ideas. Let us mention some of them. It is well known that standard cosine similarity measure is defined as a cosine of the angle between two vectors with their starting points placed at the beginning of the coordinate system. At Fig. 3 there is shown the situation with three points. The similarities of the points represented as the ending points of the vectors displayed on Fig. 3b and Fig. 3c are evaluated as the same. It is clear that the points on Fig. 3b are more similar then the points on Fig. 3c. Since cosine-base similarity measure SC defined on IFSs could give the similar results it is important to use another approach to classification of the images.

Fig. 3. Standard cosine similarity measure

In the paper [9] authors specified examples of counter-intuitive results for various types of similarity measures. There was also mentioned counter-intuitive result for SC . In the same paper authors used another approach to compare the IFSs. They computed the distance between two IFSs A and B and distance between A and BC , where BC is a complement of B and they asked the question: Is A more similar or more dissimilar to B? To answer this question authors defined function Sim in the following way:

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Definition 4. Let A, B, BC ∈ F , where BC is a complement of B and it holds A = (μ A(x), ν A(x), π A(x)), B = (μB (x), νB (x), πB (x)), BC = (νB (x), μB (x), πB (x)). Let d be the distance function defined on F , then Sim(A, B) =

d(A, B) . d(A, BC )

(7)

For function Sim it holds • • • • •

0 ≤ Sim(A, B) ≤ ∞, Sim(A, B) = 0 means identity of A and B, Sim(A, B) = 1 means A is to same extent similar to B and BC , Sim(A, B) > 1 means A is more similar to BC as to B, d(A, BC ) = 0 means complete dissimilarity of A and B.

To analyse the similarity of two IFSs A and B, the values Sim(A, B) ∈ [0, 1] are interesting. Mentioned approach was also used in this paper. Of course, there are defined various distance functions on IFSs. In this paper we used normalized Hamming distance function n  1  | μ A(xi ) − μB (xi )| + |ν A(xi ) − νB (xi )| + |π A(xi ) − πB (xi )| dH (A, B) = (8) 2n i=1 and normalized Euclidean distance function  1 n  2 1  2 2 2 (μ A(xi ) − μB (xi )) + (ν A(xi ) − νB (xi )) + (π A(xi ) − πB (xi )) . dE (A, B) = 2n i=1 (9)

3

Preparation of the Data

As it was mentioned in the introduction, we downloaded the images of tire treads from different web pages. We need to prepare them in such way that we will be able to process them by using intuitionistic fuzzy sets distance function. The process of adaptation of data into the right format is called pre-processing and in this research it could be divided into two parts. First part is pre-processing of the images, the second part is pre-processing of the data. 3.1

Pre-processing of the Images

For pre-processing of the images we developed the algorithm which consist of following four steps 1. 2. 3. 4.

Converting image into the JPEG format. Removing white pixels on all sides of the image. Converting image into the black and white format. Dividing image into 16 parts.

In the fourth step the image of tire tread is divided into 16 rectangles as it could be seen on Fig. 4. Then the image is represented by 16 coordinate vector where each coordinate represents the number of white pixels in given part.

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Fig. 4. Dividing image into 16 parts

3.2

Pre-processing of the Data

Now the second part of adaptation could start - pre-processing of the data. Since we were classifying the images, for each of seven classes mentioned in Introduction we choose so called templates. Templates represent their class while consisting of 3 images from each class. Therefore, there are 21 templates which are represented by 16 coordinate vectors. For pre-processing of the data we used the approach that was described in the paper [10]. Let us have image i (i = 1, 2, . . . , 21) represented by a 16-coordinate vector (10) Vi = (xi, 1, xi, 2, . . . , xi, 16 ). We started with the normalization of each coordinate by using formula ni, j =

xi, j − X¯j , sj

(11)

where j = 1, 2, . . . , 16, X¯j is a mean and s j is the standard deviation calculated from the j-th coordinate of all images in the template database. Then the membership degree of each template coordinate was calculated by the weighted sigmoid function rj , (12) μi, j = 1 + e−ni, j where r j is a weight value. Similarly, the non-membership degree of each template coordinate was calculated by the formula νi, j =

r j∗ 1 + eni, j

.

(13)

In the end the value of hesitance margin of each template coordinate was calculated by the formula (14) πi, j = 1 − μi, j − νi, j . We used different combination of values of the weights r j and r j∗ from unit interval (specifically combination of the values 0; 0.1; 0.2; . . . ; 1). We obtained

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the most satisfying results with the use of values r j = 1 and r j∗ = 0.6 which gave us the similar results as in the paper [10]. Since we were dividing the images into the seven classes we defined the seven patterns by the formula 

 16 (15) Pm = x¯m, j , μ¯m, j , ν¯m, j , π¯m, j j=1 , where m = 1, 2, . . . , 7 and the values μ¯m, j , ν¯m, j and π¯m, j represent the arithmetical mean of function values of the templates which belong to given class. 3.3

Classification of the Images

After pre-processing of the data we were ready to classify any image. We used following algorithm 1. Take any image. 2. Use pre-processing and characterize image by 16-coordinate vector (see Sect. 3.1). 3. For each coordinate calculate the value of membership function, nonmembership function and hesitation margin by the same formulas as were used for templates (see Sect. 3.2). 4. Calculate the distance between the image and each of seven patterns. 5. Classify image into the suitable class by using the Sim function. The image was classified into that class where the value of Sim function between the image and pattern was the lowest.

4

Experimental Results

In the paper [7] we used 326 images of tire treads which were downloaded from the web pages with the different names of tire brand. For the ability to compare the results with the mentioned experiment we took the same 326 images. We developed the software that pre-processed the images by using the above mentioned algorithms. As the result, the program creates seven folders (each folder represents one class) and moves the images into the folders as they are classified by the described process. To help us quickly identify the incorrectly classified images there is also given one template image into the each created folder. In our approach we used two distance functions. In the first step we used the normalized Hamming distance function. With the use of this distance function we didn’t reach better results than by using cosine-base similarity measure (mentioned in the paper [7]). In the second step we used the normalized Euclidean distance function. With the use of this distance function we reached partially better results than by using cosine-base similarity measure. We reached the value 91.1% of correct classified images. The comparison of the results obtained by using cosine-base similarity measure SC and by using normalized Euclidean distance function dE is presented in the Table 1.

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Table 1. Results of classification Class Class Class Class Class Class Class Class Sum

SC dE Correct Incorrect Sum Correct Incorrect Sum 1 2 3 4 5 6 7

40 34 8 48 123 7 36

5 2 0 5 6 6 6

45 36 8 53 129 13 42

41 34 8 49 122 7 36

4 3 0 4 7 6 5

45 37 8 53 129 13 41

296

30

326

297

29

326

90.8%

91.1%

On the Fig. 5 we present some examples of wrong classification of images. • Image on Fig. 5(a) was classified into the Class 1 but image on Fig. 5(b) was classified into the Class 4. • Image on Fig. 5(c) was classified into the Class 7, but the person would probably classified it into the Class 2. • Insufficiently lighted image on Fig. 5(d) was classified into the Class 1, but it belongs to the Class 5. • Insufficiently lighted image on Fig. 5(e) was classified into the Class 6 (the class which contains the images with the text) but it belongs to the Class 1.

Fig. 5. Wrong classified images

Moreover we could conclude that • By using functions SC and dE we obtained various incorrectly classified images. • Function Sim always obtains value from interval (0, 1). • Greatest values of function Sim (values between 0.715–0.788) were obtained for the images from the Class 6 (images with the text). • Class 6 was the group with the worst ratio between correctly and incorrectly classified images.

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Conclusions

In this paper, we used intuitionistic fuzzy set distance functions and the function Sim for the classification of the tire treads images. We described the way how the images of a tire treads could be represented by the vectors. Then for each vector we determine the value of membership function, non-membership function and hesitance margin. We used the distance functions defined on IFSs to calculate the distance between the image and predetermined patterns. Then we calculated the value of Sim functions as described in parts of this paper. We classified the set of images and discussed the problems of incorrect classification. The main advantage of this approach is that it has a high percentage of success and it can be used in automated processing of the images which are obtained from the web. In the future, we would like to use another approaches of soft computing to classify tire tread images, for example neural networks and genetic algorithms and their combination with fuzzy sets and intuitionistic fuzzy sets. The results could help us to answer the question: If we classified the set of images, then which images need to be checked by person and which images are classified correctly for sure? Acknowledgements. This work was supported by the grant Joint Polish - Slovak project under the agreement on scientific cooperation between the Polish Academy of Sciences and the Slovak Academy of Sciences, reg. num.15.

References 1. Atanassov, K.T.: Intuitionistic fuzzy sets. In: VII ITKR Session, Sofia, 20–23 June 1983 (1983). (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: Int. J. Bioautomation 20(S1), S1–S6 (2016) 2. Bodziak, W.J.: Forensic Footwear Evidence. CRC Press, Boca Raton (2017) 3. Lux, F.H.: Tire track identification. J. Forensic Res. 4, 198 (2013). https://doi. org/10.4172/2157-7145.1000198. ISSN 2157-7145 4. Poquet Domenech, P.: Tire print identification through registration techniques and attribute graphs. MS thesis. Universitat Polit`ecnica de Catalunya (2013) 5. Vagaˇc, M., et al.: Crawling images with web browser support. In: 13th International IEEE Scientific Conference on Informatics 2015, pp. 286–289 (2015) 6. Vagaˇc, M., Melicherˇc´ık, M., Schon, J.: Classification of tire images in order to obtain the best possible tire tread sample. In: The 5th International Scientific Conference, Applied Natural Science 2015, p. 173. UCM, Trnava (2015) 7. Michal´ıkov´ a, A.: Intuitionistic fuzzy sets and their use in image classification. In: Notes on Intuitionistic Fuzzy Sets, Sofia, Bulgaria, vol. 25, no. 2, pp. 60–66 (2019) 8. Ye, J.: Cosine similarity measures for intuitionistic fuzzy sets and their applications. Math. Comput. Model. 53(1–2), 91–97 (2011) 9. Szmidt, E., Kacprzyk, J.: Analysis of similarity measures for Atanassov’s intuitionistic fuzzy sets. In: IFSA/EUSFLAT Conference, pp. 1416–1421 (2009) 10. Intarapaiboon, P.: Text classification using similarity measures on intuitionistic fuzzy sets. ScienceAsia 42(1), 52–60 (2016)

A Study on Local Properties and Local Contrast in Fuzzy Setting Urszula Bentkowska and Barbara P¸ekala(B) College of Natural Sciences, Institute of Computer Science, University of Rzesz´ ow, Pigonia 1, 35-310 Rzesz´ ow, Poland [email protected]

Abstract. In this paper, the problem of local properties and local contrast of a fuzzy relation is considered. The importance of these two concepts which measure, in a different way, the influence of neighboring elements on the element itself is studied. Modified versions of the local properties of a fuzzy relation are proposed. Moreover, dependencies between the local properties and local contrast are examined.

1

Introduction

Many new approaches and theories treating imprecision and uncertainty have been proposed since fuzzy sets were introduced by Zadeh [12]. As one of the extensions of classical fuzzy set theory, intuitionistic fuzzy sets [2] and intervalvalued fuzzy sets [11,13] are very useful in dealing with imprecision and uncertainty (see [6] for more details). In many application areas the characteristics of neighboring data points are as important as the data itself. The concepts of local contrast and local properties of a fuzzy relation make sense in any field where it is necessary to take into account the influence of neighboring elements on the element itself, e.g.: decision making, approximate reasoning, pattern recognition, image processing (cf. [1,5,8,10]). A local contrast of a fuzzy relation is a measure of the variation among the membership degrees of elements in a specific region of a fuzzy relation. Local properties of a fuzzy relation are defined on the basis of membership degrees (suprema and infima) of the neighboring data points in a fuzzy relation. In this contribution we introduce modified versions of the local properties of a fuzzy relation (cf. [3]). We propose to study the properties a fuzzy relation not only in the whole domain but also locally, in the subregions of the domain. Moreover, we consider similarities and differences between the characteristics of local properties and local contrast of a fuzzy relation. The paper is organized as follows. In Sect. 2 we recall basic concepts related to the local properties and local contrast. In Sect. 3 we present main results of the contribution pointing out the connections between the considered local properties and local contrast of a fuzzy relation. Finally, in Sect. 4, notes on applications of both concepts in image processing are concerned. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 75–83, 2021. https://doi.org/10.1007/978-3-030-77716-6_7

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Preliminaries

We recall the notion of a fuzzy negation and an aggregation function on the unit interval [0, 1]. Definition 1 ([9]). A fuzzy negation N is a decreasing function N : [0, 1] → [0, 1] such that N (0) = 1 and N (1) = 0. A fuzzy negation is strong if N (N (a)) = a for all a ∈ [0, 1]. Definition 2 (cf. [7]). Let n  2. A : [0, 1]n → [0, 1] is said to be an aggregation function if it is increasing in each variable and fulfils boundary conditions A(0, ..., 0) = 0, A(1, ..., 1) = 1. Since the presented concepts for fuzzy relations are related to possible applications in image processing we present them on a finite domain. Definition 3 ([12]). Let X = {0, 1, ..., N − 1} and Y = {0, 1, ..., M − 1} be two finite sets. A fuzzy relation on X × Y (in short denoted by R ∈ F R(X × Y ) or if X = Y by R ∈ F R(X)) is a fuzzy set of the type R = {((x, y), R(x, y))|(x, y) ∈ X × Y }, with R : X × Y → [0, 1]. Given a fuzzy relation R, its complement is given by N R = {((x, y), N (R(x, y)))|(x, y) ∈ X × Y }, where N is a strong fuzzy negation. The converse to R ∈ F R(X × Y ) is the relation R−1 ∈ F R(Y × X), R−1 (x, y) = R(y, x), x ∈ X, y ∈ Y. In image processing an image of N × M pixels may be interpreted as a collection of N × M elements arranged in rows and columns. A numerical value representing (grayscale) intensity, chosen from the set {0, 1, 2, ..., L − 1}, is assigned to each element. An image Q is just a matrix so it may be represented as a fuzzy relation R on a finie set such that the membership degree of each element (pixel) is its intensity divided by L − 1. In the paper [5] several examples of local contrast were examined and properties required for this notion in literature were studied. This led the authors in [5] to introduce the axiomatical description of a local contrast which should fulfil the following axioms. Definition 4 ([5]). A local contrast LC associated with a strong negation N is a real function on X × Y such that: (LC1) 0 ≤ LC(x, y) ≤ 1 for all (x, y) ∈ X × Y ;

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(LC2) If the membership degrees of all the elements of the submatrix centered on (x, y) are identical, then LC(x, y) = 0. That is, if R(x − i, y − j) = q with q ∈ [0, 1] constant for all i, j = −n; ...; 0; ...; n, then LC(x, y) = 0. (LC3) If in the submatrix centered on (x, y) there is at least one element with null membership and at least one element with a membership degree of one, then LC(x, y) = 1. (LC4) The local contrast of (x, y) does not change if for all i, j = −n; ...; 0; ...; n we take N (R(x − i, y − j)) instead of R(x − i, y − j). We present some of the possible examples of a local contrast. Example 1 ([5]). Let N be a strong negation. Thus LC for (x, y) ∈ X × Y may be defined in the following way: ⎧ 1 if in the submatrix there exist ⎪ ⎪ ⎪ ⎪ ⎨ at least one element with the membership equal to 1 LCinf (x, y) = ⎪ ⎪ and another equal to 0; ⎪ ⎪ ⎩ 0 otherwise. ⎧ ⎨ 0 if the memberships of all elements in the submatrix are the same; LCsup (x, y) = ⎩ 1 otherwise. ⎧ 0 if the memberships of all elements ⎪ ⎪ ⎪ ⎪ in the submatrix are the same; ⎪ ⎪ ⎪ ⎪ 1 if in the submatrix there exist ⎪ ⎪ ⎨ at least one element LC(x, y) = with the membership equal to 1 ⎪ ⎪ ⎪ ⎪ and another equal to 0; ⎪ ⎪ n ⎪  ⎪ ⎪ R(x−i,y−j)N (R(x−i,y−j)) ⎪ ⎩ i,j=−n otherwise. size of the submatrix n

n

i,j=−n

i,j=−n

LC(x, y) = max R(x − i, y − j) − min R(x − i, y − j). Let us note that Linf and LCsup provide bounds for any other local contrast, i.e. Linf ≤ LC ≤ LCsup . The notion of local properties was discussed in detail in [3]. These properties were called the local properties, since they depend on the ‘local’, i.e. neighboring values of R (supremum or infimum) and not on the global values such as 1 and 0. Moreover, in that paper two possibilities of defining new versions of properties, i.e. local properties and weak local properties, were discussed. In this contribution we propose the modified version of the local properties which are discussed here not necessarily on the whole domain X × X of a fuzzy relation but instead they are described in a more ‘local way’, i.e. in the subregions

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of the domain of a fuzzy relation. This fact enables one to consider the modified versions of properties of a fuzzy relation such as reflexivity, connectedness etc. in the context of a domain X × Y . Similar considerations may be carried for the weak local properties. We will start with the reflexivity and irreflexivity. Definition 5. Let Z × Z ⊂ X × Y . R ∈ F R(X × Y ) is called: • locally reflexive in Z, if ∀ (R(x, x) =

x∈Z



R(x, y) and R(x, x) =

y∈Z



R(y, x)),

y∈Z

• locally irreflexive in Z, if ∀ (R(x, x) =

x∈Z

 y∈Z

R(x, y) and R(x, x) =



R(y, x)).

y∈Z

Example 2. Let card X = 3, R, S ∈ F R(X) be represented by the following matrices: ⎡ ⎤ ⎡ ⎤ 0.6 0.6 0.4 0.2 0.4 0.3 R = ⎣ 0.5 0.8 0.7 ⎦ , S = ⎣ 0.3 0.1 0.5 ⎦ . 0.4 0.7 0.9 0.3 0.3 0.3 R is locally reflexive in X and we may see that S is locally irreflexive in X. As it was explained in [3] we will not consider the local versions of properties such as symmetry (R(x, y) = R(y, x) for x, y ∈ X) or transitivity (min(R(x, y), R(y, z))  R(x, z) for x, y, z ∈ X, where operation min may be replaced by other binary operations, e.g. aggregation functions). These properties are defined ‘locally’ in their basic form, i.e. they do not depend on the boundary values 0 or 1. Now we will consider local asymmetry (local antisymmetry) and next local total connectedness (local connectedness) which may be defined dually to local asymmetry (local antisymmetry). Definition 6. Let Z × Z ⊂ X × Y . R ∈ F R(X × Y ) is called: • locally asymmetric in Z, if for Q = R ∧ R−1 and all x, y ∈ Z   Q(x, z) and Q(x, y) = Q(z, y), Q(x, y) = z∈Z

z∈Z

• locally antisymmetric in Z, if for Q = R ∧ R−1 and all x, y ∈ Z, x = y   Q(x, z) and Q(x, y) = Q(z, y). Q(x, y) = z∈Z,z=x

z∈Z,z=y

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Definition 7. Let Z × Z ⊂ X × Y . R ∈ F R(X × Y ) is called: • locally totally connected in Z, if for V = R ∨ R−1 and all x, y ∈ Z   V (x, y) = V (x, z) and V (x, y) = V (z, y), z∈Z

z∈Z

• locally connected in Z, if for V = R ∨ R−1 and all x, y ∈ Z, x = y   V (x, y) = V (x, z) and V (x, y) = V (z, y). z∈Z,z=x

z∈Z,z=y

Following the results presented in [3] we provide the characterization of local total connectedness (local connectedness) and local asymmetry (local antisymmetry) which is useful in practice for verifying these properties. Proposition 1. Let Z ×Z ⊂ X ×Y . R ∈ F R(X ×Y ) is locally totally connected in Z if and only if max{R(x, y), R(y, x)} = a for any x, y ∈ Z, where a ∈ [0, 1]. R ∈ F R(X ×Y ) is locally connected in Z if and only if max{R(x, y), R(y, x)} = a for any x, y ∈ Z, x = y, where a ∈ [0, 1] Dually we obtain the following result. Proposition 2. Let Z × Z ⊂ X × Y . R ∈ F R(X × Y ) is locally asymmetric in Z if and only if min{R(x, y), R(y, x)} = a for any x, y ∈ Z, where a ∈ [0, 1]. R ∈ F R(X ×Y ) is locally antisymmetric in Z if and only if min{R(x, y), R(y, x)} = a for any x, y ∈ Z, x = y, where a ∈ [0, 1]. Example 3. Let card X = 3, R, S ∈ F R(X) be represented by the following matrices: ⎤ ⎡ ⎤ ⎡ 0.2 0.4 0.3 0.6 0.6 0.4 R = ⎣ 0.5 0.6 0.7 ⎦ , S = ⎣ 0.1 0.1 0.5 ⎦ . 0.4 0.8 0.9 0.3 0.1 0.3 R is not locally reflexive in X but it is locally reflexive and locally totally connected (locally connected) in Z, where Z = {0, 1}. Similarly we may see, that S is not locally irreflexive in X but it is locally irreflexive, locally asymmetric (locally antisymmetric) in Z × Z  , where Z = {1, 2} and Z  = {0, 1}. Let us notice that in Example 3 we applied the notions of local properties on the square subregion Z ×Z  , where Z = Z  . In the context of image processing we abstract from the elements of domain and concentrate only on the membership values on the given domain. This is why we accept such subsets.

3

Local Contrast and Local Properties

We will provide the study on the relevance between the local properties and the local contrast. We will consider similarities and differences between the characteristics of local properties and local contrast of a fuzzy relation.

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Proposition 3. Let Z × Z ⊂ X × Y . If R is symmetric in Z and locally totally connected in Z (or locally asymmetric in Z), then local contrast in Z ×Z is zero. Proof. Let us suppose that R is locally totally connected in Z, then by Proposition 1 we have max{R(x, y), R(y, x)} = a for any x, y ∈ Z, where a ∈ [0, 1]. If additionally R is symmetric in Z, then R(x, y) = R(y, x) = a for any x, y ∈ Z. Thus LC(x, y) = 0 for any x, y ∈ Z. 

In applications it is important to know the time complexity of several procedures that should be performed. Remark 1. Time complexity of checking if a fuzzy relation is locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z is of O(n2 ) order. The same time complexity takes checking if a fuzzy relation is constant in Z (i.e. LC(x, y) = 0 for any x, y ∈ Z). It is also natural to require that a local contrast may satisfy some additional properties (cf. [5]), i.e.: • (LC5) if we multiply all the membership degrees of elements in the submatrix centered on (x, y) by a constant factor λ ∈ [0, 1], then the local contrast of the element (x, y) should also be increased by a factor λ ∈ [0, 1]; • (LC6) if we increase all the membership degrees of elements in the submatrix centered on (x, y) by the same quantity r, such that R(x − i, y − j) + r ∈ [0, 1] for all i, j = −n, ..., 0, ..., n, then the local contrast should not change. Some of the examples of local contrast used in applications fulfil one of these properties. However, it is hard to fulfil both of them (cf. [5]). We studied the behavior of a fuzzy relation which fulfils one of the local properties in the context of satisfying the conditions (LC5) and (LC6). Proposition 4. Let Z × Z ⊂ X × Y . If R ∈ F R(X × Y ) is locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z, then: • λR(x, y), for (x, y) ∈ X × Y , is also locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z; • R(x, y)+r, where R(x, y)+r ∈ [0, 1] for (x, y) ∈ X ×Y , is also locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z. Proof. Operations of multiplication and addition change in the adequate way the values of suprema and infima, but the suprema and infima are reached at the same points, so if R ∈ F R(X × Y ) is locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z, then also λR(x, y) or R(x, y) + r is locally reflexive (irreflexive, asymmetric, antisymmetric, totally connected, connected) in Z. 

Analogously to the notion of local contrast a total contrast of a fuzzy relation may be defined in the following manner.

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Definition 8 ([5]). Let N be a strong negation. A total contrast T C associated with N is a real function on F R(X × Y ) such that: (TC1) 0 ≤ T C(R) ≤ 1; (TC2) If all elements of the fuzzy relation R have the same membership degree, then T C(R) = 0; (TC3) If R is a crisp relation such that there exists at least one element with membership equal to one and another with membership equal to zero, then T C(R) = 1; (TC4) The total contrast of a fuzzy relation and that of its negation (by N ) are the same; that is, T C(R) = T C(N (R)). In the following proposition we show that aggregating local contrasts will produce a total contrast as long as we choose the aggregating function properly. Usually aggregation functions are required to be increasing (cf. Definition 2). However, we may also drop this requirement to consider aggregation function as the one which combines the single values into one value. Proposition 5 ([5]). Consider R ∈ F R(X × Y ), and let LC be a local contrast associated with a strong negation N (in the sense of definition of total contrast TC). Then let F : m∈N [0, 1]m → [0, 1] be a function such that: (1) F (0, ..., 0) = 0 and F (1, ..., 1) = 1. (2) If ai ∈ {0, 1} for all i ∈ {1, ..., m}, and there exist at least one component ap = 1 and another component aq = 0 with p, q ∈ {1, ..., m}, then F (a1 , ..., am ) = 1. Under these conditions, T C(R) = Fx=0,...,N −1,y=0,...,M −1 LC(x, y) is a total contrast associated with the strong negation N . Example 4 ([5]). Below we present functions Fg that fulfil assumptions of Proposition 5 and as a result may be used to construct adequate examples of total contrast. Namely, Fg (a1 , ..., am ) = max(a1 , ..., am ). ⎧ if ai ∈ {0, 1} for all i ∈ {1, ..., m} ⎪ ⎪ 1, ⎪ ⎪ and there are at least one ap = 1 ⎪ ⎪ ⎨ and one aq = 0 such that Fg (a1 , ..., am ) = p, q ∈ {1, ..., m} ⎪ ⎪ ⎪ m ⎪

⎪ 1 ⎪ ai , otherwise. ⎩m i=1

We recall also the results on aggregation of relations having local properties. We may observe the preservation of these properties under some assumptions. Proposition 6 (cf. [3]). Let Z × Z ⊂ X × Y . Let R, S ∈ F R(X). • If F is distributive with respect to ∨ (∧) and R, S are locally reflexive (locally irreflexive) in Z, then F preserves local reflexivity (local irreflexivity).

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• If F, ∨ : [0, 1]2 → [0, 1] are commuting and R, S are locally totally connected in Z, then F preserves local total connectedness (local connectedness). • If F, ∧ : [0, 1]2 → [0, 1] are commuting and R, S are locally asymmetric in Z, then F preserves local asymmetry (local antisymmetry). The given results may be useful in the context of construction a total contrast from a local contrast by the use of aggregation functions.

4

Notes on Application

An analysis of the properties naturally required for a contrast measure (for example, in image processing) led the authors in [5] to definitions of the local contrast and total contrast suitable for a fuzzy relation. In this paper (inspired by [5]) we study the new concept of local properties suitable to the requirements of image processing, particularly to the concept of local contrast. The matrix representing some image may satisfy some local properties and this fact may imply the adequate information about the values of a local contrast (and as a consequence a total contrast). In the future work we would like to test how the local properties of a fuzzy relation representing given image have influence on the contrast of this image. Moreover, we would like to concentrate more on the notion of local contrast of a fuzzy relation, i.e. find new examples and test them in image processing, especially in the algorithm from [5]. Some of the new examples of local contrast that we would like to apply are given below: LC(x, y) = 1 − (

n

min

i,j,k,l=−n,(i=k or j=l)

S(R(x − i, y − j), R(x − k, y − l)),

n

n

i,j=−n

i,j=−n

LC(x, y) = max I(1, R(x − i, y − j)) − min I(1 − R(x − i, y − j), 0), where (x, y) ∈ X × Y , S(a, b) = min(I(a, b), I(b, a)) and  1, if a ≤ b I(a, b) = max(1 − a, b), otherwise.

5

Conclusions

In this paper, we discussed the correlation between local properties and local contrast of a fuzzy relation. In the case of these concepts the characteristics of neighboring data points are as important as the data itself so they are called ‘local’ and we noticed that they are related to each other. In future work, we plan to consider new examples of local contrast and examine their performance in image processing. We would like to compare them with other expressions of local contrast applied in image processing issues. Acknowledgements. This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge of University of Rzesz´ ow, Poland, the project RPPK.01.03.00-18-001/10.

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References 1. Amo, A., Montero, J., Molina, E.: Representation of consistent recursive rules. Euro. J. Oper. Res. 130, 29–53 (2001) 2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 3. Bentkowska, U., P¸ekala, B., Rz¸asa, W.: The stability of local properties of fuzzy relations under ordinal equivalence. Inf. Sci. 491, 265–278 (2019) 4. Beliakov, G., Bustince, H., Calvo, T.: A practical guide to averaging functions. In: Studies in Fuzziness and Soft Computing, vol. 329. Springer (2016) 5. Bustince, H., Barrenechea, E., Fernandez, J., Pagola, M., Montero, J., Guerra, C.: Contrast of a fuzzy relation. Inf. Sci. 180, 1326–1344 (2010) 6. Bustince, H., Barrenechea, E., Pagola, M., Fern´ andez, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24(1), 174–194 (2016) 7. Calvo, T., Koles´ arov´ a, A., Komornikov´ a, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Calvo, T., et al. (eds.) Aggregation Operators, pp. 3–104. Physica-Verlag, Heidelberg (2002) 8. Jahromi, M.Z., Parvinnia, E., John, R.: A method of learning weighted similarity function to improve the performance of nearest neighbor. Inform. Sci. 179(17), 2964–2973 (2009) 9. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Acad. Publ., Dordrecht (2000) 10. Montero, J., G´ omez, D., Bustince, H.: On the relevance of some families of fuzzy sets. Fuzzy Sets Syst. 158, 2429–2442 (2007) 11. Sambuc, R.: Fonctions φ-floues: Application ´ a l’aide au diagnostic en pathologie thyroidienne. Ph.D. Thesis, Universit´ e de Marseille, France (1975). (in French) 12. Zadeh, L.A.: Fuzzy sets. Inf. Contr. 8, 338–353 (1965) 13. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inform. Sci. 8, Part I, 199–251, Part II, 301–357, Inform. Sci. 9. Part III, 43–80 (1975)

Note on the Zadeh’s Extension Principle Based on Fuzzy Variable Approach Adam Bzowski1(B) , Michal K. Urba´ nski2 , Kinga M. W´ ojcicka2 , 3 and Pawel M. W´ ojcicki 1

Department of Physics and Astronomy, ˚ Angstr¨ omlaboratoriet, Uppsala University, 751 08 Uppsala, Sweden [email protected] 2 Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland [email protected], [email protected] 3 Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland [email protected] Abstract. In the paper the extension principle was derived from the assumption that possibility measure is maxitive and fuzzy sets are defined using fuzzy variable approach. Keywords: Fuzzy numbers

1

· Fuzzy sets · Extension principle · t-norm

Introduction

Fuzzy sets were introduced by Zadeh in [14] as an extension of classical sets in order to facilitate a notion of partial inclusion in a set. As such one should be able to extend set-theoretic operations to fuzzy sets. Of particular importance is the operation of ‘range’: applying a function to a set or a product of sets. [15] proposed a specific form of such an operation, known since as the Zadeh extension principle. The extension principle was not equipped with any proof, merely declared as obeying expected properties. In this paper we want to propose a more fundamental approach to fuzzy sets and the Zadeh’s extension principle. We start with generalized measure theory as in [7]. In this way the fuzzy set can be obtained as a distribution associated with a given possibility measure. Using generalized measure theory one can uniquely determine the form of various postulates such as the Zadeh’s extension principle. As it turns out, the principle follows from the evaluation of the distribution of fuzzy variables composed with functions. This construction is analogous to derivation of the distribution of random variables in probability theory. The original Zadeh’s extension principle contained a minimum function, which can be generalized to any t-norm. As we will show the t-norm describes interactions of fuzzy variables: the fuzzy counterpart of the dependence of random variables. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 84–92, 2021. https://doi.org/10.1007/978-3-030-77716-6_8

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Fuzzy Variables

We want to define fuzzy variables as the fuzzy counterparts of random variables. Random variables are defined as measurable functions on a probability space and usually take values in real numbers. Probability space is a set endowed with a probability measure. Therefore, in the theory of fuzzy variables we first have to define notions of fuzzy space and the suitable generalized measures. Fuzzy variable approach to fuzzy sets is convenient to describe the measurement process. The probabilistic model of the measurement process is described by a random variable. This means that the numerical results of measurements appear with a certain probability, which can be described in such a way that each test object is represented by a distribution of a random variable that describes the measurement of a certain quantity. A probability measure is defined on a set of events in a probability space. Therefore we should define the set of events, and this set is a set of permissible events. In the framework of fuzzy variable concept, the possibility measure is defined on family of subsets of some set but this set is not interpreted as sample space but as the pattern space. The pattern space we understood as set of measurement standards. Possibility measure describes the degree to which the observed phenomenon fits into a particular standard which realized some value of measured quantity. 2.1

Generalized Measure Theory

The subject of this section is standard and can be found in a number of textbooks, for example [4,8,9]. We include the necessary definitions and facts for completeness. Just like in the standard measure theory, in order to define generalized measures, we must define objects such as σ-semigroups, σ-lattices and σ-fields. Definition 1. Let Ω be a set and let D be a family of subsets of Ω, D ⊆ 2Ω . Then: 1. The pair (Ω, D) is a σ-semigroup, if D closed under countable unions, and ∅, Ω ∈ D. 2. The pair (Ω, D) is a σ-lattice if D is a σ-semigroup which is closed under countable intersections. 3. The pair (Ω, D) is a σ-field, if D is a σ-lattice and for any A ∈ D its complement Ω\A ∈ D. Definition 2. Let D ⊆ 2Ω be a σ-semigroup of subsets of a set Ω. A fuzzy measure on Ω is a function μ : D → [0, ∞] such, that: 1. μ(∅) = 0, 2. for all A, B ∈ D, A ⊆ B implies μ(A) ≤ μ(B), 3. for any ascending sequence of sets (An )∞ n=1 , where An ∈ D (that is An ⊆ An+1 for all n) one has  ∞  An = lim μ(An ) (1) μ n=1

n→∞

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The name of a fuzzy measure has actually nothing to do with fuzzy sets. In fact fuzzy measures constitute a very weak concept and not many interesting theorems can be formulated. To make it more interesting, one adds assumptions regarding the behavior of the measure under set-theoretic unions. For this one introduces the following notions. Definition 3. Pseudoaddition ⊕ is a binary operator ⊕ : [0, ∞] × [0, ∞] → [0, ∞]

(2)

denoted by: x ⊕ y = ⊕(x, y), such that for all x, y, z ∈ [0, ∞] the following conditions hold: 1. 2. 3. 4. 5.

associativity: (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z), commutativity: x ⊕ y = y ⊕ x, monotonicity: x ≤ y, to x ⊕ z ≤ y ⊕ z neutral element: x ⊕ 0 = x ∞ continuity: ⊕ is continuous from below, that is, if (xn )∞ n and (yn )n are nondecreasing sequences of points in [0, ∞] with xn → x oraz yn → y, then sup(xn ⊕ yn ) = x ⊕ y. n

Commutativity and associativity allows one to drop parentheses and freely change the order of terms in an expression. Thus, for all (xn )∞ n=1 , xn ∈ [0, ∞] we define a sum of n elements: n 

xj = x1 ⊕ x2 ⊕ . . . ⊕ xn .

(3)

j=1

Monotonicty of the pseudoaddition implies that the infinite sum of terms is well-defined and independent of the summation order, ∞ 

xn = lim

n=1

n→∞

n 

xj .

(4)

j=1

Moreover, conditions 2, 3 and 4 imply that ∞⊕x = x⊕∞ = ∞ for all x ∈ [0, ∞]. Definition 4. Let D be a σ-semigroup of sets in Ω. A function μ : D → [0, ∞] is called ⊕-decomposable measure, if 1. μ(∅) = 0, 2. for any sequence of sets (An )∞ n=1 , An ∈ D such, that Ai ∩ Aj = ∅ for i = j, one has ∞  ∞   μ An = μ(An ) (5) n=1

n=1

Moreover, if μ(Ω) = 1, then μ is a normed measure.

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For the standard Lebesgue measure the pseudoaddition is the actual addition, ⊕ = +. And similarly to the Lebesgue measure, a general ⊕-decomposable measure exhibits a number of properties. In particular it is easy to verify the following Theorem 1. Let μ : F → [0, ∞] be a ⊕-decomposable measure defined on a σ-field F. Then: 1. μ is a fuzzy measure. 2. For all measurable sets (An )∞ n=1 , An ∈ F one has ∞  ∞   μ An ≤ μ(An ) n=1

(6)

n=1

Finally, as in the standard measure theory one can push measures forward through measurable functions. Definition 5. Let (Ω1 , F1 ) and (Ω2 , F2 ) be σ-fields, and μ1 be a ⊕decomposable measure on Ω1 . A function f : Ω1 → Ω2 is measurable if for every A ∈ F2 the set f −1 (A) ∈ F1 . Let μ1 be a (fuzzy, ⊕-decomposable) measure on Ω1 . A measurable function f : Ω1 → Ω2 induces the induced measure μ2 = f∗ μ1 on Ω2 defined by μ2 (A) = μ1 (f −1 (A)) for all A ∈ F2 . 2.2

From Possibility Measures to Fuzzy Sets

In the standard theory of fuzzy sets [14], a fuzzy set is defined as follows. Definition 6. Let Ω be a set. A fuzzy set A¯ in Ω is a function A¯ : Ω → [0, 1] ¯ satisfying sup A(Ω) = 1. Sometimes a fuzzy set is treated as a formally distinct object from its defining function. For this reason the function A¯ is called a membership function. Let Ω be a set and A¯ a fuzzy set in Ω. In the theory of fuzzy sets, we can call the possibility Π of a given subset B ⊆ Ω as the supremum of the membership function A¯ over this set, ¯ Π(B) = sup A(B). (7) Clearly, the possibility Π can be expressed in the language of generalized measure theory. This is the aim of this section: to derive the notion of a fuzzy set and its properties directly from generalized measure theory. Definition 7. Let (Ω, F) be a σ-field. A measure μ : F → [0, 1] is called a maxitive measure, if μ is ⊕-decomposable for ⊕ = max. Moreover, if for any family of measurable sets (Aα )α such that:  A= Aα ∈ F yields μ(A) = sup μ(Aα ) (8) α

then μ is called completely maxitive.

α

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Note that for ⊕ = max we do not have to require that the sets in the family Aα are pairwise disjoint (which is assumed for additive measures). Furthermore, in the standard Lebesgue theory the σ-field on which the measure is defined can rarely be extended to the full power set. Here, on the other hand, completely maxitive measures on the power set are the highlight of the paper. Definition 8. Let Ω be a set. A possibility measure on Ω is a completely maxitive and normed measure on the power set 2Ω . Theorem 2. The set of possibility measures on a given set Ω is in one-to-one correspondence with fuzzy sets in Ω. Proof. Indeed, if a function A¯ : Ω → [0, 1] satisfies sup A(Ω) = 1, one can ¯ define measure Π of singleton {x} as Π({x}) = A(x), for all x ∈ Ω. Complete maxitiveness of the measure defines the possibility of a given subset B ⊆ Ω by Eq. (7). Conversely, any possibility measure Π on Ω determines a function ¯ ¯ A¯ : Ω → [0, 1] with sup A(Ω) = 1, by A(x) = Π({x}).  This means that we can define a fuzzy set from the purely measure-theoretic point of view, without any reference to Definition 6. Definition 9. Let Π be a possibility measure on Ω. A function A¯ : Ω → [0, 1] such, that ¯ A(x) = Π({x}) (9) is called the distribution of Π or the fuzzy set associated with Π. Finally, we can define the fuzzy counterpart of the probability space. Definition 10. A possibility space is a pair (Ω, Π), where Ω is an arbitrary set and Π is a possibility measure on Ω. 2.3

Non-interacting Sets

Let (Ω, F, P ) be a probability space. In the standard probability theory two events A, B ∈ F are called independent if P (A∩B) = P (A)P (B). This definition can be generalized to independence of a countable number of sets as well as σfields. In the theory of possibility measures one could declare two event to be independent – or noninteracting – if Π(A ∩ B) = Π(A)Π(B), but in fact there is no good reason to choose multiplication in this formula. In the early developments of the fuzzy set theory one considered the minimum operation between Π(A) and Π(B), [15]. In general, one can choose any t-norm as a generalization of the multiplication [1,6,7,11], Definition 11. T-norm is a function T : [0, 1]2 → [0, 1] such that for any x, y, z ∈ [0, 1] the following conditions are satisfied:

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1. 2. 3. 4.

89

associativity: T (T (x, y), z) = T (x, T (y, z)), commutativity: T (x, y) = T (y, x), monotonicity: if x ≤ y, then T (x, z) ≤ T (y, z), neutral element: T (1, x) = x.

Definition 12. Let (Ω, Π) be a possibility space and let T be a t-norm. 1. Sets A, B ⊆ Ω are called T -noninteracting, if: Π(A ∩ B) = T (Π(A), Π(B)) 2. Sets A1 , . . . , An ⊆ Ω, where n ≥ 2 are called T -noninteracting, if: Π(Aj1 ∩ . . . ∩ Ajk ) = T (Π(Aj1 ), . . . , Π(Ajk )) for all k = 2, 3, . . . , n and 1 ≤ j1 < . . . < jk ≤ n. 3. A sequence of sets (Aα )α is called noninteracting, if every finite collection of its subsets is noninteracting. 4. A σ-lattice of subsets F1 , . . . , Fn , Fj ⊆ 2Ω for j = 1, . . . , n is called noninteracting, if for every Aj ∈ Fj one has Π(A1 ∩ . . . ∩ An ) = T (Π(A1 ), . . . , Π(An )) 5. An infinite sequence of σ-lattices of sets in called noninteracting, if every finite subsequence is noninteracting. Having two possibility spaces (Ω1 , Π1 ) i (Ω2 , Π2 ), one can define a product measure Π on Ω = Ω1 × Ω2 by Π(A × B) = T (Π1 (A), Π2 (B))

(10)

for any subsets A ⊆ Ω1 , B ⊆ Ω2 . Since t-norms are monotonic, this measure is completely maxitive, and so generated by a membership function. The distribution of the product measure Π is a fuzzy set: ¯ y) = T (A¯1 (x), A¯2 (y)) A(x, (11) where A¯1 , A¯2 are distributions of Π1 and Π2 respectively. Such a measure is called the product measure. 2.4

Extension Principles

Fuzzy variables used in the measurement theory are to describe a situation analogous to drawing from a set of elementary events. The idea of defining fuzzy sets by measures of possibilities and describing them as a fuzzy variable defined analogously to a random variable was presented by Nahmias in his work [7]. Fuzzy variables were described by a small group of researchers (e.g. [2,3,10]), but unfortunately (“unfortunately”; due to metrological applications) a model

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based on fuzzy random variables [5,12,13] is becoming increasingly popular in the literature. Random and fuzzy variables take values in a set E, usually a set of real numbers. From the point of view of the measurement theory E represents possible results of the measurement, numbers or any set representing states of the measuring instrument. Definition 13. Let (Ω, Π) be a possibility space. A fuzzy variable ξ with values in a set E is any function ξ : Ω → E.

(12)

Recall that the σ-field on which a possibility measure is defined is the full power set 2Ω . Hence, any function between possibility spaces is a measurable function. The mapping ξ induces a possibility measure on E defined as Πξ = Π ◦ ξ −1 . Definition 14. The fuzzy set A¯ξ associated with the possibility measure Πξ on E is called the distribution of the fuzzy variable ξ. The fuzzy set A¯ξ is described by the membership function, A¯ξ (x) = Πξ ({x}) = Π ◦ ξ −1 (x) = Π ({ω ∈ Ω : ξ(ω) = x}) .

(13)

Now one can carry out the same set of operations on fuzzy variables as can be performed on random variables. The question to address is how the distributions of fuzzy variables transform under various operations. For the possibility space (Ω, Π) consider a fuzzy variable ξ : Ω → E1 . If f : E1 → E2 is a function, then one has the induced fuzzy variable η : Ω → E2 defined as η(ω) = f (ξ(ω)). Theorem 3 (Extension principle). Let A¯ξ and A¯η be the distributions of the fuzzy variables ξ and η = f ◦ ξ as defined above. Then ¯ −1 (y)). A¯η (y) = sup A(f

(14)

A¯η (y) = Π ({ω ∈ Ω : η(ω) = y})   = Π ω ∈ Ω : ξ(ω) = f −1 (y) = sup A¯ξ (f −1 (y)).

(15)

Proof. From (13) we have

 Similarly, one can consider two fuzzy variables, ξ and η, with values in two sets E1 and E2 , consider a function f of two variables and ask what is the distribution of the fuzzy variable f (ξ, η).

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Definition 15. Let T be a fixed t-norm, and (Ω, Π) be a possibility space. Fuzzy variables (ξn )∞ n=1 with values in sets En are called T -noninteracting, if for any subsets of sets An ⊆ En , n = 1, 2, . . . , preimages ξn−1 (An ) are T noninteracting for all n ∈ N. Theorem 4 (Zadeh’s extension principle). Let T be a t-norm, ξ and η be two T -noninteracting fuzzy variables with values in E1 and E2 , and with distributions A¯ξ and A¯η respectively. Let f : E1 × E2 → E3 be an arbitrary function. Then ζ = f (ξ, η) has a distribution A¯ζ given by the Zadeh’s extension principle:  A¯ζ (z) = sup{T A¯ξ (x), A¯η (y) : (x, y) ∈ E1 × E2 , z = f (x, y)} (16) Proof. By the definition  A¯ζ (z) = Π ζ −1 (z) = Π({ω ∈ Ω : ζ(ω) = z}) = Π({ω ∈ Ω : f (ξ(ω), η(ω)) = z}) ⎛ ⎞  =Π⎝ {ω ∈ Ω : ξ(ω) = x, η(ω) = y}⎠ (x,y)∈E1 ×E2 : z=f (x,y)

=

sup (x,y)∈E1 ×E2 : z=f (x,y)

=

sup (x,y)∈E1 ×E2 : z=f (x,y)

Π({ω ∈ Ω : ξ(ω) = x, η(ω) = y}) T (Π({ω ∈ Ω : ξ(ω) = x}), Π({ω ∈ Ω : η(ω) = y}))

= sup{T (A¯ξ (x), A¯η (y)) : (x, y) ∈ E1 × E2 , z = f (x, y)}

(17)

 As we can see the Zadeh’s extension principle emerges naturally in the theory of possibility measures and fuzzy variables. Just like the definition of fuzzy sets in (9), it is a consequence rather than a postulate.

3

Conclusions

In the paper we have shown how the notion of a fuzzy set as well as the Zadeh’s extension principle emerge from the theory of possibility measures. In such an approach they are consequences rather than assumptions. It provides a robust base and more fundamental understanding for the theory of fuzzy sets. Furthermore, a significant number of results familiar in the theory of random variables can be adapted for fuzzy variables. Various notions of convergence, theorems, such as strong laws of large numbers or a notion, or a copulus, all can be transformed into the language of fuzzy variables.

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OFNBee Method Applied for Solution of Problems with Multiple Extremes Dawid Ewald1(B) , Jacek M. Czerniak1 , and Marcin Paprzycki2 1

2

Kazimierz Wielki University, Bydgoszcz, Kujawsko-Pomorskie, Poland {dawidewald,jczerniak}@ukw.edu.pl Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland [email protected]

Abstract. There are plenty of algorithms whose operation is based on the behavior of bees. Currently, the most intensively developed concept is the ABC algorithm. Its chief asset is the ease of customization and fast operation. One of the older solutions is the MBO concept. However, due to its high complexity, the algorithm had limited application. The subject of this article is the OFNbee concept - it is based on the application of the selected arithmetic of the fuzzy number in the new bee swarm optimization method. This approach can improve the efficiency of the optimization algorithm.

1

Introduction

Different optimization techniques, such as heuristics, are used for optimization. With those techniques it is possible to find the best solution in the searched set of solutions without the need to check each element. Such operation saves time and resources. Meta-heuristic algorithms are currently being developed very intensively. Methods based on bee swarm optimization constitute a large part of these algorithms. The best-known algorithm is ABC (artificial bee colony) [32– 35]. A slightly less intensively developed but well-documented solution is MBO (Marriage in honey bees optimization) and its derivatives, i.e. HBPI (Honey Bees Policy Iteration) proposed by Chang in 2006 or HBMO (honey bee mating optimization) proposed by Afshar et al. in 2007. Another version of a bee swarm-based algorithm is BBMO (bumble bees mating optimization) proposed by Marinakis Y, Marinaki M, Matsatsinis N. A in 2010 or its HBMO variant (honey bees mating optimization algorithm). A well-known expansion of MBO is also IMBO (Improved Marriage in Honey Bees Optimization) algorithm proposed in 2013 by Yuksel Celik, Erkan Ulker. In 2013 Venkata Rao R, Patel V. proposed the TLBO (Teaching–Learning-Based Optimization) method which was chosen for comparison with the above mentioned methods [3,5,6,13–15,18]. This article focuses on the main bee swarm optimization algorithms - ABC MBO and the new OFNBee method that uses the arithmetic of ordered fuzzy numbers. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 93–111, 2021. https://doi.org/10.1007/978-3-030-77716-6_9

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ABC Like other algorithms described herein, ABC is also based on herd behavior of honey bees [6,22–25,29,30]. It differs from other algorithms mostly in the use of higher number of bee types in a swarm. The algorithm consists of four periodically repeated stages, until the number of repetitions specified by the user is completed [11,37]: – – – –

employed bees stage, onlooker bees stage, scout bees stage, storage of the best solution so far.

The algorithm starts with initialization of the food source vectors xm , where m = 1 . . . SN , while SN , is the size of the population. Each of those vectors stores n values xm , i = 1 . . . n, which will be optimized during execution of that method. The vectors are initialized using the following formula: xmi = li + rand(0, 1) · (ui − li )

(1)

A characteristic property of ABC is the fact that bees adapted to different tasks participate in each stage of the algorithm operation. There are 3 types of bees involved in searching [5,20,23,26]: – Employed Bees – i.e. bees which search points near points already stored in the memory, – Onlooker Bees – objects responsible for searching neighborhood of points deemed the most attractive, – Scout Bees – (also referred to as scouts) bees of that kind explore random points not related in any way to those discovered earlier. Once initialization phase is completed, Employed Bees start their work. They are sent to places in the neighborhood of already known food sources in order to determine the amount of nectar available there [19,36]. Onlooker Bees work on the basis of information on the amount of food gained by the Employed Bees. Employed Bees randomly select a potential food source using the following formula: (2) vi = xmi + ϕmi (xmi + xki ) where: vi - vector of potential food sources, xk - randomly selected food source, ϕmi - random number from the range [−a, a] Once the fitness vector is determined, its fitting is calculated based on the formula dependent on the problem to be solved and then the fitting is compared. If the new vector fits better than the former one, then the new replaces the old one. Another phase of the algorithm operation is the Onlooker Bees stage. They

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are sent in the neighborhood of the richest food source. The probability of the Xm source selection is expressed with the formula: f itm (xm ) pm = SN m=1 f itm (xm )

(3)

where: f itm (xm ) - value of fitting functions for a given source. When onlooker bees have acquired information on the amount of nectar, such data is compared with results obtained so far and if the new food sources are better, they replace the old ones in the memory [29]. Pseudocode of ABC [4,16,17,31]: 1. random determination of the initial food sources for the employed bees involved 2. repeat 3. sending employed bees involved to places near food sources stored in the memory and determining the amount of nectar contained therein 4. calculating the probability value for the advertised benefits, according to which they will be preferred by the onlooker bees 5. sending onlooker bees to places near the selected food sources and determining the amount of nectar contained therein 6. cessation of the exploitation process for a source abandoned by the bees 7. sending scouts to randomly discover new food sources 8. storing in the memory the best source of food found so far 9. until required fulfillment. The last item of the ABC algorithm operation is commencement of exploration by scouts. Bees of that type select random points from the search space and then check nectar volumes available there. If newly found volumes are higher than the volumes stored so far, they replace the old volumes. The activity of such bees makes it possible to explore the space unavailable for other types of bees thus allowing to reduce the possibility of omission [23]. OFNBee Application of OFN notation in the bee swarm optimization seems to be a completely natural way to describe the behavioral mechanisms observed in a hive and mentioned in the above paragraphs. The input data are pieces of information carried by a single bee, i.e.: – – – –

the direction to the food location, the angle of the navigation by the sun, the length of the flight, affluence of the food source.

Figure 1 shows OFN number describing information delivered by a scout bee:

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Fig. 1. OFN number describing information delivered by a scout bee

– aF - the angle between the sun and the food source, – sF - food quantity, – dF - distance to the food. Part of the Fig. 1 shows the food which has to be reached by flying at the aF angle in relation to the position of the sun. Figure 1a includes selected information on that angle. The arm with a bee symbol represents the distance to food - dF . While the amplitude of vibrations marked with sine wave reflects the amount of food - sF . An ordered fuzzy number is determined as follows. The support(A), which is the base of the trapezoid, is determined first. Then the rising edge f (x) is drawn at the angle of 90◦ − aF . The other base of a trapezoid is laid off from the point in which the function f (x) intersects with zy = 1. At the end it is just necessary to connect two ends of the trapezoid bases using the falling edge function g(x). Pseudocode of OFNbee: 1. random determination of the initial food sources for the employed bees involved 2. repeat 3. sending employed bees involved to places near food sources stored in the memory and determining the amount of nectar contained therein 4. the process of fuzzyfication and selection of places by onlooker bees – fuzzyfication of sumyf it

sumyf itOF N fp βγ =

⎧ ⎪ ⎪ R1 = sumyf iti , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ R2 = sumyf it , i

⎪ ⎪ ⎪ R3 = sumyf itfi +|sumyf iti ∗β|, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R4 = sumyf it +|sumyf it ∗γ|, i i

(4)

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– fuzzyfication of unf easiblevalues

unf easiblevaluesOF N fi pβγ =

⎧ ⎪ ⎪ R1 = fi , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ R2 = f , i

⎪ ⎪ ⎪ R3 = fi +|fi ∗β|, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R4 = f +|f ∗γ|, i i

(5)

– calculating the probability using the roulette wheel function - prob[] probOF N =

30 i=1 unf easiblevaluesOF N fi pβγ sumyf itOF N fp βγ

(6)

– defuzzyfication of prob[] prob =

min(supp(probaOF N ))+|supp(probaOF N )| 1,618033998875

(7)

– Assignment of the values and sending the onlooker bees 5. sending onlooker bees to places near the selected food sources and determining the amount of nectar contained therein 6. cessation of the exploitation process for a source abandoned by the bees 7. sending scouts to randomly discover new food sources 8. storing in the memory the best source of food found so far 9. until requirement fulfilled. MBO Marriage in Honey-Bees Optimization (MBO) algorithm is one of many algorithms of the swarm intelligence family, which are inspired by bee colony behavior. This method was proposed by Hussein A. Abbass in 2001. That solution was mainly inspired by the underlaying idea of mating process of a certain bee genus. The composition of each MBO colony has a specific structure which includes [1,2] – queen, both biological and algorithmic, which is responsible for searching partners and looking after eggs, – drones - individuals being reproductive partners of a queen; a queen shall always randomly select, with some probability depending on the fitness, her partner and shall lay eggs as a result of mating with him, – Workers - are bees responsible for care over and improvement of the next generation, – broods are the result of mating between queens and drones as well as mutations.

