Linear Models in the Mathematics of Uncertainty [1 ed.] 3642352235, 9783642352232

Citation preview

Studies in Computational Intelligence Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected]

For further volumes: http://www.springer.com/series/7092

463

John N. Mordeson, Mark J. Wierman, Terry D. Clark, Alex Pham, and Michael A. Redmond

Linear Models in the Mathematics of Uncertainty

ABC

Authors John N. Mordeson Department of Mathematics Creighton University Omaha USA

Alex Pham Department of Mathematics Creighton University Omaha USA

Mark J. Wierman Department of Computer Science Creighton University Omaha USA

Michael A. Redmond Department of Mathematics Creighton University Omaha USA

Terry D. Clark Department of Political Science Creighton University Omaha USA

ISSN 1860-949X e-ISSN 1860-9503 ISBN 978-3-642-35223-2 e-ISBN 978-3-642-35224-9 DOI 10.1007/978-3-642-35224-9 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012953004 c Springer-Verlag Berlin Heidelberg 2013  This work is subject to copyright. All rights are reserved 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To our leader Paul P. Wang

Forward

In the soft sciences, imprecision, uncertainty, incompleteness of information and partiality of truth lie in the center rather than on the periphery in the construction of models. Lotfi Zadeh

Albert Einstein stated that development of Western science is based on two great achievements: • the invention of the formal logical system, and • the discovery of the possibility to discover causal relationships by systematic experimentation. Judea Pearl, winner of the 2011 Turing Award, has stated that in the 1990s causality underwent a major transformation. Practical problems about causal information that were long regarded as unmanageable can now be solved using elementary mathematics. Pearl stresses that basic concepts of probability theory and graph theory are all that is needed for experimenters to begin solving causal problems that are too complex for the unaided intellect. We support the notion that problems of causality can be studied profitably by using the mathematics of uncertainty and basic concepts from graph theory. We propose a model consisting of an overarching goal and components that act to achieve the overall goal. We consider the components as nodes with a directed edges emanating from them to the overarching goal. We next consider attributes making up the components as nodes with directed edges emanating from them to the components. Each attribute has at most one edge directed to a particular component, but it may have a directed edge to more than one component. The resulting figure is a directed graph, for example, Figure F.1 shows the attributes considered when the goal is choosing the best college. The importance of each component and each attribute is weighted by means of expert opinion, data, and various mathematical techniques. The weights can be used to determine linear equations, where the overarching goal is the dependent variable and the components the independent variables. Each component can also be the dependent variable for

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a linear equation of the attributes as the independent variables. By proper substitution, the overarching goal can be written in terms of a linear equation involving the attributes. These linear equations give a measure as to how well the goals are being achieved at a specific time. The independent variables can be considered as causal variables. If directed edges are drawn between attributes to denote a weighted relationship between them, then a resulting mathematical structure is determined and is known as a directed graph or network. This structure can be studied in its own right.

Fig. F.1 Factors affecting a college decision

The primary benefit of fuzzy systems theory is to approximate system behavior where analytic functions or numerical relations do not exist Ross et al. (2002). Because of this, fuzzy systems have the potential to help us understand complex systems that are devoid of an easy analytic formulation. Complex systems can be new systems that have not been previously investigated like electronic social networks. They can be hybrid systems that involve technology and the human condition like medical systems. Or they can be social, economic, and political systems, systems where the vast arrays of causes and effects can not possibly be captured analytically and the experiment is not controlled in any conventional way. Moreover, the relationship between the causes and effects of these systems is generally a subject of vigorous debate. How then is a researcher to proceed in these muddy waters?

Preface

Creighton is a Jesuit Catholic university located in the Midwest with a strong liberal arts college. The college’s tradition of excellence in teaching has in recent years combined with a commitment to undergraduate research to attract a substantial core of gifted students. We have been fortunate to tap into this intellectual resource. We are grateful for their intellectual energy, dedication, and friendship.

Outline In our applications, we consider an overarching goal. This goal is dependent of several causal factors. The factors are weighted as to their importance by experts. We use several mathematical techniques to combine these weights into one weight, one for each factor. The overarching goal is then represented as a linear combination of the factors with their associated weight as the coefficient of the factor. The overarching goal is the dependent variable and the factors are the independent variables in the linear equation. We present three main techniques to determine these linear equations, namely, the Analytic Hierarchy Process, the Guiasu method, and the Yen method. Normalized data is then substituted into the independent variables to determine a score for the overarching goal. In Chapter 1, we present some basic material concerning fuzzy set theory. This material includes the notions of membership functions, alpha cuts, and standard as well as generalized fuzzy set operations. In Chapter 2, we present evidence theory. We place special attention on mixed evidence theory and fuzzy set theory. Two of our methods of determining the representation of the overarching goal as a linear equation of causal variables, the Guiasu and Yen methods, rest on Dempster-Shafer theory. In this chapter we present enough of Dempster-Shafer theory to develop the Guiasu method and the Yen method. We also focus on data fusion.