2

OFN Ordered Fuzzy Numbers

Definition 1 An ordered fuzzy number A is an ordered pair of functions where xup , xdown : [0, 1] - !R are continuous functions. Respective parts of the functions

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are called: up and down parts. The arrow in the figure below illustrates the order of the reversed functions and the orientations of a fuzzy number [7–10,12,21,27,28]. Continuity of both of those parts shows that their images are limited by specific intervals. They are named respectively: UP and DOWN (Fig. 2).

Fig. 2. OFN

3

Test Functions

Broadly known mathematical functions were selected to effectively compare the operation of algorithms - shown below, which contains the function name, optimum and range in which it is sought. 1. Sphere Function: n – Formula: f (x) = i=1 x2i – Input Domain: [−5.12, 5.12]n – Optimum f (x): 0 2. Rosenbrock Function: n – Formula: f (x) = i=1 [100(xi+1 − x2i )2 + (xi − 1)2 ] – Input Domain: [−2.04, 2.048]n – Optimum f (x): 0 3. Rastrigin Function: n – Formula: f (x) = 10n + i=1 (x2i − 10cos(2πxi )) – Input Domain: [−5.12, 5.12]n – Optimum f (x): 0 4. Griewank Function: n n x2i xi – Formula: f (x) = 1 + i=1 4000 − i=1 cos( √ ) i n – Input Domain: [−600, 600] – Optimum f (x): 0 5. Schwefel Function:  n – Formula: f (x) = 418.9829n − i=1 xi sin( |xi |) – Input Domain: [−500, 500]n – Optimum f (x): −418.9829 6. Ackley Function:   n n – Formula: f (x) = 20 + e − 20e(−0.2 n1 i=1 x2i ) − e( n1 i=1 cos(cxi )) – Input Domain: [−32.768, 32.768]n – Optimum f (x): 0.

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Results

The effectiveness comparison for individual algorithms was based on the analysis of test results. The result for OFNbee was generated using the R environment. The remaining results come from the available publications. For a proper comparison, each algorithm was executed 30 times for ABC and OFNBee for each function. Table 1, 2, 3, 4, 5, 6 and 7 shows a comparison of the results of the ABC and OFNBee operation. Table 1. The results of the experiment conducted for ABC: NP – the size of a colony (10), SD: standard deviation, AV: global minimum average. Function

ABC 10

OFNbee

SD AV SD time AV time

0 0 0,0129942516 0,0563333333

0 0 0,0158295519 0,0733333333

Rosenbrock SD AV SD time AV time

0,015435356 0,0065841095 0,0107425462 0,0653333333

0,0111787799 0,0050520758 0,0074278135 0,08

Rastrigin

SD AV SD time AV time

0 0 0,0199453276 0,0723333333

0 0 0,0098552746 0,0783333333

Griewank

SD AV SD time AV time

0 0 0,0191815285 0,081

0 0 0,0169651434 0,0786666667

Schwefel

SD AV SD time AV time

51,1822376527 28,044323635 0,0102833422 0,0833333333

50,2750418227 30,3723629905 0,0104166092 0,1053333333

Ackley

SD AV SD time AV time

0 4,44E−16 0,0189524511 0,0716666667

0 4,44E−16 0,0084486277 0,081

Sphere

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Table 2. The results of the experiment conducted for ABC: NP – the size of a colony (50), SD: standard deviation, AV: global minimum average. Function

ABC 50

OFNbee

SD AV SD time AV time

0 0 0,0154659433 0,2423333333

0 0 0,0187082869 0,325

Rosenbrock SD AV SD time AV time

0,0035322964 0,0026732212 0,0208000884 0,2946666667

0,0031858532 0,002336541 0,0187696257 0,3683333333

Rastrigin

SD AV SD time AV time

0 0 0,0198268366 0,28

0 0 0,018695995 0,3423333333

Griewank

SD AV SD time AV time

0 0 0,0182700056 0,278

0 0 0,0206336406 0,3546666667

Schwefel

SD AV SD time AV time

0,0069411007 0,0028278171 0,0261494379 0,393

0,0590362232 0,0126121427 0,0299808368 0,4366666667

Ackley

SD AV SD time AV time

0 4,44E−16 0,0226644691 0,3036666667

0 4,44E−16 0,0209432731 0,376

Sphere

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Table 3. The results of the experiment conducted for ABC: NP – the size of a colony (100), SD: standard deviation, AV: global minimum average. Function

ABC 100

OFNbee

SD AV SD time AV time

2,53E−18 2,42E−18 0,0304393122 0,619

2,03E−18 2,35E−18 0,0460496983 0,8603333333

Rosenbrock SD AV SD time AV time

0,0007667024 0,0015398727 0,0321365621 0,585

0,001252226 0,001516892 0,029048117 0,761

Rastrigin

SD AV SD time AV time

0 0 0,0408867233 0,702

4,99E−14 9,12E−15 0,0651752721 0,9406666667

Griewank

SD AV SD time AV time

4,31E−06 1,16E−06 0,0467335317 0,6943333333

0,000028885 1,09E−05 0,0598551892 0,8863333333

Schwefel

SD AV SD time AV time

0,000395318 9,97E−05 0,0552070294 0,7826666667

0,0006950621 0,0002142501 0,0565522848 0,8746666667

Ackley

SD AV SD time AV time

0 4,44E−16 0,0393919299 0,92

0 4,44E−16 0,1623958552 1,21

Sphere

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Table 4. The results of the experiment conducted for ABC: NP – the size of a colony (300), SD: standard deviation, AV: global minimum average. Function

ABC 500

OFNbee

SD AV SD time AV time

7,07E−19 8,30E−19 0,1834562806 1,873

6,31E−19 8,25E−19 0,0911516357 2,675

Rosenbrock SD AV SD time AV time

0,0003109263 0,0003495873 0,0356692984 1,6696666667

0,0003925343 0,00040443 0,0593460531 2,3456666667

Rastrigin

SD AV SD time AV time

0 0 0,6845775608 2,4613333333

0 0 0,0881880175 2,8823333333

Griewank

SD AV SD time AV time

9,34E−09 3,64E−09 0,2196538552 2,0873333333

1,28E−08 5,16E−09 0,147276033 2,8216666667

Schwefel

SD AV SD time AV time

2,37E−06 0,000025893 0,1307823497 2,3883333333

9,82E−08 2,55E−05 0,1313781941 2,8346666667

Ackley

SD AV SD time AV time

0 4,44E−16 0,1912559818 2,7706666667

0 4,44E−16 0,1420413571 3,6163333333

Sphere

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Table 5. The results of the experiment conducted for ABC: NP – the size of a colony (500), SD: standard deviation, AV: global minimum average. Function

ABC 500

OFNbee

SD AV SD time AV time

3,81E−19 4,13E−19 0,3087477488 3,173

4,52E−19 5,08E−19 0,1410571743 4,6783333333

Rosenbrock SD AV SD time AV time

0,0001780305 0,0001767066 0,1039589884 2,8516666667

0,0002794633 0,0002222803 0,089324412 4,2026666667

Rastrigin

SD AV SD time AV time

0 0 0,1215399277 3,4073333333

0 0 0,1556406855 5,0403333333

Griewank

SD AV SD time AV time

4,08E−09 1,82E−09 0,1428607553 3,2666666667

8,12E−09 2,36E−09 0,2076327992 5,043

Schwefel

SD AV SD time AV time

0,000000026 2,55E−05 0,1814260243 3,8786666667

1,99E−07 0,000025513 0,2395359498 5,0213333333

Ackley

SD AV SD time AV time

0 4,44E−16 0,1650774809 4,4466666667

0 4,44E−16 0,7527122605 6,5703333333

Sphere

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Table 6. The results of the experiment conducted for ABC: NP – the size of a colony (800), SD: standard deviation, AV: global minimum average. Function

ABC 800

OFNbee

SD AV SD time AV time

3,03E−19 3,05E−19 0,3897898048 5,1753333333

4,37E−19 3,76E−19 0,2637919766 8,23

Rosenbrock SD AV SD time AV time

2,72E−05 6,02E−05 0,254811402 4,5323333333

2,81E−05 6,00E−05 0,1015562808 7,1463333333

Rastrigin

SD AV SD time AV time

0 0 0,1581941917 5,4856666667

0 0 0,2766883387 8,7576666667

Griewank

SD AV SD time AV time

7,86E−10 3,79E−10 0,2549239237 5,39

1,08E−08 2,71E−09 0,4275814099 8,555

Schwefel

SD AV SD time AV time

2,88E−08 2,55E−05 0,4099094377 6,4913333333

1,95E−08 2,55E−05 0,3530693006 8,802

Ackley

SD AV SD time AV time

0 4,44E−16 0,6301969223 7,4696666667

0 4,44E−16 0,3751413527 11,23

Sphere

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Table 7. The results of the experiment conducted for ABC: NP – the size of a colony 1000, SD: standard deviation, AV: global minimum average. Function

ABC 1000

OFNbee

SD AV SD time AV time

2,87106145737421E−019 3,09170252300814E−019 0,1951919185 6,0463333333

1,67143365658109E−019 2,17327638389649E−019 0,4182845372 11,0103333333

Rosenbrock SD AV SD time AV time

9,53712187610447E−005 8,81411953523153E−005 0,116166745 5,7353333333

9,86472388911669E−005 8,88684794341611E−005 0,1510579549 9,5656666667

Rastrigin

SD AV SD time AV time

0 0 0,2261728055 6,821

0 0 0,4041741971 11,5446666667

Griewank

SD AV SD time AV time

3,63297167141123E−010 2,12517662726934E−010 0,3196731448 6,7256666667

7,8642539747665E−010 3,86326233966135E−010 0,4765297207 11,4443333333

Schwefel

SD AV SD time AV time

1,18225071721224E−009 2,54554042802132E−005 0,3273019275 7,799

0,00000006 2,54714554330349E−005 1,8083592932 13,1743333333

Ackley

SD AV SD time AV time

0 4,44089209850063E−016 0,2898957403 8,8853333333

0 4,44089209850063E−016 2,8237452915 17,1306666667

Sphere

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Fig. 3. Results of 30 independent executions of OFNBee algorithm for the Rosenbrock function

Fig. 4. Results of 30 independent executions of OFNBee algorithm for the Schwefe function

The results presented in Table 1, 2, 3, 4, 5 and 6 show the effect of the colony size on the accuracy of the algorithm result. One can observe that, in several cases, the accuracy of the result achieved increases with increasing colony size. However, the increase in accuracy is not so high that it can be considered significant. When comparing the results from Tables 1, 2, 3, 4, 5, 6 and 7 the best results are obtained with a total population of less than 100 individuals. A result consists not only in finding a more accurate optimum but also a shorter time. Therefore, the colony size of 50 individuals will be used for further research.

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Table 8. The table compares the results for 6 functions - OFNBee, ABC, MBO, IMBO. Function

OFNBee

ABC

MBO

IMBO

0

5.21E−18

3.36E+01

0

Sphere

SD AV

0

4.88E−17

0

0

Rosenbrock

SD

0,0031858532

0,008658828

8.67E−03

0,142502861

AV

0,002336541

0,013107593

0

0

Rastrigin

SD

0

4.40E−18

0

3.18323E−14

AV

0

4.76E−17

0

4,55E−15

Griewank

SD

0

1.93E−19

0,32563139

0

AV

0

5.10E−19

0

0

Schwefel

SD

0,0590362232

9.09E−13

1.94E+16

3.52272E+30

AV

0,0126121427

−4189.828873

−2,96E+211 −6.17561E+29

Ackley

SD

0

3.57E−17

0,0178353

AV

4,44089209850063E−016

1.71E−16

8.23045E−15 1.57E−14

2.30E−15

Table 9. The table compares the results for 6 functions - TLBO, HBMO, BBMO Function

HBMO

BBMO

0,67

0

46,07

24,37

SD 3.56E−01 AV 2,03E−12

4,03

1.59E−08

Griewank

SD 0 AV 0

1.44E−02 0

Schwefel

SD 1.48E+02 AV −20437.84

Ackley

SD 8.32E−31 AV 3.55E−15

Sphere

TLBO SD 0 AV 0

Rosenbrock SD 3.56E−01 AV 47,0162 Rastrigin

Table 10. The comparison of OFNBee with ABC and MBO, IMBO. Function

OFNBee - ABC OFNBee - MBO OFNBee - IMBO OFNBee ABC OFNBee MBO OFNBee IMBO

Sphere

x

x

Rosenbrock x

x

x

x x

x

x

Rastrigin

x

x

x

x

Griewank

x

x

x

x

Schwefel

x

Ackley Suma

5

x x

x

1

5

x x

x 4

5

4

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Table 11. The comparison of OFNBee with TLBO as well as HBMO and BBMO Function

OFNBee - TLBO OFNBee - HBMO OFNBee - BBMO OFNBee TLBO OFNBee HBMO OFNBee BBMO

Sphere

x

x

Rosenbrock x Rastrigin

x

Griewank

x

Schwefel

x

Ackley

x

Suma

6

x

x

x

x

x

x

x

x

x

x

x

2

4

4

2

0

The repeatability of the algorithm’s operation is demonstrated by the results of individual independent executions of the algorithm. As the Table 1 and 2 indicates, the results are always the same for the Sphere, Rastrigin, Griewank and Ackley functions. However, for the Schwefe and Rosenbrock functions, the results of individual executions are shown in Fig. 3 and Fig. 4. As one can see for the Rosenbrock function, the distribution of individual results relative to the value sought is diverse. The scatter of results is smaller for Schwefe function. Summing up the scatter of results for all the six functions, we can conclude that the new OFNBee method is stable. The analysis of the results of the ABC and OFNBee algorithms for the colony population of 50 individuals shows that the results obtained using the new method are in most cases more accurate. In order to draw correct conclusions, the results available in the literature [38] for other algorithms mentioned in this article were compared and presented in Table 8 and 9. Table 10 and Table 11 summarize the comparison of the algorithms with OFNBee. As shown in the table, the OFNBee algorithm is more efficient in each case.

5

Conclusions

As a result of the experiment performed, the best-known bee swarm optimization algorithms were compared to each other. The comparison was based on the analysis of results of the operation of algorithms that optimize the most common functions. These functions are often used as a test set for optimizing algorithms. As seen here, the new method using the ordered fuzzy numbers arithmetic and their properties turns out to be as effective as the other methods. While in some cases the results are better than for the competitive methods. In future studies, the test set shall be enlarged and the group of algorithms to be compared will be extended.

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Imprecision Indexes of Oriented Fuzzy Numbers Krzysztof Piasecki(B) and Anna Łyczkowska-Han´ckowiak Institute of Economics and Finance, WSB University in Pozna´n, Pozna´n, Poland {krzysztof.piasecki,anna.lyczkowska-hanckowiak}@wsb.poznan.pl

Abstract. Oriented fuzzy number (OFN) is a kind of imprecise numbers. The main goal of this paper is propose some imprecision ratings dedicated to OFNs. Imprecision consists of ambiguity and indistinguishability. We assess these phenomena. The OFN ambiguity is evaluated by means of ambiguity index defined as a generalization of energy measure determined for fuzzy numbers (FNs). The OFN indistinctness is evaluated by indistinctness index defined as a generalization of Czogała-Gottwald-Pedrycz entropy measure determined for FNs. In this way we obtain useful tools for investigation energy and entropy measures determined for OFNs. Some basic properties of these measures are proven. Particular emphasis is placed on assessing the effects of portfolio diversification. Obtained results may be applied for financial portfolio analysis under imprecision risk.

1 Introduction Ordered fuzzy numbers are defined by Kosi´nski et al. [5] who in this way were going to introduce a fuzzy number (FN) supplemented by orientation. For some formal reason [4], the original Kosi´nski’s theory was revised in [12]. If ordered fuzzy number is linked to the revised theory, then it is called Oriented Fuzzy Number (OFN). Any FN is interpreted as imprecise information of real number. An increase in information imprecision reduces usefulness of this information. Therefore, it is logical to consider the problem of imprecision assessment. After Klir [3] we understand imprecision as a superposition of ambiguity and indistinctness of information. Ambiguity can be interpreted as a lack of a clear recommendation between one alternative among various others. We measure the FN ambiguity by applying the energy measure defined by de Luca and Termini [8]. Indistinctness is understood as a lack of explicit distinction between recommended and not recommended alternatives. The FN indistinctness of can be measured by entropy measure generally defined by de Luca and Termini [7] and by Piasecki [12]. The particular forms of entropy measure are proposed in [11, 19] and [6]. Due to a good synthetic substantiation and universalism of the applied formula, the entropy measure proposed by Kosko [6] is now widely used. The imprecision ratings obtained by Yager’s entropy measure [19] are equivalent to imprecision ratings obtained by means of entropy measure proposed in [11].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 112–124, 2021. https://doi.org/10.1007/978-3-030-77716-6_10

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The main aim of this paper is to extend the notions of energy and entropy measures to the case of all OFNs space. The paper is organised as follows. Section 2 presents the basic concept of OFNs. Section 3 briefly discusses the idea of imprecision [3]. The same chapter describes ambiguity index and indistinctness index as some ratings for evaluation of OFN imprecision. In Sect. 4 the authors present some properties of imprecision ratings determined for trapezoidal OFN. Section 5 shows some possibilities of application of obtained results for financial portfolio analysis. Concluding remarks and the future direction of research are presented in Sect. 6.

2 Oriented Fuzzy Numbers – Basic Facts Objects of any considerations may be given as elements of a predefined space X. The basic tool for an imprecise classification of these elements is the notion of fuzzy sets introduced by Zadeh [20]. Any fuzzy set A is unambiguously determined by means of its membership function μA ∈ [0, 1]X . From the point-view of multi-valued logic, the value μA (x) is interpreted as the truth value of the sentence “x ∈ A”. By the symbol F(X) we denote the family of all fuzzy sets in the space X. Dubois and Prade [2] have introduced fuzzy numbers (FNs) as such a fuzzy subset in the real line which may be interpreted as imprecise approximation of a real number. We have that any FN can be equivalently defined as follows: Theorem 1 [1]: For any FN L there exists such a non-decreasing sequence (a, b, c, d ) ⊂ R that L(a, b, c, d , LL , RL ) = L ∈ F(R) is determined by its membership function μL (·|a, b, c, d , LL , RL ) ∈ [0, 1]R described by the identity. ⎧ 0, x∈ / [a, d ], ⎪ ⎪ ⎨ LL (x), x ∈ [a, b], μL (x|a, b, c, d , LL , RL ) = (1) ⎪ 1, x ∈ [b, c], ⎪ ⎩ RL (x), x ∈ [c, d ], where the left reference function LL ∈ [0, 1][a,b] and the right reference function RL ∈ [0, 1][c,d ] are upper semi-continuous monotonic ones meeting the conditions: LL (b) = RL (c) = 1, ∀x∈]a,d [ : μL (x|a, b, c, d , LL , RL ) > 0.

(2) 

(3)

The family of all FNs we denote by the symbol F. For any z ∈ [b, c], a FN L(a, b, c, d , LL , RL ) is a formal model of linguistic variable “about z”. Understanding the phrase “about z” depends on the applied pragmatics of the natural language. The ordered FNs were intuitively introduced by Kosi´nski et al. [5] as an extension of the FNs concept. Ordered FNs usefulness follows from the fact that it is interpreted as FNs with additional information about the location of the approximated number. Currently, ordered FNs defined by Kosi´nski are often called Kosi´nski’s numbers [14, 17, 18]. A significant drawback of Kosi´nski’s theory is that there exist such

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Kosi´nski’s numbers which, in fact, are not FNs [4]. For this reason, the Kosi´nski’s theory was revised by Piasecki [13]. If an ordered FN is determined with use of the revised definition, then it is called Oriented FN (OFN). Definition 1 [13]: For any monotonic sequence (a, b, c, d ) ⊂ R OFN −→ L (a, b, c, d , SL , EL ) is defined as the pair of orientation a, d = (a, d ) and FN L ∈ F(R) determined by its membership function μL (·|a, b, c, d , SL , EL ) ∈ [0; 1]R given by the identity ⎧ 0, x∈ / [a, d ] = [d , a], ⎪ ⎪ ⎨ SL (x), x ∈ [a, b] = [b, a], μL (x|a, b, c, d , SL , EL ) = , (4) ⎪ 1, x ∈ [b, c] = [c, b], ⎪ ⎩ EL (x), x ∈ [c, d ] = [d , c]

↔

where the starting-function SL ∈ [0; 1][a,b] and the ending-function EL ∈ [0; 1][c,d ] are upper semi-continuous monotonic functions satisfying the conditions SL (b) = EL (c) = 1. ∀x∈]a,d [ μL (x|a, b, c, d , SL , EL ) > 0.

(5) (6)



The space of all OFN is denoted by the symbol K. Any OFN describes an imprecise number with additional information about the location of the approximated number. This ↔

information is given as orientation of OFN. If a < d then OFN L(a, b, c, d , SL , EL ) −→ has the positive orientation a, d . For any z ∈ [b, c], the positively oriented OFN ↔

L(a, b, c, d , SL , EL ) is a formal model of linguistic variable “about or slightly above −→ ← → z”. If a > d , then OFN L (a, b, c, d , SL , EL ) has the negative orientation a, d . The family of all positively oriented OFNs and the family of all negatively oriented OFNs we respectively denote by the symbols K+ and K− . For any z ∈ [c, b], the negatively ↔

oriented OFN L (a, b, c, d , SL , EL ) is a formal model of linguistic variable “about or slightly below z”. Understanding the phrases “about or slightly above z” and “about or slightly below z” depends on the applied pragmatics of the natural language. If a = d , describes un-oriented real number a ∈ R. It implies then OFN that (7) ↔

Moreover, for any OFN L (a, b, c, d , SL , EL ) we define its core as follows (8) where μL (·|a, b, c, d , SL , EL ) ∈ [0; 1]R is its membership function. In [13], the dot product of any real number and OFN is defined equivalently, as in [5]. The addition of OFNs also is defined in [13]. These arithmetic operations have a high level of formal complexity [9, 10]. Due to that, in many practical applications researchers limit the use of OFN only to a form presented below.

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Definition 2 [13]. For any monotonic sequence (a, b, c, d ) ⊂ R the trapezoidal OFN ← → (TrOFN) Tr (a, b, c, d ) is defined as the OFN determined by its membership function → (·|a, b, c, d ) ∈ [0; 1]R given by the identity. μ← Tr ⎧ ⎪ 0, x ∈ / [a, d ] = [d , a], ⎪ ⎪ ⎨ x−a , x ∈ [a, b[ = ]b, a], b−a → (x|a, b, c, d ) = μ← (9) Tr ⎪ 1, x ∈ [b, c] = [c, b], ⎪ ⎪ ⎩ x−d , x ∈ ]c, d ] = [d , c[ c−d The symbol KTr denotes the space of all TrOFNs. The family of all positively oriented TrOFNs and the family of all negatively oriented TrOFNs are respectively − denoted by the symbols K+ Tr and KTr . Moreover, any a ∈ R is represented by TrOFN It implies that (10)   ↔ In [5] and [13], the dot product · of any pair β, L ∈ R × KTr is determined as follows ↔ ← → ← → β  L = β  Tr (a, b, c, d) = Tr (β · a, β · b, β · c, β · d)

In [13], the sum

(11)

is determined as follows

(12)

Moreover, for any pair condition

in

[13]

it

is

shown that their sum fulfil the (13)

The unary minus operator “−” on R is extended to the minus operator the identity

by (14)

3 Evaluation of Imprecision for Oriented Fuzzy Numbers After Klir [3] we understand imprecision as a superposition of ambiguity and indistinctness of information. Ambiguity can be interpreted as a lack of a clear recommendation between one alternative among various others. Indistinctness is understood as a lack of explicit distinction between recommended and not recommended alternatives.

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Any OFN is a particular kind of imprecision information. An increase in information imprecision reduces suitability of this information. Therefore, it is logical to consider the problem of imprecision assessment. The increase in the ambiguity of an OFN suggests a higher number of alternative recommendations to choose from. This leads to an increase in the risk of choosing an incorrect assessment from recommended alternative ones. This may result in making a decision, which will be ex post associated with a loss of chance. Therefore, increase in the ambiguity of OFN implies the decrease in the utility of information described by OFN. The proper tool for measuring the ambiguity of FN is an energy measure defined by de Luca and Termini [8]. For any FN L = L(a, b, c, d , LL , RL ) ∈ F, de Luca and Termini [8] propose to use energy measure determined as follows (15) where μL ∈ [0, 1]R is the membership function determining FN L. In this paper, we propose to generalize energy measure (15) to the ambiguity index ↔

↔

assessing the ambiguity of any OFN L = L (a, b, c, d , LL , RL ) in following way (16) ↔

where μL ∈ [0, 1]R is the membership function determining OFN L. For any negatively oriented OFN its ambiguity index is negative and for any positively oriented OFN its ambiguity index is positive. For the case a ≤ d the member↔

ship function of OFN L (a, b, c, d , SL , EL ) is equal to the membership function of FN L(a, b, c, d , SL , EL ). This fact implies the existence of isomorphism For this reason, the mapping given by the identity (17) is an extension of energy measure to the domain of all OFNs. We propose to use this extension as energy measure of any OFN. It all means that ambiguity index stores the information on energy measure and additionally also about the orientation of assessed OFN. The right tool for measuring the indistinctness of an FN, is the entropy measure  F , proposed also by de Luca and Termini [7] and modified by Piasecki [12]. e ∈ R+ 0 The most widely kind of entropy measure is described by Kosko [6]. On the other hand, in [15, 16], it is shown that Kosko’ entropy measure is not convenient for portfolio analysis. Therefore, we propose to evaluate indistinctness of arbitrary FN by Czogała–Gottwald–Pedrycz entropy measure introduced in [11]. For any FN L = L(a, b, c, d , LL , RL ) ∈ F, Czogała–Gottwald–Pedrycz entropy measure is determined as follows

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(18) where μL ∈ [0, 1]R is the membership function determining FN L. In this paper, we propose to generalize entropy measure (18) to the indistinctness index ↔

↔

assessing the indistinctness of any OFN L = L (a, b, c, d , LL , RL ) in following way (19) ↔

where μL ∈ [0, 1]R is the membership function determining OFN L. For any negatively oriented OFN its indistinctness index is negative and for any positively oriented OFN its indistinctness index is positive. In analogous way as above,  K given by the identity we can justify that the mapping e ∈ R+ 0 (20)  F is an extension of entropy measure e ∈ R+ to the domain of all OFNs. We propose 0 to use this extension as entropy measure of any OFN. It all means that indistinctness index stores the information on energy measure and additionally also about the orientation of assessed OFN. The notions of ambiguity index and of indistinctness one give new perspectives for imprecision management.

4 Imprecision Evaluation for Trapezoidal Oriented Fuzzy Numbers ← → For any TrOFN Tr (a, b, c, d ), its ambiguity index and indistinctness one are determined basing on the following relations (21) (22) For any monotonic sequence (a, b, c, d ) ⊂ R, we have (23)  

←

← → → ∀(α,δ)∈[a,b]×[c,d ] : e Tr (α, b, c, δ) ≤ e Tr (a, b, c, d )

(24)

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very easy  It  is ↔ β, L ∈ R × KTr we have

to

check

that

for

any

pair

(25) (26) (27) (28) by

Moreover, for any pair using the identity (13) we can easily get

(29) (30) (31) (32) In other cases, analogous imprecision assessments are a little more complicated. We have here: Theorem 2: For any pair

we have

(33)

Proof: Let us take into account The addition is a pair commutative is commutative. Therefore, we can restrict our considerations to the value Then we have

(34)

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where ⎧ ⎪ ⎪ ⎨

(α, ε) ∈ {(a, e), (b, f )} α + ε = min{a + e, b + f } . ⎪ (δ, χ ) ∈ {(d , h), (c, g)} ⎪ ⎩ δ + χ = min{d + h, c + g} If

(35)

then by using (23) and (34) we get

We see, that condition (33) is fulfilled for any sum then we have

We see, that condition (33) is also met for any sum Theorem 3: For any pair have.

If

QED. we (36)

Proof: As we know, we can restrict our considerations to the case Then we have

(37)

where the sequence (α, δ, ε, χ ) is determined by (35). then by using (24) and (37) we get If

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Moreover, here we have (δ + χ − c) − g + f − (α + ε − b) ⎧ + h − c) − g + f − (a + e − b) ε = e, χ (d ⎪ ⎪ ⎨ ε = f ,χ (δ + χ − c) − g + f − f = ⎪ g − g + f − + ε − b) ε = e, χ (α ⎪ ⎩ g−g−f −f ε = f ,χ

=h =h =g =g

It implies that

←  1 → · ((δ + χ − c) − g + f − (α + ε − b)) ≤ g Tr (e, f , g, h) ≤ 0 2 Therefore, we get

(38)

We see, that condition (36) is fulfilled for any sum then we have

Moreover, then we have and (38), we obtain.

We see, that condition (40) is also met for any sum

If

Therefore, by using (14), (22)

QED.

5 Portfolio Diversification All the results presented above may be presented in a form suitable for financial portfolio analysis. Let the oriented portfolio function π : KTr × KTr × [0, 1] → KTr be given by the identity. (39) This function describes the effects of financial portfolio diversification for the case when any profit indexes1 are given as OFNs. We can easily prove the following theorem: 1 For example, return rate, discount factor, present value.

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Theorem 4: For any real number λ ∈ [0, 1] we have: • for any pair (40) (41) • for any pair

(42)

• for any pair (43) Proof:

All

above

conditions

are obtained by means of replacing The identities (26) and (30) implies the identity (40). The identity (41) follows from (28) and (32). The inequality (42) results from (26) and (33). The inequality (36) together with the identity (28) implies the inequality (43). QED. ↔ ↔ ← → ← → Example 1: Let us consider OFNs K = Tr (0, 4, 8, 12), L = Tr (18, 16, 10, 4) and ↔ ↔ ↔ ← → ← → M = Tr (4, 10, 16, 18). OFNs K and M are positively oriented. OFN L is negatively oriented. Let us take into account the values.

By using (21) and (17), we get

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Therefore, we have

We see that if index profits have different orientation then portfolio diversification significantly reduces the ambiguity of portfolio profit index. On the other hand, if index profits have identical orientation then portfolio diversification only averages the ambiguity of portfolio profit index. By using (22) and (20), we get

Therefore, we have

We see that if index profits have different orientation then portfolio diversification significantly reduces the indistinctness of portfolio profit index. On the other hand, if index profits have identical orientation then portfolio diversification only averages the indistinctness of portfolio profit index. In this section it is proven that if index profit values have different orientation then portfolio diversification reduces the imprecision ratings of portfolio profit index. The above example shows that this reduction may be significant. On the other hand, if index profits have identical orientation then portfolio diversification only averages the imprecision ratings of portfolio profit index. Presumably, the differently oriented profit indexes can play in portfolio analysis the same role, which play negatively correlated return rates.

6 Final Remarks In this work, four functionals were proposed to assess OFN imprecision: ambiguity index, indistinctness index, energy measure and entropy measure. The pair of ambiguity and indistinctness indexes may be considered as two-dimensional imprecision index. The pair of energy and entropy measures can be applied as two dimensional vector measure of imprecision.

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The imprecision formal index is very suitable formal tool which may be applied for formal analysis of properties of imprecision measure. Imprecision risk is a possibility of negative consequences of taking actions under the influence of imprecise information. We propose an imprecision measure which is very efficient tool for management of imprecision risk. Section 5 gives an example of using this measure for management of financial assets portfolio. The results presented there can be directly applied to the financial portfolio model described in [9]. The results shown in the numerical example show the possibility of finding smaller dominant of portfolio imprecision risk measure. In our opinion, the subject of further research should be a more accurate estimate of the portfolio imprecision measure for the case when the portfolio components are characterized by index profit with different orientation.

References 1. Delgado, M., Vila, M.A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets Syst. 93, 125–135 (1998) 2. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–629 (1978). https://doi.org/10.1080/00207727808941724 3. Klir, G.J.: Developments in uncertainty-based information. Adv. Comput. 36, 255–332 (1993). https://doi.org/10.1016/s0065-2458(08)60273-9 4. Kosi´nski, W.: On fuzzy number calculus. Int. J. Appl. Math. Comput. Sci. 16(1), 51–57 (2006) ´ ezak, D.: Drawback of fuzzy arithmetics – new intuitions 5. Kosi´nski, W., Prokopowicz, P., Sl˛ and propositions. In: Burczy´nski, T., Cholewa, W., Moczulski, W. (eds.) Methods of Artificial Intelligence, Gliwice, pp 231–237 (2002) 6. Kosko, B.: Fuzzy entropy and conditioning. Inf. Sci. 40, 165–174 (1986). https://doi.org/10. 1016/0020-0255(86)90006-x 7. de Luca, A., Termini, S.: A definition of a nonprobabilistic entropy in the settings of fuzzy set theory. Inform. Control 20, 301–312 (1972) 8. de Luca, A., Termini, S.: Entropy and energy measures of fuzzy sets. In: Gupta, M.M., Ragade, R.K., Yager, R.R. (eds.) Advances in Fuzzy Set Theory and Applications, New York, pp. 321–338 (1979) 9. Łyczkowska-Han´ckowiak, A., Piasecki, K.: The present value of a portfolio of assets with present values determined by trapezoidal ordered fuzzy numbers. Oper. Res. Dec. 28(2), 41–56 (2018). https://doi.org/10.5277/ord180203 10. Łyczkowska-Han´ckowiak, A., Piasecki, K.: Two-assets portfolio with trapezoidal oriented fuzzy present values. In: Váchová, L., Kratochvíl, V. (eds.) 36th International Conference Mathematical Methods in Economics Conference Proceedings, Prague, pp 306–311 (2018) 11. Pedrycz, W., Gottwald, S., Czogała, E.: Measures of fuzziness and operations with fuzzy sets. Stochastica 3, 187–205 (1982) 12. Piasecki, K.: Some remarks on axiomatic definition of entropy measure. J. Intell. Fuzzy Syst. 33(3), 1945–1952 (2017) 13. Piasecki, K.: Revision of the kosi´nski’s theory of ordered fuzzy numbers. Axioms 7(1), 116 (2018). https://doi.org/10.3390/axioms7010016 14. Piasecki, K.: Relation “greater than or equal to” between ordered fuzzy numbers. Appl. Syst. Innov. 2(3), 26 (2019). https://doi.org/10.3390/asi2030026

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15. Piasecki, K., Siwek, J.: Two-asset portfolio with triangular fuzzy present values—an alternative approach. In: Choudhry, T., Mizerka, J. (eds.) Contemporary Trends in Accounting, Finance and Financial Institutions. SPBE, pp. 11–26. Springer, Cham (2018). https://doi.org/ 10.1007/978-3-319-72862-9_2 16. Piasecki, K., Siwek, J.: Multi-asset portfolio with trapezoidal fuzzy present values. Przegl˛ad Statystyczny LXV 2, 183–199 (2018) 17. Prokopowicz, P., Pedrycz, W.: The directed compatibility between ordered fuzzy numbers a base tool for a direction sensitive fuzzy information processing. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2015. LNCS (LNAI), vol. 9119, pp. 249–259. Springer, Cham (2015). https://doi.org/10.1007/9783-319-19324-3_23 18. Prokopowicz, P.: The directed inference for the kosinski’s fuzzy number model. In: Abraham, A., Wegrzyn-Wolska, K., Hassanien, A.E., Snasel, V., Alimi, A.M. (eds.) Proceedings of the Second International Afro-European Conference for Industrial Advancement AECIA 2015. AISC, vol. 427, pp. 493–503. Springer, Cham (2016). https://doi.org/10.1007/978-3-31929504-6_46 19. Yager, R.R.: On the fuzziness measure and negation, Part I: membership in the unit interval. Internat. J. Gen. Systems 5, 221–229 (1979) 20. Zadeh, L.A.: Fuzzy sets. Inf. Contr. 8, 338–353 (1965). https://doi.org/10.1016/S0019-995 8(65)90241-X

Detailed Evaluation of Fuzzy Sets in Rule Conditions as a Key for Accurate and Explainable Rule-Based Systems (B) Sebastian Porebski 

Department of Cybernetics, Nanotechnology and Data Processing, Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Gliwice, Poland [email protected]

Abstract. This work is about research looking for a compromise between accuracy and interpretability of decision support systems. The most accurate of these systems are characterized by high incomprehensibility. On the other hand, rule systems are interpretable but their weakness is accuracy. The search for balanced solutions is a necessary task if we want to have tools that can be the subject of cooperation between a knowledge engineer and a human expert (or finally, a user). This work presents the results of research focused on the evaluation of fuzzy sets as a basic element of rule sets. The evaluation of fuzzy set matching to training data allows you to choose the best components at the beginning of the rule extraction. This approach results in high testing accuracy and maintains satisfactory interpretability.

Keywords: Interpretable rules decision support

1

· Fuzzy rule extraction · Explainable

Introduction

The growing interest in explainable artificial intelligence (XAI) methods means that researchers are trying to transform formerly known methods to meet conditions of the usefulness for a human user [1]. Unfortunately, usually the most accurate methods are characterized by the low interpretability of both the inference mechanism and the knowledge base they extract [2]. However, among computational intelligence methods there is a branch of solutions that have always been characterized by high interpretability, e.g. decision trees or rule sets (one can be transformed into the other and vice versa) [7]. Nonetheless, interpretability of the knowledge base as well the explainable inference mechanism are associated with lowering of these method’s accuracy. It is not easy task to find compromise between these two aims (interpretability/explainability and accuracy). That is, detailed and accurate knowledge must be of large size (and it is less interpretable). On the other hand, more general knowledge has not the highest c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 125–135, 2021. https://doi.org/10.1007/978-3-030-77716-6_11

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reliability, but can easily be written in the form of a small decision tree or several/a dozen of rules (high interpretability). The case of this compromise cannot be neglected. One could put forward the thesis that in approaches such as rule-based, the problem of poor explainability does not occur, which, however, will not be entirely true. Also in this area, researchers are trying to improve the performance of rule systems by using deciated methods to optimize rule sets. However, these approaches need to be done with carefully so as not to lose explainability [3]. This work presents the preliminary results of new research on extracting fuzzy conditional rules where decision support is done using the belief measure from Dempster-Shafer theory [10]. The combination of both theories serves to capture imprecision and uncertainty [11]. This idea was formerly utilized in research of diagnosis support [8]. However, in this paper it is focused on the new idea which is to evaluate every single fuzzy set in the extracted rules to later choose only the most suited ones. The ideas presented here try to reconcile the explainability vs. reliability compromise with satisfaction and are a good premise for developing rule-based methods in decision support issues.

2

Methods

In the study, attention is focused on data-driven rule-based system. So, let us to introduce some information regarding training data. Whole data set with N cases can be described by the following matrix   x1 x2 · · · xi · · · x N D= , (1) c1 c2 · · · ci · · · c N where x i is i-th data case which is a vector of p features   x i = xi, 1 xi, 2 · · · xi, j · · · xi, p ,

(2)

and ci is the class index of the i-th data case ci ∈ {1, · · · , C},

(3)

where C is the considered number of classes. We can also take all x i from one class and present the l-th class data as follows   X (l) = x 1, · · · , x i, · · · , x Nl , . (4) where Nl is the number of l-th class data cases and ci = l for i = 1, · · · , Nl . 2.1

Design of Fuzzy Membership Functions

When matrix D is divided into class data X (l) (4), for each row of X (l) we can calculate its mean value m j,l and it will relate to mean value of j-th feature in l-th class. To provide semantically intuitive membership functions [4], the mean

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values of j-th feature in all classes (l = 1, · · · , C) can be sorted in ascending way. Hence, let us assume in this point that m j,l < m j,l+1 for l = 1, · · · , C − 1. In this way, fuzzy membership functions μ(l) j (l = 1, · · · , C) can be calculated. Due to the fact that membership functions for classes with a minimal and maximal μ(l) j can describe left- and right-opened fuzzy sets, respectively, their definition depends on the class index:   2 − x j − m j, 1 (1) μ j = t + (1 − t) · exp , (5) 2σ1 

∀1 0 and k=1

on the idea of possibility measure [12] and calculated as [11] M k(l) (x i ) = min [μ(l) j (xi, j )] (l)

(18)

j ∈Jk

is a matching level of x i and membership functions μ(l) j included in the rule premise. Set denoted as J(l) contains indexes of features in R k(l) premise conditions, k

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(l) e.g. J(l) 1 = 1 and J17 = {1, 7, 12}. To calculate the final output of the rule-based system, in the study, belief measures for different classes are compared and the final class is chosen as this related to the maximal belief value, i.e. [8].

class(x i ) =

3

⎧ ⎪ ⎨ class(l) ⎪

if ∃! l = arg max [BE L (x i, l)] ,

⎪ ⎪ not classified ⎩

otherwise.

1 ≤l ≤C

(19)

Experiments

The popular benchmark database Cleveland Heart Disease 1 was chosen for the experiments. It contains information about 306 patients (N = 306) diagnosed for heart disease based on 13 symptoms (p = 13). In this dataset, two diagnoses are specified (C = 2). A little more than a half of patients (165) were not diagnosed with presence of heart disease, and the rest (141 patients) was diagnosed with a certain heart disease. Fuzzy sets of membership functions created according to the studied data are presented in Fig. 2.

Fig. 2. Initial fuzzy sets used in the rule premises

The first experiment about studied idea is to compare the fuzzy set evaluation measures that will allow obtaining the most balanced result in the term of interpretability and accuracy of extracted rule sets. All experiments are done with 10-fold cross validation approach [6]. The experiment can be described by following steps: 1. Calculation of the membership function shapes according to (5–7). 2. Evaluation of all fuzzy sets correspondence to training data with chosen evaluation measure Ei (10), (14) or (15). 3. For fuzzy sets designed for different classes average and maximal evaluation values are obtained. They would be used as a search interval of the best fuzzy set. 4. Set threshold τE(l) = E l . 5. Delete all fuzzy sets for which E1 ( j, l) < τE(l) . 1

(https://archive.ics.uci.edu/ml/datasets/Heart+Disease).

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6. Use remained part of fuzzy sets as the basis of creation simple rule premises. Perform classification of testing data according to (19). Calculate testing accuracy (E), interpretability measure [5] (Q I NT ) and final evaluation (FE) as follows: N − Ne · 100%, (20) E= N Q I NT = 1 − QCOM P, (21) where QCOM P is the mean value of three complexity measures of the fuzzy rule sets proposed and described in detail in [5] and FE =

E − QCOM P 100%

(22)

7. Incrementally increase τE from average and maximal evaluation value (found in step 3) and repeat the steps 5 and 6. Table 1. Comparison of 10-fold (distribution optimally balanced) cross-validation results of rule extraction for different fuzzy set evaluation methods τE

E (%) E1

Ei ( j, l)

80.9 81.9 81.9 81.5 80.9 81.8 82.5 82.8 82.8 82.5 80.2 78.9 75.4 65.6 63.9 54.7 46.8 46.8 max(Ei ( j, l)) 39.4

Q I NT

FE

E2

E3

E1

E2

E3

E1

E2

E3

83.5 82.5 83.8 84.8 85.1 83.5 83.2 81.2 81.6 72.8 66.0 60.7 60.4 59.4 59.4 59.4 59.4 59.4 59.4

81.8 82.5 82.5 81.9 82.5 84.5 85.8 86.1 85.1 85.1 84.8 85.1 84.4 84.1 82.2 77.7 77.0 76.0 75.7

0.58 0.59 0.60 0.61 0.63 0.65 0.67 0.70 0.71 0.73 0.76 0.78 0.82 0.86 0.87 0.90 0.92 0.92 0.94

0.60 0.52 0.62 0.65 0.70 0.74 0.79 0.84 0.86 0.89 0.91 0.94 0.95 0.96 0.96 0.96 0.96 0.96 0.96

0.50 0.61 0.61 0.64 0.66 0.70 0.74 0.75 0.78 0.79 0.82 0.83 0.86 0.86 0.87 0.90 0.93 0.95 0.97

0.39 0.41 0.42 0.42 0.44 0.47 0.50 0.53 0.54 0.56 0.57 0.57 0.58 0.52 0.51 0.44 0.39 0.39 0.34

0.43 0.35 0.46 0.50 0.55 0.57 0.62 0.65 0.68 0.62 0.57 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55

0.32 0.44 0.44 0.46 0.49 0.54 0.60 0.62 0.63 0.64 0.67 0.69 0.70 0.70 0.69 0.68 0.70 0.71 0.73

mean results 72.2 71.8 82.5 0.75 0.82 0.78 0.47 0.54 0.61

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Results of this experiment (for different evaluation measures) is presented in Table 1. According to the Table 1 it is clear that E3 allows obtaining best results in the term of all considered mean values of E, Q I NT and FE. Although E2 = MCC allows obtaining the most interpretable rule sets, it worsen the accuracy. Naturally, each testing accuracies is getting low for the highest τE(l) values (number of rules is getting low) but even then, results for E3 remains on the satisfactory level. That is way weighted difference of TP and FP cases (E3 ) is chosen as the most promising evaluation measure. When E3 is chosen as the most promising fuzzy set evaluation technique in the additional experiment is performed to study whether the described technique of threshold-based fuzzy set refinement allows keeping generalization quality on satisfactory level? To answer this question results of experiment described by steps 1 to 7 are presented on accuracy vs. interpretability diagram [1]. Each result is described by training & testing accuracies together with obtained Q I NT value. Due to the fact that it is also possible to create rules with complex premises, step 6 of experiment is modified to allow creating rules with maximal 1, 2, 3 and 4 conditions. This four results are presented separately in Fig. 3. Training and testing results are connected pair to explain their difference and evaluate the generalization quality of the rule sets. As we can see in Fig. 3, majority of approaches remain stable in the view or training/testing accuracies. If accuracy reduces in the testing step, this reduction is not extremely high. However, we can observe significant part of the results where testing accuracy is even better than the training one (noted by green triangular marks). Although we have found that single and double premise conditions can lead to better testing accuracy than the training one (see. Figure 3), we can see that only very few results lead to significant reduction of the accuracy. Exemplar result of rule sets extracted for considered data set are presented as natural language statements: 1. diagnosis “absence of heart disease”: • ordinary chest pain (x3 ) has occurred, m3(1) = 0.2158, • exercise-induced asthma (x9 ) has not occurred, m9(1) = 0.2755, (1) = 0.2626, • no. of fluoroscopic-positive major vessels (x12 ) is low, m12 (1) • isotope labelling results (x13 ) is negative, m13 = 0.2460. 2. diagnosis “presence of heart disease”: • asymptomatic chest pain (x3 ) has occurred, m3(2) = 0.2680, (2) = 0.2570, • ST slope during exercise (x11 ) has not increased, m11 (2) • no. of fluoroscopic-major vessels (x12 ) is high, m12 = 0.2259, (2) = 0.2491, • isotopic labelling (x13 ) is positive, m13

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Fig. 3. Comparison of training and testing percentage accuracies. Results with good and weak generalization quality are mark as green triangles and red crossed circles, respectively. Dotted lines express the difference between training and testing accuracies.

Fig. 4. Fuzzy sets that have been evaluated and selected to create the final set of rules

4

Discussion and Conclusions

This paper presents a result of experiments of extracting interpretable rules, in which special attention is focused on the appropriate evaluation of the basic elements of the rules, i.e. fuzzy sets in their premises. These fuzzy sets are used in the simple and complex rule premises and provide explainability in the real decision support issues. The inference mechanism used and the selection of rules are very important in decision making by the rule system. Nevertheless, in the study,

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evaluation of matching the fuzzy sets with features (symptoms in the medical area) of training data has a primary importance and influence the accuracy of classification. That is why in the conducted experiments focus the greatest attention to the evaluation of fuzzy sets in the premises of the rules. These fuzzy sets are created based on a simple analysis of training data to ensure the intuitiveness of created membership functions and explainability of the diagnostic rules. Then, each fuzzy set described by membership function was evaluated in terms of matching to objects of the class for which it was designed and in terms of matching to objects of a different classes. The latter aspect is treated as an undesirable situation, therefore studied evaluation methods used the so-called confusion matrix in which all possible, desired and undesired matches are taken into account in a fuzzy way (i.e. using fuzzy confusion matrix). Such a precise evaluation of fuzzy sets influenced the size of rule sets that are characterized by high accuracy of diagnosis support, high stability of accuracy during training and testing (i.e. good generalizing quality) and above all a satisfactory level of interpretability. During the experiments, rules with the simplest premises were extracted. Although the extraction of rules with higher number of logical conditions was also tested, they did not achieve such good results reconciling the aforementioned compromise. It should also be noted that the proposed method is not parametric, i.e. all tuning of the evaluation value threshold and the selection of the best rules (best solution searching) takes place in the training procedure and yet the high testing accuracy is maintained. Acknowledgements. This research is financed from the statutory activities (BK & BKM 2019/2020) of the Faculty of Automatic Control, Electronics and Computer Science of the Silesian University of Technology.

References 1. Adadi, A., Berrada, M.: Peeking inside the black-box: a survey on explainable artificial intelligence (XAI). IEEE Access 6, 52138–52160 (2018). https://doi.org/ 10.1109/ACCESS.2018.2870052 2. Esfandiari, N., Babavalian, M.R., Moghadam, A.-M.E., Tabar, V.K.: Knowledge discovery in medicine: current issue and future trend. Expert Syst. Appl. 41(9), 4434–4463 (2014). https://doi.org/10.1016/j.eswa.2014.01.011 3. Fernandez, A., Herrera, F., Cordon, O., del Jose Jesus, M., Marcelloni, F.: Evolutionary fuzzy systems for explainable artificial intelligence why, when, what for, and where to? IEEE Comput. Intell. Mag. 14(1), 69–81 (2019). https://doi.org/ 10.1109/MCI.2018.2881645 4. Gacto, M.J., Alcala, R., Herrera, F.: Interpretability of linguistic fuzzy rule-based systems: an overview of interpretability measures. Inf. Sci. 181, 4340–4360 (2011). https://doi.org/10.1016/j.ins.2011.02.021 5. Gorzalczany, M.B., Rudzinski, F.: A multi-objective genetic optimization for fast, fuzzy rule-based credit classification with balanced accuracy and interpretability. Appl. Soft Comput. 40, 206–220 (2016). https://doi.org/10.1016/j.asoc.2015.11. 037

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6. Moreno-Torres, J.G., Saez, J.A., Herrera, F.: Study on the impact of partitioninduced dataset shift on k-fold cross-validation. IEEE Trans. Neural Networks Learn. Syst. 23(8), 1304–1312 (2012). https://doi.org/10.1109/TNNLS. 2012.2199516 7. Liu, X., Feng, X., Pedrycz, W.: Extraction of fuzzy rules from fuzzy decision trees: an axiomatic fuzzy sets (AFS) approach. Data Knowl. Eng. 84, 1–25 (2013). https://doi.org/10.1016/j.datak.2012.12.001 8. Porebski, S., Porwik, P., Straszecka, E., Orczyk, T.: Liver fibrosis diagnosis support using the Dempster-Shafer theory extended for fuzzy focal elements. Eng. Appl. Artif. Intell. 76, 67–79 (2018). https://doi.org/10.1016/j.engappai.2018.09.004 9. Powers, D.M.W.: Evaluation from precision recall and f-measure to ROC informedness, markedness and correlation. J. Mach. Learn. Technol. 2(1), 37–63 (2011). https://doi.org/10.9735/2229-3981 10. Shafer, G.: The Mathematical Theory of Evidence. Princeton University Press, Hoboken (1976) 11. Straszecka E.: Diagnostic inference with the dempster-shafer theory and a fuzzy input. In: Kacprzyk J., Szmidt E., Zadrozny S., Atanassov K., Krawczak M. (eds.) Advances in Fuzzy Logic and Technology 2017. EUSFLAT 2017, IWIFSGN 2017. Advances in Intelligent Systems and Computing, vol. 643. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66827-7 33 12. Zadeh L.A.: Fuzzy logic = Computing with words. In: Zadeh L.A., Kacprzyk J. (eds.) Computing with Words in Information/Intelligent Systems 1: Foundations. Heidelberg, vol. 3, Physica-Verlag (1999). https://doi.org/10.1007/978-37908-1873-4 1

Tests for Estimates of the Tolerance Relation Based on Pairwise Comparisons in Binary and Multivalent Form Leszek Klukowski(B) Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland [email protected]

Abstract. The paper presents original tests for verification of estimates of the tolerance relation, obtained of the basis of multiple independent pairwise comparisons, in binary and multivalent form, with random errors. The tolerance relation is the equivalency relation with non-empty intersections. The estimates are obtained with the use of the idea of the nearest adjoining order (NAO), i.e. minimization of differences between relation form and pairwise comparisons. The tests are based on exact or limiting distributions of proposed statistics. The estimates do not require simulations, as the tests presented in Klukowski (2011), i.e. their computational cost is not high.

1 Introduction The estimators of the tolerance relation based on multiple pairwise comparisons in binary an multivalent form, with random errors, proposed in Klukowski (2011 Chapt. 4, 6), have good statistical properties under weak assumptions about comparisons errors. The idea of these estimators have been introduced by Slater (1961) for the preference relation and developed by the author. The properties of the estimators are valid under the principal assumption that the tolerance relation is the true model of data. Therefore, existence of the relation has to be verified with the use of statistical tests. The paper presents such the tests for comparisons in binary and multivalent form. They verify probabilistic properties of comparisons which are valid in the case of existence of tolerance relation in the set under consideration. Any binary comparison states that two elements belong to a common subset including its intersections with other subsets. Any multivalent comparison determines number of subsets of intersection which includes a compared pair of elements. Similar tests for verification of estimates of the preference relation on the basis of binary and multivalent comparisons have been presented in Klukowski (2021a, b). However, their construction and properties are not the same as in the case of the tolerance relation. The approach based on the NAO idea requires optimal solutions of an appropriate discrete optimization problems. They can be obtained with the use of exact or heuristic algorithms (see Gordon 1999, Chapt. 5, Garfinkel and Nemhauser 1972, Chapt. 8, 9). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 136–148, 2021. https://doi.org/10.1007/978-3-030-77716-6_12

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The paper consists of five sections. The second section presents the theoretical basis of the estimation problem: assumptions about distributions of errors of pairwise comparisons, form of the estimators and their statistical properties. In the next section are formulated original tests for binary comparisons – they are based on properties of true relation and distributions of errors of comparisons. The fourth section present tests for multivalent comparisons – constructed in similar way. Last section summarizes the results.

2 Estimation Problem, Form of Estimators and Their Properties for Binary Comparisons 2.1 Estimation Problem for Binary Comparisons We are given a finite set of elements X = {x1 , . . . , xm } (3 ≤ m < ∞). It is assumed that there exists in the set X the tolerance relation, i.e. equivalence relation with non-empty intersections. The relation generates some family of subsets χ1∗ , . . . , χn∗ (2 ≤ n < m). The family χ1∗ , . . . , χn∗ have the property: n q=1

χq∗ = X,

(1)

and the properties: ∃r, s(r = s) such that χq∗ ∩ χs∗ = {0}, where{0} − the empty set,

(2)

each subset χq∗ (1 ≤ q ≤ n) includes an element xi such that xi ∈ χq∗ and xi does not belong to any other subset χs∗ (s = q)

(3)

The tolerance relation defined by (1)–(3) can be expressed by values Tb (xi , xj ) defined as follows: Tb (x i , xj )  0 if exists r, s(r = s not excluded ) such that (xi , xj ) ∈ χq∗ ∩ χs,∗ = 1 othervise.

(4)

The value Tb (xi , xj ) is equal zero for a pair (xi , xj ) belonging to a common subset χq∗ (also intersections of this subset), the condition (3) guarantee uniqueness of the relation. The family χ1∗ , . . . , χn∗ is to be estimated on the basis of N (N ≥ 1) pairwise comparisons gk(b) (xi , xj )(1 ≤ k ≤ N ) of each pair (xi , xj ) ∈ X × X; any pair evaluates a value Tb (xi , xj ), i.e. assumes values from the set {0, 1} and can be disturbed by random error, with known binomial distribution. The following assumptions are made about the number of subsets n and distributions of errors of comparisons. A1. The number of subsets n is unknown. A2. The probabilities of errors satisfy the conditions: 1 (b) P(g k (xi , xj ) − Tb (xi , xj ) = 0) = 1 − δ(δ ∈ (0, ), k = 1, . . . , N ), 2

(5)

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1

(b)

P(gk (xi , xj ) − Tb (xi , xj ) = l) = 1.

l=0

(6)

(b)

A3. The comparisons gk (xi , xj )(k = 1, . . . , N ; (xi , xj ) ∈ X × X) are independent random variables. The assumptions A1–A3 reflect the following properties of distributions of errors of comparisons: the probability of correct comparison is greater than incorrect one, zero is the mode and median of each distribution, the set of all comparisons comprises realizations of independent random variables, the expected value and variance of any comparison error is equal - respectively: E(g (b) k (x i , xj ) − Tb (x i , xj )) = δ,

(7)

(b)

Var(g k (xi , xj ) − Tb (xi , xj )) = (1 − δ)δ.

(8)

2.2 The Form of Estimator and Its Properties 



The estimator χ 1 , . . . , χ n based on the NAO idea is obtained on the basis of the optimal solution of the discrete minimization problem:   N  (b)  (9) (x , x ) − t (x , x ) min gk i j b i j  , 

∈Rm

χ1 ,...,χn ∈FX

k=1

FX – the feasible set, i.e. the family of all tolerance relations χ1 , . . . , χr (r ≥ 2) in the set X, tb (xi , xj ) – the values expressing any relation χ1 , . . . , χr form the set FX , Rm – the set of the form Rm = {1 ≤ i, j ≤ m; j > i}. The estimator χ 1 , . . . , χ n can be expressed alternatively in the form T b (xi , xj ) – defined in the same way as the values Tb (xi , xj ). The minimal value of the function (9) equals zero, the number of optimal solutions can exceed one. The properties of the estimator T b (xi , xj ) are determined in Klukowski (2011), Chapt. 4. In general it is consistent for N → ∞. This fact can be expressed in the following ˜ b the following random variables: way. Let us denote by Wb∗ and W 









Wb∗ = ˜ = W b



N

∈Rm

k=1

N

 ∈Rm

k=1

   (b)  gk (xi , xj ) − Tb (xi , xj ),

(10)

   (b)  gk (xi , xj ) − T˜ b (xi , xj ),

(11) ∼

∼

where: T˜ b (xi , xj ) the values corresponding to any tolerance relation χ 1 , . . . , χ n˜ different than χ1∗ , . . . , χn∗ . Then, it can be shown, that: ˜ b ) < 0, E(W ∗b − W

(12)

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 1 1 ˜ = 0, Var Wb∗ = 0, lim Var W b N →∞ N N →∞ N

(13)

˜ b ) ≥ 1 − exp{−2N ( 1 − δ)2 }. P(W ∗b < W 2

(14)

lim





The relationships (12)–(14) indicate that the estimator χ 1 , . . . , χ n guarantee errorless estimate for N → ∞; the inequality (14), based on Hoeffding (1963) inequality for binary random variables, determines the speed of convergence. 

3 Tests for Verification of an Estimate of Tolerance Relation for Binary Comparisons Tests presented in this section verify estimates of the tolerance relation on the basis (b) of properties of comparisons gk (xi , xj ) ((xi , xj ) ∈ X × X) evaluating true relation χ1∗ , . . . , χn∗ . The null and alternative hypotheses has the general form: H0 : χˆ 1 , . . . , χˆ nˆ ≡ χ1∗ , . . . , χn∗ and the alternative H1 : χˆ 1 , . . . , χˆ nˆ ≡ χ1∗ , . . . , χn∗ . (15) It is proposed to verify the general hypothesis by the set of partial hypotheses: H01 : χ 1 = χ1∗ , H11 : χ 1 = χ1∗ , …, H0, n : χ 1,n = χ ∗ , …, H1,n : χ 1,n = χ ∗ . 1,n 1,n 























Any partial hypothesis, i.e. for any subset χ q 1(1 ≤ q ≤ n), is verified on the basis of test statistic:   N N 1  1  (b) (b) xi ∈χˆ ηq = g (xi , xj ) (1 − gk (xi , xj )) k=1 k k=1 ∈Sqˆ N νq N υq Tˆ b (xi ,xj )=1 (16) where:

  Sqˆ = |xi ∈ χˆ q , xj ∈ χˆ q (j > i) 

νq – number of comparisons of elements from the subset χ q , υq – number of pairs (xi , xj ) such, that xi ∈ χ q and T b (xi , xj ) = 1. (the second sum in (16) includes such pairs that the first element belong to χ q , while the second does not belong). The numbers νq and υq are determined in the following way:

νq = #χˆ q #χˆ q − 1 /2 (17) 









υq = #χ q (m − #χ q ), where:

(18)

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#χ q - number of elements of the subset χ q . The value νq is equal to number of pairwise comparisons of elements of the subset χ q , the vale υq – number of comparisons of elements from the subset χ q with elements not belonging to this subset. (b) The sum in (16) includes all comparisons gk (xi , xj ) of pairs belonging to the set χ q (their number is equal N νq (νq − 1)/2) and all comparisons of pairs such that the first element belongs to the subset χ q , while the second element does not belong to this subset. (b) Under H0 each random variable gk (xi , xj ) from the first sum in (16) and each









(b)

(b)

variable 1−gk (xi , xj ) from the second sum in (16) has expected value E(g k (xi , xj )) = (b)

(b)

δ and variance Var(g k (xi , xj )) = Var(1 − g k (xi , xj )) = δ(1 − δ). The number of components in both sums of the expression (16) is equal N (νq + υq ). Therefore: E(ηq ) = δ, (b)

Var(g k (xi , xj )) =

(19)

1 δ(1 − δ). N (ν q + υq )

(20) 

Under H1 two situations are possible. The first one when the set χ q comprises at least (b)

one element xr such that Tb (xr , xj ) = 1 and therefore P(g k (xr , xj ) = 1) = 1 − δ(k = 1, . . . , N ). Then, the expected value (19) is higher than δ, while the variance is the same, i.e. equal the right hand side of (20). Similar situation occurs when the number of “non-belonging” elements is higher than one. The second situation occurs when the set χ q includes only elements with the values Tb (xi , xj ) = 0, but there exists at least one element xr having also the value Tb (xr , xj ) = 0 / χ q . In this case each component of the first sum in (16) has the same for xj ∈ χ q , but xr ∈ 





(b)

expected value as under H0 , (equal to δ) but some variables 1−gk (xi , xr )(k = 1, . . . , N ) from the second in sum in (16) has expected values equal 1 − δ. Thus, the expected value of the statistic ηq is also higher than δ, while the variance is the same as under H0 . It is also true when the number of such elements is greater than one. Therefore, the verification of each partial hypothesis H0,q : χ q = χq∗ , H1,q : χ q = χq∗ (1 ≤ q ≤ n) can be done on the basis of exact binomial distribution with expected value determined by (19) and variance - by (20). The null hypothesis is E(ηq ) = δ, the alternative E(ηq ) > δ, the rejection region is right-hand side. In the case N (ν q + υq ) ≥ 200 the binomial distribution can be replaced by limiting Gaussian distribution with the same parameters, i.e. expected value and variance. The facts concerning two situations mentioned above suggest to accept the general hypothesis H0 (determined by (15)) if all partial hypotheses H0,q : χ q = χq∗ , H1,q : χ q = χq∗ (q = 1, . . . , n) are accepted. In opposite situation – to reject H0 . The partial hypotheses which are rejected indicate incorrect elements of the estimate. The probabilities of the first type error (significance level) can be determined only for partial hypotheses separately. 











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4 Estimation Problem, Form of Estimators and Their Properties for Multivalent Comparisons 4.1 Estimation Problem for Multivalent Comparisons The pairwise comparisons in multivalent case express the number of subsets of the intersection including any pair of elements. The relation (1)–(3) can be expressed by values Tμ (xi , xj ) defined as follows: Tμ (xi , xj ) = #(∗i ∩ ∗j ),

(21)

where: ∗i – the set of the form ∗i = {s|xi ∈ χs∗ }, #() – number of elements of the set . The values Tμ (xi , xj ) belong to the set = {0, 1, . . . , n}, the comparisons (μ) gk (xi , xj ) (k = 1, . . . , N ) are evaluation of the values Tμ (xi , xj ) disturbed by random errors. The distributions of errors of comparisons have to satisfy the following assumptions:

(μ) (μ) P gk (xi , xj − Tμ (xi , xj ) = 0) ≥ P gk (xi , xj − Tμ (xi , xj ) = ϑ) (ϑ = 0), (22)

(μ) (μ) P gk (xi , xj − Tμ (xi , xj ) = ϑ) ≥ P gk (xi , xj − Tμ (xi , xj ) = ϑ + 1) 1 ≤ ϑ ≤ m − 1),

(μ) (μ) P gk (xi , xj − Tμ (xi , xj ) = ϑ) ≥ P gk (xi , xj − Tμ (xi , xj ) = ϑ − 1) (−(m − 1) ≤ ϑ ≤ −1),

 (μ) P gk (xi , xj − Tμ (xi , xj ) = ϑ) > 0,5, ϑ≥0

 ϑ≤0

(23)

(24) (25)

(μ) P gk (xi , xj − Tμ (xi , xj ) = ϑ) > 0,5,

(26)

(μ) P gk (xi , xj − Tμ (xi , xj ) = ϑ) = 1.

(27)

m ϑ=−m

The conditions (22)–(27) indicate that zero is mode and median of distribution of errors of comparisons and that the distribution is unimodal. The expected value of the error of any comparison can differ from zero. 4.2 The Form of Estimator and Its Properties 



The estimator χ 1 , . . . , χ n based on the idea nearest adjoining in the case of multivalent comparisons is obtained on the basis of the optimal solution of the discrete minimization problem:   N  (μ)  (28) min (x , x ) − t (x , x ) gk i j μ i j  , 

χ1 ,...,χn ∈FX

∈Rm

k=1

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where: tμ (xi , xj ) – the values expressing any relation χ1 , . . . , χr form the set FX ; remaining symbols - the same as in the case of binary comparisons. The estimator χ 1 , . . . , χ n can be expressed alternatively in the form T μ (xi , xj ) – defined in the same way as the values Tμ (xi , xj ). The minimal value of the function (28) equals zero, the number of optimal solutions can exceed one. The properties of the estimator T μ (xi , xj ) are determined in Klukowski (2011), chapt. 6. In general it is consistent for N → ∞. This fact can be expressed in the following ˜ b the following random variables: way. Let us denote by Wb∗ and W 









Wμ∗ = ˜μ = W



N

∈Rm

k=1

N

 ∈Rm

k=1

   (μ)  gk (xi , xj ) − Tμ (xi , xj ),

(29)

   (μ)  (gk (xi , xj ) − T˜ μ (xi , xj ),

(30) ∼

∼

where: T˜ μ (xi , xj ) the values corresponding to any tolerance relation χ 1 , . . . , χ n˜ different than χ1∗ , . . . , χn∗ . Then, it can be shown, that: ˜ μ ) < 0, E(W ∗μ − W

(31)



 1 1 ˜ μ = 0, Var Wμ∗ = 0 lim Var W N →∞ N N →∞ N

(32)

˜ μ ) ≥ 1 − exp{−2N θ 2 }, P(W ∗μ < W

(33)

lim

where: ∼ ∼ θ 2 - some constant depending on χ 1 , . . . , χ n˜ and χ1∗ , . . . , χn∗ . The relationships (29)–(33) indicate that the estimator χ 1 , . . . , χ n guarantee errorless estimate for N → ∞, i.e. is consistent. Inequality (33), based on Hoeffding (1963) inequality for variables with finite set of values, indicates the speed of convergence. 





5 Tests for Verification of an Estimate of Tolerance Relation for Multivalent Comparisons Tests presented in this section verify an estimate of the tolerance relation on the basis (μ) of properties of distributions of comparisons gk (xi , xj ) ((xi , xj ) ∈ X × X), which are valid for true tolerance relation. The null and alternative hypotheses have similar general form as in the case of binary comparisons: H0 : χˆ 1 , . . . , χˆ nˆ ≡ χ1∗ , . . . , χn∗ and the alternative H1 : χˆ 1 , . . . , χˆ nˆ ≡ χ1∗ , . . . , χn∗ . (34)

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The hypothesis (34) can be verified by the set of partial hypotheses, which are appropriate to the multinomial form of comparisons:

(35) where: κ(κ ≤ n) – intersection with maximal number of subsets in an estimate χ 1 , . . . , χ n . The tests for the partial hypotheses depend on knowledge about distributions of comparisons of errors. Three cases can be taken into account: unknown distributions of errors, unknown distributions with known probabilities of mode, known distributions. In the first case is proposed some approximate approach, based on “quasi-uniform” distributions, i.e. with maximal variances. In the second case it is proposed to apply the tests for mode and median in multinomial distributions. In the case of known distributions it can be applied tests of goodness of fit, e.g. chi-square test, for individual partial hypotheses Tμ (xi , xj ) = l(l = 0, . . . , κ). The last case is the simplest one for testing, but rare in applications. Therefore, is not presented in the paper. Let us start from the case of unknown distributions of errors of comparisons with known probabilities of mode (equal median). Moreover, it is assumed that probabilities

(μ) P gk (xi , xj − T (x , xj ) = 0|Tμ (xi , xj ) = l)(0 ≤ l ≤ n) are the same for fixed l. 







μ

i

In such the case the tests for value of mode and median in multinomial distribution can be applied for the partial hypotheses (35). The test for mode (see e.g. Doma´nski 1990; Klukowski et al. 2021) can be applied in the following way. Let us denote by γl (l ≥ 0) (μ) the probability of the mode of distributions of comparisons gk (xi , xj ) evaluating values Tμ (xi , xj ) = l:

(μ) P gk (xi , xj − Tμ (xi , xj ) = 0|Tμ (xi , xj ) = l) = γl (l ∈ {l, . . . , m}). (36)

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Let us start from the case l = 0. In this case the first element any pair (xi , xj ) belongs / χq∗ . The random variables to some subset xi ∈ χ ∗q (1 ≤ q ≤ n), while the second xj ∈ (μ)

gk (xi , xj ) (k = 1, . . . , N ) assume values from the set {0, 1, . . . , m − 1} and (μ)

P(gk (xi , xj ) = 0) = γ0 > 0,5.

(37) (μ)

The null partial hypothesis H00 states that mode of all distributions gk (xi , xj ) (k = 1, . . . , N ) is equal 0 for T μ (xi , xj ) = 0(j > i) and.

(μ) P gk (xi , xj ) = 0|Tˆ μ (xi , xj ) = 0 = γ0 > 0,5, (38) 

where:

 xi ∈ χˆ q q = 1, . . . , nˆ , xj ∈ / χˆ q , while the alternative states that: mode is different than zero for some pairs (xi , xj ) satisfying the condition T μ (xi , xj ) = 0 and for such pairs

(μ) P gk (xi , xj ) = 0|T μ (xi , xj ) = 0 < 0,5 < γ0 . (39) 



(μ)

If the null hypothesis is true then comparisons gk (xi , xj ) satisfying the condition T μ (xi , xj ) = 0 (j > i) have mode equal zero in multinomial distribution. The set of values of the distribution can be assumed in the form {0, 1, . . . , κ}, where: κ – maximal (estimated) number of subsets of intersection in the estimate χ 1 , . . . , χ n . In the case of alternative H10 there exist some pairs (xi , xj ) having distributions which satisfy the inequality (39) (not (38)). Therefore, acceptation of H00 on the basis of all pairs satisfying T μ (xi , xj ) = 0(j > i) confirms the general null hypothesis for the case l = 0, while rejecting it indicates incorrect element of verified estimate. The null hypothesis that median of all comparisons, used in the test for mode in the case l = 0, is equal zero can be verified with the use of the test based on binomial distribution with probability of success (value of comparison equal zero) equal γ0 > 0, 5. The comparisons with value greater than zero are assumed as non-success (zero-one distribution). The basis for verification is the same set of comparisons as in the case of mode, i.e. satisfying T μ (xi , xj ) = 0(j > i). If null hypothesis is true then probability of success is equal γ0 and greater than probability of non-success. The test has left-hand side rejection region. Acceptation of null hypothesis for all pairs confirms hypothesis (35) for l = 0, while rejecting it indicates incorrect element of an estimate. In the case of large number of comparisons the tests can be based on the Gaussian limiting distribution. The value of median can be also verified with the use of other tests. Let us discuss next the case l = 1. In this case any pair of elements (xi , xj ) belongs to common subset χq∗ (1 ≤ q ≤ n) and does not belong to common intersection of two 











(μ)

or more subsets. The random variables gk (xi , xj ) (k = 1, . . . , N ) assume values from the set {0, 1, . . . , n} and (μ)

P(gk (xi , xj ) = 1) = γ1 ,

(40)

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where value 1 is mode and median of the distribution, i.e.:

 (μ) P gk (xi , xj ) = l|Tμ (xi , xj ) = 1 > 0,5,

(41)



(μ) P gk (xi , xj ) = l|Tμ (xi , xj ) = 1 > 0,5.

(42)

l≤1

 l≥1

Let us discuss firstly the case of the hypothesis for mode. The null partial hypothesis (μ) H01 states that mode of the distributions gk (xi , xj ) is equal 1 for T μ (xi , xj ) = 1 (j > i) and

(μ) H01 : P gk (xi , xj ) = 1|Tˆ μ (xi , xj ) = 1 = γ1 , (43) 

where:

 (xi , xj ) ∈ χˆ q 1 ≤ q ≤ nˆ , 



(xi , xj ) does not belong to intersection of any two subsets χ q ∩ χ s (s = q). The alternative H11 states that mode is different than 1 for some pairs (xi , xj ) satisfying the condition T μ (xi , xj ) = 1 and for these pairs

(μ) H11 : gk (xi , xj ) = 1|Tˆ μ (xi , xj ) = 1 < γ1 . (44) 

The probability (44) is suggested in one-sided form, because higher value of mode than γ1 is, in fact, not undesirable. Under the null hypothesis (44) the probabilities (μ) P gk (xi , xj = 1|T μ (xi , xj ) = 1)(j > i) are the same (equal γ1 ) for each pair belonging to any common subset. Positive result of the test for mode equal one in multinomial distribution with the set of values {0,1, …, κ} confirms general null hypothesis about correct form of estimate for l = 1, while opposite case indicates incorrect element of estimate. The null hypothesis that median of the distributions examined above is equal 1 can be verified also with the use of the test based on binomial distribution with probability of success (value of comparison equal zero or one 1) greater than 0,5. The comparisons with value greater than one are assumed as non-success (zero-one distribution). The test verifies the hypothesis that probability of success is greater than 0,5, under alternative that is lower than 0,5. The basis for verification is the same set of comparisons as in the case of mode. Acceptation of null hypothesis, i.e. median greater than 0,5, confirms hypothesis (35) for l = 1. Rejecting of null hypothesis indicates incorrect element of the estimate. In the case of large number of comparisons the tests can be based on the Gaussian limiting distribution. Some other tests can be applied if there exists an additional knowledge about the form

of distribution, e.g.: it is known the probability (μ) P gk (xi , xj ) = 0|T μ (xi , xj ) = 1 , distribution is symmetric around one or it is known minimal probability of median. Positive results of both tests, i.e. for mode and median, confirm partial hypothesis (35) for l = 1, negative indicates existence of incorrect elements. 



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Similar approach can be done for the next values of l(2 ≤ l ≤ κ). The case l = 2 corresponds to pairs (xi , xj (j > i)) belonging to all intersections of two subsets χq∗ ∩ χs∗ (s = q), i.e. null and alternative hypotheses for mode have a form:

(μ) H02 : P gk (xi , xj ) = 2|Tˆ μ (xi , xj ) = 2 = γ2 ), (45) under alternative:



(μ) H12 : P gk (xi , xj ) = 2|Tˆ μ (xi , xj ) = 2 < γ2 ),

(46)

(μ)

i.e. value of 2 is mode of distributions of comparisons gk (xi , xj ) satisfying the condition T μ (xi , xj ) = 2(j > i). The test for the median states that (null hypothesis):

 (μ) P gk (xi , xj ) = l|Tμ (xi , xj ) = 2 > 0,5, (47) 

l≤2

under alternative:  l≤2



(μ) P gk (xi , xj ) = l|Tμ (xi , xj ) = 2 ≤ 0,5.

(48)

Verification of the hypothesis (45)–(47) and interpretation of their results is similar as the hypothesis (43)–(44). The same tests can be applied for next values of l (l = 3, . . . , κ). However, number of comparisons in each intersection under examination has to be at least several. The case of unknown distributions of errors of comparisons requires non-parametric approach. Thus, the test for mode cannot be applied, because it requires the probability of this parameter. Nonparametric tests for median can be applied – in the same way as for known mode. Especially, it can be used the test based on binomial distribution. Efficiency of nonparametric approach, based only on the tests for median, is undoubtedly lower than based on both tests (mode and median). Some method of overcoming of it nuisance is to apply the worst case distribution – in the form of “quasi-uniform” distribution. Such the distribution can be applied instead of unknown distribution and

(μ) is obtained in the following way. For l = 0, it is assumed P gk (xi , xj = 0) =



(μ) (μ) 0, 5 + ε(0 ≤ ε ≤ 0, 5), P gk (xi , xj = ι) = (1 − P gk (xi , xj = 0)/(κ − 1) (ι = 1, . . . , κ). For l ≥ 1, the probabilities are determined by solving the system of two linear equation with two variables. For l = 1 it assumes the form:  x = (κ − 2)y . (49) 2x + (κ − 2)y = 1 The system (49) has unique solution and it is assumed:



(μ) (μ) P gk (xi , xj = 0) = P gk (xi , xj = 1) = x,

(μ) P gk (xi , xj = ι) = y(ι = 2, . . . , κ).

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The distributions for l > 1 can be determined in similar way. Such the distributions allow application of parametric approach to verification, i.e. the use the tests for mode and median. However, the probabilities of both errors in such tests are valid only in the case when quasi-uniform distribution is the same as true one. In opposite case the results are approximated, in general more conservative. However rejecting of any null partial hypothesis indicates incorrect elements of an estimate. The case of multivalent comparisons is more complicated for verification of estimates in the case of unknown distributions of errors, but application of statistical tests is still possible.

6 Summary and Conclusions The paper presents original tests for verification of estimates of the tolerance relation obtained on the basis of multiple binary and multivalent pairwise comparisons with the use of NAO method. Such the tests have been not presented in the literature of subject yet. The general null hypothesis states that the estimate obtained is the same as actual form of the relation. The hypothesis is verified by the systems of appropriate partial hypotheses. In the case of binary comparisons the tests are based on binomial distribution or limiting Gaussian distribution (for large sample). They also make use of the properties of distributions of errors of the tolerance relation. In the case of multivalent comparisons, tests are based on mode and median of multinomial distributions. They also exploits properties of distributions of errors of multivalent comparisons. Test for mode, assuming know probability of mode, has limiting Gaussian distribution, test for median has exact binomial or - limiting Gaussian distribution. Positive results of all partial null hypotheses confirm correct form of the estimate, while negative - indicates incorrect elements. In the case of multinomial distribution it is also possible nonparametric approach. It is based on the tests for mode and median with the use of the quasi-uniform distribution, having maximal variance. Estimates, which have been positively verified with the use of the tests are reliable and valuable for application.

References Bradley, R.A.: Science, statistics and paired comparisons. Biometrics 32, 213–232 (1976) David, H.A.: The Method of Paired Comparisons, 2nd edn. Ch. Griffin, London (1988) Doma´nski, C.: Statistical Tests (in Polish). PWE, Warsaw (1990) Gordon, A.D.: Classification, 2nd edn. Chapman & Hall/CRC (1999) Garfinkel, R.S., Nemhauser, G.L.: Integer Programming. Wiley, New York (1972) Hoeffding, W.: Probability inequalities for sums of bounded random variables. JASA 58, 13–30 (1963) Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, Hoboken (1980) Klukowski, L.: Methods of Estimation of Relations of: Equivalence, Tolerance, and Preference in a Finite Set. IBS PAN, Series: Systems Research, vol. 69, Warsaw (2011) Klukowski, L.: Determining an estimate of an equivalence relation for moderate and large sized sets. Oper. Res. Decis. Q. 27(2), 45–58 (2017)

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Klukowski, L.: Statistical tests for verification of estimate of the preference relation resulting from pairwise comparisons. In: Atanassov, K.T., Atanassova, V., Kacprzyk, J., Kaluszko, A., Krawczak, M., Owsinski, J.W., Sotirov, S., Sotirova, E., Szmidt, E., Zadrozny, S. (eds.) IWIFSGN 2018. AISC, vol. 1081, pp. 225–238. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-47024-1_24 Klukowski, L.: Pairwise comparisons in the form of difference of ranks - testing of estimates of the preference relation. In: Atanassov, K.T., Atanassova, V., Kacprzyk, J., Kaluszko, A., Krawczak, M., Owsinski, J.W., Sotirov, S., Sotirova, E., Szmidt, E., Zadrozny, S. (eds.) IWIFSGN 2018. AISC, vol. 1081, pp. 214–224. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-47024-1_23 Slater, P.: Inconsistencies in a schedule of paired comparisons. Biometrika 48, 303–312 (1961)

Applications in Healthcare, Medicine, and Sports

Opportunity for Obtaining an Intuitionistic Fuzzy Estimation for Health-Related Quality of Life Data Minko Minkov(B) and Evdokia Sotirova Faculty of Public Health and Health Care, Prof. Assen Zlatarov University, 1 Prof. Yakimov Street, 8010 Burgas, Bulgaria [email protected]

Abstract. In the paper a method for obtaining a degree of happiness of the residents of Burgas city using a real data of 1200 men and women for quality of life is proposed. Analyzing a health status data is very important in identifying and diagnosing diseases. For the determination the degree of happiness an intuitionistic fuzzy assessments (optimistic, strong optimistic, pessimistic and strong pessimistic) on the output of the multilayer perceptron are obtained. Keywords: Intuitionistic fuzzy set · Health-related quality of life · Neural networks · Multilayer perceptron

1 Introduction The assessment of the quality of life has been receiving significant importance in health care systems over the past years, because it reflects the trend in optimizing the effectiveness of the health services. The quality of life in public health research often is analyzed in the frame of health-related quality of life (HrQoL). HrQoL has been used to study the functional status in public health and medicine [13– 17]. Researching HrQoL at a community level could be an alternative or complementary approach in understanding health inequalities [7]. With the increasing need to evaluate the impact of socially significant diseases, it is important to use standardized instruments for evaluating HrQoL [7]. In the current analysis a health status data for measuring HrQoL of the residents of Burgas city was used [14]. The EQ-5D-3L is standardized methodology which identifies health in five dimensions including usual activities (work, housework, family, study, or leisure), mobility, self-care, pain or discomfort, and anxiety or depression. Each dimension has possible levels: “no problem”, “some problem”, or “extreme problem” [14]. The EQ-5D-3L instrument consists of two modules: 1. questionnaire with five dimensions for rating the health status, and 2. visual analogue scale type ”Thermometer” for self-rating of the health from 0 (the worst imaginable health state) to 100 (the best imaginable health state) [14]. The researchers in the public health area use HrQoL tools to measure the effects of health promotion or to monitor the self-reported health. The assessment of the HrQoL © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 151–157, 2021. https://doi.org/10.1007/978-3-030-77716-6_13

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can identify health inequalities in different subgroups and can help to guide policies or interventions to improve the population health [14]. For the survey a representative sample from 191 citizens over the age of 18 was used. The influence of the demographic, socio-economic determinants and behavioral determinants on HrQoL is investigated. The obtained data for quality of life can be analised by multiple processing techniques. Neural networks, as part of artificial intelligence, are one of the tools that use nature-inspired methods. They use the structure of the human brain and they are able to adaptively perceive information based on historical data. In this research neural networks with the learning methods are used. On the input the 15 parameters for assessment quality of life were put and the degree of happiness is obtained as a result on the output. The assessments demining the degree of happiness of the residents of Burgas city are formed on the basis of a set of intuitionistic fuzzy estimations μ, v (for Intuitionistic Fuzzy Sets, IFSs see [3–5]) of real numbers from the set [0, 1] × [0, 1], related to the EQ-5D-3L methodology [7]. These intuitionistic fuzzy estimations reflect the degree of each resident’s happiness μ, or non-happinessν, and for them is valid that μ + ν ≤ 1. The degree of uncertainty π = 1 − μ − ν represents such cases wherein there is no information for the current resident’s health status. Within the paper the ordered pairs were defined in the sense of intuitionistic fuzzy sets.

2 Multi Layer Perceptron According to [8, 9], artificial neural systems, or neural networks, are physical cellular systems, that are capable of perceiving, storing and using the perceived, while according to [6] neural networks are a part of the theory of automata, relating to the theoretical field of mathematical analysis, oftentimes based on models of the biological neural systems and being a system that may generate, encode, store and utilize information using a complex of neurons, nerve terminals and chemical synapses. P R 1 1 R

W1 1

S R

+ S n1 F 1

b1 S1 1

а1

1

1

W2 S2 S2 S1

b2 S2 1

+ Sn 1 F 2

2

W3

а2 S

2

S2 1

1

3

S2

b3 S3 1

+ Sn 1 F 3

3

3

а3 S3 1

Fig. 1. Flow chart of the MultiLayer Perceptron (MPL)

There exist neural networks that use feedback connections (recurrent neural networks) [6]. In multilayered networks, the exits of one layer become entries for the next one. The equations describing this operation are:   (1) am+1 = f m+1 wm+1 am + bm+1

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for m = 0,1, …, M − 1. On Fig. 1 and Eq. (1): • m is the current number of the layers in the MLP; • M is the number of the layers in the MLP; • P is an entry network’s vector, where    P1     P2    P =  . ;  ..    P 

(2)

R

• R is a number of inputs of the MLP; • am is the exit of the m-th layer of the MLP, where    a1     a2    a m =  . ;  ..    a 

(3)

s

• sm is a number of neurons of a m-th layer of the MLP; • wm is a matrix of the coefficients of all inputs, where   W11 W12   W21 W22  m w = . ..  .. .  W W S1 S2

 . . . W1R  . . . W2R  . . .. ; . .  ... W 

(4)

SR

• bm is neuron’s input bias, where    b1     b2    m b =  . ;  ..    b  s

• F m is the transfer function of the m-th layer exit.

(5)

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3 Discussion

Mean Squared Error (mse)

For the preparation we use MATLAB and MLP neural network structure 15:25:1 (15 inputs, 25 neurons in the hidden layer and 1 of the output) (Fig. 2). For the learning process of MLP, we set the following parameters: Performance (MSE) = 0.00001; Validation check = 15. The data set was divided into three different parts (191 altogether): Training (from 1 to 150); Validation (from 151 to 180) and Testing (from 181 to 191) (Figs. 3 and 4). Train Validation Test Best

10 4

10 2

10 0

0

1

2

3

4

5

6

7

8

9

10

11

11 Epochs

Fig. 2. The neural network structure.

Fig. 3. The neural network training.

For the testing of the neural network all 191 vectors were used. After simulation of the neural network, an regression coefficient of the MLP R = 0.89283 is obtained.

Fig. 4. The regression of the MLP.

Using the neural network output the intuitionistic fuzzy assessment for evaluation of the degree of happines ann is introduced. The values of the measurement belong to degree of affiliations μ if ann ∈ [ε2 ÷ 1], where the ε2 is the upper threshold value.

Opportunity for Obtaining an Intuitionistic Fuzzy

155

If ann ∈ [0 ÷ ε1 ], belongs to degree of the non-affiliations (ν), where the ε1 is the lower threshold value. If ann ∈ (ε1 ÷ ε2 ) the assessment, belongs to degree of the uncertainty (π ). The obtained information, are represented by ordered pairs μ, v of real numbers from the set [0,1] × [0,1]. The degree of uncertainty also represents as a π = 1 – μ – ν. At the beginning is done statistics of the 191 values that we used for learning the neural network. Initially when still no information has been obtained, all estimations are given initial values of 0, 0. When k ≥ 0, the current (k + 1)-st estimation is calculated on the basis of the previous estimations according to the recurrence relation   μk k + m νk k + n μk+1 , vk+1  = , (6) , k +1 k +1 where μk , vk  is the previous estimation, and μ, v is the estimation of the latest measurement, for m, n ∈ [0, 1] and m + n ≤ 1. We can use also optimistic formula: μk+1 , νk+1  = maxμall , minνall ,

(7)

μall = (μ0 , μ1 · · · μk ) k ∈ [0, · · · l − 1],

(8)

νall = (ν0 , ν1 · · · νk ) k ∈ [0, · · · l − 1],

(9)

where:

Strongly optimistic formula: μk+1 , νk+1  = μk+1 + μk − μk+1 .μk , νk .νk+1 ,

(10)

Pessimistic formula: μk+1 , νk+1  = minμall , maxνall ,

(11)

Strongly pessimistic formula: μk+1 , νk+1  = μk+1 .μk , νk+1 + νk − νk .νk+1 ,

(12)

4 Conclusion The authors investigate the possibility to analyze HrQoL combining different techniques like Intuitionistic fuzzy set and neural networks. This is the next article which presents the application of Neural network to HrQoL with intuitionistic fuzzy estimation [11, 12]. In the next observations we will apply the similar investigation to data connected to quality of life in patients with various abdominal anastomoses in case of colorectal cancer [1, 2, 18] and patients after surgical treatment in multilevel thrombosis [10, 19]. Intuitionistic fuzzy estimations are calculated in order to assess the degree of happiness using two different thresholds. Since, during the assessment process is not always possible to get a precise qualitative estimate, three stages of assessment are introduced: affiliations, non-affiliations and uncertainty.

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Acknowledgments. The authors are grateful for the support provided by the project DN-0210/2016 “New Instruments for Knowledge Discovery from Data, and their Modelling”, funded by the National Science Fund, Bulgarian Ministry of Education, Youth and Science.

References 1. Atanasov, A., Lefterov, E., Vasilev, V., Kehayov, A., Savov, M., Penchev, V., Dashkov, B.: Evaluation of the quality of life in patients with various abdominal anastomoses in case of colorectal cancer. Regional Scientific Conference, pp. 110–112. ISBN 978-954-397-023-0, Svilengrad, 17–19 Nov 2011 2. Atanasov, A., Lefterov, E., Vasilev, V., Kehayov, A., Savov, M., Penchev, V., Polyanov, B.: Assessment of quality of life in patients operated on for colorectal cancer. Third International Medical Congress of SEEMF, p. 42. Belgrade, Serbia, 12–15 Sept 2012 3. Atanassov, K.T.: Intuitionistic Fuzzy Relations (IFRs), On Intuitionistic Fuzzy Sets Theory, vol. 283 of Studies in Fuzziness and Soft Computing, pp. 147–193. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29127-2_8 4. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 5. Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica-Verlag, Heidelberg, Germany (1999) 6. Bishop C. M., Neural Networks for Pattern Recognition. Oxford university press, Oxford (2000) ISBN 0 19 853864 2 7. Janssen, M.F., Szende, A., Cabases, J.: Population Norms for the EQ-5D-3L: A Cross-Country Analysis of Population Surveys for 20 Countries, National and Regional EQ-5D Population Surveys, Papers, EuroQol Plenary Session, Rotterdam (2012) 8. Hagan, M.T., Demuth, H.B., Beale, M.: Neural Network Design. PWS Publishing Company, Boston (1996) 9. Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, N.J. (1999) 10. Matkov, O., Vassilev, V., Kavrakov, T., Abrashev, H.: Vascular aspects in treatment of diabetic foot. Trakia J. Sci. 12(1), 241–243 (2014). ISSN 1313-7050 11. Vankova, D., Sotirov, S., Doukovska, L.: An application of neural network to health-related quality of life process with intuitionistic fuzzy estimation. In: Atanassov, K.T., Kacprzyk, J., Kałuszko, A., Krawczak, M., Owsi´nski, J., Sotirov, S., Sotirova, E., Szmidt, E., Zadro˙zny, S. (eds.) IWIFSGN 2016. AISC, vol. 559, pp. 183–189. Springer, Cham (2018). https://doi.org/ 10.1007/978-3-319-65545-1_17 12. Sotirov, S., Vankova, D., Vasilev, V., Sotirova, E.: Clustering of intercriteria analysis data using a health-related quality of life data. In: Cuzzocrea, A., Greco, S., Larsen, H.L., Saccà, D., Andreasen, T., Christiansen, H. (eds.) FQAS 2019. LNCS (LNAI), vol. 11529, pp. 242–249. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-27629-4_23 13. Vankova, D., Kerekovska, A., Kostadinova, T., Usheva, N.: Health-related Quality of Life in the Community. Assessing the Socio-Economic, Demographic and Behavioural Impact on Health-related Quality of Life at a Community Level: Evidence from Bulgaria, Proceedings database (2013). http://www.euroqol.org/uploads/media/EQ12-P05.pdf 14. Vankova, D.: Community-cantered research in Bulgaria, a mixed-methods approach to healthrelated quality of life. Euro J. Public Health 25(Suppl 3), 291 (2015). http://dx.doi.org/ 10.1093/eurpub/ckv175.046 (8th European Public Health Conference: Proceedings) (First published online: 6 October 2015) 15. Vankova, D., Sotirova, E., Bureva, V.: An application of the InterCriteria Analysis approach to health-related quality of life. 11th International Workshop on IFSs, vol. 21, no. 5, pp. 40–48. Banská Bystrica, Slovakia, 30 Oct. 2015, Notes on Intuitionistic Fuzzy Sets (2015). ISSN 1310–4926

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16. Vankova, D., Kerekovska, A., Kostadinova, T., Feschieva N.: Health-related quality of life and social capital at a community level – a review and research. Trakia J. Sci. 10(Suppl 3), 5–12 (2012). https://doi.org/10.1016/j.ssmph.2019.100425 17. Vankova, D., Kerekovska, A., Kostadinova, T., Todorova, L.: Researching health-related quality of life at a community level: survey results from Burgas, Bulgaria, Health Promotion International, Health Promotion International, pp. 1–8 (2015). https://doi.org/10.1093/hea pro/dav016 18. Vassilev, V., Tanev, D., Kavrakov, T., Abrashev, H.: Our experience in surgical treatment of the complicated forms of colorectal cancer. Trakia J. Sci. 4, 68–70 (2015). https://doi.org/10. 3748/wjg.14.3281 19. Vassilev V., Matkov, O., Kavrakov, T., Abrashev, H.: Surgical treatment in multilevel thrombosis of lower limb‘s arteries. Trakia J. Sci. 264–266 (2014) ISSN 1313-7050

InterCriteria Analysis of the Blood Group Distribution of Patients of Saint Anna Hospital in 2015–2019 Nikolay Andreev1(B) and Vassia Atanassova2 1 Transfusion Haematology Department, Saint Anna Hospital, 1 Dimitar Mollov Street,

1750 Sofia, Bulgaria [email protected] 2 Bioinformatics and Mathematical Modelling Department, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, bl. 105, 1113 Sofia, Bulgaria [email protected]

Abstract. This paper presents findings from applying the method of InterCriteria Analysis to a dataset of 33782 patients of Saint Anna Hospital, Sofia whose blood was tested in the period 2015–2019. The dataset, after performed cleansing and anonymization, contains data of the blood groups of the tested patients, their birth year and sex and the research aims to detect dependences and changes over time of the distribution of the different blood types over the Bulgarian population, represented in this sample. The results of this study can potentially be informative also for the blood transfusion specialists in the country regarding the blood transfusion capacity of the population. Keywords: Blood groups · ABO system · InterCriteria Analysis · Intuitionistic fuzzy sets

1 Introduction Discovery of ABO system of blood groups by Landsteiner in 1901 marked the beginning of safe blood transfusion [1]. Till today, it is the most important blood group system in transfusion medicine. The system comprises four main blood groups namely A, B, AB and O, with subtypes of the A antigen being defined, namely A1 and A2. Subgroups of A antigen weaker than A2 exist, but are not frequent [2]. On this basis, the A and AB blood groups have been classified into two main subgroups each. Literature states that approximately 80% of white European having A antigen in blood belong to A1, and thus are forming either A1 or A1B subgroups, while the rest belong to A2, so as to form either A2 or A2B subgroups [3]. The aim of the present study is to assess the prevalence of A1 and A2 subgroups in the studied population of Bulgarian citizens, patients of the Saint Anna Hospital, Sofia, who were tested for blood in the period 2015–2019, and look for changes and trends in the blood type profiles over time, also employing the novel method of InterCriteria Analysis [4, 5], based on intuitionistic fuzzy sets [6, 7], which has been already used in analyzing data from the domain of transfusion hematology in Bulgaria [8]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 158–165, 2021. https://doi.org/10.1007/978-3-030-77716-6_14

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159

2 Input Data The raw input data for the present research consisted of five yearly spreadsheets of individuals, whose blood was tested in the Saint Anna Hospital, Sofia, in the period 2015–2019, in their capacity of hospital patients or blood donors. In the course of the investigation, these data had to be cleansed and anonymized in order to be used in compliance with the EU General Data Protection Regulation. The anonymization was performed as the last step of the process, as some of the personal data (the so called “Unique citizenship number” or “EGN”, which is a 10-digit string) was used in the course of data cleansing and data structuring, specifically to extract the information of year of birth and sex of the patient. Overall, the process of data cleansing comprised the following steps: 1. Removal of entries featuring invalid patient EGNs. Examples of invalid EGNs are such with less than 10 digits recorded (e.g. when emergency patients have not been identified with an ID card), as well as recorded identifiers of foreigners, instead of EGNs of local citizens. Foreigners are by definition to be removed, as the survey aims to shed light on the distribution of the blood types of the Bulgarian population. 2. Removal of duplicating patients within the yearly dataset. For all duplicating patients, check was made for consistency of the records for blood type and Rhesus factor. Any recorded inconsistencies were either corrected, or removed. 3. Removal of duplicating patients across the five yearly datasets. 4. Extracting the data about the patient’s year of birth – from the first two digits of the EGN, and data about the patient’s sex – defined by the eight digit of the EGN.

Fig. 1. Distribution by birth year of the total number of 33782 unique patients of St. Anna Hospital, Sofia, blood tested in the period 2015–2019

As a result, an aggregated dataset of 33782 unique patients tested for blood type, for the period 2015–2019 is collected. Of these, 18190 women (53.85%), 15592 men (46.15%), and according to their birth year they range from 1913 to 2019 (Fig. 1). It is noteworthy that Saint Anna Hospital in Sofia is one of the largest hospitals in the country’s capital, and due to its location and statute of regional hospital it serves multiple patients from outside Sofia, as well. While the present survey does not maintain

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information about the particular birthplace of the tested individuals, checking a randomly selected sample of the entries by EGN has shown diversity of the patients’ birthplaces, which can be further explained by the large share of the residents of the capital who have been born and migrated from elsewhere in the country. Thus, while the investigated set of patients is in a Sofia hospital, it is diverse enough to consider the conclusions drawn valid on national level, as well. The importance of identification of blood group subtypes A1, A2, A1B and A2B, for instance, for blood transfusion safety, is the reason to have these subtypes recorded. The distribution of the investigated 33782 patients across the blood types/subtypes and rhesus factors is given in Table 1 with various level of granularity from finest (a) to most general (c). Table 1. Frequency of the blood group subtype in the total number of 33782 unique patients of St. Anna Hospital, Sofia, blood tested in the period 2015–2019 Blood group

Male Female

All Patients

0 (-)

666

859

1525

0 (+)

4320

4899

9219

A1 (-)

793

1044

1837

A1 (+)

5273

6278

11551

A2 (-)

82

120

202

A2 (+)

571

570

1141

B (-)

343

419

762

B (+)

2245

2548

4793

A1B (-)

135

191

326

A1B (+)

999

1067

2066

A2B (-)

15

23

38

A2B (+)

150

172

322

(a)

Blood Patients group

%

0

10744

31.8

A1

13388

39.6

A2

1343

4.0

B

5555

16.4

A1B

2392

7.1

A2B

360

1.1

(b)

Blood Patients group

%

0

10744

31.8

A

14731

43.6

B

5555

16.4

AB

2752

8.2

(c)

On this basis, we can provide our first result. Out of 17483 donors having A antigen in blood, 15780 (90.26%) belong to the A1 (either A1 or A1B subgroups), while the rest 1703 (9.74%) belong to A2 (either A2 or A2B subgroups). More precisely, out of 14731 donors of A group, 13388 (90.88%) belong to A1 subtype and only 1343 (9.12%) belong to A2. Of 2752 donors with group AB, 2392 (86.82%) belong to A1B subtype and 360 (13.08%) belong to A2B. It can be noted that A2 in AB blood-group, as A2B, is more frequent in occurrence than presence of A2 as an A blood group. These findings differ from what is known from literature (see e.g. [3]) about the 80/20 distribution of A1 and A2 in A in Europeans, but is in accord with the data presented in [9] (based on [10, 11]), [12] and [13] of the respective frequencies in the Bulgarian population, as compared in the following Table 2.

InterCriteria Analysis of the Blood Group Distribution

161

Table 2. Comparison of the frequency of different blood group subtypes between sources [9, 12, 13] and the present study (* denotes values, calculated based on the presented data) Phenotype and ratio

Frequency, % according to [9]

Frequency, % according to [12]

Frequency, % according to [13]

Frequency, % present study

0

32.1

32.5

32.6

31.8

A

44.8

43.0

43.2

43.6

A1

39.3 *

A1/A

37.31 *

87.73

A2

5.5 *

A2/A B

12.27

A1B/AB A2B

9.72 * 8.2 6.8 *

---

7.1

83.95 * 1.3 *

--

9.1 16.4

8.1

76.4 23.6

4.0

16.11 --

1.5 *

A2B/AB

13.23 7.6

4.8 *

90.9

4.2 *

16.6

6.3 A1B

39.6

90.28 *

5.69 *

16.8

AB

39.0 *

86.77

86.6 1.1

16.05 *

13.4

More detailed comparisons with other sources and relevant reported case studies are to be done in another leg of this research. In what follows, our aim is to apply the novel (2014) intuitionistic fuzzy sets-based method of InterCriteria Analysis [4, 5].

3 Application of the InterCriteria Analysis – Results and Discussion The full dataset of 33782 registered patients has been divided in groups, according to the decade of their birth (for the purpose of even finer granularity of the results they can be divided in groups of five years, or even annually). To run the ICA, we need relative comparability regarding the data ranges, so we will use the relative frequency of a given blood subgroup within the respective decade, and not the absolute numbers, which is dictated from the objectively lower numbers of tested patients born in the 1910s, 1920s, 2000s and 2010s. Thus from the data in absolute values (Table 3) we obtain the data in relative values (Table 4). We can visualize the data from Table 4 in the following Fig. 2, demonstrating the trends in the frequency of the positive and the negative rhesus factors over the decades. As a next step, we apply InterCriteria Analysis, using the software application developed by Mavrov [14, 15]. The results of application of the InterCriteria Analysis suggest highest similarity between the blood group frequencies of patients born in 1930s and 1940s, as well as among those born in the 1950s, 1960s and 1970s. Lower similarity is detected between those born in the 1980s with those born in the 1990s and 2000s. Notably much lower levels of intercriteria correlation is seen for those born in the 2010s, which we majorly attribute to the relatively less data for this period, despite the measures taken to normalize the data or each period, in order to avoid incomparability. The results

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Table 3. Tested patients with a certain blood group and rhesus factor, by decade of patient birth (absolute numbers) 0(-)

0(+)

A1(-)

A1(+)

A2(-)

A2(+)

B(-)

B(+)

A1B(-)

A1B(+)

A2B(-)

A2B(+)

1913-1929

56

273

43

347

4

43

17

126

16

67

0

9

1930-1939

215

1294

235

1654

30

158

111

697

47

290

5

47

1940-1949

346

2033

386

2710

41

264

161

1074

67

441

10

71

1950-1959

274

1861

367

2264

42

232

151

903

56

424

7

70

1960-1969

182

1220

239

1555

16

150

97

706

46

300

3

50

1970-1979

157

1009

189

1246

20

104

84

515

27

229

6

29

1980-1989

167

823

232

976

29

101

80

426

41

175

3

26

1990-1999

102

536

115

600

17

68

48

256

21

102

3

17

2000-2010

24

146

27

171

2

16

10

80

5

36

1

1

2000-2019

2

24

4

28

1

5

3

10

0

2

0

2

Table 4. Relative frequency (%) of a certain blood group and rhesus factor, by decade of patient birth 0(-)

0(+)

A1(-)

A1(+)

A2 (-)

A2(+)

B(-)

B(+)

A1B(-)

A1B(+)

A2B(-)

A2B(+)

1913-1929

5.6

27.3

4.3

34.7

0.4

4.3

1.7

12.6

1.6

6.7

0.0

0.9

1930-1939

4.5

27.1

4.9

34.6

0.6

3.3

2.3

14.6

1.0

6.1

0.1

1.0

1940-1949

4.6

26.7

5.1

35.6

0.5

3.5

2.1

14.1

0.9

5.8

0.1

0.9

1950-1959

4.1

28.0

5.5

34.0

0.6

3.5

2.3

13.6

0.8

6.4

0.1

1.1

1960-1969

4.0

26.7

5.2

34.1

0.4

3.3

2.1

15.5

1.0

6.6

0.1

1.1

1970-1979

4.3

27.9

5.2

34.5

0.6

2.9

2.3

14.2

0.7

6.3

0.2

0.8

1980-1989

5.4

26.7

7.5

31.7

0.9

3.3

2.6

13.8

1.3

5.7

0.1

0.8

1990-1999

5.4

28.4

6.1

31.8

0.9

3.6

2.5

13.6

1.1

5.4

0.2

0.9

2000-2010

4.6

28.1

5.2

32.9

0.4

3.1

1.9

15.4

1.0

6.9

0.2

0.2

2000-2019

2.5

29.6

4.9

34.6

1.2

6.2

3.7

12.3

0.0

2.5

0.0

2.5

are shown in Table 5 and its corresponding graphical interpretation into the intuitionistic fuzzy interpretational triangle (see [16] in Fig. 3). The specific value in these results are further expressed in the determination, or specification, of the intercriteria threshold values for this very case study, which can be of later reference in the next steps of research. This result is especially significant for the practice of ICA, since different threshold values hold true for different case studies, as observed in an existing survey of the method of identifying the threshold values [17].

InterCriteria Analysis of the Blood Group Distribution

163

Fig. 2. Relative frequency (%) of a certain blood group and rhesus factor, by decade of patient birth: (a) negative rhesus factor, (b) positive rhesus factor

Table 5. Results of the application of InterCriteria Analysis on the dataset (Table 4) (a) membership parts, (b) non-membership parts (a)

1913-1929 1930-1939 1940-1949 1950-1959 1960-1969 1970-1979 1980-1989 1990-1999 2000-2009 2010-2019

1913-1929

1.0000

0.9545

0.9545

0.9545

0.9545

0.9545

0.9394

0.9242

0.9394

0.8030

1930-1939

0.9545

1.0000

1.0000

0.9848

0.9848

0.9848

0.9545

0.9394

0.9545

0.8182

1940-1949

0.9545

1.0000

1.0000

0.9848

0.9848

0.9848

0.9545

0.9394

0.9545

0.8182

1950-1959

0.9545

0.9848

0.9848

1.0000

1.0000

1.0000

0.9545

0.9394

0.9545

0.8333

1960-1969

0.9545

0.9848

0.9848

1.0000

1.0000

1.0000

0.9545

0.9394

0.9545

0.8333

1970-1979

0.9545

0.9848

0.9848

1.0000

1.0000

1.0000

0.9545

0.9394

0.9545

0.8333

1980-1989

0.9394

0.9545

0.9545

0.9545

0.9545

0.9545

1.0000

0.9697

0.9697

0.8182

1990-1999

0.9242

0.9394

0.9394

0.9394

0.9394

0.9394

0.9697

1.0000

0.9394

0.8333

2000-2009

0.9394

0.9545

0.9545

0.9545

0.9545

0.9545

0.9697

0.9394

1.0000

0.7879

2010-2019

0.8030

0.8182

0.8182

0.8333

0.8333

0.8333

0.8182

0.8333

0.7879

1.0000

(b)

1913-1929 1930-1939 1940-1949 1950-1959 1960-1969 1970-1979 1980-1989 1990-1999 2000-2009 2010-2019

1913-1929

0.0000

0.0152

0.0152

0.0303

0.0303

0.0303

0.0455

0.0303

0.0303

0.1212

1930-1939

0.0152

0.0000

0.0000

0.0000

0.0000

0.0000

0.0303

0.0152

0.0152

0.1061

1940-1949

0.0152

0.0000

0.0000

0.0000

0.0000

0.0000

0.0303

0.0152

0.0152

0.1061

1950-1959

0.0303

0.0000

0.0000

0.0000

0.0000

0.0000

0.0455

0.0303

0.0303

0.1061

1960-1969

0.0303

0.0000

0.0000

0.0000

0.0000

0.0000

0.0455

0.0303

0.0303

0.1061

1970-1979

0.0303

0.0000

0.0000

0.0000

0.0000

0.0000

0.0455

0.0303

0.0303

0.1061

1980-1989

0.0455

0.0303

0.0303

0.0455

0.0455

0.0455

0.0000

0.0000

0.0152

0.1212

1990-1999

0.0303

0.0152

0.0152

0.0303

0.0303

0.0303

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Fig. 3. Graphical representation of the results (Table 5) of the application of InterCriteria Analysis on the dataset (Table 4)

4 Conclusion The present paper aims to report findings from a collected, cleansed and anonymized dataset of 33782 unique patients of Saint Anna Hospital, Sofia whose blood was tested in the period 2015–2019. Using the recently proposed intuitionistic fuzzy sets-based method of InterCriteria Analysis certain changes were detected regarding the frequencies of the six different blood groups and subgroups (0, A1, A2, B, A1B, A2B) and patterns of similarity were outlined between the patients born in decades from 1920s to 2010s. Due to profile of the hospital and the large sample of 33782 unique patients tested, the results can be considered representative of the Bulgarian population, but the authors aim to extend this research with other available datasets from other hospital units in the country. On this basis, as another step of research, the authors further plan to analyze how the results of this study could potentially benefit the blood transfusion specialists regarding the blood transfusion capacity of the Bulgarian population. Acknowledgements. The authors are grateful for the support provided by the National Science Fund of Bulgaria under grant KP-06-N22/1.

References 1. Dacie J.V., Lewis, S.M.: In: Lewis, S.M., Bain, B.J., Bates, I. (eds.) Practical Haematology, 9th edn, pp. 444–451. Churchill Livingstone, Harcourt Publishers Limited, London (2001) 2. Thakral, B., Saluja, K., Bajpai, M., Sharma, R.R., Marwaha, N.: Importance of weak ABO subgroups. Lab. Med. 36(1), 32–34 (2005) 3. Klein, H.G., Anstee, D.J.: Mollison’s Blood Transfusion in Clinical Medicine, 11th edn., p. 115. Blackwell Publishing, London (2005) 4. Atanassov, K., Mavrov, D., Atanassova, V.: InterCriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues Intuitionistic Fuzzy Sets Gen. Nets 11, 1–8 (2014) 5. Atanassov, K., Atanassova, V., Gluhchev, G.: InterCriteria analysis: ideas and problems. Notes Intuitionistic Fuzzy Sets 21(1), 81–88 (2015) 6. Atanassov, K.: Intuitionistic fuzzy sets. In: Proceedings of VII ITKR Session, Sofia, 20–23 June 1983 (1983). (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84, in Bulgarian). Reprinted: Int. J. Bioautomation 20(S1), S1–S6 (2016)

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7. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 8. Andreev, N., Sotirova, E., Ribagin, S.: InterCriteria analysis of data from the centres for transfusion haematology in Bulgaria. Comptes rendus de l’Acade’mie bulgare des Sci. 72(7), 982–990 (2019) 9. Dobreva, A., Doychinova, N., Vasilev, N. (eds.): Transfusion Hematology, pp. 68, 114–115. State Publishing House “Medicina I Fizkultura”, Sofia (1988). (in Bulgarian) 10. Doychinova, N., Atanasov, A.: Immunohematology. State Publishing House “Medicina I Fizkultura”, Sofia (1977). (in Bulgarian) 11. Prodanov, P.: Study of the problem of qualitative or quantitative nature of the differences between A1 and A2 erythrocyte antigens. Hematology and Blood Transfusion, Bulgaria, vol. 3, pp. 21–25 (1969). (in Bulgarian) 12. Lisichkov, T. (ed.): Transfusion Hematology, pp. 94, 142–144. Publishing house “Kaynadina” (2002). (in Bulgarian) 13. Popov, R., Petrov, N., Vaseva, V.: Distribution of the blood groups from the ABO system in the Military Medical Academy immunohematological diagnostics. Bul. Med. J. VI(2), 45–48 (2012). (in Bulgarian) 14. Mavrov, D.: Software for InterCriteria analysis: implementation of the main algorithm. Notes Intuitionistic Fuzzy Sets 21(2), 77–86 (2015) 15. Mavrov, D.: Software for InterCriteria analysis: working with the results. In: Annual of “Informatics” Section, Union of Scientists in Bulgaria, vol. 8, pp. 37–44 (2015–2016) 16. Atanassova, V.: Interpretation in the intuitionistic fuzzy triangle of the results, obtained by the InterCriteria analysis. In: Proceedings of 16th World Congress of IFSA, 9th Conference of EUSFLAT, June 2015, Gijon, Spain, pp. 1369–1374 (2015) 17. Doukovska, L., Atanassova, V., Sotirova, E., Vardeva, I., Radeva, I.: Defining consonance thresholds in InterCriteria analysis: an overview. In: Hadjiski, M., Atanassov, K.T. (eds.) Intuitionistic Fuzziness and Other Intelligent Theories and Their Applications. SCI, vol. 757, pp. 161–179. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-78931-6_11

Application of the InterCriteria Analysis Method to a Data of Malignant Melanoma Disease for the Burgas Region for 2014–2018 Evdokia Sotirova1(B) , Greta Bozova2 , Hristo Bozov1,3 , Sotir Sotirov1 , and Valentin Vasilev1 1 Faculty of Public Health and Health Care, Prof. Assen Zlatarov University,

1, Prof. Yakimov Street, 8010 Burgas, Bulgaria {esotirova,ssotirov}@btu.bg, [email protected] 2 Department of Nephrology, Military Medical Academy, 3 St. George Sofiiski, Sofia, Bulgaria [email protected] 3 Oncology Complex Center – Burgas, 86 Demokratsiya blvd, 8000 Burgas, Bulgaria

Abstract. The aim of this paper is to analyze a statistical data for the registered patients with malignant melanoma in Burgas region for the period 2014–2018. The InterCriteria Analysis approach is applied. The relations between gender, marital status and the type of malignant melanoma; relations between year of registration of patient and type of malignant melanoma are studied. The obtained results by InterCriteria Analysis method are confirmed by statistical analysis according to Pearson, Kendall and Spearman. Keywords: Intercriteria analysis method · Index matrix · Intuitionistic fuzzy sets · Oncological diseases · Malignant melanoma

1 Introduction According to the Global Cancer Observatory (GCO) the number of cancer patients for 2018 have raised to 18.1 million new cases [18]. Over the last 6 years, the cancer morbidity has increased by 28% [18]. The situation in Bulgaria also shows a steady upward trend. 26,000 new cases are registered each year, which is twice more than 20 years ago. [28, 46]. Subject of the study described below are patients registered with malignant melanoma for a 5-year period. Malignant melanoma is one of the most aggressive cancer. The incidence has increased significantly in the last 30 years [17]. Malignant melanoma develops from the pigment cells of the skin and mainly affects people with pale skin [21]. The melanoma affects mostly young and middle-aged people. About half of the cases are people aged between 35 and 65 [23]. Keeping in mind the rapid development, the tendency for early metastasis in the body, the risk of recurrence and the significantly high mortality, it is especially important that malignant melanoma can be diagnosed as early as possible. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 166–174, 2021. https://doi.org/10.1007/978-3-030-77716-6_15

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The InterCriteria Analysis (ICA) approach was introduced by K. Atanassov, D. Mavrov and V. Atanassova in [8]. It is based on the apparatus of the two-dimensional Index matrices (IMs, see [5]) and the theory of the intuitionistic fuzzy sets (IFSs, see [7]). The ICA approach has been applied for analyzing data and decision making in different areas - medical investigations [20, 22, 47–50], Genetic Algorithm [1–3, 29, 31, 35, 38], Metaheuristic Algorithms [11–16, 27, 30, 36, 37, 39, 40], neural networks [41–44], and etc. In [45] the ICA approach is applied to 2018 data of newly registered and dispensary patients from Burgas with oncological diseases. The relations between marital status, gender data, relations between gender and age of the patients are analyzed. In this paper we will use the ICA method in regional databases for discovering the relations of registered patients from Burgas region that developed malignant melanoma during 2014–2018. In the next observations we will apply the ICA approach to data connected to metastatic melanoma [32–34]. Using the ICA approach, the multicriteria decision making can be applied. The matrix containing the data of the measurements of m in number evaluated objects (in our case patients with malignant melanoma) - by n in number evaluation criteria, based on pairwise comparisons of objects and criteria will be used. The applying the ICA method will be done through the developed software for ICA [19, 24–26] (freely available online at: http://intercriteria.net/software). The proposed approach calculates the degrees of correlation between all possible pairs of criteria in the form of intuitionistic fuzzy pairs of values in the [0,1]-interval [9], which means that there can be a re-discovery of relations already known from literature (and established by other methods), as well as a discovery of new connections leading to the generation of new knowledge. The correlations between the criteria are called “positive consonance”, “negative consonance” or “dissonance”. Here we use the scale used in previous studies that is shown in [4].

2 An Application of the ICA The InterCriteria Analysis approach is applied to real data for 100 (57 man and 43 woman) registered patients with malignant melanoma disease registered in Burgas region for 2014–2018. The data contains information about age of patients, name of the disease, according to International statistical classification of diseases and health problems, gender, marital status, data of the registration of the patient, etc. In the observed data there are: • 14 age groups: men up to 30 years, women up to 30 years, men 31–40, women 31–40, men 41–50, women 41–50, men 51–60, women 51–60, men 61–70, women 61–70, men 71–80, women 71–80, men over 80, women over 80. According age groups the patients are distributed the following way: 7 up to 30 years (2 men, 5 women); 10 in 31–40 (8 men, 2 women); 15 in 41–50 (8 men, 7 women); 11 in 51–60 (8 men, 3 women); 25 in 61–70 (11 men, 14 women); 9 in 71–80 (5 men, 4 women); and 10 over 80 (6 men, 4 women);

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• 8 marital status groups: men unmarried, women unmarried, men married, women married, men divorced, women divorced, men widower, women widower. According marital status the patients are distributed the following way: 7 unmarried (4 men, 3 women); 69 married (38 men, 31 women); 4 divorced (2 men, 2 women) and 7 divorced (4 men, 3 women); • 7 malignant melanoma of skin, group C43 classified according to the International Statistical Classification of Diseases and Health Problems (ICD): C43.2 (Malignant melanoma of ear and external auricular canal), C43.3 (Malignant melanoma of other and unspecified parts of face), C43.4 (Malignant melanoma of scalp and neck), C43.5 (Malignant melanoma of trunk), C43.6 (Malignant melanoma of upper limb, including shoulder), C43.7 (Malignant melanoma of lower limb, including hip) area), C43.9 (Malignant melanoma of skin, unspecified). According ICD the patients with malignant melanoma of skin are distributed the following way: 1 with code C43.2 (1 man, 0 women); 7 with code C43.3 (3 men, 4 women); 4 with code C43.4 (4 men, 0 women); 42 with code C43.5 (27 men, 15 women); 12 with code C43.6 (6 men, 6 women); 27 with code C43.7 (11 men, 16 women); 7 with code C43.9 (5 men, 2 women). 2.1 Applying ICA Approach for International Statistical Classification of Diseases and Health Problems Group for Malignant Melanoma of Skin We will use an index matrix that contains 7 rows (for ICD) and 8 columns (for gender and marital status). After the applying the ICA method we obtain index matrix (see Table 1) with intuitionistic fuzzy pairs that represents an intuitionistic fuzzy evaluation of the relations between every pair of criteria “ICD groups”. The stronger the correlation between a given pair is, the more intense the color is. Table 1. Membership parts of the Intuitionistic fuzzy pairs of the relations between “ICD groups”

μ C43.2 C43.3 C43.4 C43.5 C43.6 C43.7 C43.9

C43.2 1,000 0,571 1,000 0,357 0,464 0,429 0,429

C43.3 0,571 1,000 0,571 0,536 0,607 0,821 0,286

C43.4 1,000 0,571 1,000 0,357 0,464 0,429 0,429

C43.5 0,357 0,536 0,357 1,000 0,607 0,571 0,571

C43.6 0,464 0,607 0,464 0,607 1,000 0,536 0,464

C43.7 0,429 0,821 0,429 0,571 0,536 1,000 0,286

C43.9 0,429 0,286 0,429 0,571 0,464 0,286 1,000

2.2 Applying ICA Approach for Marital Status and Gender Data We will use an index matrix that contains 8 rows (for marital status and gender) and 7 columns (for ICD groups).

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After the applying the ICA method we obtain index matrix (see Table 2) with intuitionistic fuzzy pairs that represents an intuitionistic fuzzy evaluation of the relations between every pair of criteria “marital status and gender”. The stronger the correlation between a given pair is, the more intense the color is. Table 2. Membership parts of the Intuitionistic fuzzy pairs of the relations between “marital status and gender group”

μ man, unmarried man, married man, divorced man, widow women, unmarried women, married women, divorced women, widow

man, man, man, diunmarried married vorced

1,000 0,524 0,667 0,381 0,524 0,667 0,667 0,476

0,524 1,000 0,429 0,429 0,333 0,810 0,429 0,476

0,667 0,429 1,000 0,571 0,381 0,429 1,000 0,714

man, widow

0,381 0,429 0,571 1,000 0,095 0,429 0,571 0,619

women, women, unmarried married

0,524 0,333 0,381 0,095 1,000 0,524 0,381 0,333

0,667 0,810 0,429 0,429 0,524 1,000 0,429 0,476

women, women, divorced widow

0,667 0,429 1,000 0,571 0,381 0,429 1,000 0,714

0,476 0,476 0,714 0,619 0,333 0,476 0,714 1,000

3 Discussions 1. After applying the ICA approach for ICD groups the following conclusions can be made: 2 pairs of criteria (type of malignant melanoma) are in positive consonance by gender and marital status. The pair “C43.2 (Malignant melanoma of ear and external auricular canal) – “C43.4 (Malignant melanoma of scalp and neck)” is in a strong positive consonance - 1; 0. This shows the same tendency of morbidity of these two types of melanoma depending on sex, status, year of registration, and age group. The pair “C43.3 (Malignant melanoma of other and unspecified parts of face)” – “C43.7 (Malignant melanoma of lower limb, including hip area)” have a very similar tendency (0,821; 0,000). The other 19 couples do not show a consistent trend of finding a specific type of malignant melanoma depending on gender and marital status. In Table 3 compilations between ICA approach and correlation analysis according to Pearson, Kendall and Spearman for “ICD groups” are shown. The selected pairs, based on the four methods are identical in the first and the second rows. In the third row three of the ICD groups yield identical results (ICA, Kendall and Spearman), and the only difference is in the selected criteria as calculated by the Pearson method. In the fourth row, the situation is the same.

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Table 3. Membership parts of the Intuitionistic fuzzy pairs of the relations between “ICD groups” according ICA approach and correlation analysis according to Pearson, Kendall and Spearman ICA

Correlation coefficient Correlation coefficient Correlation coefficient according Pearson according Kendall according Spearman

1

C43.2 - C43.4: C43.2 - C43.4: 1,000 1,000; 0,000 Strong Posititve Consonance

C43.2 - C43.4: 1,000

C43.2 - C43.4: 1,000

2

C43.3 - C43.7: 0,821; 0,000 Weak Positive Consonance

C43.3 - C43.7: 0,968

C43.3 - C43.7: 0,879

C43.3 - C43.7: 0,923

3

C43.5 - C43.6: 0,607; 0,071 Dissonance

C43.6 - C43.7: 0,943

C43.5 - C43.6: 0,626

C43.5 - C43.6: 0,712

4

C43.3 - C43.6: 0,607; 0,071 Dissonance

C43.5 - C43.6: 0,937

C43.3 - C43.6: 0,542

C43.3 - C43.6: 0,624

2. After applying the ICA approach for marital status and gender group the following conclusions can be made: 2 pairs of criteria (gender and marital status) are in positive consonance by type of malignant melanoma. The “men divorced - women divorced” couple is in strong positive consonance with IF pair 1; 0, which means the exact same behavior for malignant melanoma. A similar conclusion can be made for the pair “men married - women married” with a IF pair 0,810; 0,048, which means that the trends for malignant melanoma in married men is similar to that of married women. The remaining 28 pairs of criteria did not show a consistent behavior for finding malignant melanoma depending on family status and age group, which is explained by the small number of patients in groups of unmarried male and unmarried female and widow male and widow female. In Table 4 compilations between ICA approach and correlation analysis according to Pearson, Kendall and Spearman for marital status and gender data are shown. The selected pairs, based on the four methods are identical in the first and the second rows. In the third and in the fourth rows the three of the ICD groups yield identical results (ICA, Kendall and Spearman), and the only difference is in the selected criteria as calculated by the Pearson method.

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Table 4. Membership parts of the Intuitionistic fuzzy pairs of the relations between marital status and gender data according ICA approach and correlation analysis according to Pearson, Kendall and Spearman ICA

Correlation coefficient Correlation according Pearson coefficient according Kendall

Correlation coefficient according Spearman

1

men divorced women divorced: 1,000; 0,000 Strong Posititve Consonance

men divorced women divorced: 1,000

men divorced women divorced: 1,000

men divorced women divorced: 1,000

2

men married women married: 0,810; 0,048 Weak Positive Consonance

men married women married: 0,823

men unmarried women unmarried: 0,982

men married women married: 0,889

3

men divorced women widow: 0,714; 0 Weak Dissonance

men unmarried men married women married: 0,782 women married: 0,864

men unmarried women married: 0,832

4

women divorced women widow: 0,714; 0 Weak Dissonance

men divorced women widow: 0,750 women divorced women widow: 0,750

men unmarried men married: 0,802

men unmarried women married: 0,836

4 Conclusion The InterCriteria Analysis method is applied to statistical data for newly registered and dispensary patients with oncological diseases for 2018 in Burgas. The analyzed data is from regional databases. The results are commented from different points of view: relations between gender and age of the patients, and relations between gender and marital status. Thus predictors of disease can be generated and search for dependencies between determinants of multi-criteria decision-making in oncological diseases associated with limitations such as time and resources. In the next paper, we will use the method proposed in [6] for constructing of three two-dimensional Index Matrices from the three-dimensional one aiming at detecting patterns and relationships across the data per oncology patients’ profiles, per marital status, and per year. ICA approach will be applied into the next directions: for age and gender group data, for marital status data and by years with collected statistical data for registered patients with oncological diseases in Burgas for 2014–2018.

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Acknowledgments. The authors are grateful for the support provided by the project DN-02– 10/2016 “New Instruments for Knowledge Discovery from Data, and their Modelling”, funded by the National Science Fund, Bulgarian Ministry of Education, Youth and Science. The authors declare that there is no conflict of interest regarding the publication of this paper.

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A Generalized Net Model of the Abdominal Aorta and Its Branches as a Part of the Vascular System Krassimir Atanassov1(B) , Valentin Vasilev2 , Velin Andonov3 , and Evdokia Sotirova2 1

Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bl. 105, 1113 Sofia, Bulgaria [email protected] 2 Faculty of Public Health and Health Care, Prof. Assen Zlatarov University, 1 Prof. Yakimov Street, 8010 Burgas, Bulgaria [email protected], [email protected] 3 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria velin [email protected]

Abstract. In a series of papers, Generalized Net (GN) models of the ways of functioning of the different systems and organs in the human body are described in general. Each GN model of a particular system or organ can be detailed and made more complex. In this paper a GN model of the abdominal aorta and its branches of the vascular system is proposed. Keywords: Generalized nets Vascular system

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Introduction

The heart together with the blood vessels – arteries, veins and lymph vessels – form the Cardiovascular System (CVS) [10,13,26]. The life of every human being begins with the first contraction (systole) of the heart and ends with the last. The main function of the CVS is to transport the blood which is the base of the exchange processes of the organism. Despite the fact that blood is a complex colloid and the blood vessels are much more different in terms of structure and properties than the ordinary water-pipes, to a vast degree the blood circulation is subject to the laws of hydrodynamics, but with some specific features. Because of this, the blood circulation in the blood vessels is defined as hemodynamics. The heart resembles a pump with an infinite displacement because the system is closed and inside the system a certain quantity of blood circulates but with different quality and functions – arterial or venous. As a result of the exchange c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 175–185, 2021. https://doi.org/10.1007/978-3-030-77716-6_16

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processes in the tissues, the arterial blood becomes venous and the venous blood becomes arterial by absorbing the blood from the liver (nutrients) and the oxygen in the lungs (releasing carbon dioxide at the same time). For practical purposes, the blood circulation in the arterial system is of special importance because the problems arising here are the most dangerous and oftentimes lead to irreparable consequences and death. The venous and lymph systems have to a greater degree collective and draining functions, and the diseases, connected to them, in most cases have chronic character and only in some cases emergency interventions are required. Therefore, with the aim of avoiding the unnecessary complication, our efforts are aimed at the construction of a mathematical model only of the arterial system. The venous system will be included only schematically with the aim of completing the cycle of the blood circulation and making sense of the significance of the heart as a main force behind the processes of the CVS. The CVS is a complex system working thanks to many fine mechanisms for regulation and adaptation according to the inner environment of the organism and the outer environment. Certain laws exist both at the level of separate organs, and at the organism level, which determine the correct and effective function of the CVS. Apart from this, the complexity and diversity of the CVS pathology should be taken into account. The gained empiric experience, the scientific discoveries in this field, the numerous medicaments and invasive technologies allow for a relatively effective treatment. However, medical science is still far away from the complete understanding of this complex system. Modelling the CVS gives the possibilities to evaluate its parameters and to analyze its behavior and functioning [12,14,28]. In the present paper, for the construction of the model the theory of the Generalized Nets (GNs, see [1]) is used. A lot of GN models of the different systems and organs in the human body are described in [2,4–9,15–25]. Most of them were collected in the book [3]. In this paper, a GN model of the abdominal aorta and its branches as a part of the vascular system is constructed. Using hierarchical operators from the GN theory, it can be used as a subnet of the GN model of the vascular system presented in [3]. The anatomical segmentation of the aorta as a main artery of the human body is important, because these segments are analyzed separately in the context of diagnosing aortic disease [27]. The abdominal aorta begins at the level of the diaphragm [10,11,26]. The main arteries arising from the abdominal aorta are the common and internal iliac arteries, the common and superficial femoral arteries, the anterior and posterior tibial arteries, and the peronial artery.

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A Generalized Net Model of the Abdominal Aorta and Its Branches as a Part of the Vascular System

The GN-model contains 24 transitions and 50 places. The following types of tokens are used:

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– α-tokens – they represent the blood in the arteries and capillaries, i.e., before the metabolism; – β-tokens – they represent the blood in the veins, i.e., after the metabolism; – χ-token – it represents the heart; – μ1 -, μ2 -, . . . , μ11 -tokens – they represent the human body matter in which metabolism processes flow as follows: • • • • • •

μ1 -token represents the abdominal aorta; μ2 - and μ3 -tokens represent the left and the right internal iliac artery; μ4 - and μ5 -tokens represent the left and the right profunda femoris artery; μ6 - and μ9 -tokens represent the left and the right peronial artery; μ7 - and μ10 -tokens represent the left and the right left anterior artery; μ8 - and μ11 -tokens represent the left and the right posterior artery.

Each one of these tokens obtain as a characteristic: “blood, quantity, time-moment”. The transitions of the GN represent the following functions: – – – – – – – – – – –

Z1 – the functions of the heart; Z2 – the functions of the abdominal aorta; Z3 and Z4 – the functions of the left/right common iliac artery; Z5 and Z6 – the functions of the left/right internal iliac artery; Z8 and Z9 – the functions of the left/right common femoral artery; Z11 and Z12 – the functions of the left/right profunda femoris artery; Z14 and Z15 – the functions of the left/right superficial femoral artery; Z17 and Z20 – the functions of the left/right peronial artery; Z18 and Z21 – the functions of the left/right anterior tibial artery; Z19 and Z22 – the functions of the left/right posterior tibial artery; Z7 , Z10 , Z13 , Z16 , Z23 , Z24 – the functions of the veins after metabolism. Below is a formal description of the GN-transitions. Z1 = {l2 , l3 }, {l1 , l2 }, l1 l2 l2 true true . l3 f alse true

At each time-step, token from place l3 enters place l2 and unites with token χ and simultaneously, token χ splits to two tokens - the same token χ and a token α that enters place l1 . Z2 = {l1 , l6 , l15 , l1 l6 l15 l20

20 }, {l3 , l4 , l5 , l6 },

l3 l4 l5 l6 f alse true true true true f alse f alse true . f alse f alse f alse true f alse f alse f alse true

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At each time-step, token α from place l1 splits to three α-tokens: the first of them enters place l6 and unites with token μ1 that obtains the above mentioned characteristic; the other two α-tokens enter places l4 and l5 . In the same timemoment, token μ1 splits to two tokens - the same token μ1 and a token β that enters place l3 , where it unites with the β-tokens from places l15 and l20 . Z3 = {l4 }, {l7 , l8 }, l7 l8 . l4 true true At each time-step, token from place l4 splits to two α-tokens that enter places l7 and l8 . Z4 = {l5 }, {l9 , l10 }, l9 l10 . l5 true true At each time-step, token from place l5 splits to two α-tokens that enter places l9 and l10 . Z5 = {l7 , l11 }, {l11 , l12 }, l11 l12 l7 true f alse . l11 true true At each time-step, the α-token from place l7 enters place l11 and unites with token μ2 and simultaneously, token μ2 splits to two tokens - the same token μ2 and a token β that enters place l12 . Z6 = {l10 , l14 }, {l13 , l14 }, l10 l14

l13 l14 f alse true . true true

At each time-step, the α-token from place l10 enters place l14 and unites with token μ2 and simultaneously, token μ2 splits to two tokens - the same token μ2 and a token β that enters place l13 . Z7 = {l12 , l25 }, {l15 }, l15 l12 true . l25 true At each time-step, the β-tokens from places l12 and l25 enter place l15 where they unite in a β-token (Fig. 1) .

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Fig. 1. Graphical representation of the GN model of the abdominal aorta and its branches as a part of the vascular system.

Z8 = {l8 }, {l16 , l17 }, l16 l17 . l8 true true

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At each time-step, the α-token from place l8 splits to two α-tokens that enter places l16 and l17 . Z9 = {l9 }, {l18 , l19 }, l18 l19 . l9 true true At each time-step, the α-token from place l9 splits to two α-tokens that enter places l18 and l19 . Z10 = {l13 , l32 }, {l20 }, l20 l13 true . l32 true At each time-step, the β-tokens from places l13 and l32 enter place l20 where they unite in a β-token. Z11 = {l16 , l21 }, {l21 , l22 }, l21 l22 l16 true f alse . l21 true true At each time-step, the α-token from place l16 enters place l21 and unites with token μ4 and simultaneously, token μ4 splits to two tokens - the same token μ4 and a token β that enters place l22 . Z12 = {l19 , l24 }, {l23 , l24 }, l19 l24

l23 l24 f alse true . true true

At each time-step, the α-token from place l19 enters place l24 and unites with token μ5 and simultaneously, token μ5 splits to two tokens - the same token μ5 and a token β that enters place l23 . Z13 = {l22 , l49 }, {l25 }, l25 l22 true . l49 true At each time-step, the β-tokens from places l22 and l49 enter place l25 where they unite in a β-token. Z14 = {l17 }, {l26 , l27 , l28 },

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l26 l27 l28 . l17 true true true At each time-step, token α from place l17 splits to three α-tokens that enter places l26 , l27 and l28 . Z15 = {l18 }, {l29 , l30 , l31 }, l29 l30 l31 . l18 true true true At each time-step, token α from place l18 splits to three α-tokens that enter places l29 , l30 and l31 . Z16 = {l23 , l50 }, {l32 }, l32 l23 true . l50 true At each time-step, the β-tokens from places l23 and l50 enter place l32 where they unite in a β-token. Z17 = {l26 , l33 , l35 }, {l33 , l34 }, l33 l34 l26 true f alse . l33 true true l35 true f alse At each time-step, the α-token from place l26 enters place l33 and unites with token μ6 and simultaneously, token μ6 splits to two tokens - the same token μ6 and a β-token that enters place l34 . When there is an α-token in place l35 , it enters place l33 and unites with token μ6 . Z18 = {l27 , l38 }, {l35 , l36 , l37 , l38 }, l27 l38

l35 l36 l37 l38 W27,35 f alse W27,37 true , f alse true f alse true

where W27,35 = “there is necessity for blood for the left peronial artery”, W27,37 = “there is necessity for blood for the left posterior tibial artery”. At each time-step, the α-token from place l27 enters place l38 and unites with token μ7 and simultaneously, token μ7 splits to two tokens - the same token μ7 and a β-token that enters place l36 . When W27,35 = true and/or W27,37 = true, the α-token from place l27 splits to one of two additional α-tokens that enter places l35 and/or l37 , respectively, depending on the truth-values of predicates W27,35 and W27,37 .

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Z19 = {l28 , l37 , l40 }, {l39 , l40 }, l39 l40 l28 f alse true . l40 true true l37 f alse true At each time-step, the α-token from place l28 enters place l40 and unites with token μ8 and simultaneously, token μ8 splits to two tokens - the same token μ8 and a β-token that enters place l39 . When there is an α-token in place l37 , it enters place l40 and unites with token μ8 . Z20 = {l29 , l41 , l44 }, {l41 , l42 }, l41 l42 l29 true f alse . l41 true true l44 true f alse At each time-step, the α-token from place l29 enters place l41 and unites with token μ9 and simultaneously, token μ9 splits to two tokens - the same token μ9 and a β-token that enters place l42 . When there is an α-token in place l44 , it enters place l41 and unites with token μ9 . Z21 = {l30 , l43 }, {l43 , l44 , l45 , l46 }, l30 l43

l43 l44 l45 l46 true W30,44 f alse W30,46 , true f alse true f alse

where W30,44 = “there is necessity for blood for the right peronial artery.”, W30,46 = “there is necessity for blood for the right posterior tibial artery”. At each time-step, the α-token from place l30 enters place l43 and unites with token μ10 and simultaneously, token μ10 splits to two tokens - the same token μ10 and a β-token that enters place l45 . When W30,44 = true and/or W30,46 = true, the α-token from place l30 splits to one of two additional α-tokens that enter places l44 and/or l46 , respectively, in respect of the truth-values of predicates W30,44 and W30,46 . Z22 = {l31 , l46 , l48 }, {l47 , l48 }, l47 l48 l31 f alse true . l48 true true l46 f alse true At each time-step, the α-token from place l31 enters place l48 and unites with token μ11 and simultaneously, token μ11 splits to two tokens - the same token

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μ11 and a β-token that enters place l47 . When there is an α-token in place l46 , it enters place l48 and unites with token μ11 . Z23 = {l34 , l36 , l39 }, {l49 }, l49 l34 true . l36 true l39 true At each time-step, the β-tokens from places l34 , l36 and l39 enter place l49 where they unite in a β-token. Z24 = {l42 , l45 , l47 }, {l50 }, l50 l42 true . l45 true l47 true At each time-step, the β-tokens from places l42 , l45 and l47 enter place l50 where they unite in a β-token.

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Conclusion

In the present paper, a GN model of the abdominal aorta and its branches as a part of the vascular system is presented. The purpose of the construction of the GN model of the CVS is to complement and enrich the knowledge about the CVS, or at least about a part of it. Objective criteria and parameters can be chosen which through the GN model would give us a clearer understanding about the state of the CVS, presence and weight of the pathology, the risk degree and a relatively correct prognosis for the development of one disease or another. In our future research, we intend to construct a GN model of the aortic arch branches of the CVS as a continuation of our work on the modeling of systems and organs of the human body with GNs.

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A Generalized Net Model of the Human Body Excretory System Martin Lubich1 , Velin Andonov2 , Anthony Shannon3 , Chavdar Slavov4 , Tania Pencheva5(B) , and Krassimir Atanassov5,6 1

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University Hospital “Sofiamed”, 16 G. M. Dimitrov Blvd., 1797 Sofia, Bulgaria [email protected] 2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria 3 Warrane College, The University of New South Wales, Kensington, NSW 2033, Australia 4 University Hospital “Tsaritsa Yoanna – ISUL” Medical University, 8 Bialo More Str., 1527 Sofia, Bulgaria 5 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 105, 1113 Sofia, Bulgaria [email protected], [email protected] Prof. D-r Asen Zlatarov University, 1 Prof Yakimov Str., 8010 Bourgas, Bulgaria

Abstract. The Generalized Nets (GN) have been proved as a successful tool for modelling of parallel processes. Up to now they have been used to describe different human body systems. In the present paper, GN is going to be applied for first time for the detailed description of human body excretory system. Keywords: Generalized Nets

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Introduction

The kidneys are a pair of bean-shaped organs tasked with excreting waste products and excess water in order to maintain the body’s electrolyte and fluid balance. This is achieved by means of blood filtration. Kidneys are supplied by renal arteries that branch directly from the abdominal aorta. Renal arteries divide gradually, leading to a glomerular capillary plexus. The functioning unit of the kidney is called the glomerulus. Afferent and efferent arterioles carry blood into and out of the glomerulus, respectively. In a normal physiological condition, 20% of the cardiac output flows through the 2.5–3 million glomerular capillary plexus [6]. The process of glomerular filtration is called renal ultrafiltration. The force of hydrostatic pressure in the glomerulus is the driving force that thrusts filtrate out of the capillaries and into the openings, or “fenesterations”, of the nephron. During the process filterable blood elements, such as water and nitrogenous waste, will move towards the inside of the glomerulus, and nonfilterable components, such as cells and proteins, will exit via the efferent arteriole, and enter c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 186–192, 2021. https://doi.org/10.1007/978-3-030-77716-6_17

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the bloodstream [8]. Filtration depends not only on the size of the molecule but also on its electrical charge. The proteins of the glomerular basement membrane carry a negative charge which repels negatively charged protein molecules [10]. The filterable components accumulate in the glomerulus to form the glomerular filtrate. Glomerular Filtration Rate (GFR) is the rate of filtered fluid through the kidneys per unit time. It is controlled by three factors [9]: (1) the difference in hydrostatic and oncotic pressure, (2) the renal plasma flow, and (3) the permeability of the glomerular membrane. For a young healthy man, the GFR value is 180 l/24 h or 125 ml/min [7]. A typical adult has a Blood Volume (BV) of approximately 5 l. Plasma volume comprises about 55% of the BV, or approximately 3 l. Renal Blood Flow (RBF) measures at about 1.1 l/min. Renal Plasma Flow (RBF) is therefore calculated as follows: 0.55 l/min × 1.1 l/min = 605 ml/min. This means that for every 605 ml of plasma entering the glomeruli through afferent arterioles every minute, 125 ml (20%) are filtered and the remaining 480 ml pass through the efferent arterioles and back into the bloodstream. Assessment of GFR is essential for determining renal function. The GFR rate is determined by the water permeability of its membrane, the surface area and by the Net Filtration Pressure (NFP). As such, the filtration rate is calculated by water permeability × area × NFP. Due to the fact that it is very difficult to determine the area of the entire capillary bed, an indicator called the filtration coefficient Kf is used, which is derived from the water permeability and the filtration area. NFP itself is an algebraic sum of hydrostatic and osmotic pressures driven by protein-oncotic or colloid-osmotic pressures. There are 4 pressures that counteract: 2 hydrostatic and 2 oncotic. GFR is not a constant and shows considerable variability in various physiological conditions and diseases. The main reason for GFR decline in various diseases [5] is not due to an alteration in these parameters, but rather to decrease of number of functioning nephrons and, therefore, to decrease of the glomerular surface Kf . The Generalized Nets (GN; see [1–3]) are a tool for modelling of parallel processes, extending Petri Nets and their other modifications. GN have numerous successful applications in different scientific areas, i.e. medicine, ecology, artificial intelligence, and many others. Up to now GN have been used to describe different human body systems. In the present paper, GN is going to be applied for first time for the detailed description of human body excretory system.

2

The Generalized Net Model

The GN model described the human body excretory system is presented in Fig. 1. The model contains 8 transitions, 16 places and 11 types of tokens. The transitions Z1 , Z2 , Z5 , Z6 , Z7 represent the following human body systems and organs together with their ongoing processes:

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Z1 Z2 Z5 Z6 Z7

– – – – –

Cardiovascular System (CVS); kidneys; bladder; Nervous System (NS); Muscle System (MS).

In the present model, aforementioned systems and organs are given in a simplified form. For the purposes of further research they might be described in much more detailed form. The types of tokens, their meaning and characteristics are described as follows: β – CVS, current status; κ – kidneys, current status; ω – bladder, quantity of urine; ν – NS, current status; μ – MS, current status; α – blood, quantity; γ – Primary Ultra Filtrated (PUF) blood that enters efferent artery; δ – reabsorbed products from PUF that return in the blood; ζ – a signal from bladder to NS for contraction; θ – urine; λ – a signal from NS to MS for contraction. GN model transitions are successively represented as follows: l1 l2 l2 true true . Z1 = {l2 , l5 , l8 }, {l1 , l2 }, l5 f alse true l8 f alse true At each time-step, tokens from places l5 and l8 enter place l2 and unite with token β that obtains the aforementioned characteristic. The token β splits to two tokens – the same token β and a token α that enters place l1 with a characteristic: “blood, perfusion volume”. l3 l4 Z2 = {l1 , l4 }, {l3 , l4 }, l1 f alse true . l4 true true At each time-step, the token α from place l1 enters place l4 and unites with token κ that obtains the aforementioned characteristic. The token κ splits into two tokens – the same token κ and a token α that enters place l3 without a new characteristic. Z3 = {l3 }, {l5 , l6 },

l5 l6 . l3 true true

l1

l2

 



 

Z1



l3

 

l4

 

Z2

 l5

 

l6

 

Z3

 l7

l8

 

 

Z4



l11

l10

   



l9

 

Z5

 l12

 

l13

 

Z6



l14

 

l15

 

Z7





 l16

 

Z8



GN Model of the Human Body Excretory System

Fig. 1. GN model of the human body excretory system

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At each time-step, the token α from place l3 splits to two tokens – the same token α that enters place l6 with a characteristic “filtrated blood” and a token γ that enters place l5 with a characteristic “PUF that enters afferent artery”. Z4 = {l6 }, {l7 , l8 },

l7 l8 . l6 true true

At each time-step, the token α from place l6 splits to two tokens – the same token α that enters place l7 with a characteristic “the rest products of PUF that enter renal papila” and a token δ that enters place l8 with a characteristic “reabsorb products from PUF that return in the blood system”. Z5 = {l7 , l11 }, {l9 , l10 , l11 }, l9 l10 l11 l7 f alse f alse true , l11 W11,9 W11,10 true where W11,9 = “there is enough quantity of urine in the bladder”, W11,10 = “MS is relaxed”. At each time-step, the token α from place l7 enters place l11 and unites with token ω that obtains the aforementioned characteristic. When the predicate W11,9 becomes true, the token ω splits to two tokens – the same token ω without a new characteristic and a token ζ that enters place l9 with a characteristic: “signal from stretch receptors to NS that there is enough quantity of urine in the bladder”. When the predicate W11,10 becomes true, the token ω splits to two tokens – the same token ω without a new characteristic and a token θ that enters place l10 with a characteristic: “the urine is passed from the bladder to the urethra”. l12 l13 Z6 = {l9 , l13 }, {l12 , l13 }, l9 f alse true , l13 W13,12 true

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where W13,12 = “signal from bladder that there is enough quantity of urine in it”. When the token ζ from place l9 enters place l13 , it unites with token ν that obtains the aforementioned characteristic. When the predicate W13,12 becomes true, the token ν splits to two tokens – the same token ν and a token λ that enters place l12 with a characteristic: “signal to MS for contraction”. l14 l15 Z7 = {l12 , l15 }, {l14 , l15 }, l12 f alse true , l15 W15,14 true where W15,14 = “contraction of the detrusor muscle”. When the token λ from place l12 enters place l15 , it unites with token μ that obtains the aforementioned characteristic. When W15,14 becomes true, the token μ splits to two tokens – the same token μ and a token λ that enters place l14 with a characteristic: “signal from MS for contraction”. l16 Z8 = {l10 , l14 }, {l16 }, l10 true . l14 true The token θ from place l10 and the token λ from place l14 are united in place l16 in a token θ with a characteristic: “final urine; quantity”.

3

Conclusion

GN model of human body excretory system is here developed for first time in a form more detailed than the GN model presented in [4]. As the main systems and organs are presented in a simplified form, in future, the model might be worked out in more details. Also, it may be included as a subset of the human body GN model (see [4]). It can be used for simulating of some situations for which there is enough information with a purpose to assist in decision making process, and for students training. Acknowledgements. The present research has been supported by the Bulgarian National Science Fund under Grant Ref. No. DN-02/10 “New Instruments for Knowledge Discovery from Data and Their Modelling”.

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References 1. Alexieva, J., Choy, E., Koycheva, E.: Review and bibliography on generalized nets theory and applications. In: Choy, E., Krawczak, M., Shannon, A., Szmidt, E. (eds.) A Survey of Generalized Nets, Raffles KvB Monograph No. 10, pp. 207–301 (2007) 2. Atanassov, K.: Generalized Nets. World Scientific, London (1991) 3. Atanassov, K.: On Generalized Nets Theory. “Prof. M. Drinov” Academic Publishing House, Sofia (2007) 4. Atanassov, K., Chakarov, V., Shannon, A., Sorsich, J.: Generalized Net Models of the Human Body. “Prof. M. Drinov” Academic Publishing House, Sofia (2008) 5. Aboumarzouk, O. (ed.): Blandy’s Urology, 3rd edn., pp. 107–115. John Wiley & Sons, New York (2019) 6. Kanwar, Y.S., Venkatachalam, M.A.: Ultrastructure of glomerulus and juxtaglomerular apparatus. In: Windhager, E.E. (ed.) Handbook of Physiology. Oxford University Press, Oxford, pp. 3–40 (1992) 7. Kaufman, D.P., Basit, H., Knohl, S.J.: Physiology, Glomerular Filtration Rate (GFR). In: StatPearls [Internet]. StatPearls Publishing, Treasure Island (FL) (2019) 8. Miner, J.H.: The glomerular basement membrane. Exp. Cell Res. 318(9), 973–978 (2012) 9. Mundy, A., Fitzpatrick, J.: The Scientific Basis of Urology, 3rd edn., pp. 57–82. CRC Press, Oxford (2010) 10. Shoskes, D.A.M.: Renal physiology and pathophysiology. In: Wein, A. (ed.) Campbell-Welsh Urology, 10th edn., pp. 1025–1046. Elsevier, New York (2012)

Clustering of InterCriteria Analysis Data Using a Malignant Neoplasms of the Digestive Organs Data Sotir Sotirov1(B) , Greta Bozova2 , Valentin Vasilev3 , and Maciej Krawczak4 1 Intelligent Systems Laboratory, “Prof. Dr. Assen Zlatarov” University, “Prof. Y. Yakimov”

Blvd, Burgas 8010, Bulgaria [email protected] 2 Department of Nephrology, Military Medical Academy, 3 Street George Sofiiski, Sofia, Bulgaria [email protected] 3 Faculty of Public Health and Health Care, Prof. Assen Zlatarov University, 1 Prof. Yakimov Street, Burgas, Bulgaria [email protected] 4 Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland [email protected]

Abstract. Very often, very large data tables as results of the InterCriteria Analysis are obtained. In this article, we propose using of one of the types of artificial neural networks that serve to cluster data. The aim is to obtain aggregations of such data. Our example has 9 neurons and 9 clusters along with the centers of gravity of these clusters. To analyze the patients with oncological diseases, registered in Burgas from 2014 to 2018, the InterCriteria Analysis (ICA) approach is applied. Keywords: SOM neural network · InterCriteria analysis · Fuzzy sets · Malignant neoplasms of the digestive organs

1 Introduction Malignant neoplasms of the digestive organs are entities of a malignant nature, with a localized primary lesion in one of the organs of the digestive system - esophagus, stomach, bile system, pancreas, intestine (duodenum, small intestine, large intestine) and anus [2, 13, 19]. The clinical picture and symptoms depend on the organ concerned and may include obstruction (obstruction leading to difficulty swallowing or defecation), bleeding (mellitus, haematemesis) or other organ-related symptoms. The diagnosis of malignant tumors of the digestive system requires a number of diagnostic activities, some of which are history, objective examination, endoscopy, followed by suspected tissue biopsy, X-ray, computed tomography and more. Treatment depends on the location of the tumor, the type of malignant cells, the degree of infiltration of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 193–201, 2021. https://doi.org/10.1007/978-3-030-77716-6_18

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surrounding tissues, regional lymph nodes, or other distant organs. These characteristics of the primary tumor also determine the prognosis. In some cases very large data tables are obtained when the results of the InterCriteria Analysis are obtained. This makes them difficult to perceive and process. In this article, we propose the use of one of the types of artificial neural slots that serve to cluster data. The aim is to obtain aggregations of such data. Our example has 9 neurons. They identify 9 clusters along with the centers of gravity of these clusters. For analysis the patients with oncological diseases, registered in Burgas from 2014 to 2018 the InterCriteria Analysis (ICA) approach is applied. The InterCriteria Analysis approach is applied to real data for 1772 (1036 man and 736 woman) registered patients with malignant neoplasms of the digestive organs in Burgas region for 2014–2018. The data contains information about age of patients, name of the disease, according to International statistical classification of diseases and health problems, gender, marital status, data of the registration of the patient, etc. [10, 15, 25]. In the observed data there are: • 14 age groups: men 21–30 years, women 21–30 years, men 31–40, women 31–40, men 41–50, women 41–50, men 51–60, women 51–60, men 61–70, women 61–70, men 71–80, women 71–80, men over 80, women over 80 (Fig. 1). According age groups the patients are distributed the following way: 7 in 20–30 years (3 men, 4 women); 18 in 31–40 (12 men, 6 women); 100 in 41–50 (61 men, 39 women); 336 in 51–60 (195 men, 141 women); 663 in 61–70 (417 men, 246 women); 470 in 71–80 (269 men, 201 women); and 178 over 80 (79 men, 99 women) (Fig. 1). • 8 marital status groups: men unmarried, women unmarried, men married, women married, men divorced, women divorced, men widower, women widower (Fig. 2). According marital status the patients are distributed the following way: 16 unmarried (7 men, 9 women); 1469 married (884 men, 585 women); 48 divorced (35 men, 13 women) and 239 widower (109 men, 130 women) (Fig. 2).

Distribution of the number of the patients by age groups 600 400 200

3 12

417 269 195 79 61

246 141 20199 39 4 6

0 men

women

21-30

31-40

41-50

61-70

71-80

over 81

51-60

Fig. 1. Distribution of the number of the patients by age groups

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Distribution of the number of the patients by marital status groups 884

1000 500

585 7

35 109

13

9

130

0 men unmarried

married

women divorced

widower/widow

Fig. 2. Distribution of the number of the patients by marital status groups

24 malignant neoplasms of the digestive organs, groups Malignant neoplasms of digestive organs classified according to the International Statistical Classification of Diseases and Health Problems (ICD) [21]: Abdominal part of oesophagus, Overlapping lesion of oesophagus, Malignant neoplasm of stomach, Pyloric antrum, Lesser curvature of stomach, unspecified, Overlapping lesion of stomach, Appendix, Ascending colon, Transverse colon, Descending colon, Sigmoid colon, Overlapping lesion of colon, Malignant neoplasm of rectosigmoid junction, Malignant neoplasm of rectum, Malignant neoplasm of anus and anal canal, Anal canal, Liver cell carcinoma, Intrahepatic bile duct carcinoma, Malignant neoplasm of gallbladder, Ampulla of Vater, Head of pancreas, Tail of pancreas, Overlapping lesion of pancreas, Pancreas and unspecified.

2 InterCriteria Analysis The ICA-method [1, 3, 7, 9, 14, 17, 18, 20–24, 26–29] is based on two main concepts: intuitionistic fuzzy sets [4, 6, 8] and index matrices [5]. A brief description is offered below for completeness. Let I be a fixed set of indices and let R be the set of the real numbers. An index matrix (IM) with sets of indices K and L (K, L ⊂ I) is defined by [4]

where K = {k 1 , k 2 ,…, k m }, L = {l1 , l 2 ,…, l n }, for 1 ≤ i ≤ m , and 1 ≤ j ≤ n : aki ,lj ∈ R.

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For any two IMs, a series of relations, operations, and operators have been defined. The theory behind the IMs is described in a more detail fashion in [5]. Here, following the description of the ICA approach, given by [7], we will start with the IM M with index sets with m rows {O1 , …, Om } and n columns {C 1 , …, C n }, where for every p, q (1 ≤ p ≤ m, 1 ≤ q ≤ n), Op in an evaluated object, C q is an evaluation criterion, and eOp ,Cp is the evaluation of the p-th object against the q-th criterion, defined as a real number or another object that is comparable according to relation R with all the rest elements of the IM M.

From the requirement for comparability above, it follows that for each i, j, k it holds  the relation R eOi ,Ck , eOj ,Ck . The relation R has a dual relation R, which is true in the cases when the relation R is false, and vice versa. For the requirements of the proposed method, pairwise comparisons between every two different criteria are made along all evaluated objects. During the comparison, a counter is maintained for the number of times when the relation R holds, and another counter – for the dual relation.   μ Let Sk,l be the number of cases in which the relations R eOi ,Ck , eOj ,Ck and   ν be the number of cases in R eOi ,Cl , eOj ,Cl are simultaneously satisfied. Let also Sk,l     which the relations R eOi ,Ck , eOj ,Ck and its dual R eOi ,Ck , eOj ,Ck are simultaneously satisfied. As the total number of pairwise comparisons between the objects is given by m(m – 1)/2, it can be verified that the following inequalities hold: μ

ν ≤ 0 ≤ Sk,l + Sk,l

m(m − 1) 2

For every k, l, such that 1 ≤ k ≤ l ≤ n, and for m ≥ 2 two numbers are defined: μ

μCk ,Cl = 2

Sk,l m(m − 1)

, νCk ,Cl = 2

ν Sk,l

m(m − 1)

The pair constructed from these two numbers plays the role of the intuitionistic fuzzy evaluation [2, 3, 5, 7] as an extension of the concept of reflection of sets defined by Zadde of the relations that can be established between any two criteria C k and C l . In this way, the IM M that relates evaluated objects with evaluating criteria can be transformed to another IM M* that gives the relations detected among the criteria, where stronger

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correlation exists where the first component μCk ,Cl is higher while the second component νCk ,Cl is lower.

From practical considerations, it has been more flexible to work with two IMs M μ and M ν , rather than with the IM M * of IF pairs. IM M μ contains as elements the first components of the IFPs of M *, while M ν - the second components of the IFPs of M *.

3 Self Organizing Map Neural Networks Self-learning self-organizing maps are a kind of artificial neural networks [11, 12, 16]. The purpose of self-organizing maps (SOM) is to transform the input model (signal) of a certain size into n-dimensional dimension and to transform it adaptively into a topological mode. The SOM neurons are located in a m-grid, each of the inlets being fed to the entrance of each of the neurons. The figure shows Self Organizing Map Neural Networks (Fig. 3). This network is a single-layer straight structure of neurons located in lines and columns. The training of the neural network is based on the principle of competitive training, with only one winner after the SOM training. For the learning is used unsupervised method. The number of the iteration is 1200. The results after the testing are shown at the Table 1 and Fig. 3.

4 Testing with Data for Malignant Neoplasms of the Digestive Organs Data For the learning process of SOM NN, we set the following parameters: Structure 3*3, Performance based on Mean Square Error. As an input parameters we use μ values from the results from the M* matrix from InterCriteria Analysis. The InterCriteria Analysis approach is applied to real data for 1772 (1036 men and 736 women) registered patients with malignant neoplasms of the digestive organs in Burgas region for 2014–2018. Using the ICA software applied to the results of 1772 peoples, we obtain a matrix with μ and ν values. For the learning process of the SOM neural network we use only μ values (index matrix 40 × 40). After the learning, 9 clusters are constructed. Every cluster has a center and own limits. This data are shown in the Table 1. In this table are also shown the number of vectors that fall into this cluster. For example, 134 vectors fall into cluster number 3 and the cluster center is 0.270. Every cluster with values from 0.142 to 0.285 is used. The idea is to substitute the values of the every cluster that hist in the few of the clusters with cluster centers. In the Fig. 4 is shown how many vectors are hits in every clusters.

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Fig. 3. The structure of the SOM neural network.

Table 1. Table for the individual clusters, the affected one and their boundaries Number of the cluster

Numbers fits vectors

Cluster center Start and end of cluster

1

186

0.545

0.476-0.571

2

230

0.460

0.428-0.476

3

134

0.270

0.142-0.285

4

144

0.743

0.571-0.809

5

178

0.347

0.333-0.381

6

212

0.046

0-0.095

7

194

0.941

0.857-1

8

138

0.642

0.619-0.809

9

184

0.168

0.142-0.190

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Hits 2.5

2 194

138

184

1.5

1

144

178

212

0.5

0

186

230

134

-0.5

-1 -1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 4. The hit of vectors in clusters in the neural network.

5 Conclusion Where we use big number of the criteria, we obtain very large data tables as results of the InterCriteria Analysis. For analysis we use the patients with oncological diseases, registered in Burgas from 2014 to 2018. In this article, is propose to used one of the types of artificial neural network that serve to cluster data. The aim is to obtain aggregations of such data. Our example has 9 neurons and 9 clusters along with the centers of gravity of these clusters. Acknowledgments. The authors are grateful for the support provided by the project DN-02– 10/2016 “New Instruments for Knowledge Discovery from Data, and their Modelling”, funded by the National Science Fund, Bulgarian Ministry of Education, Youth and Science. The authors declare that there is no conflict of interest regarding the publication of this paper.

References 1. Angelova, M., Roeva, O., Pencheva, T.: InterCriteria analysis of a cultivation process model based on the genetic algorithm population size influence. Notes on Intuitionistic Fuzzy Sets 21(4), 90–103 (2015) 2. Annese, V., et al.: European evidence-based consensus: inflammatory bowel disease and malignancies. J. Crohn’s Colitis 9(11), 945–965 (2015) 3. Atanassov, K.V., Atanassova, G., Gluhchev, G.: InterCriteria analysis: ideas and problems. Notes Intuitionistic Fuzzy Sets. 21(1), 81–88 (2015) 4. Atanassov, K.: Index Matrices: Towards an Augmented Matrix Calculus. Studies in Computational Intelligence Series, Vol. 573, Springer, Cham (2014) 5. Atanassov, K.: On index matrices. Part 5: 3-dimensional index matrices. Adv. Studies Contemp. Math. 24(4), 423–432 (2014)

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6. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 7. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues in Intuitionistic Fuzzy Sets Gener. Nets 11, 1–8 (2014) 8. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes Intuitionistic Fuzzy Sets 19(3), 1–13 (2013) 9. Fidanova, S., Roeva, O.: InterCriteria analysis of different metaheuristics applied to E. coli cultivation process. Numer. Methods Sci. Comput. Adv. Appl. 21–25 (2016). ISBN: 978-6197223-18-7 10. Gibson, J.A., Odze, R.D.: Tissue sampling, specimen handling, and laboratory processing. In: Clinical Gastrointestinal Endoscopy, pp. 51–68. Elsevier (2019). https://doi.org/10.1016/ B978-0-323-41509-5.00005-0 11. Hagan, M.T., Demuth, H.B., Beale, M.: Neural Network Design. PWS Publishing Company, Boston (1996) 12. Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, N.J. (1999) 13. http://gco.iarc.fr/. Global Cancer Observatory (2019) 14. Ilkova, T., Roeva, O., Vassilev, P., Petrov, M.: InterCriteria analysis in structural and parameter identification of L-lysine production model. Issues Intuitionistic Fuzzy Sets Gener. Nets, 12, 39–52 (2015/2016) 15. International Statistical Classification of Diseases and Related Health Problems 10th Revision (2016). https://icd.who.int/browse10/2016/en#/C15 16. Kohonen, T.: Exploration of very large databases by self-organizing maps. In: Proceedings of International Conference on Neural Networks (ICNN 1997), vol. 1. IEEE (1997) 17. Krumova, S., et al.: Intercriteria analysis of calorimetric data of blood serum proteome. Biochimica et Biophysica Acta – General Subjects 1861(2), 409–417 (2017) ISSN: 0304– 4165. https://doi.org/10.1016/j.bbagen.2016.10.012 18. Mucherino, A., Fidanova, S., Ganzha, M.: Introducing the environment in ant colony optimization. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization. SCI, vol. 655, pp. 147–158. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40132-4_9 19. National Center of Public Health and Analyses, Annual information. http://ncpha.govern ment.bg/index.php?lang=en 20. Pencheva, T., Angelova, M.: InterCriteria analysis of simple genetic algorithms performance. In: Georgiev, K., Todorov, M., Georgiev, I. (eds.) Advanced Computing in Industrial Mathematics. SCI, vol. 681, pp. 147–159. Springer, Cham (2017). https://doi.org/10.1007/978-3319-49544-6_13 21. Pencheva, T., Angelova, M., Vassilev, P., Roeva, O.: InterCriteria analysis approach to parameter identification of a fermentation process model. In: Atanassov, K.T., et al. (eds.) Novel Developments in Uncertainty Representation and Processing. AISC, vol. 401, pp. 385–397. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-26211-6_33 22. Roeva, O., Fidanova, S., Paprzycki, M.: Comparison of different ACO start strategies based on InterCriteria analysis. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization. SCI, vol. 717, pp. 53–72. Springer, Cham (2018). https://doi.org/10.1007/978-3-31959861-1_4 23. Roeva, O., Fidanova, S., Paprzycki, M.: InterCriteria analysis of ACO and GA hybrid algorithms. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization. SCI, vol. 610, pp. 107–126. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-21133-6_7 24. Sotirova, E., Petrova, Y., Bozov, H.: InterCriteria analysis of oncological data of the patients for the city of Burgas. Notes Intuitionistic Fuzzy Sets 25(2), 96–103 (2019) 25. State of Health in the EU, Bulgaria Health profile for the country (2017). https://ec.europa. eu/health/sites/health/files/state/docs/chp_bulgaria_bulgarian.pdf (in Bulgarian)

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26. Todinova, S., et al.: Blood plasma thermograms dataset analysis by means of InterCriteria and correlation analyses for the case of colorectal cancer. Int. J. Bioautomation 20(1), 115–124 (2016) 27. Vankova, D., Sotirova, E., Bureva, V.: An application of the InterCriteria analysis approach to health-related quality of life. Notes Intuitionistic Fuzzy Sets 21(5), 40–48 (2015) 28. Zaharieva, B., Doukovska, L., Ribagin, S., Michalikova, A., Radeva, I.: Intercriteria analysis of behterev’s kinesitherapy program. Notes Intuitionistic Fuzzy Sets 23(3), 69–80 (2017) 29. Zaharieva, B., Doukovska, L., Ribagin, S., Radeva, I.: InterCriteria approach to Behterev’s disease analysis. Notes on Intuitionistic Fuzzy Sets 23(2), 119–127 (2017). Bishop C. M., Neural networks for pattern recognition, Oxford University Press, ISBN: 0 19 853864 2 (2000)

Fuzzy-Based Algorithm for QRS Detection Tomasz Pander(B) and Tomasz Przybyla Institute of Electronics, Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Gliwice, Poland {tpander,tprzybyla}@polsl.pl

Abstract. ECG signal analysis is important task from the point of view of modern cardiological diagnostics. The accuracy and precision of ECG signal measurements or classification tasks depend on the quality of the method for determining the position of QRS complexes in the electrocardiogram. In this paper we propose a simple method of R-wave detection of QRS complex based on exceeding value of the amplitude threshold by the samples of the so-called detection function waveform. The amplitude threshold is calculated by the means of the fuzzy c-median clustering method. The study is carried out for the MIT-BIH Arrhythmia Database, Noise Stress Test Database, Fantasia Database and QT database. In the case of QRS detection performance testing for MIT-BIH Arrhythmia Database, the following results are achieved, the sensitivity value is 99.81%, the positive predictivity P + is 99.69% and the F-measure value is 99.75%. The obtained results show that the proposed method is competitive to the reference methods. Keywords: QRS detection processing

1

· Fuzzy c-median clustering · ECG

Introduction

Electrocardiogram (ECG) is one of the most popular electro-physiological signal which is used for the diagnosis of the heart diseases by physicians. This signal has been used for over 100 years and its acquisition nowadays is not a problem. The big advantage of this signal is the fact that in most cases it is recorded in a non-invasive way. For this reason, the processing and measurement of ECG signal characteristics are essential for a meaningful diagnosis of the patient. This signal is an effective non-invasive tool for wide range of biomedical applications such as measuring heart rate, examining the rhythm of heartbeats, diagnosing heart abnormalities, emotion recognition and biometric identification [1]. The ECG signal is a graphical representation of the electrical activity of the heart over a period of time which is acquired with the means of the electrodes placed on the skin body surface [2]. It is composed of basic characteristic waveforms as P wave, QRS complex and T wave. The basic waveform that represents the ECG signal is depicted in Fig. 1. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 202–215, 2021. https://doi.org/10.1007/978-3-030-77716-6_19

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Fig. 1. An artificial generated single cardiac cycle of ECG signal.

The segments listed above and the time intervals between the segments are used for ECG signal analysis. Comparing the characteristic components of the ECG signal, it can be seen that usually the QRS complex has the highest amplitude. For this reason, precise and reliable detection of the R-wave position in the QRS complex is one of the most critical importance for clinicians in diagnosing cardiac disorders. However, QRS detection is not an easy task because of the high morphological variability of the QRS complexes and disturbances. An accurate beats recognition is obstructed by the fact that electrophysiological signals are usually recorded in the presence of different kinds of noise that include power-line interference (50/60 Hz), baseline wander, muscle noise, electrodes motion and high-frequency P or T waves which are similar to QRS complexes. In the case of long-term monitoring of the heart rate, a new problem arises: the changes in electrode-skin impedance, which may even make it impossible to properly record the signal [3]. Another important issue is also the widespread use of telemedicine ECG measurements when ECG signals are recorded by people who are not trained to do so. Therefore, the above mentioned interferences can make it completely impossible to locate R-waves in the electrocardiogram [4]. The QRS complex detection is required as a first stage in almost all automatic of ECG signal processing/analysis methods. It is required to quantify the heart rate and as a reference for the classification of heartbeats types and in arrhythmia recognition. The QRS detection methods have been studied for more than thirty years. The most desired features of QRS detection methods meet the following requirements: accuracy, repeatability, robustness to various types of noise (e.g. outliers, etc.) and the ability to work with different ECG signal processing applications [5]. The main challenges arise mainly from the great variety of complex QRS waveforms (like negative QRS and low-amplitude QRS,

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etc.), abnormalities, low signal-to-noise ratio (SNR) and artefacts accompanying ECG signals. There is a wide selection of methods for detecting QRS complexes that use different digital signal processing methods. One of the most commonly used is the Pan & Tompkins (PT) method [6] which has become an unofficial benchmark for QRS detector performance [4]. Other methods use the mathematical morphology [7], the banks of digital filters [8], the wavelet transform [9], a time dependent entropy [10], an empirical mode decomposition [11], knowledge based method [12], Hilbert transform [13] to name just a few of the most famous or that ensure low computing effort [14]. The motivation for this work is to increase the accuracy of QRS complex detector in ECG signals with arrhythmia. The proposed method is based on the threshold detection applied on the detection function waveform. The amplitude threshold value is determined by using a fuzzy c-median clustering method.

2

Overview of Algorithm

A schematic representation of the intermediate steps for finding the R-peaks in ECG signal is drawn in Fig. 2. In general, the overall detection process can be divided into a following stages: (1) preprocessing, (2) the amplitude threshold determination with the fuzzy c-median clustering method, (3) peaks detection and (4) QRS localization. The detection function is created by typical way, e.g. band-pass filtering, baseline drift removing, non-linear operation, differentiation and smoothing of the obtained waveform [15].

Fig. 2. Block diagram presenting main stages of QRS detector.

Fuzzy-Based Algorithm for QRS Detection

2.1

205

The Preprocessing Stage

Preprocessing is basic and necessary element of this algorithm, and this stage is responsible for determination of the smoothed detection function waveform. At first the DC-component is removed as well as the base-line wandering. The low frequency wandering of base-line is removed with application of the median filter of the length equalled f s/2 [4]. In order to suppress any kind of impulsive noise or outliers the cascaded OWA filter is applied with short length of the moving windows and with the Gaussian weighted function [16]. The pass-band filtering is realized with zero-phase forward and reverse digital filtering in the frequency band 0.7–20 Hz and the filter’s coefficients are determined with the Kaiser window of the 1.25 · f s length. The obtained waveform is denoted as x0 (n). The next step is differentiation and squaring. The first derivative is determined with the 2-point central difference x0 (n) =

1 (x0 (n + 1) − x0 (n − 1)). 2

(1)

And then the each sample of x0 (n) is subjected to a squaring operation as x1 (n) = [x0 (n)]2 .

(2)

The last stage is the smoothing realized with the moving average (MA) filter as x(n) =

N1 1  x1 (n − k), N1

(3)

k=1

where N1 = fs ×QRSwidth  and the QRS duration is assumed as 150 ms in this study. The correct width of the MA filter is significant for the proper R-peak detection process. Too narrow selection can cause that numerous unnecessary peaks arise in the QRS complex. If the filter width is too wide, it can lead to incorrect detection of R-peaks because there can occur a T-wave in the window [14]. This completes formation of the detection function waveform. In the case of the lack of disturbances, the detection function waveform includes characteristic peaks that correspond the QRS complexes locations in ECG signal, and in other places the waveform of the detection function is approximately zero. The next stage is the amplitude threshold value estimation. However, if the signal-to-noise ratio is low, then the detection function waveform may contain additional peaks that make it difficult to detect those that are expected (from QRS complexes). Therefore, the value of the amplitude threshold should be selected so that only those with high amplitude values can be recognized and those with low amplitude values should be ignored in the process of locating peaks [18]. The process of peaks locating of the detection function waveform is performed applying its first derivative. Moreover, the first derivative signal is then smoothed by the MA filter of the length 0.1·f s and the events of zero-crossings are detected. If the amplitude of the selected sample of the detection function

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Fig. 3. The ECG (record 118 from MIT-BIH Arrhythmia Database, lower plot) and the detection function waveform (upper plot).

waveform is greater than the threshold Ath then the QRS complex in the ECG signal is identified [18]. The appropriate value of Ath is therefore the key to the correct location identification of QRS complexes. Careful analysis of the shape of the detection function waveform allow to identify several different levels of samples amplitudes with the lowest value corresponding to distortions. The amplitude levels of the detection function waveform samples vary depending on the ECG signal, and as a result, the use of a fixed amplitude threshold Ath does not provide sufficient accuracy for automatic recognition. Hence, the identification of groups of signal’s samples can be accomplished by applying fuzzy clustering method [17,18]. 2.2

Estimation of the Amplitude Threshold with Fuzzy c-Median Clustering Method

Data clustering consists of a partition of a set of N elements (objects) into c groups (subsets) of similar objects. Our objective is to find a subset of the detection function samples with similar values. The similarity criterion is usually defined on the basis of comparison of the selected object properties, represented by the so-called feature vector ∀ x (n) ∈ Rp . 1≤n≤N

The class of clustering methods is defined as a challenge to minimize a certain criterion function (a scalar index) representing the quality of the partitioning of elements of subsets. Among them, the subclass of algorithms based on the idea of fuzzy sets [19] can be distinguished. Fuzzy clustering allows a partial membership of objects into groups and the degree of membership of the n-th object into the i-th group is described as the element uin ∈ [0, 1] of a partition matrix U ∈ Rc×N . Zero value of uin indicates that the object x (n) is not a

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207

member of the i-th group, while uin = 1 denotes the full membership. Each group is represented by the so-called prototype ∀ v (i) ∈ Rp . 1≤i≤c

To find the amplitude of the detection function waveform that corresponds to distortions and thereby to estimate the amplitude threshold for the correct identification of QRS complex location we applied the robust fuzzy c-median clustering (FCMED) [17,18,20,21]. In the FCMED method the group prototypes v (i) are determined as fuzzy (weighted) medians. The median weights are m defined as the m-th power of membership values (uin ) , where m ∈ (1, +∞). Fuzzy medians ensure the robustness to outliers. The classic approach for fuzzy median calculating requires sorting the entire set of objects. In this work, for shortening the computational time we applied the bisection method to estimate the fuzzy medians [21]. The partition is determined as a result of alternating calculations of prototypes and the partition matrix (Picard algorithm). The degree of membership of the object into the particular group is a function of the distance of the feature vector from the group prototype. The closer object x (n) is to the prototype v (i), the higher is its membership uin to the i-th group. In the proposed approach, the feature vectors are directly the values of detection function samples, i.e. p = 1, hence 1

∀

∀

|v (i) − x (n)| 1−m

1≤i≤c 1≤n≤N

uin = c

j=1

1

|v (j) − x (n)| 1−m

.

(4)

The algorithm starts with an initial partition matrix U(0) obtained from clas(0) sic fuzzy c-mean algorithm (FCM). On the  basis of U , the group prototypes  (0)

(0)

(0)

are calculated V(0) = v1 , v2 , · · · , vc

as fuzzy medians. The new location

of prototypes provides the new degrees of membership U(1) , calculated on the basis of (4). The process is repeated until the maximum number of iterations (tmax ) is reached, or if the change of the scalar index J=

c N  

um in |x (n) − v (i)| ,

(5)

n=1 i=1

in the two subsequent iterations is less than an assumed value ε, i.e. |J (t + 1) − Jr (t)| < ε, where t is the iteration index. In the proposed solution ε = 10−5 was assumed. Since the R-peak of ECG signal must correspond to maximum of the detection function waveform, only the K < N local maxima of the detection function are clustered (6) x (n − 1) ≤ xl (k) = x (n) ≤ x (n + 1) , where xl (k) is the k-th local maximum of the detection function waveform. This approach reduces the computational time of the amplitude threshold estimation. In this approach, a comprehensive analysis of the detection function waveform allowed us to distinguish three different levels of samples amplitudes and for that

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reason the number of groups c for the FCMED is set to 3. Because the values of the detection function waveform related to noise are the smallest thus, the prototype representing a group of samples from noise is determined as v (η) = min (v (1) , v (2) , v (3)) .

(7)

The amplitude threshold is set to exceed the maximum value of the detection function samples, that are identified as originating from noise. However, the sample is considered as representing the noise component only if its membership degree to the η-th group is higher than specified value δ. Consequently, the amplitude threshold of the detection function is defined as   (8) Ath = max xl (n) |uηn >δ . 1≤n≤K

If there are no samples of the detection function that are characterized by the high membership degree to the group η, i.e. ∀1≤n≤K uηn < δ, then Ath is calculated using scaled Uη = [sη1 , sη2 , · · · , sηK ], where ∀

1≤n≤K

sηn =

uηn . max (uηn )

(9)

1≤n≤K

3 3.1

Numerical Experiment and Results Data

The necessity of QRS complex accurate detection is one of the challenges facing the ECG signal processing algorithms. This is the reason why an ECG signal should be used for testing purposes in which the localization of QRS complexes are known and annotated. Three annotated ECG databases are used here: MITBIH Arrhythmia Database (MITDB) [22], MIT-BIH Noise Stress Test Database (NSTDB) [22], Fantasia [23] and QT database [24]. The use of well-annotated and validated databases provides the same conditions of numerical experiments and comparable results in terms of accuracy. These databases contain a large variety of selected ECG signals that allow to provide various tests of the proposed algorithm under different conditions, from clear ECG signal to others with lots of artefacts. A brief summary of the used databases is presented in the Table 1 [4]. The MIT-BIH Arrhythmia Database (MITAD) contains 48 half-hour excerpts of two-channel ambulatory ECG recordings. The recordings were sampled at 360 samples per second per channel with 11-bit resolution over a 10 mV range and have a diagnostic bandwidth of 0.1–100 Hz. Two or more cardiologists independently annotated each record; disagreements were resolved to obtain the computer-readable reference annotations for each beat (approximately 110,000 annotations in all) included with the database [22,25,26]. Each recording comprises two ECG leads, one lead is modified-lead II (MLII) and other lead is mainly lead V1, sometimes V2, V4 or V5 [27]. The standard of ANSI/AAMI/ISO

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Table 1. Summary of used ECG signal databases Record

Number QRS Duration (minutes)

Per record

Database (fs )

Type

Amount

Mean

SD

Total

Mean

SD

MITDB (360 Hz)

Clinical

48

30.09

0

109494

2281.1

451.96

NSTDB (360 Hz)

Clinical+noise

12

30.09

0

25590

2132.5

151.97

Fantasia (250 Hz)

Clinical

39

119.73

6.77

278440

7139.49

1070.08

QT (250 Hz)

Clinical

82

15

0

87919

1072.18

274.07

EC57:1998/(R)2008 claims that the QRS detection algorithm has to supply the reporting of statistics from the MIT-BIH Arrhythmia Database [28]. The MIT-BIH Noise Stress Test Database (NSTDB) includes 12 half-hour ECG recordings and 3 half-hour recordings of noise typical in ambulatory ECG recordings [22].The ECG recordings were created using two clean recordings (118 and 119) from the MIT-BIH Arrhythmia Database, to which calibrated amounts of noise from record ’motion artifact’ were added. Since the original ECG recordings are undisturbed, the correct beat annotations are known even when the noise makes the recordings visually unreadable. The reference annotations for these records are simply copies of those for the original clean ECGs [29]. Fantasia database includes 2 groups having rigorously-screened healthy subjects of twenty young (21–34 years old) and twenty elderly (68–85 years old) [23]. However, one record in this work was excluded (record f2y02 ) due to the corruption. QT database consists of ECG recordings selected to show a wide spectrum of QRS and ST-T morphology for challenging QT detection algorithms with real world variability [14,24]. This database contains 105 15-min of two-channel ECG recordings and since 23 of 24 sudden death ECG recordings did not have annotations files, they were not considered in numerical experiments [14]. For that reason the performance of the proposed method was obtained using 82 of 105 ECG files from QT database. The detection accuracy can be assessed using numbers of: • true-positive detections (TP – specifies the number of correctly identified peaks), • false positive detections (FP – specifies the number of incorrectly detected peaks), • false negative detections (FN – specifies the number of undetected peaks). Consequently, the detection efficacy can be evaluated on the basis of the following performance measures: • sensitivity SEN = TP/(TP+FN)·100%, • positive predictivity P+ = TP/(TP+FP)·100%,

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• F-measure F = (2 · SEN · P+ )/(SEN + P+ ). SEN value represents the algorithm’s detection capability of real heart beats. P+ value represents the algorithm’s distinguishing capability between true and false beats. The F-measure combines the information about the detection SEN and P+ , being the basic measure of the detection efficacy. 3.2

Results

The performance results obtained from the numerical experiment of the MITBIH Arrhythmia Database based on the proposed method are given in Table 2. As Table 2 shows, the overall SEN = 99.81 %, P+ = 99.69 % and F = 99.75 %. For 19 cases of detection the SEN = 100 % and for 31 cases the P+ = 100%. The worst results of the proposed method are obtained for record 207 that contains 472 annotated episodes of a ventricular flutter wave. The second most difficult case for the proposed method arises in the record 203. In other cases, the number of errors does not differ significantly from other QRS detection methods. The proposed method is compared with the reference QRS detection methods in Table 3. In the case of MIT-BIH AD the proposed method, comparing to the reference methods, leads to similar results in the detection of QRS complexes. The performance results obtained from the numerical experiment of the MITBIH Noise Stress Test Database based on the proposed method are given in Table 4. The method described in this paper detected 24530 TP, 1794 FP and 1060 FN beats of total 25590 beats taken from the MIT-BIH NSTDB, resulting in overall QRS detection SEN= 95.86%, P+ = 93.18% and F= 94.50% as depicted in Table 4. The comparison results of the performance of the proposed method with the reference algorithms are presented in Table 5. It is observed that obtained results are pretty good and are not worse than the reference methods. The performance results obtained from the numerical experiment of the Fantasia Database based on the proposed method are given in Table 6. The method described in this paper detected 278501 TP, 423 FP and 459 FN beats of total 278960 beats taken from the Fantasia Database, resulting in overall QRS detection SEN = 99.83%, P+ = 99.85% and F = 99.84% as depicted in Table 6. These results are slightly worse than the best reference methods but better than the Pan and Tompkins algorithm [6]. The performance results obtained from the numerical experiment of the QT Database (82 records from the first channel) based on the proposed method are given in Table 7. The results are worse compared to the reference methods except for the Pan and Tompkins method [6]. The most problems with detection were encountered in the case of the sel231 record, which produced the result of 291 FN. All numerical experiments were conducted in Matlab 2016b in Windows 10 of computer equipped with Intel(R) Xeon(R) CPU @ 3.50 GHz with 8 GB RAM.

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Table 2. QRS detection performance for the MIT-BIH Arrhythmia Database (parameter δ = 0.7, m = 2.5) Number

Record

Channel

TB

TP

FP

FN

SEN (%)

P+ (%)

F (%)

1

100

1

2273

2272

0

1

99.96

100.00

99.98

2

101

1

1865

1864

1

1

99.95

99.95

99.95

3

102

1

2187

2187

0

0

100.00

100.00

100.00

4

103

1

2084

2084

0

0

100.00

100.00

100.00

5

104

1

2229

2228

4

1

99.96

99.82

99.89

6

105

1

2572

2558

18

14

99.46

99.30

99.38

7

106

1

2027

2026

1

1

99.95

99.95

99.95

8

107

1

2137

2135

0

2

99.91

100.00

99.95

9

108

1

1763

1759

2

4

99.77

99.89

99.83

10

109

1

2532

2531

0

1

99.96

100.00

99.98

11

111

2

2124

2122

6

2

99.91

99.72

99.81

12

112

1

2539

2539

0

0

100.00

100.00

100.00

13

113

2

1795

1794

0

1

99.94

100.00

99.97

14

114

2

1879

1878

0

1

99.95

100.00

99.97

15

115

1

1953

1952

0

1

99.95

100.00

99.97

16

116

2

2412

2411

1

1

99.96

99.96

99.96

17

117

1

1535

1535

0

0

100.00

100.00

100.00

18

118

2

2278

2278

1

0

100.00

99.96

99.98

19

119

2

1987

1987

0

0

100.00

100.00

100.00

20

121

1

1863

1862

0

1

99.95

100.00

99.97

21

122

1

2476

2476

0

0

100.00

100.00

100.00

22

123

1

1518

1518

0

0

100.00

100.00

100.00

23

124

1

1619

1619

0

0

100.00

100.00

100.00

24

200

1

2601

2598

0

3

99.88

100.00

99.94

25

201

1

1963

1962

0

1

99.95

100.00

99.97

26

202

2

2136

2136

4

0

100.00

99.81

99.91

27

203

2

2980

2903

60

77

97.42

97.98

97.69

28

205

1

2656

2640

0

16

99.40

100.00

99.70

29

207

2

1860

1855

233

5

99.73

88.84

93.97

30

208

1

2955

2935

2

20

99.32

99.93

99.63

31

209

1

3005

3005

0

0

100.00

100.00

100.00

32

210

1

2650

2625

1

25

99.06

99.96

99.51

33

212

1

2748

2748

0

0

100.00

100.00

100.00

34

213

1

3251

3248

0

3

99.91

100.00

99.95

35

214

1

2262

2258

0

4

99.82

100.00

99.91

36

215

1

3363

3357

0

6

99.82

100.00

99.91

37

217

1

2208

2205

0

3

99.86

100.00

99.93

38

219

1

2154

2154

0

0

100.00

100.00

100.00

39

220

1

2048

2047

0

1

99.95

100.00

99.98

40

221

1

2427

2426

1

1

99.96

99.96

99.96

41

222

2

2483

2481

1

2

99.92

99.96

99.94

42

223

1

2605

2605

0

0

100.00

100.00

100.00

43

228

2

2053

2053

1

0

100.00

99.95

99.98

44

230

2

2256

2256

0

0

100.00

100.00

100.00

45

231

1

1571

1571

0

0

100.00

100.00

100.00

46

232

1

1780

1780

2

0

100.00

99.89

99.94

47

233

1

3079

3074

0

5

99.84

100.00

99.92

48

234

1

2753

2753

0

0

100.00

100.00

100.00

Overall

109494

109290

339

204

99.81

99.69

99.75

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Table 3. Comparison of the performance of the proposed method with other algorithms for the MIT-BIH AD. FP FN SEN (%) P+ (%) F (%)

QRS detector

Cases TB

This work (δ = 0.7)

48

109494 339 204 99.81

99.69

99.75

Yakut et al. [14]

48

109494 182 184 99.83

99.83

99.83

Elgendi et al. [12]

48

109985 124 247 99.78

99.87

99.82

Hamilton and Tompkins [30] 48

109267 248 340 99.69

99.77

99.73

Pan and Tompkins [6]

109809 507 277 99.75

99.54

99.64

48

Table 4. QRS performance results for the MIT-BIH Noise Stress Test Database (channel 2, m = 2.5). SNR [dB] δ

File

TP

FP

FN

SEN (%) P+ (%) F (%)

24

0.80 118e24

2278

0

0

100.00

100.00

100.00

18

0.60 118e18

2278

0

0

100.00

100.00

100.00

12

0.40 118e12

2278

1

0

100.00

99.96

99.98

6

0.50 118e06

2276

55

2

99.91

97.64

98.76

0

0.65 118e00

2157

262

121

94.69

89.17

91.85

−6

0.98 118e 6

1722

526

556

75.59

76.60

76.09

24

0.80 119e24

1987

0

0

100.00

100.00

100.00

18

0.60 119e18

1987

0

0

100.00

100.00

100.00

12

0.40 119e12

1986

0

1

99.95

100.00

99.97

6

0.50 119e06

1985

53

2

99.90

97.40

98.63

0

0.65 119e00

1917

329

70

96.48

85.35

90.57

−6

0.98 119e 6

1679

568

308

84.50

74.72

79.31

93.18

94.50

Overall 24530 1794 1060 95.86

Table 5. Comparison of the performance of the proposed method with the reference algorithms for the MIT-BIH Noise Stress Test Database. QRS detector

TB

SEN (%) P+ (%) F (%)

This work (evolving δ, m = 2.5) 25590 95.86

93.18

94.50

Yakut et al. [14]

25590 93.62

94.52

94.07

Elgendi et al. [12]

26370 95.39

90.25

92.75

Pan and Tompkins [6]

26370 74.46

93.67

82.97

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213

Table 6. Comparison of the performance of the proposed method with other algorithms for the Fantasia Database (N/R - Not Reported). FP

FN

SEN (%) P+ (%) F (%)

39

278960 423

459

99.83

99.85

99.84

Yakut et al. [14]

40

283747 148

152

99.94

99.98

99.96

Elgendi et al. [12]

40

278996 315

50

99.98

99.97

99.98

Pan and Tompkins [6]

40

278996 N/R N/R 89.16

99.89

94.22

QRS detector

Cases TB

This work (δ = 0.6, m = 2.5)

Table 7. Comparison of the performance of the proposed method with other algorithms for the QT Database. QRS detector

4

TB

SEN (%) P+ (%) F (%)

This work (channel 1, δ = 0.75, m = 2.5)

87919 98.86

99.90

99.38

Yakut et al. [14]

86741 99.89

99.96

99.92

Elgendi et al. [12]

111201 99.99

99.67

99.83

Pan and Tompkins [6]

111201 97.99

99.05

98.52

Conclusion

In this work a fuzzy-based algorithm of QRS complex detection is proposed. The proposed method belongs to the family of QRS detectors based on the threshold level. In the first stage the detection function waveform is created. Therefore, the fuzzy c-median clustering method is used to group values of samples of the detection function waveform into three clusters. The highest value from the smallest amplitude cluster is determined as a value of the amplitude threshold. However this value is also controlled by the tunning parameter δ ∈ [0, 1]. Exceeding the determined amplitude threshold by the detection function samples causes the start of searching for the maximum value of the detection function wave, the location which corresponds to the occurrence of the QRS unit in the ECG signal. An important advantage of the presented algorithm is the fact that it works immediately on all samples of the whole signal. The proposed method is tested on standard databases such as MIT-BIH Arrhythmia, Fantasia, MIT-BIH NoiseStress Test and QT. Satisfying good results are obtained from the novel proposed method and the performance indices are higher or not worse to similar algorithms in the scientific literature. Acknowledgements. This research is financed from the statutory activities of the Institute of Electronics at Faculty of Automatic Control, Electronics and Computer Science in the Silesian University of Technology.

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19. Zadeh, L.A.: Fuzzy sets. Inform. Control. 8, 338–353 (1965) 20. Kersten, P.R.: Fuzzy order statistics and their application to fuzzy clustering. IEEE Trans. Fuzzy Syst. 7, 708–712 (1999) 21. Kersten P.R., Implementation issues in the fuzzy c-medians clustering algorithm. In: Proceedings of the Sixth IEEE International Conference on Fuzzy Systems, vol. 2, pp. 957–962 (1997) 22. Goldberger, A.L., Amaral, L.A.N., Glass, L., et al.: PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101, e215–e220 (2000) 23. Iyengar, N., Peng, C.-K., Morin, R., Goldberger, A.L., Lipsitz, L.A.: Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am. J. Physiol. 271, 1078–1084 (1996) 24. Laguna, P., Mark, R.G., Goldberger, A.L., Moody, G.B.: A database for evaluation of algorithms for measurement of QT and other waveform intervals in the ECG. Comput. Cardiol. 24, 673–676 (1997) ´ Novel real-time 25. Guti´errez-Rivas, R., Garc´ıa, J.J., Marnane, W.P., Hern´ andez, A.: low-complexity QRS complex detector based on adaptive thresholding. IEEE Sens. J. 15, 6036–6043 (2015). https://doi.org/10.1109/JSEN.2015.2450773 26. Moody, G.B., Mark, R.G.: The impact of the MIT-BIH arrhythmia database. IEEE Eng. Med. Biol. Mag 20, 45–50 (2001) 27. Xiang, Y., Zhitao, L., Jianyi, M.: Automatic QRS complex detection using twolevel convolutional neural network. BioMed. Eng. OnLine 17, 13 (2018). https:// doi.org/10.1186/s12938-018-0441-4 28. American National Standard ANSI/AAMI EC 57:2012, Testing and Reporting Performance Results of Cardiac Rhythm and ST-Segment Measurement Algorithms. https://www.amazon.com/AAMI-EC57-Performance-MeasurementAlgorithms/dp/1570204780. 25 Nov 2019 29. Moody, G.B., et al.: A noise stress test for arrhythmia detectors. Comput. Cardiol. 11, 381–384 (1984) 30. Hamilton, P.S., Tompkins, W.J.: Quantitative investigation of QRS detection rules using the MIT-BIH arrhythmia database. IEEE Trans. Biomed. Eng. 12, 1157– 1165 (1986)

Influence of the Indoor Hockey “Push & Flick” Methodology on the Ball Speed During the Penalty Corner Shooting Antonio Antonov1 , Dafina Zoteva2,3 , and Olympia Roeva2(B) 1 Department Football and TennisSector Hockey, National Sports Academy “Vassil Levski”,

Sofia, Bulgaria 2 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences,

Acad. G. Bonchev Street, bl. 105, 1113 Sofia, Bulgaria [email protected] 3 Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 Blvd “James Bourchier”, 1164 Sofia, Bulgaria [email protected]

Abstract. Indoor hockey is an official non-Olympic discipline, regulated by International Hockey Federation (FIH), practiced in a hall with a handball size pitch. The main objective of the game is a victory achieved by marking more goals than the opponent. The format of the game was created in the 1950s and 1960s. Later, in 1966, the first Indoor hockey rules were published. In 1968, the World Headquarters officially recognized the discipline as an integral part of hockey. The penalty corner is one of the most important game situations in hockey (both outdoor and indoor field hockey) with 40% of all goals resulting from this tactical situation. This number may reach 46% or even 68% . The aim of this paper is to study the influence of the indoor hockey “Push & Flick” methodology on the ball speed improvement during the penalty corner shooting in the potentially effective goal zones. Using variation analysis and InterCriteria Analysis the research team has sought to establish values and possible relations and dependencies between indicators reflecting the ball speed of zone shooting. Four elite indoor hockey players from the team of the National Sports Academy in Bulgaria, participants in the European Indoor Hockey Clubs Challenge, have been involved in the experiment. According to the requirements of the experimental “Push & Flick” methodology, the duration of the specialized training has been set to 16 weeks. Each player has performed 4,800 shootings, or approximately 300 shootings each week. Tests have been carried out at the beginning (the first week) and at the end (the sixteenth week) of the experiment in order to determine the speed of the ball during the shooting – push/flick from a penalty corner spot (9 m, central from the goal line). The speed of the ball has been measured with a sports radar Ra-Vid Pro Sport™ (Accuracy: ±0.1 km/h, Speed range: 1–480 km/h, Stopwatch within 1/100 s, 10 m sec acquisition time, 12-degree radar beam, 1200 to 38.4 K baud, Available in mph or km/h, Maximum Range, Sports: 400–500 ft., Autos: 1.75 miles) located just behind the net and the corresponding shooting areas. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 216–229, 2021. https://doi.org/10.1007/978-3-030-77716-6_20

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This report will demonstrate the effectiveness of the specialized methodologies related to the preparation of penalty corners “specialists”. In addition, InterCriteria Analysis applied for processing the data reveals important dependencies related to the refinement of the technique of pushing and flicking. Keywords: Flick · Indoor hockey · Penalty corner · Speed · Shooting · InterCriteria analysis

1 Introduction The penalty corner is one of the most important game situations in field hockey, both outdoor and indoor. A number of studies show that more than 40% of the scored goals are a result of this tactical situation [1, 3, 4, 15, 17, 21, 22, 24, 25]. The drag-flick technique is between 1.4 and 2.7 times more efficient than hitting or push-shooting the ball towards the goal, when a penalty corner is performed. The penalty corner depends upon three different technical skills: (1) push – pass from back line, (2) stop and (3) drag flick or hit shooting for outdoor hockey and flick or push shooting for indoor hockey. Four key factors influence the effectiveness of the penalty corner – the speed of execution, the accuracy of shooting, the speed of shooting and the delusion/surprise of shooting. While the first factor is functionally dependent on the synchrony between the executions of the three basic technical elements, the other three factors depend entirely on the mastery of the shooter – the performer of the push, flick, drag flick or hit. The high speed of the ball when shooting a penalty corner is of the utmost importance, because the slow movements reduce the possibility of scoring – the effect of well-directed and/or covert shooting is lost [3, 4]. In relation to technical training, authors in [14] have applied different approaches to the push and the flick in indoor hockey twice per week, during six weeks. They have obtained very heterogeneous results [17, 22]. Only a few studies have been conducted up to now regarding the training of the drag-flick skill in field hockey. Therefore, following the research [22], the aim of this study is to develop and implement a training method for improving the drag-flick skill of young top-class field hockey players. In this regard, any study on the impact of the “Push & Flick” methodology on the performance of indoor hockey penalty corner (the ball shooting speed in effective areas (see Fig. 1), as well as the establishment of significant dependencies is of primary importance for the training process and competitive activity of high performance indoor hockey players. The aim of this paper is to study the influence of the experimental “Push & Flick” methodology on the dynamics in the development of the ball shooting speed when performing penalty corner in the potentially effective goal zones. Through the application of variation analysis and InterCriteria Analysis, the research team has sought to establish values, relations and dependencies between indicators reflecting the ball speed of zone shooting. The approach, named InterCriteria Analysis (ICrA) [10], based on the apparatus of the index matrices (IM) [5–7], and the intuitionistic fuzzy sets (IFS) [8] has been successfully applied to decision making in different areas of knowledge [12, 13, 18, 20,

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23]. ICrA has been applied here in order to determine the possible relationships and dependences between the indicators, characterizing the ball speed during a shooting.

2 Methodology In the course of sixteen weeks, an experimental methodology, “Push & Flick”, has been implemented in order to train “narrow specialists” performing a penalty corner indoors. One of the main tasks of the methodology is to improve the speed of shooting. The subject of this research are four competitors (VB, PM, SI and TI) from the National Sports Academy hockey club, whose training involves practicing “Push & Flick” techniques. The experimental methodology includes 32 specialized training sessions, twice a week, with a duration of 90 to 120 min each. According to the set up plan, each competitor’s training session includes direct execution of 150 shootings from a 9-m distance in three predetermined zones with the outer dimensions of the squares 66 × 66 cm. During each training session, the competitors work on improving their shooting skills in zones either to the left or to the right, alternating consequently (Fig. 1). Throughout the experiment, each competitor has performed 4,800 shootings – an average of 300 per week – 2,400 to the left and 2,400 to the right zones, respectively.

3

4

2

5

1

6 Fig. 1. Model of a training hockey goal (300 × 200 cm)

The accuracy of the shooting and the speed of the ball during a penalty corner execution in training conditions have been determined at the beginning and the end of the experiment through pedagogical observation and a radar. The level of accuracy has been established by a technique involving 10 consecutive shootings into a spherical target with a diameter F – 40 cm, located symmetrically in each zone (Fig. 1). A shot in the target is awarded 2 points, while a hit outside the target, but within the zone, including the beam – 1 point. The white outlines of the squares of each zone are part of the zones.

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3 InterCriteria Analysis Background Following [8], an Intuitionistic Fuzzy Pair (IFP) [11] as an estimation of the degrees of “agreement” and “disagreement” between two criteria applied to different objects is obtained. An IFP is an ordered pair of real non-negative numbers a, b such that: a + b ≤ 1. Consider an IM [5–7] whose index sets consist of the criteria (for rows) and objects (for columns). The elements of this IM are further assumed to be real numbers. An IM with index sets consisting of the criteria (for rows and for columns) with elements IFPs corresponding to the degrees of “agreement” and “disagreement” between the respective criteria is then constructed. Let O denotes the set of all objects O1 , O2 , . . . , On being evaluated, and C(O) be the set of values assigned to the objects by a given criterion C, i.e., def

def

O = {O1 , O2 , . . . , On }; C(O) = {C(O1 ), C(O2 ), . . . , C(On )}. def

Then, let C ∗ (O) = {x, y| x = y & x, y ∈ C(O) × C(O)}. In order to find the “agreement” between two criteria, the vector of all internal comparisons of each criteria, which fulfil exactly one of the following three relations: R, R and  R, is constructed. In other words, it is required that for a fixed criterion C and any ordered pair x, y ∈ C ∗ (O) it is true: x, y ∈ R ⇔ y, x ∈ R,

(1)

x, y ∈  R ⇔ x, y ∈ / (R ∪ R),

(2)

R = C ∗ (O). R ∪ R ∪

(3)

Only a subset of C(O) × C(O) needs to be considered for the effective calculation of the vector of internal comparisons (further denoted by V (C)) since from Eqs. (1)–(3) it follows that if the relation between x and y is known, then so is the relation between y and x. Thus, only the lexicographically ordered pairs x, y are of interest. Denote for brevity Ci,j = C(Oi ), C(Oj ). Then, for a fixed criterion C, the vector with n(n − 1)/2 elements is obtained: V (C) = {C1,2 , C1,3 , . . . , C1,n , C2,3 , C2,4 , . . . , C2,n , C3,4 , . . . , C3,n , . . . , Cn−1,n }. 

Let V (C) be replaced by V (C), where for the k th component (1 ≤ k ≤ n(n − 1)/2): ⎧ ⎨ 1, iff Vk (C) ∈ R Vk (C) = −1, iff Vk (C) ∈ R . ⎩ 0, otherwise 

When comparing two criteria, the degree of “agreement” μC,C is determined as the number of matching components of the respective vectors, divided by the length of the vector for normalization purposes. The degree of “disagreement” νC,C is the

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number of components of opposing signs in the two vectors, again normalized by the length. Ordered pair μC,C , νC,C  is an IFP. In the most of the obtained pairs, the sum μC,C + νC,C is equal to 1. However, there may be some pairs, for which this sum is less than 1. The difference πC,C = 1 − μC,C − νC,C is considered as a degree of “uncertainty”.

4 Results and Discussion from the Study on the Speed of the Ball When Executing a Penalty Corner Using Flick 4.1 Variation Analysis of the Speed of the Ball When Executing a Penalty Corner in Goal-Scoring Areas The results of the variation analysis of the indicators characterizing the speed of the ball in case of shooting in the goal-scoring areas (zones), defined by the methodology, for each competitor, as well as the summarized data are presented in Tables 1 and 2. The analysis of the individual results of the competitors shows that at the beginning of the experiment the minimum average speed values of flicking into the zones range from 60 to 68 km/h (Difference R – 8 km/h), while the maximum average speed values are from 67 to 75 km/h (R – 8 km/h). At the end of the experiment, the minimum average speed values are between 67 and 72 km/h (R – 5 km), while the maximum values are from 70 to 79 km/h (R – 9 km/h). The values of the established individual average speed, based on shooting in all the six zones, at the beginning of the experiment range from 63.7 km/h (PM) to 70.8 km/h (SI). At the end of the experiment these values are from 68 km/h (PM) to 74 km/h (SI). The difference between the values of the speed indicators in the first and the second testing shows that the methodology has a positive impact on the speed development, with the highest average increase rate being established for PM – 7%, with a difference of 4.4 km/h between the average speed values in the first and second testing, and the lowest for VB – 2.62%, with a difference of 1.8 km/h between the average speed values in the first and the second testing. The information reflecting the average speed of flicking in the most effective goalscoring areas in the first and the second testing of the competitors, the average speed by zones in the two tests, the difference between the indicators in the two tests, as well as the established percentage increase (%) are summarized in Table 2. It is evident that the competitors have achieved the highest average shooting speed in the left high zone – Zone 3, both during the first and the second testing (V 1 – 70.5 km/h and V 2 – 73.3 km/h). This fact has been also confirmed by the highest average speed of all competitors, when the total number of shootings is considered during the first and second testing, V avg is 71.9 km/h for Zone 3. The competitors have almost equal shooting speed values for the first and the sixth zone, with positions exchanged between the first and the second testing, but with absolutely equal average shooting speed values of V avg – 70 km/h for both tests. The increase in the values of the indicators between the first and the second testing shows that the methodology has a stronger influence on the development of the flicking speed in the right zones: Zone 5 (6.4%), Zone 6 (5.1%) and Zone 4 (5%). However,

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Table 1. Summarized results of variation analysis of speed indicators Competitors Test stage 1. PM

2. SI

3. TI

4. VB

Speed values by zones 3

Min Max R

4

5

Avg

1

2

6

Start

63.0

63.0 64.0 65.0 60.0 67.0 60.0 67.0 7.0 63.7

End

67.0

67.0 68.0 69.0 68.0 70.0 67.0 70.0 3.0 68.0

Difference

4.0

4.0

4.0

4.0

8.0

3.0 3.0

8.0 5.0

4.4

6.0

6.0 13.0

5.0 5.0

13.0 8.0

7.0

Increase, % 6.0

6.0

Start

72.0

71.0 75.0 68.0 69.0 70.0 68.0 75.0 7.0 70.8

End

73.0

73.0 79.0 72.0 73.0 76.0 72.0 79.0 7.0 74.3

Difference

1.0

2.0

4.0

4.0

4.0

6.0 1.0

6.0 5.0

3.5

Increase, % 1.4

2.8

5.3

5.9

5.5

6.6 1.4

6.6 5.2

4.5

Start

64.0 69.0 65.0 64.0 67.0 64.0 69.0 5.0 66.0

67.0

End

71.0

68.0 71.0 68.0 67.0 69.0 67.0 71.0 4.0 69.0

Difference

4.0

4.0

2.0

3.0

3.0

4.0 2.0

4.0 2.0

3.0

2.9

4.6

4.7

6.0 2.9

6.0 3.1

5.02

Increase, % 6.0

5.9

Start

72.0

72.0 74.0 68.0 68.0 69.0 68.0 74.0 6.0 70.5

End

75.0

72.0 75.0 70.0 70.0 72.0 70.0 75.0 5.0 72.3

Difference

3.0

0

1.0

2.0

2.0

3.0 0

3.0 3.0

1.80

Increase, % 4.2

0

1.4

2.9

2.9

4.3 0

4.3 4.3

2.62

V avg , Start/T1

68.5

67.5 70.5 66.5 65.3 68.3 65.3 70.5 5.2 67.9

V avg , End/T2

71.5

70

73.3 69.8 69.5 71.8 69.5 73.3 3.8 71.4

Difference, R (End/T2 – Start/T1)

3.0

2.5

2.8

3.3

4.2

3.5

2.5 4.2

1.7

3.50

Total increase in %

4.4

3.7

4.0

5.0

6.4

5.1

3.7 6.4

2.3

4.77

Table 2. Summarized results of the indicators reflecting the speed of the shooting by zones Zone

V 1 , km/h

V 2 , km/h

V avg , km/h

Difference, (R)

Increase, %

1 – low-left

68.5

71.5

70.0

3.0

4.4

2 – middle-left

67.5

70.0

68.8

2.5

3.7

3 – high-left

70.5

73.3

71.9

2.8

4.0

4 – high-right

66.5

69.8

68.1

3.3

5.0

5 – middle-right

65.3

69.5

67.4

4.2

6.4

6 – low-right

68.3

71.8

70.0

3.5

5.1

X average

67.8

71.0

69.4

3.2

4.8

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despite the high speed increase in Zone 5, where the highest values of the development indicators have been found (Difference R – 4.2 km, Increase – 6.4%), the values of the maximum speed remain at their lowest level, both during the first and the second testing (Table 2). A number of authors have performed biomechanical studies of the main technique for shooting when executing a penalty corner – drag flick. In the studied resources, however, there is not sufficiently reliable information corresponding to the speed characteristic of the main means of shooting indoors – “Push & Flick” [2–4, 19, 24]. The maximum speed of the ball in case of the drag flick technique being used both in playing and in laboratory conditions, has been estimated in the range of 110–130 km/h. In the research [17] authors have found that the speed of the ball, when the drag flick is performed, is functionally dependent on the acceleration of the stick at the end of the flick phase, when the ball separates from the stick – the average speed is 32–33 m/s (115–120 km/h). These high speed values of the ball, when the drag flick is performed, can be explained with the long uniformly accelerated motion of the stick and the ball of approximately 2.5–2.7 m during the two execution phases: drag and flick [17]. The dragging technique is not allowed when shooting indoors, which requires the use of classic flick in case of a penalty corner. The speed of the ball after the flick should be substantially lower than the drag flick, since the path and the contact time between the stick and the ball during the uniformly accelerating execution phase is much shorter. The effective development and application of specialized methodologies is impossible without any available information. Therefore, it is important for the speed of the ball during the execution of a penalty corner using flick to be studied. The analysis of the results of the first and the second testing, as well as the application of the ICrA allows revealing the influence of the experimental “Push & Flick” methodology on the development of the speed of the ball when executing a penalty corner, objectifying the speed and establishing important relations between the speed indicators when shooting in different zones. The established average speed of the ball when executing a penalty corner indoors using flick (ranging from 68 to 71 km/h (±2.3)) is more than 50 km/h lower than the highest recorded shooting values using drag flick (120–130 km/h). The methodology has shown a positive increase of the ball speed, when shooting in all of the 6 zones is considered, with an average difference R of 3.2 km/h and average increase of 4.8%. The methodology has a greater impact on the development of the speed of flicking in the right zones 5 (6.4%), 6 (5.1%) and 4 (5%) compared to the others. Despite this fact, however, the maximum speed in these zones remains lower than the shooting speed in the left zones, both during the first and the second testing (Table 2). In Zone 5, where the highest values of the development indicators (R – 4.2 km, Increase – 6.4%) have been found, the lowest maximum speed has been recorded both during the first (V 1 – 65,3 km/h) and the second testing (V 2 – 69.5 km/h during). Although present, these differences are considered minor and may be explained with some biomechanical factors which are not explored in this article.

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4.2 InterCriteria Analysis of the Speed of the Ball When Executing a Penalty Corner in Goal-Scoring Areas Two IMs have been constructed in order to perform the ICrA of the indicators characterizing the speed of shooting in the most effective goal-scoring areas (zones):

IM1 corresponds to the results of all tested athletes in the first testing, IM2 – in the second testing. A cross-platform software implementing ICrA, called ICrAData [16], has been used. The obtained results are presented in the next IM: First testing Degree of agreement μC,C and disagreement νC,C , based on IM1 .

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Second testing Degree of agreement μC,C and disagreement νC,C , based on IM2 .

The scale for defining the corresponding dissonance and consonance values, proposed in [9], has been used in the analysis of the results obtained here (see Table 3).

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Table 3. Definition of consonance and dissonance between each pair of criteria μC,C

Meaning

[0.00–0.05] Strong negative consonance, SNC (0.05–0.15] Negative consonance, NC (0.15–0.25] Weak negative consonance, WNC (0.25–0.33] Weak dissonance, WD (0.33–0.43] Dissonance, D (0.43–0.57] Strong dissonance, SD (0.57–0.67] Dissonance, D (0.67–0.75] Weak dissonance, WD (0.75–0.85] Weak positive consonance, WPC (0.85–0.95] Positive consonance, PC (0.95–1.00] Strong positive consonance, SPC

The analysis of the ICrA results, during the first testing of the athletes, shows the following correlations: • The greatest correlation (positive consonance) has been found between the achievements of the athletes in Zones 5 and 6: μ5,6 , ν5,6  = 0.91, 0.04, with degree of “uncertainty” π5,6 = 0.05. The observed degree of “uncertainty” is too small and does not affect the results. This means that those athletes who show good speed of the ball (or bad) shooting in Zone 5 in most cases show the same results in Zone 6. • The next two zones with correlation (weak positive consonance) between the recorded results are Zone 5 and Zone 3: μ5,3 , ν5,3  = 0.78, 0.22, with degree of “uncertainty” π5,3 = 0. • The results for the rest of zones show that there is no correlation. The obtained results for μC,C are in dissonance, i.e., the athletes do not show similar behaviour when performing the shooting in the rest of the zones. The highest positive consonance with a small degree of uncertainty between the results characterizing the speed of the ball when flicked in Zones 5 and 6 is due to the close biomechanical characteristics of the kinematic chain: body-stick-ball-target, the external and internal forces influencing the flicking in these zones. Increasing the speed parameters of the ball when flicked in Zone 6 will increase to a significant degree the speed of the ball also when shooting in Zone 5, and vice versa. The weak positive consonance between the speed of the ball when the flick is used in Zones 3 and 5 is due to our approved sports training methodology for improvement of the shooting of a penalty corner indoors, rather than any biomechanical factors. The most frequently targeted zones when executing a penalty corner indoors with the ball is passed to the left of the defence are Zone 3 (high left) or Zone 5 (middle right).

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Shooting in the high left Zone 3 is extremely effective in case of a slow attacking goalkeeper, especially when the goalie lies down before shooting. The semi-high shooting in the middle right Zone 5 is extremely effective as it is not guarded by the goalkeeper and is difficult to protect against the last left defender in the goal – responsible for the protection of Zones 4, 5 and 6. Shooting in the low right Zone 6 is ineffective as it is easily guarded by a defender with a horizontally positioned stick. The last left defender can easily react also to a shooting in the high right Zone 4. The most difficult and unexpected for the defence is the low shooting in the right Zone 5. Taking this into consideration, the weak positive consonance between the speed indicators characterizing the shootings in Zones 3 and 5 can be explained with the long training of the athletes to perform flick in these zones, rather than with any biomechanical factors. The absence of other significant positive dependences, as well as the above discussion, are a reason to believe that in order to improve the penalty corner execution, the focus should be on a particular aim (zone). Work on improving the speed of the shooting in the low Zones 1 and 6 cannot be relied upon to significantly improve the speed in the high Zones 3 and 4. Prolonged and focused work on improving the shooting speed in the low Zones 1 and 6 would logically affect to some extent the speed of the low-flying balls, which fall both in Zones 1 and 6 and in Zones 2 and 5. Prolonged work to improve shooting speed in high Zones 3 and 4 would logically affect to some extent the results in middle Zones 2 and 5, and vice versa. These assumptions are based on the similar biomechanical models of the flick execution techniques in the same direction, with small differences in shooting height. Based on the results during the second testing of the athletes the following correlations are found: • Again the greatest correlation (weak positive consonance) among all results, has been found between the achievements of the athletes in Zones 5 and 6: μ5,6 , ν5,6  = 0.80, 0.16, π5,6 = 0.04. The observed degree of “uncertainty” is too small and does not affect the results. Here, the correlation between the zones is weaker compared to the first testing. This mean that some of the athletes have improved their performance in one of the two zones – Zone 5 or Zone 6. • Weak positive consonance has been observed also between Zone 5 and 1, Zone 3 and 2, and Zone 5 and 4, respectively: μ5,1 , ν5,1  = 0.78, 0.20, π5,1 = 0; μ3,2 , ν3,2  = 0.78, 0.13, π3,2 = 0.07 (The observed degree of “uncertainty” is too small and does not affect the results.); μ5,4 , ν5,4  = 0.78, 0.20, π5,4 = 0. After training, the athletes who have shown good speed of shooting in Zone 5, show good speed of shooting in Zone 1 and in Zone 4. An improvement in the shooting has been observed in Zones 2 and 3, too. • After training the correlation between Zone 5 and Zone 3 has not been observed. This proves the assumption that this dependence at the beginning of the study is due to the sports training methodology applied to the subjects prior to the beginning of the experiment. This connection has disappeared when the experimental methodology has been applied because of proportionally distributed work between the zones. • The results for the rest of zones show that there is no correlation. The obtained results for μC,C are in dissonance. ICrA analysis shows that there is no correlation between the rest of the zones.

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The lack of dependencies, or the presence of low positive consonance between the indicators characterizing speed in the different zones, confirms the discussion from the beginning of the experiment – the work to improve the speed parameters of the shooting from a penalty corner should be specialized and focused on the specific zone. The weak positive consonance found at the end of the experiment between the indicators in Zones 2 and 3 and Zones 4 and 5 confirm that the continuous and focused work on improving the shooting speed in the high Zones 3 and 4 would also logically affect the results in the middle Zones 2 and 5 and vice versa. The weak positive consonance is based on similar biomechanical models of the flick techniques in the same direction, with small differences in the height of the shooting. The weak positive consonance between Zones 1 and 5 is somehow illogical and is due to random, unknown or unexplored reasons. In order to refine these results, a larger number of subjects should be explored.

5 Conclusion In conclusion, the estimated average speed of the ball during the first (67.8 km/h or 18.9 m/s) and the second testing (71 km/h or 19.7 m/s), the average positive difference of 3.2 km/h with average increase rate of 4.8% are a reason to believe that the experimental methodology has shown a positive impact on the speed of the ball when flick is used for a penalty corner shooting. The performed ICrA of the considered indicators during the first and the second testing, shows some correlation (positive consonance) of the athletes’ results, especially regarding the Zones 5 and 6. In the second testing the results show refinement of the pushing (flicking) technique in the low-left Zone 1 and high-right Zone 4. After training, the athletes manage to achieve higher ball speed not only shooting in Zone 5, but also shooting in Zone 1 and 4. The techniques also significantly affect the shooting speed in the medium-left and high-left zones (Zone 2 and Zone 3). The ICrA results are a reason to claim that if a higher efficiency of shooting indoors from a penalty corner is sought – focused on the speed of the ball, the methodology should be directed to improving the performance in a specific effective, pre-planned zone. The methodology should build up narrow specialists rather than complex competitors. It should also be pointed out that specialized performance in the high Zones 3 and 4 would positively increase also the shooting speed in the middle Zones 2 and 5, and vice versa. To a certain extent, this weak positive consonance should logically be observed also when shooting in the low and middle Zones 1 and 2, 6 and 5. Acknowledgements. The work presented here is partially supported by the Bulgarian National Scientific Fund under Grant KP-06-N22/1 “Theoretical Research and Applications of InterCriteria Analysis”.

References 1. Ansari, N., Bari, M.A., Ahmad, F., Hassain, I.: Three dimensional analysis of drag-flick in the Field Hockey of University Players. Adv. Phys. Theories Appl. 29, 87–93 (2014)

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2. Antonov, A., Zoteva, D., Roeva, O.: Influence of the “Push & Flick” methodology on the accuracy of the Indoor hockey penalty corner shooting. J. Appl. Sports Sci. 1, 64–76 (2020) 3. Antonov, A., Mindov, T., Igov, V.: Drag flick – phase structure of the technique. In: Third International Scientific Conference of the Department of Football and Tennis, National Sports Academy “Vassil Levski”, pp. 6–14. Avangard Prima, Sofia (2006) 4. Antonov, A., Mindov, T., Chavdarov, S.: Analysis of annual training of the field Hockey Penalty Corner Drag Flick Specialists. In: Third International Scientific Conference of the Department of Football and Tennis, pp. 112–121. Avangard Prima, Sofia (2006) 5. Atanassov, K.: Generalized index matrices. Comptes rendus de l’Academie Bulgare des Sci. 40(11), 15–18 (1987) 6. Atanassov, K.: On index matrices, part 1: standard cases. Adv. Stud. Contemp. Math. 20(2), 291–302 (2010) 7. Atanssov, K. (ed.) Index Matrices: Towards an Augmented Matrix Calculus. SCI 573. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10945-9 8. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Heidelberg (2012). https://doi. org/10.1007/978-3-642-29127-2 9. Atanassov, K., Atanassova, V., Gluhchev, G.: InterCriteria analysis: ideas and problems. Notes Intuitionistic Fuzzy Sets 21(1), 81–88 (2015) 10. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues IFSs GNs 11, 1–8 (2014) 11. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes Intuitionistic Fuzzy Sets 19(3), 1–13 (2013) 12. Atanassov, K., Vassilev, P.: On the Intutitionistic fuzzy sets of n-th type . In: Gaw˛eda, A.E., Kacprzyk, J., Rutkowski, L., Yen, G.G. (eds.) Advances in Data Analysis with Computational Intelligence Methods. SCI 738. Springer, Cham (2018). https://doi.org/10.1007/978-3-31967946-4 13. Atanassova, V., Doukovska, L., Atanassov, K. Mavrov, D.: Intercriteria decision making approach to EU member states competitiveness analysis. In: Proceedings of the International Symposium on Business Modeling and Software Design – BMSD 2014, pp. 289–294 (2014) 14. Beckamnn, H., Winkel, C., Shöllhorn, W.I.: Optimal Range of variation in hockey technique training. Int. J. Sport Psychol. 41(4), 5–45 (2010) 15. Eskiyecek, C.G., Bingul, B.M., Bulgan, C., Aydin, M.: 3D Biomechanical analysis of targeted and non-jargeted drag flick shooting technique in field hockey. Acta Kinesiolodica 12(2), 13–19 (2018) 16. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData – Software for InterCriteria analysis. Int. J. Bioautom. 22(1), 1–10 (2018) 17. Ibrahim, R., Faber, G.S., Kingma, I., van Dieen, J.H.: Kinematic analysis of the drag flick in field hockey. Sports Biomech. 16(1), 45–57 (2017) 18. Ilkova, T., Petrov, M.: Application of intercriteria analysis to the Mesta river pollution modelling. Notes Intuitionistic Fuzzy Sets 21(2), 118–125 (2015) 19. de Subijana, C.L., Juárez, D., Mallo, J., Navarro, E.: The application of biomechanics to penalty corner drag-flick training: a case study. Madrid Spain J. Sports. Med. 10, 590–595 (2011) 20. Krawczak, M., Bureva, V., Sotirova, E., Szmidt, E.: Application of the InterCriteria decision making method to universities ranking. Adv. Intell. Syst. Comput. 401, 365–372 (2016) 21. Meulman, H.N., Berger, M.A.M., van der Zande, M.E., Kok, P.M., Ottevanger, E.J.C., Crucq, M.B.: Development of tool for training the drag flick penalty corner in field hockey. In: 9th Conference of the International Sports Engineering Assosiation (ISEA), Procedia Engineering, vol. 34, pp. 508–513 (2012)

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22. Palaniappan, R., Sungar, V.: Biomechanical analysis of penalty cornar drag flick in field hockey. In: 36th Conference of the International Society of Biomechanics in Sports, Auckland, New Zealand, 10–14 September, pp. 690–694 (2018) 23. Todinova, S., et al.: Blood plasma thermograms dataset analysis by means of InterCriteria and correlation analyses for the case of colorectal cancer. Int. J. Bioautom. 20(1), 115–124 (2016) 24. https://chrisfryperformanceanalyst.wordpress.com/2013/04/09/drag-flick-technique-ana lysis/. Accessed 22 Jan 2020 25. http://www.fih.ch/media/12236439/fih-rules-of-indoor-hockey-2019-new-cover-final-editson-website.pdf. Accessed 22 Jan 2020

InterCriteria Analysis of Data Obtained from University Students Practicing Sports Activities Simeon Ribagin1,2(B) and Spas Stavrev3 1 Bioinformatics and Mathematical Modelling Department, Institute of Biophysics and

Biomedical Engineering, Bulgarian Academy of Sciences, 105 Acad. Georgi Bonchev Street, 1113 Sofia, Bulgaria 2 Department of Health and Social Care, Medical College, University “Prof. D-r Asen Zlatarov”, Burgas, Bulgaria 3 Department of “Physical Culture and Sport, University of National and World Economy, 1700 Sofia, Student Town, Bulgaria [email protected]

Abstract. Intellectual qualities of the university students are important factors for the successful participation in different academic activities. In view of this it is important to evaluate the initial level of intellectual development by applying a group of specific tests. In this paper we propose the application of the approach of InterCriteria Analysis to data of intellectual status parameters, obtained from university students practicing sports activities in order to evaluate the appropriateness of the used tests. Keywords: Intellectual development · University students · InterCriteria analysis · Intuitionistic fuzzy sets

1 Introduction One of the core aspects of higher education is to prepare the students for the future employment and for the society. University students represent a specific subgroup in the period of adolescence and emerging adulthood as they are particularly affected by changing (structural) life circumstances with the start of their studies [6]. It is a time of physical, social, psychological, and structural changes, which may influence the way of lifestyle. It should also be noted that it is an important stage of life with great changes in behaviors and responsibilities derived from emancipation and the demands developed by a new university degree [5]. Academic performance is affected by a host of factors. These include individual characteristics such as student ability and motivation. The gender of the student may also be a factor in determining student performance. In view of this it is important to evaluate the initial level of intellectual development of the male and female students by applying a group of specific tests. A number of indicators can be used to determine the level of intellectual development and the future © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 230–237, 2021. https://doi.org/10.1007/978-3-030-77716-6_21

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academic achievements of university students. Students’ personal characteristics may predict study outcome. Among other things, a wide range of qualities appropriate to the personality with its specific individual temperament is necessary for successful professional fulfillment. Personal qualities are of high importance for economics profession. It is therefore important to research and if possible to influence on them by means of physical education and sport. We have investigated 13 of such qualities and compared the results by male and female students, attending sports activities from “Physical Culture” subject. Never the less it is of great importance to evaluate the usefulness of the applied method of testing and to establish a well-structured test battery. In this paper we propose the application of the approach of InterCriteria Analysis to data, obtained from university students practicing sports activities. The data was analyzed in search of correlations between the results from the method of testing for the level of development of the motive qualities, as well as tests for assessment of the psychic and personality qualities in order to predict the reliability of the proposed test battery on male and female participants.

2 Presentation of the Input Data The investigation and the data collection process, has been carried out in the educational year 2011/12. Contingent of the investigation are 70, 1st year students (males) and 47, 1st year students (females) in UNWE, attending sports trainings from “Physical Culture” subject. We have chosen a set of several parameters defining the level of cognitive and emotional well-being in a test battery of 13 control tasks for intellectual status as follows: – Operational thinking. This type of thinking repeats and illustrates action, preceding or following it within a limited temporal span (C1), – Analytical thinking. This type is a critical component of visual thinking that gives one the ability to solve problems quickly and effectively (C2), – Logical thinking. This is the process, in which one uses reasoning consistently to come to a conclusion (C3), – Visual memory. It is a form of memory which preserves some characteristics of our senses pertaining to visual experience (C4 and C5), – Willpower. This is the cognitive process, by which an individual decides on and commits to a particular course of action (C7), – Personal activity. This is the skill of a person to participate actively in different activities (C8), – Communication skills. This skill allows you to understand and be understood by others (C9), – Organizational skills. Refers to our ability to stay focused on different tasks, and use your time, energy, strength, mental capacity, physical space, etc. effectively and efficiently in order to achieve the desired outcome (C10), – Morality. It is the differentiation of intentions, decisions and actions between those that are distinguished as proper and those that are improper (C6). – Neuroticism. It is one of the Big Five higher-order personality traits in the study of psychology. Individuals who score high on neuroticism are more likely than average to be moody and to experience such feelings as anxiety, worry, fear (C13),

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– Extraversion and Introversion. Extraversion tends to be manifested in outgoing, talkative, energetic behavior, whereas introversion is manifested in more reserved and solitary behavior (C12), – Anxiety. It is a normal emotion that causes increased alertness, fear, and physical signs, such as a rapid heart rate (C11). The results from the tests, as an input data are obtained from two control groups of students (males and females) and are listed below in Table 1. Table 1. Resultant table of data from cognitive and emotional well-being tests obtained from male and female students (min/max values). males females

C1 1.70 4.38 1.60/ 5.04

C2 94.3 152.7 92.8/ 144.0

C3 2/8

C4 5 / 10

C5 3 / 13

C6 2 / 14

C7 7 / 28

1/9

3 / 10

2 / 13

3 / 14

14 / 28

C8 47.72/ 77.335 44.89/ 77.97

C9 4 / 19

C10 6 / 20

C11 22 / 57

C12 6 / 21

C13 1 / 21

6 / 20

6 / 19

26 / 68

8 / 22

1 / 22

3 Application of the InterCriteria Analysis The InterCriteria Analysis (ICA) method is introduced in [2] by Atanassov et al. It can be applied to multiobject multicriteria problems, where measurements according to some of the criteria are slower or more expensive, which results in delaying or raising the cost of the overall process of decision-making. Various aspects of its theoretical investigation are given in papers [3]. In general, the concept of ICrA is based on the apparatus of index matrices (IMs), [4] and intuitionistic fuzzy sets (IFSs), [1]. Comparison between elements of two IFSs, involves pairwise comparisons between the degrees of membership and non-membership of the elements of both sets. Let us have an index matrix (IM) with elements ap,q , p = 1, . . . , m, q = 1, . . . , n,

... ...

aCi ,On

...

aC j ,Ol

...

aC j ,On ...

...

... ...

... aCi ,Ol

...

aCm ,O j

...

...

...

...

On aC1 ,On ,

...

... aC j ,Ok

...

...

...

...

...

Ol aC1 ,Ol

...

aCm ,O1

...

...

Cm

aCi ,Ok

...

aC j ,O1

... ...

...

Cj

...

...

...

aCi ,O1

...

Ci

...

Ok aC1 ,Ok

...

...

...

O1 aC1 ,O1

...

M =C 1

...

aCm ,Ol

...

aCm ,On

where Cp is a criterion, taking part in the evaluation; Oq is an object, being evaluated; ap,q is the evaluation of the q-th object against the p-th criterion, and it is defined as a real number or another object that is comparableaccording to relation R with all the rest   elements of the index matrix M, so the relation R aCk Oi , aCk Oj holds for each i, j, k.

InterCriteria Analysis of Data Obtained from University Students

233

The relation R has dual relation R, which is true in the cases when relation R is false, and vice versa.      If the number of cases for which the relations R aCk Oi , aCk Oj and R aCl Oi , aCl Oj μ are simultaneously satisfied is Sk,l and the number of cases for which the relations     v , then R aCk Oi , aCk Oj and its dual R aCl Oi , aCl Oj are simultaneously satisfied is Sk,l μ

v ≤ 0 ≤ Sk,l + Sk,l

n(n − 1) , 2

Since the total number of pairwise comparisons between the object is n (n – 1)/2. For every k, l, such that 1 ≤ k ≤ l ≤ m, and for n ≥ 2 two numbers are defined μ

μCk ,Cl = 2

Sk,l n(n − 1) μ

, vCk ,Cl = 2

v 0 ≤ Sk,l + Sk,l ≤

v Sk,l

n(n − 1)

n(n − 1) 2

...

Cm

μC ,C ,ν C ,C

...

μC ,C ,ν C ,C

...

C1

1

Cm

μC

1

m ,C1

1

1

,ν C1 ,Cm

...

1

m

1

. m

...

C1

...

M* =

...

The pair, constructed from these two numbers, plays the role of the intuitionistic fuzzy evaluation of the relations that can be established between any two criteria Ck and Cl . In this way the index matrix M that relates evaluated objects with evaluating criteria can be transformed to another index matrix M* that gives the relations among the criteria.

μC

m ,Cm

,ν Cm ,Cm

Alternatively, it is practical to work with two index matrices M μ and M v , rather than with the index matrix M ∗ of IF pairs. The final step of the algorithm requires defining the values of thresholds for both the membership and the non-membership, against which we evaluate the precision of the ICA decision making. We call that two criteria are in relation of either ‘positive consonance’, or ‘negative consonance’, or ‘dissonance’, depending on their InterCriteria pair’s comparison with these two defined threshold values. Let α, β ∈ [0; 1] be the threshold values, against which we compare the values of μCk ,Cl and vCk ,Cl . We call these criteria Ck and Cl are in: • (α, β)-positive consonance, if μCk ,Cl > α and vCk ,Cl > β; • (α, β)-negative consonance, if μCk ,Cl > β and vCk ,Cl > α; • (α, β)-dissonance, otherwise.

Here we are interested in detecting patterns and trends between the investigated indicators of the intellectual status selected in our test-battery of male and female university students. For this aim, we use the developed software for ICrA [7] (freely available

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online at: http://intercriteria.net/software) and feed it with the data from the two control groups of students. The complete resultant tables with InterCriteria intuitionistic fuzzy membership pairs are given on Table 2, Table 3 and Table 4. Table 2. Results of the application of the InterCriteria Analysis on the aggregated data from Table 1 (male students), IF membership parts. C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C1

1

0.3938

0.3959

0.3035

0.4025

0.4754

0.4331

0.4302

0.4153

0.4559

0.4932

0.4609

0.4828

C2

0.3938

1

0.5764

0.4058

0.5019

0.3176

0.4108

0.5176

0.4853

0.4671

0.4277

0.4969

0.4948

C3

0.3959

0.5764

1

0.3706

0.4629

0.3031

0.3826

0.4319

0.4708

0.4199

0.3387

0.4816

0.3921

C4

0.3035

0.4058

0.3706

1

0.3752

0.2729

0.3188

0.4021

0.4029

0.3988

0.3743

0.4447

0.3992

C5

0.4025

0.5019

0.4629

0.3752

1

0.3901

0.4327

0.4232

0.4373

0.5193

0.4041

0.4389

0.4385

C6

0.4754

0.3176

0.3031

0.2729

0.3901

1

0.4915

0.4497

0.4087

0.5135

0.4874

0.3027

0.4638

C7

0.4331

0.4108

0.3826

0.3188

0.4327

0.4915

1

0.5164

0.4832

0.5491

0.4389

0.3317

0.4509

C8

0.4302

0.5176

0.4319

0.4021

0.4232

0.4497

0.5164

1

0.5321

0.5354

0.4455

0.4178

0.4592

C9

0.4153

0.4853

0.4708

0.4029

0.4373

0.4087

0.4832

0.5321

1

0.5723

0.3072

0.5325

0.3271

C10

0.4559

0.4671

0.4199

0.3988

0.5193

0.5135

0.5491

0.5354

0.5723

1

0.395

0.4203

0.4443

C11

0.4932

0.4277

0.3387

0.3743

0.4041

0.4874

0.4389

0.4455

0.3072

0.395

1

0.3698

0.6712

C12

0.4609

0.4969

0.4816

0.4447

0.4389

0.3027

0.3317

0.4178

0.5325

0.4203

0.3698

1

0.4265

C13

0.4828

0.4948

0.3921

0.3992

0.4385

0.4638

0.4509

0.4592

0.3271

0.4443

0.6712

0.4265

1

Table 3. Results of the application of the InterCriteria Analysis on the aggregated data from Table 1 (female students), IF membership parts. C1

C2

C3

C4

C5

C6

C7

C8

C9

C11

C12

C13

C1

1

0.5375

0.4061

0.3432

0.3525

0.4764

0.4653

0.445

0.42

0.42

C10

0.4625

0.3774

0.4625

C2

0.5375

1

0.4931

0.3673

0.4302

0.3117

0.4329

0.4496

0.4699

0.5291

0.383

0.5356

0.3599

C3

0.4061

0.4931

1

0.2858

0.4403

0.3478

0.3571

0.4413

0.3811

0.3876

0.3904

0.3654

0.4033

C4

0.3432

0.3673

0.2858

1

0.3608

0.3228

0.3728

0.3543

0.37

0.3293

0.3154

0.4764

0.2618

C5

0.3525

0.4302

0.4403

0.3608

1

0.3525

0.3876

0.3663

0.3719

0.4385

0.4329

0.506

0.395

C6

0.4764

0.3117

0.3478

0.3228

0.3525

1

0.5264

0.4625

0.4191

0.4311

0.4237

0.3321

0.4755

C7

0.4653

0.4329

0.3571

0.3728

0.3876

0.5264

1

0.5467

0.4866

0.543

0.3451

0.4107

0.3728

C8

0.445

0.4496

0.4413

0.3543

0.3663

0.4625

0.5467

1

0.6115

0.5634

0.4209

0.4385

0.383

C9

0.42

0.4699

0.3811

0.37

0.3719

0.4191

0.4866

0.6115

1

0.605

0.2692

0.5458

0.247

C10

0.42

0.5291

0.3876

0.3293

0.4385

0.4311

0.543

0.5634

0.605

1

0.2942

0.4616

0.321

C11

0.4625

0.383

0.3904

0.3154

0.4329

0.4237

0.3451

0.4209

0.2692

0.2942

1

0.3793

0.7521

C12

0.3774

0.5356

0.3654

0.4764

0.506

0.3321

0.4107

0.4385

0.5458

0.4616

0.3793

1

0.3728

C13

0.4625

0.3599

0.4033

0.2618

0.395

0.4755

0.3728

0.383

0.247

0.321

0.7521

0.3728

1

InterCriteria Analysis of Data Obtained from University Students

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Table 4. Results of the application of the InterCriteria Analysis on the aggregated data from Table 1 (male and female students), IF membership parts. C1

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

1

0.443

0.3886

0.3337

0.3814

0.4781

0.4519

0.449

0.4238

0.4487

0.4805

0.4307

0.467

C2

0.443

1

0.5432

0.3892

0.4771

0.3129

0.4202

0.49

0.4834

0.4975

0.4046

0.5136

0.4237

C3

0.3886

0.5432

1

0.3322

0.4586

0.3229

0.378

0.436

0.4382

0.4105

0.3598

0.4388

0.3909

C4

0.3337

0.3892

0.3322

1

0.3733

0.3028

0.3484

0.3871

0.3919

0.3696

0.3508

0.4619

0.3426

C5

0.3814

0.4771

0.4586

0.3733

1

0.3648

0.4163

0.4012

0.4097

0.4865

0.4157

0.4676

0.4169

C6

0.4781

0.3129

0.3229

0.3028

0.3648

1

0.5096

0.455

0.4106

0.4808

0.4615

0.3085

0.4763

C7

0.4519

0.4202

0.378

0.3484

0.4163

0.5096

1

0.5229

0.4885

0.54

0.4069

0.3615

0.4217

C8

0.449

0.49

0.436

0.3871

0.4012

0.455

0.5229

1

0.5688

0.54

0.4543

0.4241

0.4397

1

0.5765

0.3075

0.5376

0.2982

0.3408

0.4369

0.3898

C9

0.4238

0.4834

0.4382

0.3919

0.4097

0.4106

0.4885

0.5688

C10

0.4487

0.4975

0.4105

0.3696

0.4865

0.4808

0.54

0.54

0.5765

1

C11

0.4805

0.4046

0.3598

0.3508

0.4157

0.4615

0.4069

0.4543

0.3075

0.3408

1

0.3718

0.7105

C12

0.4307

0.5136

0.4388

0.4619

0.4676

0.3085

0.3615

0.4241

0.5376

0.4369

0.3718

1

0.4004

C13

0.467

0.4237

0.3909

0.3426

0.4169

0.4763

0.4217

0.4397

0.2982

0.3898

0.7105

0.4004

1

The results from the application of ICrA supports our previous findings [see 8], while not strictly positive consonances are detected, we see that the strongest available one, i.e. those with smallest distance from the intuitionistic fuzzy truth [1, 0], is those detected between the C13 (Anxiety) - C11 (Neuroticism), both in male and female students. These pairs are formed between logically related qualities. The strongest negative consonance is detected between the C13 (Anxiety) – C9 (Communication skills) and between the C13 (Anxiety) – C4 (Visual memory) only in female participants. The strongest negative consonance in male participants is detected between the C1 (Operational thinking) - C4 (Visual memory) and C12 (Extraversion and Introversion) – C6 (Morality). All other pairs score intuitionistic fuzzy dissonance values and are independent of each other. The results represented on Table 4 from the application of the ICrA on the aggregated data from both control groups of students are showing similar behavior. The strongest positive consonance is detected between C13 (Anxiety) - C11 (Neuroticism) and the strongest negative consonance is detected between C13 (Anxiety) – C9 (Communication skills). The graphical representation of the results is presented on Figs. 1, 2 and 3.

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Fig. 1. InterCriteria pairs between the 13 criterions (Table 2), as plotted as points (black) on the intuitionistic fuzzy interpretational triangle.

Fig. 2. InterCriteria pairs between the 13 criterions (Table 3), as plotted as points (black) on the intuitionistic fuzzy interpretational triangle.

Fig. 3. InterCriteria pairs between the 13 criterions (Table 4), as plotted as points (black) on the intuitionistic fuzzy interpretational triangle.

4 Conclusions This study describes processing of data obtained from cognitive and emotional wellbeing tests of students. After the application of ICrA several conclusions can be made. The strong relation between C13 (Anxiety) - C11 (Neuroticism) suggests that we can easily exclude one of the two criteria from the test battery, without complementing the overall evaluation both in male and female participants. When evaluating the overall psychological status of students, it is important to take into account gender differences particularly in specific personal qualities like anxiety, communication skills, extraversion and introversion and visual memory. In general, the results show that the chosen tests in the test battery are very well combined to evaluate the initial level of intellectual

InterCriteria Analysis of Data Obtained from University Students

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development of the students. Naturally, given the relatively small size of the considered data it is not possible to claim with absolute certainty that our interpretations are doubtlessly valid but they provide a starting point for further investigations. Acknowledgements. The first author is grateful for the support provided under Grant No. KP-06N-22/1/2018 “Theoretical research and applications of InterCriteria Analysis”.

References 1. Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-7908-1870-3 2. Atanassov, K., Mavrov, D., Atanassova, V.: InterCriteria decision making. A new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues IFS and GN 11, 1–8 (2014) 3. Atanassov, K., Atanassova, V., Gluhchev, G.: InterCriteria analysis: ideas and problems. NIFS 21(1), 81–88 (2015) 4. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues IFS GN 11, 1–8 (2014) 5. Chacón-Cuberos, R., Zurita-Ortega, F., Puertas-Molero, P., et al.: Relationship between healthy habits and perceived motivational climate in sport among university students: A structural equation model. Sustainability 10, 938 (2018). https://doi.org/10.3390/su10040938 6. Diehl, K., Fuchs, A., Rathmann, K., Hilger-Kolb, J.: Students’ motivation for sport activity and participation in university sports: a mixed-methods study. BioMed Res. Int. 2018, 1–7 (2018) 7. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData – software for InterCriteria analysis. Int J Bioautomation 22(1), 1–10 (2018) 8. Ribagin, S., Stavrev, S.: (2019) InterCriteria Analysis of data from intellectual and physical evaluation tests of students practicing sports activities. NIFS 25(4), 83–89 (2019)

Applications in Industry, Business and Critical Infrastructure

Forest Fire Analysis Based on InterCriteria Analysis Dafina Zoteva1(B) , Olympia Roeva1 , and Hristo Tsakov2 1 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, bl. 105, 1113 Sofia, Bulgaria [email protected] 2 Forest Research Institute, Bulgarian Academy of Sciences, 132 St. Kliment Ohridski Blvd., 1756 Sofia, Bulgaria [email protected]

Abstract. Forest fires annually affect large areas all over the world. They are one of the reasons for changes in the forest ecosystems, which lead to unpredictable consequences in long term. The present research extends and builds up an analysis of the forest fires risk assessment in Bulgaria over the last 20 years. Two methodologies, different in their essence, are used in the study: a common approach (Lubenov’s methodology) and InterCriteria Analysis (ICrA). Lubenov’s methodology classifies the different regions of Bulgaria in groups according to the risk of forest fires. ICrA, which seeks to find relations between some predefined criteria, is used as an additional approach to refine this classification. The research is based on the number of the forest fires occurred in different regions of Bulgaria and the size of the burned areas over the past 20 years. Expanding the period over which the study is conducted aims for the Lubenov’s methodology to confirm the results of ICrA. Keywords: Forest fire risk · InterCriteria Analysis · Intuitioninstic fuzzy sets · Index Matrices

1 Introduction InterCriteria Analysis (ICrA) [5] has been developed in support of the decision making processes in multicriteria environment. The aim is to analyse the nature of predefined criteria by seeking to establish certain relations between them. Evaluations of multiple objects against these predefined criteria, presented in the form of Index Matrices (IM) [1], are used in the process. Intuitionistic Fuzzy Sets (IFS) [2] are employed in the process to handle the uncertainties.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 241–253, 2021. https://doi.org/10.1007/978-3-030-77716-6_22

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ICrA has been successfully applied in different fields of science and practice: for analysis of the EU member states competitiveness [6], river pollution modelling [8], ranking of universities [9] or parameter identification of fermentation processes [11]. Forest fires have been a significant environmental, economic and social problem in the world and in Bulgaria for the past 20 years. The fires destroy valuable natural resources, disrupt the forest plantations, worsen their protective function, slow down the phases of natural restoration of forests and burned areas. The term “fire season” is complex and is mainly influenced by meteorological factors (rainfall, air temperature, length of the dry period, wind speed), economic and agricultural activity. Data on the frequency of occurring of forests fires in Bulgaria show that the fire activity is observed almost all year round with expected peaks during the spring (February, March and April) and the summer (July, August and September). Over the last few years Bulgaria has seen a significant number of fires in the winter which should be considered as a new phenomenon. Forest fires data have been studied in order to establish some dependencies in occurring of the fires in different parts of Bulgaria. The regions in Bulgaria have been classified in specific fire risk groups based on forest fires data for the period 2009 to 2018 in [12]. ICrA has been applied there in order to confirm and to refine the results shown by a classic methodology [10]. ICrA has shown high values of degrees of “agreement” for those regions which unconditionally fall in a particular group, while others are classified with certain degree of uncertainty. Following the results and the conclusions of [12], the present work extends the assessment of the forest fire risk over a period of 20 years (1999–2018), expecting the Lubenov’s methodology to confirm the ICrA results over a longer period of time. The paper is organized as follows. The data used for the analysis are presented in Sect. 2. A brief description of the background of the Lubenov’s methodology and ICrA is given in Sect. 4. The results from both of the analysis are shown and discussed in Sect. 5. The concluding remarks are made in Sect. 6.

2 Forest Fires Data The territory of Bulgaria is divided into sixteen Regional Forest Directorates (RFD): Berkovitsa, Blagoevgrad, Burgas, Kardzhali, Kyustendil, Lovech, Pazardzhik, Plovdiv, Ruse, Sofia, Shumen, Sliven, Smolyan, Stara Zagora, Varna and Veliko Tarnovo. Each year in its Annual Reports [13], the Bulgarian Executive Forest Agency provides statistics for the fires occurred on the RFD territories. The data include the number of the forest fires, size of the burned territories, type of the fires, type of woods affected by the fires, reasons for the fires and others.

Forest Fire Analysis Based on InterCriteria Analysis

243

The present research uses data on the number of the forest fires and the size of the burned territories in each RFD for the past 20 years, from 1999 to 2018. These data are presented in Table 1 and Table 2, respectively. For better visualization and interpretation, the data are further illustrated in Fig. 1 and Fig. 2. Table 1. Number of fires by RFD (1999–2018)

2007

2005

2004

2001

2000

15

30

146

11

8

18

60

76

70

150

6

16

12

44

84

21

25

31

24

16

60

92

7

Burgas

8

32

45

37

4

22

60

49

3

31

44

55

11

9

9

13

19

43

41

2

Varna

21

33

48

17

12

43

102

34

5

34

34

117

34

6

10

20

24

117

84

2

Veliko Tarnovo

4

11

9

10

3

8

27

19

6

19

24

83

6

6

5

20

7

14

41

2

Kardzhali

23

44

91

48

8

58

59

36

13

32

75

84

74

27

37

68

29

105

152

30

Kyustendil

22

47

36

35

11

29

54

65

16

12

50

119

32

19

26

36

12

12

117

17

Lovech

20

37

30

36

29

18

34

37

63

27

80

163

17

37

18

20

66

43

98

12

Pazardzhik

8

28

28

17

4

35

74

59

11

34

37

91

40

11

26

27

32

46

90

4

Plovdiv

14

27

43

30

6

32

55

31

12

26

33

101

45

8

27

37

22

56

65

3

Ruse

10

16

14

5

14

5

28

19

8

3

20

37

13

7

2

29

4

54

45

0

Sliven

7

20

33

20

2

14

62

27

8

22

23

66

15

2

6

21

3

51

62

11

Smolyan

8

27

34

16

4

18

34

18

7

8

12

45

9

14

20

15

27

28

58

12

Sofia

4

17

27

26

7

18

50

29

1

10

19

53

15

51

50

50

42

38

247

37

Stara Zagora

18

54

60

82

26

57

62

78

27

24

53

185

41

7

5

7

16

62

82

20

1999

2008

2002

2009

24

69

Total

2003

2010

45

72

2011

78

32

2012

13

12

2013

5

32

2015 19

51

2014

29

70

2016

40

36

2017

12

Blagoevgrad

2018

Berkovitsa

Shumen

2006

Number of fires RFD

7

10

6

9

4

6

27

20

2

5

4

50

8

4

4

5

7

26

29

2

222

513

584

439

151

408

878

635

222

314

582

1479

392

241

294

452

402

825

1453

167

The total number of fires and the size of the burned territories during the first decade of the 20-year period (1999–2008) exceed significantly the numbers during the second decade (2009–2018).

27.8

667.3

8.2

60.4

29.4

58.3

14.6

1.2

29.4

Kyustendil

Lovech

Pazardzhik

Plovdiv

Ruse

Sliven

Smolyan

Sofia

Stara Zagora

Total

286.6

9.2

249.9

272.2

40.4

116.9

127.6

179.1

113.1

194.1

4569.4

10.7

Kardzhali

93

1487.2

6.6

Veliko Tarnovo

31.5

11.2

26.4

Varna

538.2

1694.1

612.3

2017

7

19.6

Burgas

Shumen

419.1

101.2

Blagoevgrad

2018

Berkovitsa

RFD

2016

6340.2

4.9

154.4

6.2

695.5

414.7

29.8

77.2

146.3

402.9

40

3494.9

15.5

154.5

497.5

58.1

147.8

530

2015

4315

11

872.3

55.2

13.9

207.7

8.8

139.5

26.4

1471.5

98

268.7

138.2

44.1

259.6

170.1

2014

916

6.5

371.3

2.6

3.2

2.7

32.3

3.7

35.7

337.7

23.5

12

2.4

21.7

12.3

20.1

28.3

2013 3313.9

1.5

307.5

10.4

513.8

417.5

1.9

176

92.6

181.1

230.1

796.3

6.3

114.5

304.9

80.2

79.3

2012 12729.8

239.4

623.8

42.2

337.4

1965.9

179.1

185

802.8

942.3

717

372.4

268.6

474.2

2304.9

543.3

2731.5

2011 6882.6

28.3

980.8

22.9

67.5

250.9

56.5

380.7

225.9

1214.8

749.4

303.4

157.5

98.2

804.7

514.5

1026.6

2010 6526

2.6

2507

0.1

20.8

15

27.8

66.1

13.8

3000.6

76.2

26.3

123.5

7.4

41.8

21

576

2009 2276.4

4.4

171.5

4.5

8.2

260.3

15.3

32.3

146.9

420.5

18.8

753.1

118.2

57.1

133.1

13.8

118.4

250.6

2008 5289.2

5.4

443.8

14.4

58

389.3

130.5

138.2

279.3

1490.4

245.7

730.8

213.7

212.2

536.8

150.1

42999.7

230

2717.1

387.1

5231

4403

306.2

1228.4

1638.5

6653.3

1369.8

6745.2

1544.9

663.9

1296.8

490.2

8094.3

2007

Size of the burned territories

3537.4

33.8

97

33.6

39.7

363.7

15.7

138

123

159.1

70.5

2052.1

12.8

67.2

91.1

71.5

168.6

2006

Table 2. Burned territories by RFD (1999–2018)

60.9

2005 1455.7

34.3

17.3

386.3

6.5

7

16

11.8

91.3

268.3

115.3

78.2

106

27.3

68.9

160.3

2004 1136.5

2.4

3.3

370

56.5

59

4

34.4

102.2

40.7

59.6

172.1

4.6

7.8

58.8

43.4

117.7

2003 5000.5

31.1

36

103.4

20

550.1

103.4

250.9

203.2

168.9

100.9

1684.9

63.3

110.4

147.9

108.2

1317.9

2002 6513.6

59.2

101.1

962.5

12.3

5.5

13.2

236.3

306.3

1989

46

79

160.9

132.6

24.1

46.4

2339.2

1245

2001 20152.3

175.6

1338.7

147.6

56

2168.7

379.1

548.9

271.6

579.4

23.1

8090.1

81.3

630.6

4259.4

157.2

2000 54540.3

200.6

5385.1

4625.3

136.1

7456.6

556.7

1896.3

1662.2

3435.8

4081.8

16945.4

672

495.9

2624

867

3499.5

8481.1

27

200.4

839.8

1618

690

0

43

338

811

1012

1884.6

5.6

0.7

318

234

459

1999

244 D. Zoteva et al.

Forest Fire Analysis Based on InterCriteria Analysis

245

Fig. 1. Number of fires in Bulgaria (1999–2018)

Fig. 2. Burned territories in Bulgarian (1999–2018)

3 Methods Theoretical Background The study on the forest fires and the classification of the regions in Bulgaria in certain risk groups are conducted using two different in their essence methodologies: Lubenov’s methodology and ICrA. In this case, however, both methodologies complement each other. 3.1 Methodology of Lubenov The methodology of Lubenov which classifies a given region in a certain forest fire risk group is introduced in [10]. Based on both the number of forest fires occurred on a given

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territory and the size of the burned area, a certain region can be classified in one of the following risk groups: low, average, high and very high. The risk of occurring of forest fires in a certain region Rffo is defined as: Rffo = Rdens × Rrba

(1)

where Rdens is the average annual numerical value of the fire density on the given region, defined as the average annual number of fires (N i ) occurred over a given period of time (n) on an area of 1000 ha (10 km2 ) of the total area of the region (F reg.ter ) in the following way: 1000 Rdens =

n 

Ni

i=1

n × Freg.ter.

(2)

Rrba is the real burned area is defined as the average annual burned area (F ba ) in hectares (ha) over a given period of time (n) on 1000 ha (10 km2 ) of the total area of the region (F reg.ter ) as follows: 1000 Rrba =

n 

Fba

i=1

n × Freg.ter.

(3)

A scale of the degrees of forest fires risk, based on the values of Rffo , is defined in [10] and refined in [12].

4 InterCriteria Analysis InterCriteria Analysis is used to support the process of making decisions in a multicriteria environment [5]. ICrA is employed in order to establish certain relations between some predefined criteria. The input data for the analysis are presented in the form of an IM with index sets the criteria for the rows and the objects for the columns. The elements are the evaluations of the objects against the predefined criteria, which usually are real numbers. The result of the ICrA is an output IM with the criteria as index sets of the rows and the columns. The elements of the resulting IM are Intuitionistic Fuzzy Pairs (IFP) [4] that correspond to the degree of “agreement” and “disagreement” between the relevant criteria pair. Here is an explanation of the way the resulting matrix is constructed. def

Let O = {O1 , O2 , . . . , On } denotes the set of objects being evaluated, and def

C(O) = {C(O1 ), C(O2 ), . . . , C(On )} be the set of objects evaluations by a given criterion C. elements is constructed as: Then for a fixed criterion C a vector with n(n−1) 2 V (C) = {C(O1 ), C(O2 ), C(O1 ), C(O3 ), . . . , C(O1 ), C(On ), . . . , C(On−1 ), C(On )}.

Forest Fire Analysis Based on InterCriteria Analysis

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The elements of the vector V (C) are then replaced by 1 if C(Oi ) > C(Oj ), −1 if C(Oi ) < C(Oj ) and 0 if C(Oi ) = C(Oj ), ∀i ∈ [1; n − 1], ∀j ∈ [i + 1; n]. The degree of “agreement” and “disagreement” between two criteria C and C  are denoted as μC,C  and νC,C  , respectively. Depending on the specific ICrA algorithm [7], μC,C  (μC,C  = μC  ,C ) and νC,C  (νC,C  = νC  ,C ) are calculated. For example, in case of µ-biased algorithm [7], the degree of “agreement” is defined as the number of matching components of the respective vectors, normalized by the length of the vector. Respectively, the degree of “disagreement” νC,C  is the number of components of opposing signs in the two vectors, again normalized by the length of the vector. As a result, μC,C  , νC,C   is an IFP. For most of the cases the sum μC,C  + νC,C  is equal to 1. However, there might be some pairs, for which this sum is less than 1. The difference πC,C  = 1 − μC,C  − νC,C  is considered as a degree of “uncertainty”.

5 Results and Discussion Lubenov’s methodology has been applied to 20-year period data (1999–2018). Based on the obtained results, the RFD in Bulgaria have been classified in forest fires risk groups, presented in Table 3. These fire risk evaluations are compared with the results in [12] (Table 4), for the period 2009–2018. To facilitate the comparison, the resulting groups of Table 3 are summarized in Table 5. Table 3. Fire risk assessment according to Lubenov, 20-year period (1999–2018) RFD

Fire density

Real burned area

Forest fires risk

Risk degree

Berkovitsa

0.18

4.90

0.86

Very high

Blagoevgrad

0.10

0.68

0.07

Low

Burgas

0.08

2.15

0.17

Average

Varna

0.21

0.91

0.19

Average

Veliko Tarnovo

0.07

0.88

0.07

Low

Kardzhali

0.15

6.21

0.94

Very high

Kyustendil

0.15

1.86

0.29

Average

Lovech

0.20

5.46

1.08

Very high

Pazardzhik

0.13

1.27

0.17

Average

Plovdiv

0.16

1.36

0.22

Average

Ruse

0.10

0.56

0.05

Low

Sliven

0.10

4.24

0.43

High

Smolyan

0.08

1.87

0.16

Average

Sofia

0.09

0.95

0.09

Low

Stara Zagora

0.27

4.72

1.29

Very high

Shumen

0.06

0.31

0.02

Low

248

D. Zoteva et al. Table 4. RFD classification according Lubenov’s methodology (2009–2018) [12]

RFD

Degree of forest fires risk

Blagoevgrad, Sofia, Shumen, Ruse, Veliko Tarnovo, Smolyan, Pazardzhik, Plovdiv

Low

Berkovitsa, Burgas, Kardzhali, Kyustendil, Sliven, Varna

Average

Lovech

High

Stara Zagora

Very high

Table 5. RFD classification according to Lubenov’s methodology [10] (1999–2018) RFD

Degree of forest fires risk

Blagoevgrad, Sofia, Shumen, Ruse, Veliko Tarnovo

Low

Burgas, Kyustendil, Pazardzhik, Plovdiv, Smolyan, Varna

Average

Sliven

High

Berkovitsa, Kardzhali, Lovech, Stara Zagora

Very high

In order to apply ICrA to the data on forest fires for the period 1999–2018, the input index matrix presented in Table 6 is constructed. The RFD are considered as criteria (in the IM rows) and the years – as objects in the IM columns. The elements of the input matrix are the ratios between the number of forest fires occurred on the territory of a given RFD and the size of the burned areas there. The results of the ICrA conducted on thus constructed input IM are shown in Table 7. The elements of the resulting IM are the obtained degrees of “agreement” μC,C  between the RFDs. The scale presented in [3] is used to interpret the results of the ICrA. For completeness it is shown in Table 8. The Lubenov’s methodology for the whole 20-year period has shown differences for five regions: • Smolyan has been classified in the group of average risk of occurring of forest fires when data of the period 1999–2018 are analyzed, in comparison to low risk group over the shorter period of time (Table 4). • Sliven has moved to the high risk group instead of the average one. • Berkovitsa is in the group of very high risk, while for the 10-year period it has been classified in the average risk group (Table 4). • Kardzhali has moved from the average to the very high risk group. • The degree of risk for Lovech has been increased to very high when data on the forest fires over the period of 20 years have been considered (Table 5). These result could be explained with the much higher number of fires occurred in these regions in the first 10 years (1999–2009) of the 20-years period (Fig. 1). The reasons

1.03

4.31

2.94

8.33 13.61 12.57 10.39 1.35 29.82 31.71 9.29

1.83

0.30

1.63

1.00

Pazardzhik

Plovdiv

Ruse

Sliven

Smolyan

Sofia

Stara Zagora

Shumen

4.65

1.55

2.13 1.76

1.80

5.23

1.12

5.31

0.54

1.81 2.66

2.12

4.65 2.89

1.48

6.40 2.97

0.37 0.58

0.84 0.79

0.80 28.54 9.92 3.75

2.31 0.38

1.25 5.10 6.53

1.24 4.19

4.32 7.55

0.82

1.22

1.63 0.25

8.87 1.42

2006

5.67

2001

2002

2003

2004

2.90 2.62

4.55 0.78 5.52

5.53 5.39

2.13 17.67 0.92 3.17 22.99 5.81

1.98

6.41 1.40 4.51

9.42

33.43

16.39

5.90

2.80

0.35

7.66 6.53 11.38 1.27 99.06 64.00 159.00

8.28

12.16

18.01

11.51

1.21

3.07

3.08

9.36

2.20

3.83 1.93

34.89 9.57 5.90 2.29 2.00 3.57

3.30 7.02

1.48 1.27 6.78 10.74 9.80

8.30 3.93 7.53

12.37

29.17

18.47

7.25 2.26 8.45 30.14 13.47 35.06

6.07 2.29 2.80

0.00

14.33

84.50

67.58

59.53

80.30 27.73 2.90 4.65 24.78 2.72 77.05 111.48 62.82

18.61

8.28

3.40

76.50

1999

0.10

2.97

0.45 0.76

1.30

0.88 1.35

2.24 4.60

4.23

14.69 2.37

7.30

1.03 4.83 116.24 4.41

0.46 2.00

8.58 0.60 6.22

2.47 0.66 5.14

2.35 18.73 8.46 6.75

6.92

6.32 21.59 65.67

7.57 7.40 2.07 22.92 3.88

0.46 2.83 1.33

13.50

10.02

22.70

134.83

1.88 11.83 16.93 66.71 24.25 3.50 9.83 26.20 1.83 42.52 120.27 62.73

3.48

3.36 12.28 5.51

8.93 2.65 10.85 3.83

1.57 4.91

2.02 23.53 9.74

2.14 7.93 13.28 11.53 4.76

1.50 13.73 6.31 8.43

0.62 5.50

1.68 6.24

9.95 8.29 20.58 6.22 8.90

2.57 10.64 14.28 5.39 10.06 12.57 92.85 7.15 8.37

0.23

9.26 20.46 0.87

2.53

4.33

4.56

2.80

5.84

33.37 4.84 13.43 40.88 11.64 10.06 27.71 32.83 47.63 15.57 18.63 40.82

1.11

1.15 3.41

Lovech

2.41

4.41 38.41 5.60

1.31

1.26

2.59

1.72 13.82 0.80 0.79

3.22

7.55 7.46

3.08 13.86 38.42 16.42 13.93 4.29 12.20 23.58

1.68 2.51

0.47

8.45

0.95

5.32

2000

55.44 15.33 7.61 6.54 21.97 30.78 17.79 23.33

Kyustendil

2018

Kardzhali

2017

1.65

2016

1.26

2015

Veliko Tarnovo

2014

Varna

2013

2.45 16.82 11.06 7.02

2012

2.81 24.20 1.14

2011

Burgas

2010

34.93 15.31 5.10 27.89 5.66 6.10 35.02 22.81 24.00 7.89 8.35

2009

Blagoevgrad

2008

Burned area/ Number of fires 2007

Berkovitsa

RFD 2005

Table 6. Input IM for ICrA over 20-year period (1999–2018)

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250

D. Zoteva et al. Table 7. ICrA results for the period 1999–2018

for these fires are not simply the meteorological conditions, but also some unpredictable and uncontrollable human factors. The Lubenov’s methodology results for the whole 20-year period have confirmed also some of the results of ICrA applied to the numbers of the occurred fires and the corresponding sizes of burned territories in each of the RFDs for the period 2009–2018. Some regions, like Pazardzhik, Plovdiv, Shumen and Varna have shown there ambiguity when they have been classified [12]. These results are shown again here in Table 9 for completeness.

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Table 8. Definition of consonance and dissonance μC,C 

Meaning

[0.00–0.05] Strong negative consonance (0.05–0.15] Negative consonance (0.15–0.25] Weak negative consonance (0.25–0.33] Weak dissonance (0.33–0.43] Dissonance (0.43–0.57] Strong dissonance (0.57–0.67] Dissonance (0.67–0.75] Weak dissonance (0.75–0.85] Weak positive consonance (0.85–0.95] Positive consonance (0.95–1.00] Strong positive consonance

Table 9. Results for fire risk according to the ICrA (2009–2018) [12] RFD

Degree of forest fires risk

Sofia, Ruse, Veliko Tarnovo, Smolyan, Blagoevgrad

Low

Pazardzhik, Plovdiv, Shumen, Varna

Low/Average

Kyustendil, Kardzhali, Berkovitsa, Burgas, Sliven

Average

Lovech

High

Stara Zagora

Very high

Pazardzhik and Plovdiv, classified in the group of low risk of occurring of forest fires by the Lubenov’s methodology for the 10-year period (Table 4), rather should have been classified in the average risk group (Table 9). This inference has been confirmed by the results of Lubenov’s methodology applied to the data for 20-year period (Table 5). Shumen has been confirmed to be part of the low risk group (Table 5) as in the case of data on the 10-year period (Table 4). There has been uncertainty about Varna being classified in the low risk group rather than the average in the case of 10-year period data (Table 9). Now, when all 20 years are taken in consideration, Varna is classified again in the average risk group (Table 5). The results of ICrA for the first period 2009–2018 has shown some doubts about the belonging of Veliko Tarnovo to the group of low risk of occurring of forest fires [12], since this region has shown some relations to the regions of Shumen, Stara Zagora and Lovech. The Lubenov’s methodology for the extended period 1999–2018 has confirmed this low risk group. Overall, the distribution of the rest of the RFD into forest fires risk groups has been preserved between the periods.

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The resulting degrees of “agreement” μC,C  in Table 7 show that Sliven is in weak positive consonance with Kardzhali. The reasoning behind this could be explained by the fact that the two RFD has been classified in the same group, the group of average risk of occurring of forest fires, when the 10-year period (2009–2018) is analyzed (Table 9). The bigger number of forest fires and the larger area of burned territories in the previous ten years (1999–2008) (Fig. 1 and Fig. 2) imply that these regions will be classified in groups with higher risk. According to the Lubenov’s methodology for the whole 20-year period (Table 5), Sliven is in the group of high risk, while Kardzhali moves to the group of very high risk. Sliven and Burgas are in weak positive consonance too, with a degree of “agreement” right in the middle of the interval according to Table 8. The reason is similar to the above. The weak positive consonance found between Shumen and Pazardzhik leads to the conclusion that Shumen should be considered as a region with average risk of fires rather than low. Here, when the whole 20-year period is considered, a hesitation about the belonging of Shumen in the group of low risk arises again, similarly to the results of the 10-year period (Table 9). ICrA has shown weak positive consonance also for Blagoevgrad and Kyustendil. There is a clear tendency of increasing the relation between those two regions in comparison to the results of the 10-year period, weak dissonance is found there [12]. However, there are still insufficient data to consider Blagoevgrad as a region with average risk of occurring of forest fires. The classification of the RFD, refined and confirmed by the results of the ICrA, is presented in Table 10. The RFD Blagoevgrad and Shumen should be examined with caution. The measures which are usually taken to prevent fire occurring should be applied to these regions as well as the regions in the group of average risk of forest fires. Table 10. Results for fire risk according to the ICrA (1999–2018) RFD

Degree of forest fires risk

Sofia, Ruse, Blagoevgrad, Veliko Tarnovo

Low

Shumen

Low/Average

Burgas, Kyustendil, Pazardzhik, Plovdiv, Smolyan, Varna

Average

Sliven

High

Berkovitsa, Kardzhali, Lovech, Stara Zagora

Very high

6 Conclusion Forest fires are one of the most common causes of forest destruction, which causes hazards that have severe and long-term impact on human life. The present research uses data on the number of forest fires occurred in Bulgaria, as well as the size of the burned territories. The research period are the past 20 years, 1999– 2018. The regions in Bulgaria are classified in a predefined groups of forest fires risk,

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consequently using two essentially different methodologies for the analysis – Lubenov’s methodology and InterCriteria Analysis. ICrA is used as a complementary approach to the Lubenov’s methodology in order to refine and improve the classification of the regions in Bulgaria into forest fires risk groups. ICrA is able to identify regions which are classified in a particular risk group with ambiguity. Thus, certain areas identified as out of risk will not be overlooked. Acknowledgement. The paper is partially funded by the Project DN16/6/2017 “Integrated Approach for Modeling of the Forest Fires Spread” funded by the National Science Fund of Bulgaria.

References 1. Atanassov, K.T.: Index Matrices: Towards an Augmented Matrix Calculus. Studies in Computational Intelligence, vol. 573. Springer, Cham (2014). https://doi.org/10.1007/978-3-31910945-9 2. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Heidelberg (2012). https://doi. org/10.1007/978-3-642-29127-2 3. Atanassov, K., Atanassova, V., Gluhchev, G.: InterCriteria analysis: ideas and problems. Notes Intuit. Fuzzy Sets 21(1), 81–88 (2015) 4. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes on Intuit. Fuzzy Sets 19(3), 1–13 (2013) 5. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues IFSs GNs 11, 1–8 (2014) 6. Atanassova, V., Doukovska, L., Atanassov, K. Mavrov, D.: Intercriteria decision making approach to EU member states competitiveness analysis. In: Proceedings of the International Symposium on Business Modeling and Software Design – BMSD 2014, pp. 289–294 (2014) 7. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData – software for InterCriteria analysis. Int. J. Bioautom. 22(1), 1–10 (2018) 8. Ilkova, T., Petrov, M.: Application of intercriteria analysis to the Mesta river pollution modelling. Notes Intuit. Fuzzy Sets 21(2), 118–125 (2015) 9. Krawczak, M., Bureva, V., Sotirova, E., Szmidt, E.: Application of the InterCriteria decision making method to universities ranking. In: Atanassov, K.T., et al. (eds.) Novel Developments in Uncertainty Representation and Processing. AISC, vol. 401, pp. 365–372. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-26211-6_31 10. Lubenov, K.: Methodology for determining the risk of forest fires on the territory of Bulgaria for the purposes of measure 8 “Investments in forest areas – development and improvement of the viability of forests” (2014–2020), Final Report PD 10-80/14.08.2015 (2015). www. iag.bg/data/docs/Metodika_16.doc 11. Pencheva, T., Angelova, M., Vassilev, P., Roeva, O.: Intercriteria analysis approach to parameter identification of a fermentation process model. In: Atanassov, K.T., et al. (eds.) Novel Developments in Uncertainty Representation and Processing. AISC, vol. 401, pp. 385–397. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-26211-6_33 12. Zoteva, D., Roeva, O., Delkov, A., Tsakov, H.: Intercriteria analysis of forest fire risk. In: Proceedings of the SYMCOMP 2019, Porto, 11–12 April 2019, pp. 215–229 (2019) 13. Bulgarian Executive Forest Agency, Annual Reports. http://www.iag.bg/docs/lang/1/cat/1/ index

Generalized Net Model of Overall Telecommunication System with Queuing Velin Andonov(B) , Stoyan Poryazov, and Emiliya Saranova Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria {velin andonov,stoyan,emiliya}@math.bas.bg

Abstract. Generalized Net (GN) model of overall telecommunication system with queuing is proposed. It is based on the classical conceptual model of overall telecommunication system which considers user’s behaviour, finite number of users and terminals, losses due to abandoned and interrupted dialing, blocked and interrupted switching, unavailable intent terminal, blocked and abandoned ringing and abandoned communication. A queuing system with finite capacities of the server and buffer and FIFO discipline of service of the requests is included in the Switching stage. The proposed model can be used in the analytical modeling of overall telecommunication systems. Keywords: Generalized Nets Conceptual modeling

1

· Overall telecommunication system ·

Introduction

The classical conceptual model of overall telecommunication system considers user’s behaviour, finite number of users and terminals, losses due to abandoned and interrupted dialing, blocked and interrupted switching, unavailable intent terminal, blocked and abandoned ringing and abandoned communication. The traffic of the calling (denoted by A) and the called (denoted by B) terminals and user’s traffic are considered separately but in their interrelation. It is described in [7] and developed in more details in [8]. The model uses the concepts of Service Systems Theory. Generalized Nets (GNs, see [3]) have been used as an alternative approach to the conceptual modeling of telecommunication systems starting with the paper [6]. More recently, a GN model of the Switching stage of overall telecommunication network is proposed in [9] and it is compared to a model based on Service Systems Theory. The research on the GNs as a tool for conceptual modeling of service systems continues in [10,12] where GN models of queuing systems are proposed. In [1], for the first time, a queuing system is included in a GN model of a part of overall telecommunication system. The first goal of the present paper is to construct a GN model of overall telecommunication system with queuing in the Switching stage. The second goal c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 254–279, 2021. https://doi.org/10.1007/978-3-030-77716-6_23

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is to demonstrate how the GN model can be used to derive some analytical expressions for the parameters of the system. In Sect. 2, the classical conceptual model of overall telecommunication system which is the base of the new GN model is presented briefly. The GN model of the overall telecommunication system with queuing is divided formally into four parts: GN model of the Dialing stage, GN model of the Switching stage, GN model of the Ringing stage and GN model of the Communication stage. They are described in Subsect. 3.1–3.4, respectively. In Sect. 4, classification of the parameters of the system is proposed and assumptions about the system are formulated. Analytical expression for the traffic intensity of the called terminals is derived.

2

Classical Conceptual Model of Overall Telecommunication System

The classical model of overall telecommunication system is proposed in [7] and developed in more details in [8]. It is a detailed conceptual traffic model of an overall (virtual) circuit switching telecommunication network, like PSTN and GSM, including users’ behaviour, with: BPP (Bernoulli-Poisson-Pascal) input flow; repeated calls; limited number of homogeneous terminals; losses due to abandoned and interrupted dialing, blocked and interrupted switching, not available intent terminal, blocked and abandoned ringing and abandoned communication. The described approach is applicable directly to every (virtual) circuit switching telecommunication system (like GSM and PSTN) and may help considerably for ISDN, BISDN and most of core and access networks traffic modelling. For packet switching systems, like Internet, the proposed approach may be used as a comparison basis. The traffic of the calling (denoted by A) and the called (denoted by B) terminals and user’s traffic are considered separately but in their interrelation. Two types of virtual devices are included in the model: base and comprise devices. 2.1

Base Virtual Devices Representation and Their Parameters

At the bottom of the structural model presentation, we consider base virtual devices that do not contain any other virtual devices. A base virtual device has a general graphical representation as shown in Fig. 1. The parameters of the base virtual device x are the following (see [5] for terms definitions): – Fx - intensity or incoming rate (frequency) of the flow of requests (i.e. the number of requests per time unit) to device x; – Px - probability of directing the requests towards device x; – Tx - service time (duration of service of a request) in device x; – Yx - traffic intensity [Erlang]; – Vx - traffic volume [Erlang - time unit]; – Nx - number of lines (service resources, positions, capacity) of device x.

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Fig. 1. Graphical representation of a base virtual device x.

For better understanding of the model and for a more convenient description of the intensity of the flow, a special notation including qualifiers (see [5]) is used. For example dem.F stands for demand flow; inc.Y stands for incoming traffic; of r.Y for offered traffic; rep.Y for repeated traffic. 2.2

Types and Names of the Base Virtual Devices

The graphic representations of the base virtual devices together with their names and types are shown in Fig. 2 (see [7]). Each base virtual device belongs to one of the following types: Generator, Terminator, Modifier, Server, Enter Switch, Switch and Graphic connector. With the exception of the Switch, which has one or two entrances and one or two exits, every other virtual device has one entrance and/or one exit. In the conceptual model, each virtual device has a unique name. The names of the devices are constructed according to their position in the model. The model is partitioned into service stages (dialing, switching, ringing and communication). Every service stage has branches (enter, abandoned, blocked, interrupted, not available, carried), corresponding to the modeled possible cases of ends of the calls’ service in the branch considered. Every branch has two exits (repeated, terminated) which show what happens with the calls if they can not be serviced by some device. Users may make a new bid (repeated call), or stop the attempts (terminated call). In the virtual device name construction, the corresponding bold first letters of the names of stages, branches and exits are used in the order shown below.

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Fig. 2. Classical conceptual model of an overall telecommunication system (see [7]).

Virtual Device Name = The parameter’s name of a virtual device is a concatenation of parameter’s name letter and virtual device name. For example, “Yid” means “traffic intensity in interrupted dialing case”; “Fid” means “flow (calls) intensity in interrupted dialing case”; “Pid” means “probability for interrupted dialing”; Tid = “mean duration of the interrupted dialing”; “Frid” = “intensity of repeated flow calls, caused by (after) interrupted dialing”. 2.3

Comprise Virtual Devices

The following comprise virtual devices denoted by a, b, s (see Fig. 2) and ab (not shown in Fig. 2) are considered in the model. – a comprises all calling terminals (A-terminals) in the system. It is shown with continuous line box in Fig. 2; – b comprises all called terminals (B-terminals) in the system. It is shown in box with dashed line in the down right corner in Fig. 2; – ab comprises all the terminals (calling and called) in the system. It is not shown in Fig. 2; – s virtual device corresponding to the switching system. It is shown with dashed line box into the a-device in Fig. 2.

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Generalized Net Model of Overall Telecommunication System with Queuing

Using the classical model of overall telecommunication system described in the previous section and the GN representations of the basic elements of Service Systems Theory proposed in [2], we shall construct a GN model of overall telecommunication system with queuing. The buffer of the queuing system we denote by ws (from “waiting for switch”). The queuing system considered can be represented in Kendall’s notation (see [4,11]) as M |M |Ns|Ns + Nws|Nab|F IF O, where M stands for exponential distribution, Ns is the capacity of the Switching system (number of equivalent internal switching lines), Nws is the capacity of the buffer device and Nab is the total number of active terminals which can be calling and called. This is related to the analytical modeling of the system. The GN consists of 29 transitions and 77 places. Among these 77 places, 28 correspond to base virtual devices of the classical model. Their labels are in the form lx where “x” is a name of a base virtual device (see Fig. 2). Due to the huge number of transitions and places, the GN model is formally divided into 4 parts corresponding to the four stages of the classical model: Dialing, Switching, Ringing and Communication. 3.1

Generalized Net Model of the Dialing Stage

The GN model of the Dialing is shown in Fig. 3. It consists of 7 transitions and 25 places. The transitions represent functions of base virtual devices as follows: – – – – – – –

Z1 Z2 Z3 Z4 Z5 Z6 Z7

represents represents represents represents represents represents represents

the the the the the the the

function of cd device; function of function of function of function of function of

the Modifier from Fig. 2; the the the the the

ad device; rad device; id device; rid device; cd device.

Among the 25 places, 7 are not shown in Fig. 3. They are output places of transitions from the other three stages as follows: – places l15 , l19 and l23 are output places of transitions of the Switching stage (see Fig. 4, Sect. 3.2); – places l28 and l33 are output places of transitions of the Ringing stage (see Fig. 5, Sect. 3.3); – places l44 and l47 are output places of transitions of the Communication stage (see Fig. 6, Sect. 3.4).

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The places corresponding to base virtual devices are lF o , led , lad , lrad , lid , lrid , lcd . In place lF o token of type α stays in the initial time moment with characteristic “formula for generating the demand flow of requests”. Tokens αed , αad , αrad , αid , αrid , αcd stay in the initial time moment in places led , lad , lrad , lid , lrid , lcd , respectively. Each of them has initial characteristic “initial values of Yx , Px , Fx , Tx , Nx ”, where x is the name of the corresponding base virtul device.

Fig. 3. Generalized net model of the Dialiang stage.

Below is a formal description of the transitions. Z1 = {lF o }, {l1 , lF o }, r1 , where r1 =

l1 lF o lF o WF o,1 true

and – WF o,1 = “A demand flow of requests is generated”. When the truth value of the predicate WF o,1 is “true”, the α token splits into two tokens – the same α token which stays in place lF o and α1 which enters place l1 with characteristic “intensity of the demand flow of requests”. The token α

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in place lF o obtains the characteristic “current formula for generation of the demand flow of requests”. Z2 = {l1 , l6 , l10 , l15 , l19 , l23 , l28 , l33 , l44 , l47 , led }, {l2 , l3 , led }, r2 , where l1 l6 l10 l15 l r2 = 19 l23 l28 l33 l44 l47 led

l2 f alse f alse f alse f alse f alse f alse f alse f alse f alse f alse Wed,2

l3 f alse f alse f alse f alse f alse f alse f alse f alse f alse f alse Wed,3

led true true true true true true true true true true true

and – Wed,2 = “the current request is not abandoned (with a given probability)”; – Wed,3 = “the current request is abandoned (with a given probability)”. The tokens from all input places of the transition Z2 merge with the token αed in place led . Depending on the truth values of the predicates Wed,2 and Wed,3 token αed may split into two or three tokens. If both predicates have truth value “true”, it splits into three tokens – the same αed , α1 and α2 . Token αed obtains the characteristic “current value of Yed ”. Tokens α1 and α2 enter places l2 and l3 respectively without obtaining new characteristic. Z3 = {l3 , lad }, {l4 , lad }, r3 , where

l4 lad r3 = l3 f alse true . lad true true

The α2 token from place l3 merges with the αad token in place lad . The αad token splits into two tokens – the same αad token which stays in playce lad with characteristic “current value of Yad ” and token α3 which enters place l4 without obtaining new characteristic. Z4 = {l4 , lrad }, {l5 , l6 , lrad }, r4 , where r4 = l4 lrad and

l5 l6 lrad W4,5 f alse W4,rad , f alse true true

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– W4,5 = “the current call is terminated (with a given probability)”; – W4,rad = “the current call is repeated (with a given probability)”. When the truth value of predicate W4,5 is “true” token α3 enters place l5 without obtaining new characteristic. When the truth value of W4,rad is “true” token α3 enters place lrad where it merges with token αrad . Token αrad splits into two tokens – the same αrad which remains in place lrad with characteristic “current value of Yrad ” and token α4 which enters place l6 without obtaining new characteristic. Z5 = {l2 , lid }, {l7 , l8 , lid }, r5 , where

l7 l8 lid r5 = l2 W2,7 f alse W2,id , lid f alse true true

and – W2,7 = “the current call is carried (with a given probability)”; – W2,id = “the current call is interrupted (with a given probability)”. When the truth value of predicate W2,7 is “true” token α1 enters place l7 without obtaining new characteristic. When the truth value of W2,id is “true” token α1 enters place lid where it merges with token αid . Token αid splits into two tokens – the same αid which remains in place lid with characteristic “current value of Yid ” and token α5 which enters place l8 without obtaining new characteristic. Z6 = {l8 , lrid }, {l9 , l10 , lrid }, r6 , where

l9 l10 lrid r6 = l8 W8,9 f alse W8,rid , lrid f alse true true

and – W8,9 = “the current call is terminated (with a given probability)”; – W8,rid = “the current call is repeated (with a given probability)”. When the truth value of predicate W8,9 is “true” token α5 enters place l9 without obtaining new characteristic. When the truth value of W8,rid is “true” token α5 enters place lrid where it merges with token αrid . Token αrid splits into two tokens – the same αrid which remains in place lrid with characteristic “current value of Yrid ” and token α6 which enters place l10 without obtaining new characteristic. Z7 = {l7 , lcd }, {l11 , lcd }, r7 , where

l11 lcd r7 = l7 true true . lcd f alse true

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Token α1 from l7 splits into two tokens one of which merges with token αcd in place lcd and the other one enters place l11 without obtaining new characteristic. Token αcd obtains the characteristic “current value of Ycd ”. 3.2

Generalized Net Model of the Switching Stage

The graphical representation of the GN model of the Switching stage with queuing is shown in Fig. 4.

Fig. 4. Generalized net model of the Switching stage of an overall telecommunication system with queuing.

It consists of 8 transitions and 23 + Nws places where Nws is the capacity of the buffer of the queuing system. The transitions represent functions of base virtual devices as follows: – Z8 represents the function of the Enter Switch device before the comprise virtual device s in Fig. 2; – Z9 represents the function of Switch device after the bws device. The bws is a new device analogous to the bs device in Fig. 2; – Z10 represents the function of the buffer device; – Z11 represents the function of the Switch device before the is device; – Z12 represents the function of the Switch device after the is device; – Z13 represents the function of the Switch device before the cs device; – Z14 represents the function of the Switch device after the ns device. – Z15 represents the function of the cs device.

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Among the 23 + Nws places, 8 places correspond to base virtual devices. These are the places with labels lbws , lrbws , lws , lis , lris , lns , lrns , lcs . Tokens αbws , αrbws , αws , αis , αris , αns , αrns , αcs stay in the initial time moment in places lbws , lrbws , lws , lis , lris , lns , lrns , lcs , respectively. Each of them has initial characteristic “initial values of Yx , Px , Fx , Tx , Nx ”, where x is the name of the corresponding base virtul device. The queuing system in the Switching stage is represented in the GN by transitions Z10 and Z11 . There are other possible representations of queuing systems by GNs (see [10,12]). The one presented here shows explicitly the two ways of servicing of the requests by the queuing system – with waiting and without waiting. When the Switching system has not reached its capacity the tokens enter the place lzw (zero waiting) which corresponds to the service without waiting. If the Switching system has reached its capacity, the tokens enter some of the places l10,1 , ..., l10,N ws which corresponds to the waiting in the buffer. Three of the places: l15 , l19 and l23 , are input places of transition Z2 from the Dialing stage (see Fig. 3 from Sect. 3.1). This is shown in the graphical representation in Fig. 4 by arcs which leave these places but do not end in transitions. Below is a formal description of the transitions of the GN model of the Switching stage. Z8 = {l11 , lbws }, {l12 , l13 , lbws }, r8 , where

l12 l13 lbws r8 = l11 W11,12 f alse W11,bws , lbws f alse true true

and – W11,12 = “Yws < Nws”; – W11,bws = ¬W11,12 . When the truth value of predicate W11,12 is “true” token α1 enters place l12 without obtaining new characteristic. When the truth value of W11,bws is “true” token α1 enters place lbws where it merges with token αbws . Token αbws splits into two tokens – the same αbws which remains in place lbws with characteristic “current value of Y bws” and token α7 which enters place l13 without obtaining new characteristic. Z9 = {l13 , lrbws }, {l14 , l15 , lrbws }, r9 , where

l14 l15 lrbws r9 = l13 W13,14 f alse W13,rbws , lrbws f alse true true

and – W13,14 = “the current request is terminated (with given probability)”;

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– W13,rbws = “the current request becomes repeated request (with given probability)”. When the truth value of predicate W13,14 is “true” token α7 enters place l14 without obtaining new characteristic. When the truth value of W13,rbws is “true” token α7 enters place lrbws where it merges with token αrbws . Token αrbws splits into two tokens – the same αrbws which remains in place lrbws with characteristic “current value of Yrbws” and token α8 which enters place l15 without obtaining new characteristic. Z10 = {l12 , lws }, {lzw , l10,1 , ..., l10,N ws , lws }, r10 , where

lzw l10,1 ... l10,N ws lws r10 = l12 W12,zw W12,101 ... W12,10N ws W12,ws , lws f alse f alse ... f alse true

and – W12,zw = “Ys < Ns”; – W12,10i = ¬W12,zw & “place l10,i is the highest priority empty place among places l10,1 , ..., l10,N ws ” for i = 1, 2, ..., N ws; – W12,ws = W12,101 ∨ W12,102 ∨ .... ∨ W12,10N ws . When the truth value of predicate W12,zw is “true” token α1 enters place lzw without obtaining new characteristic. When the truth value of the predicate W12,10i for i = 1, 2, ..., N ws is “true” token α1 splits into as many tokens as the number of the predicates with truth value “true” which enter the corresponding buffer places without new characteristic, and a token α9 which enter place lws where it merges with token αws . Token αws splits into two tokens – the same αws which remains in place lws with characteristic “current value of Yws; list of all tokens in places l10,1 , ..., l10,N ws and the duration of tokens stay in the places”. Z11 = {lzw , l10,1 , ..., l10,N ws , lis }, {l16 , l17 , lis }, r11 , where

r11 =

lzw l10,1 .. . l10,N ws lis

l16 l17 lis Wzw,16 f alse Wzw,is W101,16 f alse W101,is , .. .. .. . . . W10N ws,16 f alse W10N ws,is f alse true true

and – Wzw,is = “the current request is interrupted (with a given probability)”; – Wzw,16 = ¬Wzw,is ;

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– W10i,is = “the current request is interrupted (with a given probability)” & “the token in place l10i,is has stayed in the place more time compared to the tokens in the rest of the input places l10,1 , l10,2 , ..., l10,N ws ” for i = 1, 2, ..., Nws; – W10i,16 = “the current request is not interrupted (with a given probability)” & “the token in place l10,i has stayed in the place more time compared to the tokens in the rest of the input places l10,1 , l10,2 , ..., l10,N ws ” for i = 1, 2, ..., Nws; When the truth value of predicate Wzw,16 is “true” token α1 from place lzw enters place l16 without obtaining new characteristic. When the truth value of some of the predicates W101,16 , W102,16 , ..., W10N ws,16 is “true” the corresponding α1 tokens enter place l16 without obtaining new characteristic. When the truth value of the predicate Wzw,is is “true” token α1 from place lzw enters place lis where it merges with the αis token. When the truth value of some of the predicates W10i,is for i = 1, 2, ..., N ws is “true” the corresponding token α1 enters place lis where it merges with the αis token. Token αis splits into two tokens – the same αis which remains in place lis with characteristic “current value of Yis” and token α9 which enters place l17 without obtaining characteristic. Z12 = {l17 , lris }, {l18 , l19 , lris }, r12 , where

l18 l19 lris r12 = l17 W17,18 f alse W17,ris lris f alse true true

and – W17,18 = “the current request is terminated (with a given probability)”; – W17,ris = “the current request becomes repeated request (with a given probability)”. When the truth value of predicate W17,18 is “true” token α9 from place l17 enters place l18 without obtaining new characteristic. When the truth value of W17,ris is “true” token α9 enters place lris where it merges with token αris . Token αris splits into two tokens – the same αris which remains in place lris with characteristic “current value of Yris” and token α10 which enters place l19 without obtaining new characteristic. Z13 = {l16 , lns }, {l20 , l21 , lns }, r13 , where

and

l20 l21 lns r13 = l16 W16,20 f alse W16,ns , lns f alse true true

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– W16,ns = “not available switching for the current request (with a given probability)”; – W16,20 = ¬W16,ns . When the truth value of predicate W16,20 is “true” token α1 from place l16 enters place l20 without obtaining new characteristic. When the truth value of W16,ns is “true” token α1 enters place lns where it merges with token αns . Token αns splits into two tokens – the same αns which remains in place lns with characteristic “current value of Yns” and token α11 which enters place l21 without obtaining new characteristic. Z14 = {l21 , lrns }, {l22 , l23 , lrns }, r14 , where

l22 l23 lrns r14 = l21 W21,22 f alse W21,rns lrns f alse true true

and – W21,22 = “the current request is terminated (with a given probability)”; – W21,rns = “the current request becomes repeated request (with a given probability)”. When the truth value of predicate W21,22 is “true” token α11 from place l21 enters place l22 without obtaining new characteristic. When the truth value of W21,rns is “true” token α11 enters place lrns where it merges with token αrns . Token αrns splits into two tokens – the same αrns which remains in place lrns with characteristic “current value of Yrns” and token α12 which enters place l23 without obtaining new characteristic. Z15 = {l20 , lcs }, {l24 , lcs }, r15 , where

l24 lcs r15 = l20 true true . lcs f alse true

Token α1 from place l20 splits into two tokens – the same α1 which enters place l24 without obtaining new characteristic and token α13 which enters place lcs where it merges with token αcs . Token αcs obtains the characteristic “current value of Ycs”. 3.3

Generalized Net Model of the Ringing Stage

The graphical representation of the GN model of the Ringing stage is shown in Fig. 5. It consists of 10 transitions and 27 places. The transitions represent functions of base virtual devices as follows:

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– Z16 represents the function of the Enter Switch device after the base virtual device cs in Fig. 2; – Z17 represents the function of Switch device after the br device in Fig. 2; – Z18 represents the function of the COPY device; – Z19 represents the function of the Switch device after the COPY device; – Z20 represents the function of the Switch device after the ar device; – Z21 represents the function of the cr device at the end of the Ringing stage in Fig. 2; – Z22 represents the function of the Switch device at the beginning of the comprise virtual device b of the called terminals in Fig. 2; – Z23 represents the function of the cr device in the comprise virtual device b; – Z24 represents the function of the Switch device after the cr device in the comprise virtual device b; – Z25 represents the function of the cc device in the comprise virtual device b. Among the 27 places, 9 places correspond to base virtual devices. These are     , lcr , lac , lcc . Tokens αbr , αrbr , αar , the places with labels lbr , lrbr , lar , lrar , lcr , lar     αrar , αcr , αar , αcr , αac , αcc stay in the initial time moment in places lbr , lrbr , lar ,     , lcr , lac , lcc , respectively. Each of them has initial characteristic “inilrar , lcr , lar tial values of Yx , Px , Fx , Tx , Nx ”, where x is the name of the corresponding base virtual device. Transitions Z22 , Z23 , Z24 and Z25 represent the comprise virtual device b of the called terminals. Although Z24 and Z25 are part of the Communication stage, they are included in the Ringing stage to avoid unnecessary complication of the graphical representation of the GN model. Places l28 and l33 are input places of transition Z2 from the Dialing stage (see Fig. 3 from Sect. 3.1). This is shown in the graphical representation in Fig. 5 by arcs which leave these places but do not end in transitions. Below is the formal description of the transitions. Z16 = {l24 , lbr }, {l25 , l26 , lbr }, r16 , where

l25 l26 lbr r16 = l24 W24,25 f alse W24,br lbr f alse true true

and – W24,br = “the current request is blocked (with a given probability)”; – W24,25 = ¬W24,br . When the truth value of predicate W24,25 is “true” token α1 from place l24 enters place l25 without obtaining new characteristic. When the truth value of W24,br is “true” token α1 enters place lbr where it merges with token αbr . Token αbr splits into two tokens – the same αbr which remains in place lbr with characteristic “current value of Ybr ” and token α14 which enters place l26 without obtaining new characteristic.

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Fig. 5. Generalized net model of the Ringing stage of an overall telecommunication network with queuing.

Z17 = {l26 , lrbr }, {l27 , l28 , lrbr }, r17 , where

l27 l28 lrbr r17 = l26 W26,27 f alse W26,rbr lrbr f alse true true

and – W26,rbr = “the current request is repeated (with given probability)”; – W26,27 = “the current request is terminated (with given probability)”. When the truth value of predicate W26,27 is “true” token α14 from place l26 enters place l27 without obtaining new characteristic. When the truth value of W26,rbr is “true” token α14 enters place lrbr where it merges with token αrbr . Token αrbr splits into two tokens – the same αrbr which remains in place lrbr with characteristic “current value of Yrbr ” and token α15 which enters place l28 without obtaining new characteristic. Z18 = {l25 }, {l29 , lcopy }, r18 ,

Generalized Net Model of Overall Telecommunication

where r18 =

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l29 lcopy . l25 true true

Token α1 from place l25 splits into two identical tokens – the same token α1 and token α16 – which enter places l29 and lcopy , respectively, without obtaining new characteristics. Z19 = {l29 , lar }, {l30 , l31 , lar }, r19 , where

l30 l31 lar r19 = l29 W29,30 f alse W29,ar lar f alse true true

and – W29,ar = “the current request is abandoned (with given probability)”; – W29,30 = ¬W29,ar . When the truth value of predicate W29,30 is “true” token α1 from place l29 enters place l30 without obtaining new characteristic. When the truth value of W29,ar is “true” token α1 enters place lar where it merges with token αar . Token αar splits into two tokens – the same αar which remains in place lar with characteristic “current value of Yar ” and token α17 which enters place l31 without obtaining new characteristic. Z20 = {l31 , lrar }, {l32 , l33 , lrar }, r20 , where

l32 l33 lrar r20 = l31 W31,32 f alse W31,rar lrar f alse true true

and – W31,rar = “the current request is repeated (with given probability)”; – W31,32 = “the current request is terminated (with given probability)”. When the truth value of predicate W31,32 is “true” token α17 from place l31 enters place l32 without obtaining new characteristic. When the truth value of W31,rar is “true” token α17 enters place lrar where it merges with token αrar . Token αrar splits into two tokens – the same αrar which remains in place lrar with characteristic “current value of Yrar ” and token α18 which enters place l33 without obtaining new characteristic. Z21 = {l30 , lcr }, {l34 , lcr }, r21 , where

l34 lcr r21 = l30 true true . lcr f alse true

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Token α1 from place l30 splits into two tokens – the same α1 which enters place l34 without new characteristic and token α19 which enters place lcr where it merges with token αcr . Token αcr obtains the characteristic “current value of Ycr ”.   }, {l35 , l36 , lar }, r22 , Z22 = {lcopy , lar

where r22 = lcopy  lar

 l36 lar Wcopy,35 f alse Wcopy,ar , f alse true true

l35

and – Wcopy,ar = “the current request is abandoned (with given probability)”; – Wcopy,35 = ¬Wcopy,ar . When the truth value of predicate Wcopy,35 is “true” token α16 from place lcopy enters place l35 without obtaining new characteristic. When the truth value   where it merges with token αar . of Wcopy,ar is “true” token α16 enters place lar    Token αar splits into two tokens – the same αar which remains in place lar with characteristic “current value of Yar ” and token α20 which enters place l36 without obtaining new characteristic.   }, {l37 , lcr }, r23 , Z23 = {l35 , lcr

where r23 = l35  lcr

 l37 lcr true true . f alse true

Token α16 from place l35 splits into two tokens – the same α16 which enters  where place l37 without new characteristic and token α21 which enters place lcr   it merges with token αcr . Token αcr obtains the characteristic “current value of Ycr ”.   }, {l38 , l39 , lac }, r24 , Z24 = {l37 , lac where r24 = l37  lac

 l38 l39 lac W37,38 f alse W37,ac f alse true true

and – W37,ac = “the current request is abandoned (with given probability)”; – W37,38 = ¬W37,ac . When the truth value of predicate W37,38 is “true” token α16 from place l37 enters place l38 without obtaining new characteristic. When the truth value   where it merges with token αac . of W37,ac is “true” token α16 enters place lac

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   Token αac splits into two tokens – the same αac which remains in place lac with characteristic “current value of Yar ” and token α22 which enters place l39 without obtaining new characteristic.   Z25 = {l38 , lcc }, {l40 , lcc }, r25 ,

where r25 = l38  lcc

 l40 lcc true true . f alse true

Token α16 from place l38 splits into two tokens – the same α16 which enters  where place l40 without new characteristic and token α23 which enters place lcc   it merges with token αcc . Token αcc obtains the characteristic “current value of Ycc”. 3.4

Generalized Net Model of the Communication Stage

The graphical representation of the GN model of the Communication stage of an overall telecommunication network with queuing is shown in Fig. 6. It consists of 4 transitions and 12 places. The transitions represent functions of base virtual devices as follows: – Z26 represents the function of the Switch device after the base virtual device cr in Fig. 2;

Fig. 6. Generalized net model of the Communication stage of an overall telecommunication system with queuing.

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– Z27 represents the function of Switch device after the ac device in Fig. 2; – Z28 represents the function of the cc device in Fig. 2; – Z29 represents the function of the Switch device after the cc device. Among the 12 places, 4 places correspond to base virtual devices. These are the places with labels lac , lrac , lcc , lrcc . Tokens αac , αrac , αcc , αrcc stay in the initial time moment in places lac , lrac , lcc , lrcc respectively. Each of them has initial characteristic “initial values of Yx , Px , Fx , Tx , Nx ”, where x is the name of the corresponding base virtual device. Places l44 and l47 are input places of transition Z2 from the Dialing stage (see Fig. 3 from Sect. 3.1). This is shown in the graphical representation in Fig. 6 by arcs which leave these places but do not end in transitions. Below follows the description of the transitions. Z26 = {l34 , lac }, {l41 , l42 , lac }, r26 , where

l41 l42 lac r26 = l34 W34,41 f alse W34,ac lac f alse true true

and – W34,ac = “the current request is abandoned (with a given probability)”; – W34,41 = ¬W34,ac . When the truth value of predicate W34,41 is “true” token α1 from place l34 enters place l41 without obtaining new characteristic. When the truth value of W34,ac is “true” token α1 enters place lac where it merges with token αac . Token αac splits into two tokens – the same αac which remains in place lac with characteristic “current value of Yac” and token α24 which enters place l42 without obtaining new characteristic. Z27 = {l42 , lrac }, {l43 , l44 , lrac }, r27 , where

l43 l44 lrac r27 = l42 W42,43 f alse W42,rac lrac f alse true true

and – W42,43 = “the current request is terminated (with a given probability)”; – W42,rac = “the current request becomes repeated request (with a given probability)”. When the truth value of predicate W42,43 is “true” token α24 from place l42 enters place l43 without obtaining new characteristic. When the truth value of W42,rac is “true” token α24 enters place lrac where it merges with token αrac .

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Token αrac splits into two tokens – the same αrac which remains in place lrac with characteristic “current value of Yrac” and token α25 which enters place l44 without obtaining new characteristic. Z28 = {l41 , lcc }, {l45 , lcc }, r28 , where

l45 lcc r28 = l41 true true . lcc f alse true

Token α1 from place l41 splits into two tokens – the same α1 which enters place l45 without new characteristic and token α26 which enters place lcc where it merges with token αcc . Token αcc obtains the characteristic “current value of Ycc”. Z29 = {l45 , lrcc }, {l46 , l47 , lrcc }, r29 , where

l46 l47 lrcc r29 = l45 W45,46 f alse W45,rcc lrcc f alse true true

and – W45,46 = “the current request is terminated (with given probability)”; – W45,rcc = “the current request becomes repeated request (with given probability”. When the truth value of predicate W45,46 is “true” token α1 from place l45 enters place l46 without obtaining new characteristic. When the truth value of W45,rcc is “true” token α1 enters place lrcc where it merges with token αrcc . Token αrcc splits into two tokens – the same αrcc which remains in place lrcc with characteristic “current value of Yrcc” and token α27 which enters place l47 without obtaining new characteristic.

4

Analytical Modeling of the Overall Telecommunication System Using the GN Model

In the GN model described in the previous section the functions of at least 36 important virtual devices are included. Of them 32 are base virtual devices and 4 (a, b, s, ab) are comprise (not shown in the graphical representation). They are of interest because the values of their parameters characterize the state of the overall telecommunication system. Every device has five parameters: P, F, T, Y and N . Therefore the total number of parameters is 180.

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Static and Dynamic Parameters of the Model

In order to construct relatively simple analytical model of the overall telecommunication system, the notions of system tuple and base tuple have to be introduced. Definition 1. A system tuple is a finite set of distinguishable (by name and/or position) parameters’ values, which fulfills simultaneously the three following requirements: 1. All parameters (parameters’ set), evaluated by the system tuple, correspond to one considered (observed, modeled) system; 2. All the values of a system tuple correspondent to one and the same time interval of measurements or considerations; 3. The instant of beginning and duration of this time interval are elements of the system tuple set. Every subset of a system tuple is called subtuple. There are many obvious dependencies in a system tuple, corresponding to the full parameters’ set of the conceptual model. For example, the sum of probabilities of outgoing transitions in every virtual switch devices has value one; in stationary state Little’s formula (Y = F T ) can be applied to every virtual device; we assume that most of the devices have infinite capacity. As a result, there are sets of base parameters (sub-tuples), with the following property: if we know the values of the base parameters, we may calculate the values of all other parameters of the same system tuple. Several different base parameters’ sets may exist. After careful analysis and some assumptions (see below) we have chosen a base parameters’ set with 46 parameters. The values of these parameters we call base tuple. The base tuple is a sub-tuple of a system tuple. In this paper, we propose a short term classification of the chosen base parameters’ set into static and dynamic parameters. For the static parameters we assume that their values do not depend on the state of the system and correspondingly on the intensity of the input flow. They may depend on other factors, e.g. the time of the day; seasons, human temperament, Telecom Administration and so on, but for the observed and modeled time interval we consider them as constants. – The static parameters are: M, N ab, Ns, T ed, P ad, T ad, P rad, P id, T id, P rid, T cd, T bws, P rbws, P is, T is, P ris, P ns, T ns, T ∗ cs, P rns, T br, P rbr, P ar, T ar, P rar, T cr, P ac, T ac, P rac, T cc, P rcc. – The dynamic parameters, with mutually dependent values are: F o, Yab, F a, dem.F a, T cs, rep.F a, P bws, P br, of r.F ws, T ws, Y s, crr.F s, F s, T s. 4.2

Main Assumptions

Due to the complexity of the model, in order to obtain relatively simple analytical expressions about the parameters, after a careful analysis the following assumptions are formulated:

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A-1. (Closed System Structure) We consider a closed telecommunication system with functional structure shown in Fig. 3, 4, 5 and Fig. 6; A-2. (Device Capacity) All base virtual devices in the model have unlimited capacity. Comprise devices are limited: ab-device contains all the active terminals N ab ∈ [2, ∞]; switching system (s) has capacity of N s calls (every internal switching line may carry only one call); every terminal has capacity of one call, common for both incoming and outgoing calls; A-3. (A-Terminal Occupation) Every call, from the flow incoming in the telecommunication system (inc.F a), falls only on a free terminal. This terminal becomes a busy A-terminal; A-4. (Stationarity) The system is in stationary state. This means that in every virtual device in the model (including comprising devices like switching system), the intensity of input flow F (0, t), call holding time T (0, t) and traffic intensity Y (0, t) in the observed interval (0, t) converge to the correspondent finite numbers F, T and Y , when t → ∞. In this case we may apply the theorem of Little (1961) and for every device: Y = F T ; A-5. (Calls’ Capacity) Every call occupies one place in a base virtual device, independently from the other devices (e.g. a call may occupy one internal switching line, if it find free one, independently from the state of the intent B-terminal (busy or free)); A-6. (Environment) The calls in the communication systems’ environment don’t occupy any telecommunication systems’ device and therefore they do not create communication systems’ load. (For example, unsuccessful calls, waiting for the next attempt, are in “the head” of the user only. The calls and devices in the environment form the intent and repeated calls flows). Calls leave the environment (and the model) in the instance they enter a Terminator virtual device; A-7. (Parameters’ independence) We consider probabilities for direction of calls to, and holding times in the base virtual devices as independent of each other and from the intensity F a = inc.F a of the incoming flow of calls. The values of these parameters are determined by users’ behavior and technical characteristics of the communication system. (Obviously, this is not applicable to the devices of type Enter Switch, corresponding to P bws and P br.); A-8. (Randomness) All variables in the analytical model may be random and we are working with their mean values, following the Theorem of Little. A-9. (B-Terminal Occupation) Probabilities of direction of calls to, and duration of occupation of devices ar, cr, ac and cc are the same for A and B-calls; A-10. (Channel Switching) Every call occupies simultaneously places in all the base virtual devices in the telecommunication system (comprised of devices a or b) it passed through, including the base device where it is in the moment of observation. Every call releases all its occupied places in all base virtual devices of the communication system, in the instant it leaves comprising devices a or b.

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A-11. (Homogeinity of the terminals) All terminals are homogenous, i.e., for every terminal all corresponding characteristics are equal. A-12. (Direction of the A-calls) Every A-terminal generates all call attempts only towards other terminals, not towards itself. A-13. (Ordinadirity of the B-flow) The flow directed to the B-terminals (F b) is ordinary. The only exception is when two or more calls reach a free B-terminal simultaneously. A-14. (Probability of repeated calls blocking) The probability P br for finding the B-terminal busy is one and the same for the first and all of the next repeated attempts. A-15. The probabilities P ad, P id, P bws, P is, P ns, P br, P ar, P ac preserve their values during the repeated call attempts. A-16. The probabilities of entering of repeated call attempts due to abandoned dialing P rad, interrupted dialing P rid, interrupted switching P ris, not available switching P rns, abandoned ringing P rar, abandoned communication P rac, blocked waiting for switch (P rbws) and blocked ringing (P rbr) which characterize the users’ behavior also preserve their values. A-17. Parameters characterizing the users’ behavior are the mean service time of the separate devices: T ed, T ad, T id, T cd, T bws, T is, T ns, T ∗ cs, T br, T ar, T cr, T ac and T cc. They remain the same for the repeated call attempts. 4.3

Equation for the Traffic Intensity of the Called Terminals

We shall illustrate how equations for the dynamic parameters of the system can be obtained using the GN model and the assumptions. As an example, the traffic intensity of the called terminals is chosen. Theorem 1. The traffic intensity of the B-terminals (Yb) can be determined from the equation Yb = FbTb, (1) where F b is the flow intensity of the B-terminals and T b is the mean holding time of calls in a B-terminal, and: F b = F a(1 − P ad)(1 − P id)(1 − P bws)(1 − P is)(1 − P ns)(1 − P br), T b = P arT ar + (1 − P ar)[T cr + P acT ac + (1 − P ac)T cc].

(2) (3)

Proof: Equation (1) is the Little’s formula for device b in stationary state (A-4). Equation (2) expresses the fact that the A-calls have to avoid the six modeled losses before occupying the intent B-terminals, with mean intensity of calls F b. This is seen from the graphical representation of the GN models of the four stages. Equation (3) is direct corollary from the GN model in which the Bterminals are represented by transitions Z35 , Z36 , Z37 and Z38 , the closed system structure (A-1), calls’ capacity (A-5), excluding calls in the environment (A-6)

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parameters independance (A-7), randomness (A-8) and B-terminal occupation assumption (A-9). We may derive the expression (3) for the B-terminals holding time (T b) from the following considerations. From the graphical representation of the GN model, parameters independence (A-7) and channel switching (A-10), it follows that Yb is a sum of the traffics of the base blocks, comprised in it. The assumption for B-terminal occupation (A-9), implies that Yar , Ycr , Yac and Ycc are the same traffic intensities for A and B-terminals, so: Yb = Yar + Ycr + Yac + Ycc.

(4)

On the other hand, we may express the traffic intensities using the Little’s formula and presenting every flow intensity in the base devices as a function of Fb: Yar = F arT ar = F bP arT ar; (5) Ycr = F crT cr = F b(1 − P ar)T cr;

(6)

Yac = F acT ac = F b(1 − P ar)P acT ac;

(7)

Ycc = F ccT cc = F b(1 − P ar)(1 − P ac)T cc.

(8)

After replacing (5), (6), (7) and (8) in (4), taking in consideration A-9 and using (1) we obtain (3).  Equations for the rest of the dynamic parameters can be obtained in a similar way using the approach from [8].

5

Conclusions

The GN model of overall telecommunication system with queuing described in the present paper has the following important features: – The virtual devices are represented only by three objects: transitions, arcs and places of the GNs. – The repeated flows of requests are represented graphically by the arcs of the GN and the path of the requests can be easily tracked. – The graphical representation of the GN can be used for the derivation of equations about the parameters of the telecommunication system. Some problems related to the use of GNs as a tool for conceptual modeling of telecommunication systems which arise are: – The functions of the virtual devices are not evident from the graphical representation of the GN. They are described in the transitions’ components. This requires knowledge of the theory of the GNs in order to understand the model. – The problem for representation of comprise virtual devices in GNs should be studied. Different representations are possible: using additional transitions; additional places; including the parameters of the comprise devices as characteristics of specific tokens, etc.

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– The huge number of virtual devices requires the graphical representation of the GN model to be divided into smaller parts. Some of the problems above can be solved using extensions of the ordinary GNs. Acknowledgements. The work of Velin Andonov was partially funded by the Bulgarian NSF under grant DM 12/2 – “New Models of Overall Telecommunication Networks with Quality of Service Guarantees”. The work of Stoyan Poryazov is partially supported by the joint research project “Symbolic-Numerical Decision Methods for Algebraic Systems of Equations in Perspective Telecommunication Tasks” of IMI-BAS, Bulgaria and JINR, Dubna, Russia. The work of Emiliya Saranova was supported by the Task 1.2.5. “Prediction and Guaranteeing of the Quality of Service in Human-Cyber-Physical Systems” of National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICT in SES)” financed by the Bulgarian Ministry of Education and Science.

References 1. Andonov, V., Poryazov, S., Otsetova, A., Saranova, E.: Generalized net model of a part of an overall telecommunication system with a queue in the switching stage. In: Proceedings of the 16th International Workshop on Generalized Nets, Sofia, 10 February 2018, pp. 75–84 (2018) 2. Andonov, V., Poryazov, S., Saranova, E.: Generralized net representations of elements of service systems theory. Adv. Stud. Contemp. Math. 29(2), 179–189 (2019) 3. Atanassov, K.: On Generalized Nets Theory. Prof. M. Drinov Academic Publ. House, Sofia, Bulgaria (2007) 4. Haverkort, B.R.: Performance of Computer Communication Systems: A ModelBased Approach. Wiley, New York (1998) 5. ITU E.600, ITU-T Recommendation E.600: Terms and Definitions of Traffic Engineering, Melbourne, 1988; revised at Helsinki (1993) 6. Poryazov, S., Atanassov K.: Generalized net subscribers’ traffic model of communication switching systems. J. Adv. Modell. Anal. 37(1–2), 27–35 (1997) 7. Poryazov, S.A., Saranova, E.T.: Some general terminal and network teletraffic equations for virtual circuit switching systems. In: Nejat, I.A., Topuz, E. (eds.) Modeling and Simulation Tools for Emerging Telecommunication Networks, pp. 471–505. Springer, Boston (2006) 8. Poryazov, S., Saranova, E.: Models of Telecommunication Networks with Virtual Channel Switching and Applications. Academic Publishing House “Prof. M. Drinov”, Sofia (2012) 9. Poryazov S., Andonov V., Saranova E.: Comparison of conceptual models of overall telecommunication systems with QoS guarantees. In: Christiansen, H., Jaudoin, H., Chountas, P., Andreasen, T., Legind, Larsen, H. (eds.) Flexible Query Answering Systems. FQAS 2017, LNCS, vol. 10333. Springer, Cham (2017) 10. Poryazov, S., Andonov, V., Saranova, E.: Comparison of four conceptual models of a queuing system in service networks. In: Proceedings of the 26th National Conference with International Participation – TELECOM 2018, Sofia, 25–26 October, pp. 71-77 (2018)

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11. Schneps, M.: Systems for Distribution of Information. Svyaz Publishing House, Moscow (1979). (in Russian) 12. Tomov, Z., Krawczak, M., Andonov, V., Dimitrov, E., Atanassov, K.: Generalized net models of queueing disciplines in finite buffer queueing systems. In: Proceedings of the 16th International Workshop on Generalized Nets, 10 February 2018, Sofia, pp. 1–9 (2018)

Generalized Net Model of Information Security Activities in the Automated Information Systems Veselina Bureva(B) Intelligent Systems Laboratory, “Prof. Dr. Assen Zlatarov” University, 1 “Prof. Yakimov” Blvd., 8010 Burgas, Bulgaria [email protected]

Abstract. A generalized net model of information security activities in automated information systems is constructed. The processes of information security are investigated, and the separate steps are analyzed. The constructed generalized net model describes the procedures of detecting security vulnerabilities, calculating the risk of attack, implementation of threats and security policy execution. Keywords: Information system · Generalized nets · Security

1 Introduction Computer security provides protection of automated information system for preserving the integrity, availability and confidentiality of the information resources (hardware, software, data, telecommunications). Integrity refers to the information protection from any unauthorized modification or destruction. Availability represents the procedure of providing responsible access in time to the information. Confidentiality includes the methods for defending the system from an unauthorized disclosure of information. The terms “vulnerability”, “threat” and “attack” are used in the description of the standard security policy applied in automated systems. Vulnerability is an availability of a weakness, error in the design or implementation caused by an unexpected and undesired event. The threat is an action that can violate the security of the system. Threats must be identified, classified by category, and evaluated to calculate their damage potential. Therefore, the attack is an assault on system security that applies intelligent threats to make violations. The set of rules providing security services for protecting system resources are defined as security policy. The risk is expressed by the expectation of loss provoked by a threat that will use a vulnerability of the system to cause harmful result. The risk can be classified in several types: physical damage, human interaction, equipment malfunction, inside and outside attacks, misappropriation of data and loss of data. The system resource or asset includes hardware (computer systems, data storage, data communications devices), software (operating system, system utilities, applications), data (files and databases), communication facilities and networks (network communication routers, bridges and etc.) [5, 8–10, 14, 16–18].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 280–288, 2021. https://doi.org/10.1007/978-3-030-77716-6_24

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2 Generalized Net Model of Information Security Activities in Automated Information Systems The concept of Generalized nets is introduced in [2–4]. The processes that are modeled by generalized nets are presented in [1, 6, 7, 11–13, 15, 19]. The Generalized net model of the information security activities in the automated information systems contains 6 transitions and 25 places (Fig. 1). The set of transitions A has the following form: A = {Z1 , Z2 , Z3 , Z4 , Z5 , Z6 },

Fig. 1. Generalized net model of the information security activities in the automated information systems

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where the transitions describe the following processes: • Z 1 - “defining the set of data recources for protecting (information resources, assets)”; • Z 2 - “source of threats (generating threats and determining their properties: nature of occurrence, character, relation to the objects of IS)”; • Z 3 - “analyzing the security vulnerabilities”; • Z 4 - “determining the probability for realization (risk) of attacks”. • Z 5 - “determining the risk for the selected information resources”; • Z 6 - “defining the strategy of defence from threats (security policy)”; Initially, there is one α 2 -token that is located in place L 4 with initial characteristic: "set of data resources".

In the next time-moments this α 2 -token splits into two or more tokens. The original α 2 -token will continue to stay in place L 4 , while the other α 4 -token will move to transition Z3 via place L 2 . Initially, there is one β 2 -token that is located in place L 9 with initial characteristic: "source of threats".

In the next time-moments this β 2 -token splits into two tokens. The original β 2 token will continue to stay in place L 9 , while the other β-tokens will move to the next transitions. Token α 1 enters the net via place L 1 with initial characteristics: "information resources".

The transition Z 1 has the form: Z1 = {L1 , L3 , L4 }, {L2 , L3 , L4 }, R1 , ∨(L1 , L3 , L4 ), where

and • • • •

W 4,3 W 4,4 W 3,2 W 3,3

= “there are information resources for determining the level of authority”; = ¬W 4,3 ; = “there are information resources”; = ¬W 3,2 .

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The α 1 -token, moving from place L 1 to place L 4 does not obtain new characteristic. The α 2 -token in place L 4 generates a new token that enters place L 3 with characteristic: "information resources for determining the level of authority".

At the second activation of the transition, theα 3 -token from place L 3 generates a newα 4 -token that enters place L 2 with the characteristic: "determined information resources".

Token β 1 enters the net via place L 5 with initial characteristics: “threats”. The transition Z 2 has the form: Z2 = {L5 , L8 , L9, L22 , L23 }, {L6 , L7 , L8 , L9 }, R2 , ∨(L5 , L8 , L9, L22 , L23 ), where

and • W 9,8 = “there are threats for characteristics identification”; • W 9,9 = ¬W 9,8 ; • W 8,6 = “there are identified threats for determining the risk of the selected information resources”; • W 8,7 = “there are identified threats for determining the probability for realization (risk) of attacks”; • W 9,9 = ¬W 9,8 . The tokens, entering place L 9 do not obtain new characteristics. The β 2 -token in place L 9 generates a new token that enters place L 8 with the characteristic: " threats for characteristics identification".

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At the second activation of the transition the β 2 -token from place L 8 generates β 4 and β 5 -tokens that enter places L 6 and L 7 with characteristics: "identified threats for determining the risk of the selected information resources" in place L6 and "identified threats for determining the probability for realization (risk) of attacks" in place L7.

The transition Z 3 has the form: Z3 = {L2 , L12 , L14 , L20 }, {L10 , L11 , L12 }, R3 , ∨(L2 , L12 , L14 , L20 ), where

and • W 12,10 = “there are vulnerabilities for determining the risk of the selected information resources”; • W 12,11 = “there are vulnerabilities for determining the probability for realization (risk) of attacks”; • W 12,12 = ¬(W 12,10 ∧W 12,11 ). The α-tokens, entering place L 12 do not obtain new characteristics. The α 5 -token in place L 12 generates two new α 7 - and α 6 -tokens that enter places L 10 and L 11 with the characteristics: "vulnerabilities for determining the risk of the selected information resources" in place L10, and "vulnerabilities for determining the probability for realization (risk) of attacks" in place L11.

Token β 6 enters the net via place L 13 with initial characteristics: "methods for attacks realization".

The transition Z 4 has the form: Z4 = {L7 , L11 , L13 , L17 , L24 }, R4 , ∨(∧(L7 , L11 , L13 ), L17 , L24 ), where

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and • W 17,14 = “there are impacts for increasing vulnerability”; • W 17,15 = “there are threats implementations with damages for determining the risk or the selected information resources”; • W 17,16 = “there are impacts reducing the level of security services”; • W 17,17 = ¬(W 17,14 ∧W 17,15 ∧W 17,16 ). The tokens, entering place L 17 do not obtain new characteristics. The β 7 -token in place L 17 generates three new β 10 -, β 9 - and β 8 - tokens that enter places L 14 , L 15 and L 16 with the characteristics: "impacts for increasing vulnerability" in place L14, "threats implementations with damages for determining the risk for the selected information resources" in place L15 and "impacts reducing the level of security services" in place L16.

The transition Z 5 has the form: Z5 = {L6 , L10 , L15 , L19 }, {L18 , L19 }, R5 , ∨(∧(L6 , L10 , L15 ), L19 ), where

and • W 19,18 = “the risk for the selected information resources is calculated”; • W 19,19 = ¬W 19,18 .

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The tokens, entering place L 19 do not obtain new characteristics. Theα 8 - token in place L 19 generates a new token that enters place L 18 with the characteristics: "calculated risk for the information resources".

The transition Z 6 has the form: Z6 = {L16 , L18 , L25 }, {L20 , L21 , L22 , L23 , L24 , L25 }, R6 , ∨(∧(L16 , L18 ), L25 ), where

and W 25,20 = “security arrangements for reducing vulnerabilities are prepared”; W 25,21 = “the quality of the information resources is estimated”; W 25,22 = “the threats are avoided”; W 25,23 = “the threats are dropped out”; W 25,24 = “security policy mechanisms are applied for adaptation of threats and minimizing the damage from an attack”; • W 25,25 = ¬(W 25,20 ∧W 25,21 ∧W 25,22 ∧W 25,23 ∧W 25,24 ).

• • • • •

The tokens, entering place L 25 do not obtain new characteristics. The α 15 - token in place L 25 generates five new α 10 -, α 11 -, α 12 -, α 13 - and α 14 -tokens that enter places L 20 , L 21 , L 22 , L 23 and L 24 with characteristics: "security arrangements for reducing vulnerabilities" in place L20, in place L21,

"estimated quality of the information resources " "avoided threats"

in place L22,

"dropped out threats" in place L23 and "applied security policy mechanisms for adaptation of threats and minimizing the damage from an attack" in place L24.

3 Conclusion The constructed Generated net model can be used for description and simulation of the process of information security activities in automated information systems. It can

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be helpful in analyzing, managing and optimizing the processes of computer security. In future research, the research work of the constructed Generalized net model of the process of information security activities in automated information systems will be extended with a detailed representation of the threat classification. Acknowledgment. The authors are grateful for the support provided by the project “New Instruments for Knowledge Discovery from Data, and Their Modelling” funded by the National Science Fund, Bulgarian Ministry of Education, Youth and Science (no. DN-02-10/2016) for section 1 and the project “Intelligent Instruments for Knowledge Discovering and Processing” (No. NIH-418/2018) for section 2.

References 1. Atanassov, K.: Generalized nets as a tool for the modelling of data mining processes. In: Sgurev, V., Yager, R.R., Kacprzyk, J., Jotsov, V. (eds.) Innovative Issues in Intelligent Systems. SCI, vol. 623, pp. 161–215. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-272 67-2_6 2. Atanassov, K.: Generalized Nets. World Scientific, Singapore (1991) 3. Atanassov, K.: On Generalized Nets Theory. “Prof. Marin Drinov”Academic Publishing House, Sofia (2007) 4. Atanassov, K., Sotirova, E.: Generalized Nets. Prof. M. Drinov Academic Publishing House, Sofia (2017). (in Bulgarian) 5. Atymtayeva, L., Kozhakhmet, K., Bortsova, G.: Building a knowledge base for expert system in information security. In: Cho, Y.I., Matson, E.T. (eds.) Soft Computing in Artificial Intelligence. AISC, vol. 270, pp. 57–76. Springer, Cham (2014). https://doi.org/10.1007/978-3319-05515-2_7 6. Bureva, V., Sotirova, E., Popov, S., Mavrov, D., Traneva, V.: Generalized net of cluster analysis process using STING: a statistical information grid approach to spatial data mining. In: Christiansen, H., Jaudoin, H., Chountas, P., Andreasen, T., Legind Larsen, H. (eds.) FQAS 2017. LNCS (LNAI), vol. 10333, pp. 239–248. Springer, Cham (2017). https://doi.org/10. 1007/978-3-319-59692-1_21 7. Chountas, P., Kolev, B., Rogova, E., Tasseva, V., Atanassov, K.: Generalized Nets in Artificial Intelligence. Volume 4: Generalized Nets, Uncertain Data and Knowledge Engineering. “Prof. Marin Drinov” Publishing House, Sofia (2007) 8. Harkins, M.: Managing Risk and Information Security Protect to Enable, 2nd edn. Apress, New York (2016) 9. Harris, S., Maymi, F.: CISSP Exam Guide, 8th edn. McGraw-Hill Education, New York (2019) 10. Kostopoulos, G.: Cyberspace and Cybersecurity. CRC Press, New York (2013) 11. Krawczak M., Sotirov, S., Sotirova, E.: Generalized net model for parallel optimization of multilayer neural network with time limit. In: Proceedings of 6th IEEE International Conference on Intelligent Systems, IS 2012, pp. 173–177 (2012) 12. Roeva, O., Shannon, A., Pencheva, T.: Description of simple genetic algorithm modifications using generalized nets. In: Proceedings of the 6th IEEE International Conference Intelligent Systems, pp. 178–183 (2012) 13. Roeva, O., Pencheva, T., Shannon, A., Atanassov, K.: Generalized Nets in Artificial Intrelligence. Volume 7: Generalized Nets and Genetic Algorithms. Prof. Marin Drinov Academic Publishing House, Sofia (2013)

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14. Salomon, D.: Foundations of Computer Security. Springer, London (2006). https://doi.org/ 10.1007/1-84628-341-8 15. Sotirova, E., Orozova, D.: Generalized net model of the phases of the data mining process. In: Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics. Vol. II: Applications, Warsaw, Poland, pp. 247–260 (2010) 16. Stallings W., Brown L.: Computer Security Principles and Practice, 3rd edn. Pearson, New York 17. Stamp, M.: Information Security: Principles and PracticeSecond Edition. Wiley, New Jersey (2015) 18. Wittkop, J.: Building a Comprehensive IT Security Program: Practical Guidelines and Best Practices. Apress, New York (2016) 19. Zoteva, D., Roeva, O., Atanassova, V.: Generalized net model of artificial bee colony optimization algorithm with intuitionistic fuzzy parameter adaptation. Notes Intuit. Fuzzy Sets 24(3), 79–91 (2018)

Use of OFN in the Short-Term Prediction of Exchange Rates Hubert Zarzycki1(B) , Wojciech T. Dobrosielski2 , Jacek M. Czerniak2 , and Dawid Ewald2 1

2

Tadeusz Kosciuszko Military Academy of Land Forces in Wroclaw, Poland, ul. Piotra Czajkowskiego 109, 51-147 Wroclaw, Poland [email protected] Institute of Computer Science, Kazimierz Wielki University in Bydgoszcz, ul. Chodkiewicza 30, 85-064 Bydgoszcz, Poland {wdobrosielski,jczerniak,dewald}@ukw.edu.pl

Abstract. The article deals with the search for short-term trends based on the exchange rates of the leading currencies. A novel method of detecting patterns in trends written in a linguistic manner has been proposed. Linguistic variables take their values as a result of applying calculations in the Ordered Fuzzy Numbers notation. The solution is based on the application of fuzzification of the source data, using the transposition of the parameters (change direction, max, min, opening and closing prices) of the daily exchange rates of the EUR to USD currency pair. These exchange rate data for each trade day were recorded using single OFN numbers. This is an innovative application to the description of the Forex exchange rate notation allowing for the processing of five standard numerical data in one OFN. Then the data takes the form of linguistics. Based on the parameters set, the level of trend sequence similarity is determined. Parameters include: percent sequence compatibility of the trend with the frame set at the beginning, frame size, threshold and frequency. Undoubtedly, the analyzed patterns have different coefficients of similarity of the whole and individual elements. The study used a dedicated computer program that searches for patterns. The euro-dollar exchange rate from 1999–2019 served as the research material. These promising research are part of the rule-based forecasting methodology for financial applications.

1

Introduction

Rule-based forecasting (RBF) allows you to create more and more precise forecasting methods based on a combination of domain knowledge and statistical data. RBF can be seen as an expert system using the properties of time series as well as data extrapolation techniques [1–3,10,11,46,47]. The article shows the methods of detecting patterns in the time series of exchange rates [39,49]. The actual data was initially fuzzified [26–28]. Patterns of literal sequences were detected, which in a fuzzy manner described the repeated tendencies in the examined sequence of given exchange rates. Next, the rules for predicting short-term c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 289–301, 2021. https://doi.org/10.1007/978-3-030-77716-6_25

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trends were formulated. At the entrance we have a set of linguistic data [33– 35,51] created using fuzzy logic. The set describes the course of the EUR/USD exchange rate trend. The euro-dollar exchange rate from 1999–2019 served as the research material. The following figure shows this data range.

Fig. 1. Euro/Dollar exchange rate from 1999 to 2019

1 Euro exchange rate means how many dollars you have to pay per euro. When the exchange rate of the euro against the dollar is 1.6000, it means that for one euro it was necessary to pay as much as USD 1.60.

2

Exchange Rates and Currency Markets

The most popular currency pair on the currency markets is undoubtedly the euro/dollar exchange rate - EUR-USD. The most important currency exchange rate on the forex market is recorded by as much as 28% of the transaction. It owes its popularity to the fact that the US dollar (USD) is extremely important due to the fact that this currency constitutes a large part of foreign exchange reserves of the majority of countries in the world. The Euro zone alongside the United States and China is the largest economy in the world. A common European currency was introduced at the end of the 20th century. Since then, the Euro exchange rate has been closely watched by all kinds of speculators as well as by financial market observers. The euro dollar exchange rate is a reference point for other pairs related to EUR and USD. Depending on whether there is a strengthening or weakening of this pair, we can see a strong impact on the quotations of other currency pairs. A pair are characterized by a low transactional spread, which is crucial for traders. A number of factors affect the exchange rate of the euro against the US dollar, such as statements of central

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banks, macroeconomic data, non-farm payrolls, and trade between the Euro Zone and the United States. Both for the first and second currency are strongly influenced by readings concerning Gross Domestic Product (GDP). Therefore, at the time of deteriorating data from the Euro Zone, investors are heading towards the dollar and vice versa. The most important data from the Euro zone are data from Germany and France. These countries generate the largest GDP in the Euro Area and act as hegemons in the European Union. It is the data concerning, for example, Germany’s GDP which is the most important in terms of aversion or optimism to the euro currency than other European Union countries. In 2009, the US Fed announced a quantitative easing (QE) plan. This action was to increase the money supply and depreciate the currency. Therefore, in the graph of Fig. 1 one can observe how the then weakening dollar influenced the growing EUR/USD exchange rate The euro is legal tender in 19 countries that make up the euro area in the European Union and covers around 341 million Europeans. The eurozone is a monetary union creating a common space of EU countries that have introduced the euro as their currency, where the European Central Bank conducts an independent monetary policy. With an outstanding value of over 751 billion euros, the currency with the highest cash value in the world is 27% of global foreign exchange reserves. On 1 January 1999, the euro was inaugurated in non-cash transactions in 11 countries (excluding Greece), and from January 1, 2002, this currency was introduced in cash in twelve EU countries The US dollar - the official currency of the United States, Puerto Rico, Micronesia, Northern Marinas, Palau, Marshall Islands, Panama, Ecuador (from 2000), El Salvador (from 2001), East Timor, Zimbabwe (from 2009) and Bonaire, Saby and Sint Eustatius from 2011. Every US dollar, produced since the beginning of this currency, remains a legal tender in the US, the dollar’s value generally falls, especially in the wider time horizon. However, the abolition of the exchange for the money ore in 1971 shows a smaller decline in the purchasing power of fiat money to an analogous decrease of the same money in relation to bullion. In 1972, 1 ounce was worth $ 42.74. In 2012, 1 ounce of gold is $ 1692,40. What is the value of 1 dollar in gold at about 0.025 dollar from 1972. According to the values of the counterpart of purchasing power, 1 dollar from 2011 is 0.034 dollar from 1774, and the dollar from 1970 is 0.20 dollar from 1774. Ficiual decline is ok 6 times, and in relation to the metal 40 times. In the 21st century, the decline in foreign currency reserves in that currency by Russia and other countries, as well as the change from dollar to other currencies in the settlement of oil transactions by several countries (including Iran, China, Venezuela) [31].

3

Fuzzy Observation of the Exchange Rate of Euro in Relation to USD

The table below shows the source data of the Euro exchange rate against the USD. They contain standard values from the day of the session, ie the opening price, the maximum and minimum value as well as the closing price, or the change itself in relation to the value from the previous day. The actual values and the

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corresponding linguistic data are shown in the Table 1. Alphabet characters have been added to each of the data lines describing exchange rate values for a specific day. The table contains a fragment of the Forex market data from May 1, 2019 to May 31, 2019. In a standard format for this type of time series Each of the rows with daily data is described in letters of the alphabet. Table 1. Selected Forex EUR/USD historical data. The dataset covers the time period from 1 May 2019 to 31 May 2019, based on investing.com. Index Date

Open

High

Low

Close

Change %

Y

May 31, 2019

1.01.1134

1.01.1181

1.01.1125

1.01.1169

0.34%

X

May 30, 2019

1.01.1130

1.01.1144

1.01.1115

1.01.1131

0.00%

W

May 29, 2019

1.01.1162

1.01.1175

1.01.1124

1.01.1131

−0.26%

U

May 28, 2019

1.01.1198

1.01.1202

1.01.1158

1.01.1160

−0.35%

T

May 27, 2019

1.01.1197

1.01.1218

1.01.1186

1.01.1199

−0.04%

S

May 24, 2019

1.01.1183

1.01.1215

1.01.1175

1.01.1204

0.21%

R

May 23, 2019

1.01.1150

1.01.1187

1.01.1106

1.01.1180

0.24%

P

May 22, 2019

1.01.1166

1.01.1180

1.01.1145

1.01.1153

−0.06%

O

May 21, 2019

1.01.1171

1.01.1189

1.01.1142

1.01.1160

−0.10%

N

May 20, 2019

1.01.1158

1.01.1177

1.01.1148

1.01.1171

0.12%

M

May 17, 2019

1.01.1176

1.01.1186

1.01.1155

1.01.1158

−0.14%

L

May 16, 2019

1.01.1202

1.01.1227

1.01.1166

1.01.1174

−0.25%

K

May 15, 2019

1.01.1205

1.01.1226

1.01.1177

1.01.1202

−0.02%

J

May 14, 2019

1.01.1224

1.01.1247

1.01.1200

1.01.1204

−0.18%

I

May 13, 2019

1.01.1241

1.01.1266

1.01.1221

1.01.1224

−0.10%

H

May 10, 2019

1.01.1215

1.01.1255

1.01.1214

1.01.1235

0.12%

G

May 09, 2019

1.01.1194

1.01.1254

1.01.1174

1.01.1222

0.27%

F

May 08, 2019

1.01.1189

1.01.1215

1.01.1181

1.01.1192

0.01%

E

May 07, 2019

1.01.1198

1.01.1222

1.01.1165

1.01.1191

−0.08%

D

May 06, 2019

1.01.1173

1.01.1212

1.01.1160

1.01.1200

−0.02%

C

May 03, 2019

1.01.1178

1.01.1208

1.01.1135

1.01.1202

0.21%

B

May 02, 2019

1.01.1202

1.01.1221

1.01.1172

1.01.1179

−0.15%

A

May 01, 2019

1.01.1215

1.01.1267

1.01.1188

1.01.1196

Highest: 1.1267

Lowest: 1.1106

Difference: Average: 0.0161 1.1184

−0.19% Change %: −0.4279

In Fig. 2, the OHLC (Open, High, Low, Close) chart of the EUR/USD pair’s daily rates for the month of May 2019 is presented. The chart contains all the necessary attributes. For each trading day we have the price of opening, closing, the highest and the lowest. This data together with the percentage change of the exchange rate are presented in the Table 1. In the High-Low graph Fig. 2, graphical interpretation of the A, B and E records and the entire series I, J, K, L, M should be interpreted as the successive daily decreases of the EURUSD pair. In contrast, visualizations of C, D, F, G, H records show exchange rate increases. The T, F and X records are interesting, which despite the fluctuations during the trading day, do not experience a significant change between

Use of OFN in the Short-Term Prediction of Exchange Rates

293

the opening and closing values. In the further part of the study, it is necessary to apply the logic of Ordered Fuzzy Numbers. The above data and graphical interpretation can be easily processed into the OFN logic. Table 2 summarizes the attributes of the daily quotations used in the Table 1 with the characteristic points of the fuzzy number in the OFN notation.

Fig. 2. Forex OHLC chart for the period of 1 May 2019 to 31 May 2019 based on investing.com

Table 2. Characteristic points Orderd OFN number f0 Exchange rate

f1

g1

g2

OFN number orientation

Low Open Close High Change

Figure 4 shows exemplary parameters (min, max, open, close) of the exchange rate for a single day. There was an increase in quotations - the daily closing rate is higher than the opening one. Table 3 shows how the translation of attributes

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Fig. 3. Graphical representation of positively directed OFN number with characteristic points

Fig. 4. Positive value of the change parameter as displayed in Forex 1 Day chart Table 3. Positively directed OFN number for the Forex exchange rate data Orderd OFN number f0 Exchange rate

f1

g1

g2

OFN number orientation

Low Open Close High Change

occurs in this case. After translation, we get the fuzzy number in the OFN notation - as shown in Fig. 3. An arrow pointing towards the growing values of daily rates means that the OFN number is positively directed. Figure 6 shows the parameters (min, max, open, close) of the exchange rate for a single day. In these data a decrease in quotations was recorded - the daily closing rate is lower than the opening price. Table 4 demonstrates how to translate attributes in this case. After translation, we receive a negatively directed fuzzy number in the OFN notation; presented in Fig. 5.

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295

Fig. 5. Graphical representation of negatively directed OFN number with characteristic points

Fig. 6. Negative value of the change parameter as displayed in Forex 1 Day chart Table 4. Negatively directed OFN number for the Forex exchange rate data Orderd OFN number g0 Exchange rate

4

g1

h1

h2

OFN number orientation

Low Close Open High Change

Formalizing the Description of Fuzzy Observation

Data are daily currency exchange rates Euro to USD denoted as R1 ÷ Rm . On the R set we make fuzzy observations of changes of independent parameters of min, max, open, close and one dependent parameter which is direction of change. The OFN notation is used to record these observations. For each observed day the number R ∈ {R1 ÷ Rm } is created on the basis of four values obtained at a specific time. ti will mean the day of measurement and i means the day number,

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while the times tOP EN , tM AX , tM IN and tCLOSE indicate the moments in which the exchange rate has assumed; opening, maximum, minimum and closing values. Also index (1) stands for open and close values, while index (0) for the highest and lowest daily values. The fuzzy observation of EURUSD exchange rates on the given day ti is a set of R/ti ∈ {R(0) /tOP EN , R(1) /tM IN , R(1) /tM AX , (1) R(0) /tCLOSE } where tCLOSE ≥ {tM IN , tM AX } ≥ tOP EN fR (0) ≤ fR (1) ≤ gR (1) ≤ gR (0) The order of measurements remains important, the measurement time t decides about ordering in OFN. This means that for the order t ∈ {tM IN , tOP EN , tCLOSE , tM AX } the direction of the R number is positive. The direction of the OFN number, indicated on the graph by means of an arrow, also shows the daily trend of EUR/USD exchange rates. ⎧ RCLOSE > ROP EN ⎨ (2) Rpositive = RM IN , ROP EN , RCLOSE , RM AX ⎩ fR (0), fR (1), gR (1), gR (0) and in the opposite case Rnegative

⎧ RCLOSE ≤ ROP EN ⎨ = RM IN , RCLOSE , ROP EN , RM AX ⎩ gR (0), gR (1), fR (1), fR (0)

(3)

The currency pair most often traded by investors in the forex market is the euro (EUR) in combination with the US dollar (USD). The EUR/USD currency pair is the most bought and sold currency pair in the world. This pair is characterized by the lowest transaction costs (spread), as well as the highest liquidity, which allows for the effective conclusion of transactions almost at any time. A large transaction volume means greater volatility, which in turn translates into more opportunities to effectively occupy a position. We can use the volatility from past periods of currency pair quotations as the sum of OFN numbers. Fuzzy observation of the Sm aggregate for n days of exchange rates of EUR/USD components of exchange rate changes is expressed by the aggregate formula of OFN numbers. Definition sm

 n   Rpositive Rnegative = Ri | −Ri

(4)

i=1

where n is the specified number of days from past currency pairings. At set n we can examine short-term changes in the trend (daily, weekly, monthly), medium-term and long-term (quarterly, semi-annual, annual, multi-year)

Use of OFN in the Short-Term Prediction of Exchange Rates

297

The forecast of a possible change in the trend can be considered a situation when currency Sm aggregate representing a short-term period of component data (eg from one week) indicates a different trend by direction of the sum of OFN numbers than the trend visible for a longer period (eg month). This example rule can be described as follows: IF Sm(week) is positive AND Sm(month) is negative THEN P ossible continuation in trend

(5)

Table 5. Prediction using the rule, with marginal values of the euro dollar pair for the months of 2019 First day of the month

First day of the last week

Last day of the month

LW trend

Month trend

Prediction

January 2019

1,1397

1,1346

1,1488

P

P

Not applicable

February 2019

1.1471

1,1354

1.1416

P

N

Correct

March 2019

1,1383

1,1302

1,1235

N

N

Not applicable

April 2019

1,1236

1,1209

1,1218

P

N

Correct

May 2019

1,1212

1,1187

1,1151

N

N

Not applicable

June 2019

1,1185

1,1316

1,1380

P

P

Not applicable

July 2019

1,1349

1,1140

1,1151

P

N

August data not available

The Table 5 above shows the effectiveness of the method for an example rule. The Sm value is checked for data from the last week of the month and for the entire month. If the week is positive and the month is negative, the next month is expected to be negative. This is the case for the months of February and April. For the remaining months, the rule cannot be applied.

5

Conclusion

Effective investments in currency markets and forecasting trends in these markets are among the most complex issues. This is due to the fact that the FX exchange market phenomena are extremely dynamic and exhibit features of nonlinear chaotic behavior. One of the most important issues in the technical analysis is the detection and pro-per use of recognizable trends. A safe investment strategy therefore requires building a solution that will search for strong trends and support investment decisions. However, in moments of unstable behavior of courses, it will allow investors to minimize losses. Currently, it is possible to create such solutions and model knowledge of experts written in intuitive rules using fuzzy logic. Contemporary indicators and analytical methods based on classical technical analysis algorithms in order to identify a trend have already exhausted development opportunities to a large extent. Acceleration of calculations through the use of faster computers does not always mean better results. Only the creation of

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systems [9,48,58] using new calculation methods - including those based on fuzzy logic can significantly improve the quality of obtained solutions and the efficiency of trend determination. The method presented in the article allows using one number in the OFN notation to process as many as five attributes describing the daily Euro to USD exchange rates. The applied solution based on ordered fuzzy numbers has the property of relatively early detection of a possible change in the trend. The authors have already used similar methods based on OFN [28,56] in issues of investments in financial markets [15,57] and intend to continue research and development systems in this field [52,53]. More and more methods based on OFN are slowly paving the way for practical investment applications. According to the authors presented solution is competitive to other contemporary fuzzy/neural networks based currency price prediction systems.

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Ordered Fuzzy Numbers for IoT Smart Home Solution L  ukasz Apiecionek(B) , Wojciech T. Dobrosielski, and Dawid Ewald Kazimierz Wielki University, Bydgoszcz, Kujawsko-Pomorskie, Poland {lapiecionek,wdobrosielski,dewald}@ukw.edu.pl

Abstract. Computer systems are presently a common element of everyday life. There is currently an era of Internet of Things in the computer systems, which consists in connecting all possible devices to the Internet in order to provide them with new functionalities and thus – to improve the user’s life standard. One of such solutions could be Intelligent Homes called Smart Home. For such solutions there is a possibility of monitoring inner environment which provides potential for e.g. better heating control. The authors of this paper propose to use Ordered Fuzzy Numbers for some heating and cooling method. The proposed solution was tested in a special climate chamber. The authors provided conclusions at the end of the paper.

1

Introduction

Computer systems are presently a common element of everyday life. In this domain the world has entered the Internet of Things era, which consists in connecting all possible devices to the Internet in order to provide them with new functionalities and thus – to improve the user’s life standard [7]. Its objective is to ensure access to any desired service to anyone at any place and by any possible transmission medium. The Internet of Things can currently be described as a solution under development. New architectures are being constantly developed and at the same time the technology allows development of new services. The simplest general example of the idea behind the Internet of Things is the system in which a fridge with an access to the Internet recognizes on its own accord the amount and the volume of the products inside and in the case of any shortage it places an order in a shop, makes a payment with the saved credit card data and arranges the delivery. After picking up the supply and placing the new products into the fridge, it identifies them and recalculates their quantity or volume in order to forecast their run-out time. One of the presently available and commonly used services is the monitoring of all kinds of physical parameters of a house. They are called Smart Homes when they provide the opportunity to make some decisions depending on the conditions prevailing inside them. One of such conditions is the temperature which requires the heating and cooling system control. The authors of the paper presented an attempt of developing an IoT (Internet of Things) solution for heating and cooling control in the house. Section 2 of c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 302–310, 2021. https://doi.org/10.1007/978-3-030-77716-6_26

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the article concerns the Smart Home idea of the IoT. Section 3 describes the concept of the proposed solution. Section 4 presents the test results for the proposed model. Section 5 provided conclusions of the paper.

2

The Idea of IoT and Intelligent - Smart Homes

Figure 1 illustrates the overall concept of the IoT, where every domain-specific application interacts with domain-independent services, whereas in each domain, sensors and actuators communicate directly with each other [1]. It is assumed that, in a course of time, more and more devices will be connected to the Internet and altogether they will create an intelligent environment. The synchronization of the solutions will make it possible, for example, to open the garage door in advance when the car is approaching the premises. Use of intelligent transport systems will enable more efficient traffic control, including preventing congestions or ensuring the emergency vehicles right-of-way by appropriate traffic lights control. However, it requires overcoming numerous obstacles. Frequent problems in this matter include: – the necessity of providing power supply for all elements in the IoT solution; – the necessity of transmitting the data to remote destinations – within the area the solution covers; – the necessity of developing data transmission protocols – how to send the data, how to transmit them efficiently; – creating a datacenter for collecting and processing the data – where and how to store big amounts of data, how to share them; – developing the algorithms for data analysis – how to analyze the data, how to draw adequate conclusions; – the necessity of connecting various devices which may have been incompatible before – how to connect various elements, what converters and what gateways are to be developed; – addressing the data – how to address and identify the devices. The IoT solutions are also described by their elements. Figure 2 shows the following elements of the solution. One of the most interesting concepts is a Smart Home. There are lots of concepts how to build it. Some authors provided examples where they proposed the management systems using IEEE 802.15.4, zigbee or PLC [5,6]. Agent-based architecture with Markov models of decision making were presented in some studies [4]. Some authors present the concept of using Software Oriented Architecture for creating smart home functionalities [9]. There are some heating solutions proposed for this process using Gaussian Process Prediction [10]. Some methods use clustering techniques for the same purposes [8]. And, finally, some solutions are covered by the patents [3].

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Fig. 1. The overall picture of IoT emphasizing the vertical markets and the horizontal integration between them.

Fig. 2. The elements of IoT [1].

3

Ordered Fuzzy Numbers for Smart Home Heating and Cooling System

Figure 3 shows the diagram of the house temperature control system: heating and cooling. In the presented system, the temperature set-point of the enclosed building is controlled by means of: – setting the temperature set-point by setting the oven temperature using function FK by the controller or the cooling system control using function hK also by the controller. – the oven controls the temperature by supplying heat to the heating system, which gives up the heat to ambient air (in a closed room, in a house),

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– the controller acquires to its control system at least two temperature values, one of which is the internal temperature A, while the other one is the external temperature B.

Fig. 3. System diagram

The A and B temperature sensors connected to the control system measure the ambient temperature and return its value in analog form. The analog value is converted to a digital value by the controller. The temperature is measured at specified time intervals t0 set in the controller The target/set temperature of the system is TZ. Measurement values measured by each sensor at specific time intervals generate a series of results. The control for each sensor from four consecutive measurement moments, i.e.: ti , ti−1 , ti−2 , ti−3 for t0 = ti − ti−1 where ti is a consecutive measurement moment, creates an ordered fuzzy number. According to the theory of ordered fuzzy numbers, where for OFN: – – – –

fA (0) fA (1) gA (1) gA (1)

corresponds corresponds corresponds corresponds

to to to to

the the the the

moment moment moment moment

ti−3 , ti−2 , ti−1 , ti .

Then the ordered fuzzy number (OFN) is created as shown in Fig. 4. An OFN is created for each temperature sensor and marked as for the sensor, respectively: – for temperature sensor A as AOF N , – for temperature sensor B as BOF N . The control is translated from the domain of real numbers R to the domain of ordered fuzzy numbers OFN. In other words, real values of temperatures TA and

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Fig. 4. Ordered fuzzy number

TB are fuzzyfied to their extended representation containing information about the change in the value of OFN, i.e. to the numbers AOF N and BOF N : FOF N (tA , tB ) → {AOF N , BOF N } Using the properties of OFN numbers, the controller calculates and anticipates the settings for the stove fK , which supplies the heating system, and for the cooling system hK . The calculations are performed as follows: – – – –

If AOF N and BOF N increase then FOF N down high and HOF N up high If AOF N rises and BOF N decreases then FOF N down high and HOF N up low If AOF N decreases and BOF N rises then FOF N up low and HOF N down high If AOF N and BOF N decreases then FOF N up high and HOF N down high

Whereas high means high control setting of the stove and the cooling system, while low a small one. It is characteristic that the use of OFN allows to determine the temperature value and its trend (positive or negative). The value of the function FOF N should therefore be adjusted depending on: – stove type and cooling type, – temperature values, – temperature trend (this is characteristic for this control). The values high and low are therefore dependent on the set temperature. The list of set-points is presented by the following rules divided in three sections:

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– Section one: • If the internal temperature TA is rising and the external temperature TB is rising, while the TZ − TA difference is positive, then the stove power output is set to 40%. • If the internal temperature TA is rising and the external temperature TB is falling, while the TZ − TA difference is positive, then the stove power output is set to 70%. • If the internal temperature TA is falling and the external temperature TB is rising, while the TZ − TA difference is positive, then the stove power output is set to 60%. • If the internal temperature TA is falling and the external temperature TB is falling, while the TZ − TA difference is positive, then the stove power output is set to 90%. – Section two: • If the internal temperature TA is rising and the external temperature TB is rising, while the TZ − TA difference is negative, then the stove power output is set to 10%. • If the internal temperature TA is rising and the external temperature TB is falling, while the TZ − TA difference is negative, then the stove power output is set to 30%. • If the internal temperature TA is falling and the external temperature TB is rising, while the TZ − TA difference is negative, then the stove power output is set to 20%. • If the internal temperature TA is falling and the external temperature TB is falling, while the TZ − TA difference is negative, then the stove power output is set to 50%. – Section three: • If the internal temperature TA is rising and the external temperature TB is rising, while the TZ − TA difference is zero, then the stove power output is set to 0% and the cooling system is set to 50%. • If the internal temperature TA is rising and the external temperature TB is falling, while the TZ − TA difference is zero, then the stove power output is set to 10%. • If the internal temperature TA is falling and the external temperature TB is rising, while the TZ − TA difference is zero, then the stove power output is set to 10%. • If the internal temperature TA is falling and the external temperature TB is falling, and the TZ − TA difference is zero, then the stove power output is set to 20%. The authors predict that using Ordered Fuzzy Numbers (OFN in short) let them to obtain lower temperature oscillation in the room with regard to the set-point than traditional controllers with one thermometer and thermostat. It is because OFN let to predict the trend and make a decision faster [2].

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The Real Test of Proposed Method

In order to verify the proposed method, a Smart Home simulation device was built. The problem of this device was that it was built from steel box. So this simulator do not keep all hot inside, but all the walls was a kind of radiator. The control unit with method using OFN implemented is shown in Fig. 5. The controller was built on the basis of the Arduino solution. The heating and cooling system is shown on Fig. 6. The device was placed in a climate chamber made available by TELDAT Sp. z o.o. sp.k. (Fig. 7).

Fig. 5. The control unit for the Smart Home simulator

Fig. 6. Heating and cooling system for the Smart Home simulator

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Fig. 7. The Smart Home simulator in a climate chamber

The controller was designed in two variants. Variant A included control without the use of Fuzzy Logic, while variant B using the proposed solution. Figure 8 shows a temperature chart for both controller variants. In the actual system i.e. a house - the results obtained will differ in terms of the temperature increase and decrease rate. This is due to the fact that in the simulation the model had a metal housing, which allowed to obtain a higher response of the internal temperature to changes in the external temperature. Actual houses are equipped with better thermal insulation, but such simulations would require longer research. As illustrated in the graph, variant B allows to achieve smaller temperature oscillation with reference to the set-point inside to be maintained.

Fig. 8. The results of internal temperature control

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Conclusion

The IoT solutions are not only a matter of the future, but also of the presentday life. Numerous new technologies and capacities, as well as virtualization methods are developed nowadays with regards to them. Existing technologies and open source software solutions allow quick achievement of new capabilities and launching new services for the users. By combining them with fuzzy logic, especially Ordered Fuzzy Numbers, it is possible to achieve better energy control using the heating system of the Smart Home. The results achieved by means of the proposed solution have been confirmed by tests in the climatic chamber simulating the external conditions.

References 1. Al-Fuqaha, A., Aledhari, M., Ayyash, M., Guizani, M., Mohammadi, M.: Internet of Things: a survey on enabling technologies, protocols, and applications. IEEE Commun. Surv. Tutor. 17, 2347–2376 (2015) 2. Apiecionek, L.: Fuzzy observation of DDoS attack. In: Prokopowicz, P., Czerniak, J., Mikolajewski, D., Apiecionek, L  ., Slezak, D. (eds.) Theory and Applications of Ordered Fuzzy Numbers. A Tribute to Professor Witold Kosinski. International Publishing Studies in Fuzziness and Soft Computing, 1434–9922, pp. 241–254. Springer, Cham (2017) 3. Bell, I.: Self-programmable temperature control system for a heating and cooling system. Patent Number: 5,088,645, Date of Patent: 18 February 1992 4. Cook, D., Youngblood, M., Heierman, E., Gopalratnam, K., Rao, S., Litvin, A., Khawaja, F.: MavHome: an agent-based smart home. In: Proceedings of the First IEEE International Conference on Pervasive Computing and Communications (2003) 5. Han, D., Lim, J.: Smart home energy management system using IEEE 802.15.4 and ZigBee. IEEE Trans. Consum. Electron. 56(3), 1403–1410 (2010) 6. Han, J., Choi, C., Park, W., Lee, I., Kim, S.H.: Smart home energy management system including renewable energy based on ZigBee and PLC. IEEE Trans. Consum. Electron. 60(2), 198–202 (2014) 7. Harald, P.F., Sundmaeker, Guillemin, P., Woelffl´e, S.: Vision and challenges for realis-ing the internet of things. pub. office eu (2010) 8. Iglesias, F., Kastner, W.: Clustering methods for occupancy prediction in smart home control. In Proceedings of 2011 IEEE International Symposium on Industrial Electronics, pp. 1321–1328 (2011) 9. Ricquebourg, V., Menga, D., Durand, D., Marhic, B., Delahoche, L., Log´e, C.: The smart home concept: our immediate future. In: 1st IEEE International Conference on E-Learning in Industrial Electronics, pp. 23–28 (2006) 10. Rogers, A., Maleki, S., Ghosh, S., Ghosh, J., Nicholas, R.: Adaptive home heating control through Gaussian process prediction and mathematical programming. In: Second International Workshop on Agent Technology for Energy Systems, pp. 71– 78 (2011)

When Two-Constraint Binary Knapsack Problem is Equivalent to Classical Knapsack Problem? Krzysztof Szkatula1,2(B) 1

Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland 2 Siedlce University of Natural Sciences and Humanities, ul. Konarskiego 2, 08-110 Siedlce, Poland [email protected]

Abstract. The goal of the paper is to show when Two-Constraint 0-1 Knapsack Problem is equivalent to the single constraint Classical Knapsack Problem. In other words when one constraint is “active”, and another constraint is “inactive” - the “active” one fulfills it. It is assumed that some of the problem coefficients are realizations of mutually independent random variables. For the considered asymptotical random model of the problem, the case when corresponding Lagrange multiplier is equal to zero is identified which in turn means that the corresponding constraint is redundant. Keywords: Combinatorial optimization · Knapsack problems · Probabilistic analysis · Lagrange function · Constraints activity

1

Introduction

Let us consider the Two-Constraint 0-1 Knapsack Problem in the following formulation: zOP T (n) = max subject to

n 

ci · xi

i=1 n  i=1 n 

where

i=1

a1i · xi  b1 (n)

(1.1)

a2i · xi  b2 (n)

xi = 0 or 1, i = 1, . . . , n

It is assumed that: ci > 0, aji > 0, 0 < bj (n)
0, 0 < bj (n) < i=1 aji , i = 1, . . . , n,  j = 1, 2, n are made to avoid the trivial and degenerated problems. When bj (n) ≥ i=1 aji then the corresponding constraint is always fulfilled and therefore it may be removed from the problem formulation, otherwise if bj (n) = 0 then (1.1) has only the trivial solution i.e. xi = 0, i = 1, . . . , n and zOP T (n) = 0. Two-Constraint 0-1 Knapsack Problem, see Martello and Toth [6] is special case of the 0-1 Multi-Constraint Knapsack problem, also known as m-constraint knapsack problem, see Nemhauser and Wolsey [8], Martello and Toth [5] and Kellerer et al. [3], where in general case there is arbitrary number m of constraints, i.e. bj (n), j = 1, . . . , m. In the Szkatula’s papers, cf. [9] and [10], probabilistic analysis results of the different special cases of the 0-1 Multi-Constraint Knapsack problem were presented. Another particular important case of the multiconstraint knapsack problem is a classical (single constraint) or, in other words, 0-1 Knapsack Problem, which have only one constraint, i.e. j = 1. The Multi-Constraint Knapsack Problem is well known to be N P-hard and moreover, when m  2, it is strongly N P-hard, see Garey and Johnson [2]. More precisely Single Constraint Knapsack problem could be solved by the so called fully polynomial-time approximation scheme (FPTAS) while when m  2 then exists only polynomial-time approximation scheme (PTAS) but not FPTAS. It means that there is a substantial difference between Two- and Single- Constraint 0-1 Knapsack Problems from the computational complexity point of view. The aim of this paper is to identify the cases when the second constraint is inactive, i.e. it is fulfilled by the first constraint in the asymptotical random model of the Two-Constraint 0-1 Knapsack Problem, where coefficients ci , aji are considered as realizations of the random variables and bj (n) are deterministic, i = 1, . . . , n, j = 1, 2, n → ∞. Activities of constraints are represented by Lagrange multipliers, which are assumed to be the deterministic functions. For the constraints, a positive value of the corresponding Lagrange multiplier indicates activity while a value of zero means inactivity (redundancy). When the second constraint is inactive, then Two-Constraint 0-1 Knapsack Problem is reduced to the Single-Constraint one. In the general case classical, Single-Constraint 0-1 Knapsack Problem is much easier to be solved than the Two-Constraint one. Therefore, the fact of the activity or inactivity of the constraints is essential from both theoretical and practical points of view.

2

Definitions

The following definitions are necessary for the further presentation: Definition 1. We denote Vn ≈ Yn , where n → ∞, if Yn · (1 − on (1))  Vn  Yn · (1 + on (1))

When Two-Constraint Binary Knapsack Problem

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when Vn , Yn are sequences of numbers, or lim P {Yn · (1 − on (1))  Vn  Yn · (1 + on (1))} = 1

n→∞

when Vn is a sequence of random variables and Yn is a sequence of numbers or random variables, where on (1) is function fulfilling: on (1) ≥ 0 and limn→∞ on (1) = 0. The following random model of (1.1) will be considered in the paper: • ci , aji are realizations of mutually independent random variables uniformly distributed over (0, 1], i = 1, . . . , n, j = 1, 2; n → ∞. • δ  b1 (n)  b2 (n)  n/2, bj (n)  bj (n + 1), for every n  1 and all bj (n), j = 1, 2, are deterministic, where δ > 0 is a constant. Under the assumptions made about ci , aji and bj (n) the following always hold n n   0  zOP T (n)  ci  n, δ  bj (n) < aji  n, j = 1, 2. (1.3) i=1

i=1

Moreover, from the strong law of large numbers it follows that n  i=1

ci ≈ E(c1 ) · n = n/2,

n 

aji ≈ E(a11 ) · n = n/2.

i=1

Therefore, it is justified reformulate formula (1.3) in the following way: 0  zOP T (n)  n/2, 0 < δ  b1 (n)  b2 (n) < n/2

(1.4)

Formula (1.4) shows that random model of the Two-Constraint 0-1 Knapsack Problem (1.1) is complete in the sense that nearly  all possible instances of the n problem are considered. Taking into account that i=1 aji ≈ n/2 assumption that bj (n)  bj (n + 1), j = 1, 2, for all n  1, is quite logical. The activity of the constraints of the problem (1.1) may be influenced by the problem coefficients, namely: n, ci , aji , b1 (n), b2 (n), where i = 1, . . . , n. It is assumed that ci , aji are realizations of the random variables and therefore their impact on the activity of the constraints is in this case indirect. The aim of the probabilistic analysis is to investigate in the asymptotic case, when n → ∞, how the problem coefficients influence the activity of the constraints in the case of the problem (1.1) and to identify the cases when the second constraint is inactive, i.e. redundant. The impact of the constraints right-hand-side values - b1 (n), b2 (n) - is well illustrated by the Lagrange function and the problem dual to (1.1).

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Lagrange and Dual Estimations

When the general knapsack type problem, with one or more constraints, is considered then Lagrange function and the corresponding dual problems, see Averbakh [1], Meanti et al. [7], Szkatula [9] and [10] are often used to perform various kind of analyses of the original problem. In the specific case of the TwoConstraint 0-1 Knapsack Problem Lagrange function of the problem (1.1) may be formulated as follows:   n 2 n    ci · xi + λj · bj (n) − aji · xi Ln (x) = i=1

j=1

i=1 n 

= λ1 · b1 (n) + λ2 · b2 (n) +

(ci − λ1 · a1i − λ2 · a2i ) · xi

i=1 n

where x = [x1 , . . . , xn ], x ∈ X, X = {0, 1} , and Λ = [λ1 , λ2 ] is the vector of Lagrange multipliers. Moreover, let for every Λ, λj ≥ 0, j = 1, 2: ⎧ ⎛ ⎞ ⎫ 2 n 2 ⎨ ⎬   ⎝ci − λj · bj (n) + λj · aji ⎠ xi . φn (Λ) = max Ln (x, Λ) = max x∈X x∈X ⎩ ⎭ j=1

i=1

j=1

Using the following notation: ⎧ 2 ⎨ 1 if c −  λj · aji > 0 i , (1.5) xi (Λ) = j=1 ⎩ 0 otherwise. ⎧ ⎧ 2 2 ⎨ a if c −  ⎨ c if c −  λj · aji > 0 λj · aji > 0 ji i i i aji (Λ) = , ci (Λ) = j=1 j=1 ⎩ ⎩ 0 otherwise. 0 otherwise. we receive for every Λ, λj ≥ 0, j = 1, 2: φn (Λ) =

=

2 

λj · bj (n) +

n 

j=1

i=1

2 

n 

λj · bj (n) +

j=1

⎛ ⎝ci − ⎛

2 

λj · aji ⎠ · xi (Λ)

j=1

⎝ci (Λ) −

i=1

⎞

2 

⎞ λj · aji (Λ)⎠

j=1

Obviously for i = 1, . . . , n, j = 1, 2, ci (Λ) = ci · xi (Λ), aji (Λ) = aji · xi (Λ). Dual problem to the Two-Constraint 0-1 Knapsack Problem (1.1) may be formulated as follows: Φ∗n = min φn (Λ). (1.6) Λ≥0

When Two-Constraint Binary Knapsack Problem

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Let us denote: zn (Λ) =

n 

ci · xi (Λ) =

i=1

Sn (Λ) =

2 

n 

ci (Λ), sj (Λ) =

i=1

n 

aji · xi (Λ) =

i=1

λj · sj (Λ), B(Λ) =

j=1

2 

n 

aji (Λ),

i=1

λj · bj (n).

j=1

For every Λ ≥ 0 the following holds: zOP T (n) ≤ Φ∗n ≤ φn (Λ) = zn (Λ) +

2 

λj (bj (n) − sj (Λ)).

(1.7)

j=1

By definition of ci (Λ) and aji (Λ), see (1.5), we have: ci (Λ) ≥

2 

λj · aji (Λ), i = 1, . . . , n,

j=1

and therefore zn (Λ) ≥ Sn (Λ).

(1.8)

For certain Λ, xi (Λ) given by (1.5) may provide feasible solution of (1.1), i.e.: s1 (Λ) ≤ b1 (n) and s2 (Λ) ≤ b2 (n).

(1.9)

If the above holds then: zn (Λ) ≤ zOP T (n) ≤ Φ∗n ≤ φn (Λ) = zn (Λ) + B(Λ) − Sn (Λ).

(1.10)

So, if (1.9) holds, then the below inequality also holds: B(Λ) − Sn (Λ) ≥ 0. From (1.8) we get: φn (Λ) zn (Λ) B(Λ) − Sn (Λ) B(Λ) − Sn (Λ) = + ≤1+ . zn (Λ) zn (Λ) zn (Λ) Sn (Λ) Therefore if (1.9) holds, then the following inequality also holds: 1≤

Φ∗n φn (Λ) B(Λ) zOP T (n) ≤ ≤ ≤ . zn (Λ) zn (Λ) zn (Λ) Sn (Λ)

(1.11)

Formulas (1.9) and (1.11) may allow to provide the asymptotical approximation of the zOP T (n) i.e. the optimal solution value of the (1.1) problem. Namely: If lim

n→∞

B(Λ(n)) zOP T (n) = 1 and sj (Λ) ≤ bj (n), j = 1, 2, then lim = 1. n→∞ zn (Λ(n)) Sn (Λ(n)) (1.12)

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Therefore if (1.12) holds then xi (Λ(n)), i = 1, . . . , n, given by (1.5), provides the asymptotically sub-optimal solution of the Two-Constraint 0-1 Knapsack Problem (1.1). Moreover the value of zn (Λ(n)) is an asymptotical approximation of the optimal solution value of the Two-Constraint 0-1 Knapsack Problem i.e. zOP T (n).

4

Probabilistic Analysis

In this section certain probabilistic properties of the Two-Constraint 0-1 Knapsack Problem (1.1) will be investigated. It is assumed that ci , aji , i = 1, . . . , n, j = 1, 2, are realizations of mutually independent random variables uniformly distributed over (0, 1]. It is also assumed that constraints right-hand sides b1 (n), b2 (n) and Lagrange multipliers λ1 (n), λ2 (n), Λ(n) = (λ1 (n), λ2 (n)), are deterministic and that 0 < δ  b1 (n)  b2 (n)  n/2, bj (n)  bj (n + 1). Monotonicity of the constraints right-hand sides, b1 (n)  b2 (n), is in this case determining monotonicity of the Lagrange multipliers, i.e. λ2 (n) ≤ λ1 (n). Similar probabilistic models are often used in the literature for the various types of knapsack problems including the Two-Constraint 0-1 Knapsack Problem (1.1), see Kellerer et al. [3]. Let us first observe that probabilistic distribution functions of the random variables ci , aji , i = 1, . . . , n, j = 1, 2, are as follows: ⎧ ⎧ ⎨ 0 when x  0 ⎨ 0 when x  0 P (aji < x) = x when 0 < x  1 , P (ci < x) = x when 0 < x  1 .(1.13) ⎩ ⎩ 1 when x  1 1 when x  1 In order to proceed with the probabilistic analysis of the Two-Constraint 0-1 Knapsack Problem (1.1) it is necessary to consider distribution functions of the following random variables λj · aji , j = 1, 2 and

2 

λj · aji .

j=1

Let

|x| + x = (x)+ = 2



x if x > 0 , j∗ = 0 otherwise



2 if j = 1 . 1 if j = 2

Then for i = 1, . . . , n, j = 1, 2, the following holds: 1 ((x)+ − (x − λj )+ ), j = 1, 2, λj 1 1 F2 (x, Λ) = P {λ1 · a1i + λ2 · a2i < x} = F1 (x − λj ∗ t, λj )dt λj

F1 (x, λj ) = P {λj · aji < x} =

0

(1.14)

 1  2 = (x)+ − (x − λ1 )2+ − (x − λ2 )2+ + (x − λ1 − λ2 )2+ λ1 · λ2

When Two-Constraint Binary Knapsack Problem

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Then distribution functions of the random variables aji (Λ), i = 1, . . . , n, j = 1, 2 are as follows: Gji (x, Λ) = P {aji (Λ) < x} =   2   λk · aik ≥ ci = P aji < x aji ≥ x

(1.15)

k=1

1 1 F1 (r − λj · t, λj ∗ )drdt

=1− x

0

Using above formula (1.15) expectations of the aji (Λ) could be expressed as follows: 1 E(aji (Λ)) = 0

1 1 xdGji (x, Λ) = x F1 (r − λj · x, λj ∗ )drdx 0

⎛

(1.16)

0

⎞ 1 1 1 ⎝ = x ((r − x · λj )+ − (r − x · λj − λj ∗ )+ )drdx⎠ . λj ∗ 0

0

Probabilistic, or in other words average case, analysis consists in determining such Lagrange multipliers λ1 (n), λ2 (n) that when n → ∞, xi (Λ(n)), i = 1, . . . , n, defined by (1.5) will provide asymptotically feasible solutions of the Two-Constraint 0-1 Knapsack Problem (1.1), in the sense of convergence in probability, see Loeve [4]. It means that sj (Λ(n)) is satisfying (1.9) and moreT (n) over Sn (Λ(n)) is fulfilling (1.12). Then, due to (1.11), limn→∞ zznOP (Λ(n)) = 1 and zn (Λ(n)) is the suboptimal solution of the (1.1) and moreover zOP T (n) ≈ zn (Λ(n)) ≈ E(zn (Λ(n))). The above goal may be achieved by determining Λ(n) as the solution of the following system of equations: E(s1 (Λ(n))) = b1 (n),

E(s2 (Λ(n))) = b2 (n),

(1.17)

where bj (n) = bj (n)−j (n), j (n) = on (bj (n)) and j (n) → ∞ when bj (n) → ∞. In the Szkatula’s papers, cf. [9] and [10], j (n) were proposed which satisfy sj (Λ(n)) ≈ bj (n) ≈ bj (n), sj (Λ(n))  bj (n), j = 1, 2, in the sense of convergence in probability and therefore Λ(n) is asymptotically fulfilling both (1.9) and (1.12). It may also happen that the system of equations (1.17) has no solutions, e.g., when asymptotical difference between b1 (n) and b2 (n) is too large or both b1 (n), and b2 (n) are very close to the limit value of n2 . In this case, λ1 (n) > 0 and λ2 (n) = 0 which means that second constraint in the Two-Constraint 0-1 Knapsack Problem (1.1) formulation is inactive (redundant) and could be removed. If

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this occurs then problem (1.1) reduces to the classical Single-Constraint Knapsack problem. In the Szkatula paper [10] results of the probabilistic analysis of the classical Single-Constraint Knapsack problem for all possible right-hand constraints values were presented. In particular behavior of the optimal solution is established as follows:    zOP T (n) ≈ zn (Λ(n)) ≈ zOP T (n) ≈ zn (Λ(n)) ≈ E(zn (Λ(n)))

and

⎧ ⎨ 2·n·b1 (n)   3  E(zn (Λ(n))) = ⎩ 1 · n + 6 · b (n) · 1 − 1 4 2

b1 (n) n



n 6, ≤ n2 .

if δ ≤ b1 (n) ≤ if

n 6

≤ b1 (n)

(1.18)

In the remaining part of the paper, it is by default assumed that ci , aji , are realizations of mutually independent random variables uniformly distributed over (0, 1), so this assumption will be not repeated. Lemma 1. If there does not exist 0 < λ2 (n) ≤ λ1 (n), being solution of the (1.17) for all n ≥ 1 then second constraint in (1.1) is redundant (excessive) i.e., it is always fulfilled by the first one. In this case λ2 (n) = 0 and problem is equivalent to the single constraint knapsack problem, and ⎧ ⎨ 2·n·b1 (n) if δ ≤ b1 (n) ≤ n6 ,   3  zOP T (n) ≈ (1.19) ⎩ 1 · n + 6 · b1 (n) · 1 − b1 (n) if n6 ≤ b1 (n) ≤ n2 . 4 2 n Proof. In this case the system of equations (1.17) has no solution and therefore Two-Constraint 0-1 Knapsack Problem (1.1) is equivalent to the single constraint knapsack problem, cf. Szkatula [10]; (1.19) follows immediately from (1.18). We have E(sj (Λ(n))) = n · E(aj1 (Λ(n))), E(zn (Λ(n))) = n · E(c1 (Λ(n))). It is easy to observe that (1.16) may take different formulations, depending on the mutual relations between λ1 (n), λ2 (n) and x, r since some items in the formulae (1.16) may remain strongly positive or become 0, due to the function (.)+ properties. Lemma 2. If there exist constant δ > 0 such that δ ≤ b1 (n) ≤ then (1.17) has no solution, λ2 (n) = 0, and  2 · n · b1 (n) zOP T (n) ≈ 3

n 24

< b2 (n) ≤

n 6

(1.20)

When Two-Constraint Binary Knapsack Problem

319

Proof. In this case 0 ≤ λ2 (n) < 1 ≤ λ1 (n) and  1/λ1 (n)  1 1 E(a1i (Λ(n))) = x (r − x · λ1 (n))drdx (1.21) λ2 (n) 0 x·λ1   1−λ  1 2 (n) λ1 (n) − x (r − x · λ1 (n) − λ2 (n))drdx (x·λ1 (n)+λ2 (n))

0

=

1 24

+ 4λ22 (n) − 6λ2 (n) + 4 , λ21

−λ32 (n)

1 E(a2i (Λ)) = λ1 (n) =





1

x 0



1

x·λ2 (n)

(r − x · λ2 (n))drdx

1 3λ22 (n) − 8λ2 (n) + 6 , 24 λ1 (n)

(1.22)

(1.17) has no solution and therefore only first constraint is active: λ1 (n) > 0 and λ2 (n) = 0. From (1.21) we obtain  n and λ2 (n) = 0, λ1 (n) = 6 · b1 (n) (1.20) follows from the Lemma 1 and (1.19). Finally there is also situation when 0 < λ1 (n) ≤ 1 and λ2 (n) = 0: Lemma 3. If

n n < b1 (n) ≤ b2 (n) ≤ and 6 2 3 n 3 5·n either b2 (n) > · b1 (n) + or b2 (n) < · b1 (n) + 4 8 4 96 

then λ1 (n) = 3 ·

1 b1 (n) − 2 n

 and λ2 (n) = 0,

is the optimal set of Lagrange multipliers and    n 1 b1 (n) zOP T (n) ≈ · + 6 · b1 (n) · 1 − 4 2 n Proof. In this case λ1 (n) ≤ 1, moreover λ2 (n) + λ1 (n) ≤ 1, and 1 1 − λ1 (n) − 2 3 1 1 E(a2i (Λ(n))) = − λ2 (n) − 2 3 E(a1i (Λ(n))) =

1 λ2 (n), 4 1 λ1 (n). 4

(1.23)

320

K. Szkatula

Under assumptions of Lemma 3 system of equations (1.17) has either no solutions or λ2 (n) > 12 which is violating condition λ2 (n) ≤ λ1 (n). Therefore λ2 (n) = 0 and Two-Constraint 0-1 Knapsack Problem (1.1) is equivalent to the Single Constraint Knapsack Problem. (1.23) follows immediately from the Lemma 1 and (1.19). Conjecture 1. Lemmas 2 and 3 are identifying the cases when λ1 (n) > 0 and λ2 (n) = 0 i.e. the second constraint in (1.1) is redundant and as the consequence Two-Constraint 0-1 Knapsack Problem (1.1) is reduced to the classical Single-Constraint 0-1 Knapsack Problem. In the all other cases of the mutual relations between b1 (n) and b2 (n), δ  b1 (n)  b2 (n)  n/2, there exist a nontrivial solution, i.e. 0 < λ2 (n) ≤ λ1 (n), of equations (1.17). In other words, both constraints are active, and problem solution substantially depends on both of them, which means that in this case, Two-Constraint 0-1 Knapsack Problem (1.1) is N P-hard in the strong sense.

5

Concluding Remarks

Results presented in this paper are describing probabilistic properties of the TwoConstraint 0-1 Knapsack Problem (1.1) in the case when Lagrange multiplier λ2 (n) = 0 and Two-Constraint 0-1 Knapsack Problem is reduced to the SingleConstraint 0-1 Knapsack Problem, which belongs to the possibly simpler class of N P-hard but not in the strong sense combinatorial optimization problems. In turn, this fact may have a serious positive impact on the efficiency of the solution process. The future research will be aimed at investigation of all possible mutual relations between λ1 (n) and λ2 (n) and, as a consequence, all possible relations between b1 (n) and b2 (n). This in turn will allow analysis of the growth of the optimal solution values zOP T (n), when n → ∞ in the considered probabilistic model of the Two-Constraint 0-1 Knapsack Problem (1.1) for the full spectrum of the constraints left hand sides.

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321

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Author Index

A Andonov, Velin, 175, 186, 254 Andreev, Nikolay, 158 Antonov, Antonio, 216 Apiecionek, Łukasz, 302 Atanassov, Krassimir, 3, 175, 186 Atanassova, Vassia, 158 B Bentkowska, Urszula, 75 Bozov, Hristo, 166 Bozova, Greta, 166, 193 Bureva, Veselina, 280 Bzowski, Adam, 84 C Castillo, Oscar, 26 ˇ Cunderlíková, Katarína, 54 Czerniak, Jacek M., 93, 289 D Dobrosielski, Wojciech T., 289, 302 E Ewald, Dawid, 93, 289, 302 H Hryniewicz, Olgierd, 46 K Klukowski, Leszek, 136 Krawczak, Maciej, 193

L Landowski, Marek, 16 Lubich, Martin, 186 Łyczkowska-Han´ckowiak, Anna, 112 M Martínez, Gabriela E., 26 Melin, Patricia, 26 Michalíková, Alžbeta, 66 Minkov, Minko, 151 N Nowak, Piotr, 46 P Pander, Tomasz, 202 Paprzycki, Marcin, 93 P¸ekala, Barbara, 75 Pencheva, Tania, 186 Piasecki, Krzysztof, 112 Piegat, Andrzej, 16 Por˛ebski, Sebastian, 125 Poryazov, Stoyan, 254 Przybyła, Tomasz, 202 R Ribagin, Simeon, 230 Roeva, Olympia, 216, 241 S Saranova, Emiliya, 254 Shannon, Anthony, 186 Slavov, Chavdar, 186 Sotirov, Sotir, 166, 193

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. T. Atanassov et al. (Eds.): IWIFSGN 2019, AISC 1308, pp. 323–324, 2021. https://doi.org/10.1007/978-3-030-77716-6

324 Sotirova, Evdokia, 151, 166, 175 Stavrev, Spas, 230 Szkatuła, Krzysztof, 311 T Tsakov, Hristo, 241 U Urba´nski, Michał K., 84

Author Index V Vasilev, Valentin, 166, 175, 193 W Wójcicka, Kinga M., 84 Wójcicki, Paweł M., 84 Z Zarzycki, Hubert, 289 Zoteva, Dafina, 216, 241