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This chapter is key to our applications. The techniques in this chapter allow us to combine expert opinion. In Chapter 3, 4, and 5, we describe the problems that are analyzed in the book. The areas of interest concern issues in international relations, issues in comparative politics, and issues concerning deaf and hard of hearing children. We present the expert opinion as to the importance of the causal factors of the overarching goal. Counterintuitive outcomes occur in politics and other forms of human interaction. In Chapter 4, we introduce the use of intuitionistic fuzzy sets as an approach to deal with counterintuitive behavior. In Chapter 6, we present just enough of the analytic hierarchy process for the purposes of our book. The analytic hierarchy process is also central to our applications. We examine such country attributes as economic freedom, failed states, quality of life, and political stability. Countries are ranked with respect to these attributes. We use intuitionistic fuzzy sets to model the notion of counterintuitive opinion. In Chapter 7, we use the Guiasu method to estimate the relative weight of a set of causal factors of the overarching goals of various applications. If the basic probability assignments are determined by expert opinion, say factors making up a goal, then the belief of a subset of the set of all factors can be interpreted as the degree of importance of that subset making up the overarching goal. Also, if the basic probability assignments are positive only in singleton sets, say for example single factors, then the combination of these basic probability assignments into a single basic probability assignment can be used as coefficients for the factors of the linear equation of the overarching goal expressed in terms of its factors. Belief functions are used to determine a degree of belief of the factors. We compare the Guiasu method and the analytic hierarchy process in this chapter. In Chapter 8, we use Jeffery’s Rule to determine a score for the overarching goal. Jeffrey’s rule can be seen as a special case of the Guiasu method. We also present a method based on Jeffrey’s Rule that allows for an estimate of an outcome variable given prior probability assignments over a set of input factors estimating the degree of correlation between fuzzy variables. We then use it to answer the question “How much confidence should we place in our democracy model?” In Chapter 9, we use the Yen method to estimate the relative weight of the causal factors. Yen’s method is a generalization of the Dempster-Shafer theory to allow for fuzzy sets. It addresses the issue of managing imprecise and vague information in evidential reasoning by combining the DempsterShafer theory with fuzzy set theory. Some extensions of Dempster-Shafer to deal with vague information did not preserve an important principle that the belief and the plausibility measures are lower and upper probabilities. Yen’s method preserves this principle by employing normalized fuzzy measures of credibility. It also preserves the property that the belief of a (fuzzy) subset is the difference of one and the plausibility of the subset’s complement, It is

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also more responsive to a change to a focal element’s membership function than some approaches. In Chapter 10, we use fuzzy subset methods to accomplish similar purposes as those for the AHP, Guiasu, and Yen methods. We use a method developed by Yen that involves various measures of subsethood and extends DempsterShafer theory by including a measure of subsethood, the degree to which a fuzzy subset is included an another fuzzy subset. In Chapter 11 we present the set–valued statistical method. This uses a very subjective method to construct fuzzy membership grades. The method turns out to have similarities to the Borda count method used in political science. In Chapter 12, we present intuitionistic fuzzy sets to study political stability. We also present a weighted average method for MADM problems and the extended GOWA operator based MADM method. We are indebted to Deng-Feng Li for many results that are utilized in this chapter. In Chapter 13 and 14, we examine our outcomes concerning Quality of Life and Failed States statistically. In Chapter 15, We compare our methods of ranking countries in various application areas. We also present commentary concerning our results. The Appendix contains a reprint of the seminal article by Guiasu from Advances in Fuzzy Theory and Technology Vol. II, Durham Press 1994, with the permission of the author, Silviu Guiasu, and editor, Paul P. Wang.

Dedications The authors wish to express their thanks to those persons who played a special role in the creation of this book. • John N. Mordeson would like to dedicate this book to his parents, Margaret and Charles Mordeson. • Mark J. Wierman dedicates this book to his parents, Margaret and William Wierman. • Terry D. Clark dedicates his work to the greatest empirical evidence in his life of the love and grace of God: Marnie Clark, his wife. • Alex Pham dedicates the book to his parents, George and Minh Pham. • Michael A. Redmond dedicates this book to his parents, James and Barbara Redmond. We also acknowledge the involvement of a large number of student researchers who collaborated with us on the applications and Creighton faculty, Drs. Lynne Houtz, Lynn Olson, and Beverly Doyle from the Education Department and Drs. Mallenby and Fong from the Mathematics Department. We are also indebted to the following staff of Omaha Hearing School, Karen Rossi,

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Lisa Burton, Barbara Walker, Marilee Kelly, Nicole Lanum, Jenna Voss, Adina Bell and staff from Madonna School, Cathy Hirchert and Sister Michelle Faltus. We are indebted to Dean Robert Lueger and the administration in general for their support of undergraduate research. We are particularly indebted to Dr. George and Mrs. Sally Haddix for their generous endowments to the Department of Mathematics at Creighton University.

Omaha, NE, September, 2012

John N. Mordeson Mark J. Wierman

Contents

Part I Mathematics of Uncertainty 1

Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Membership Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Alpha-cut or α-cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Standard Fuzzy Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Subsethood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Generalized Fuzzy Set Operations . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Fuzzy Set Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Fuzzy Set Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Residuum Operator – Omega Operator . . . . . . . . . . . . . . 1.7 Fuzzy Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Averaging Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Aggregation Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 OWA Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Classical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Types of Classical Relations . . . . . . . . . . . . . . . . . . . . . . . 1.11 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 sup–t Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Properties of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.1 Types of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.2 Transitive Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 6 9 11 11 12 13 15 16 17 17 19 22 23 23 24 26 27 29 30 30 32 33 35 36 37 39

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Evidence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Evidence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Possibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Data Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dempster’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Combining n Special Bodies of Evidence . . . . . . . . . . . . 2.4.2 Probabilistic Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Alternatives to Dempsters’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Yager’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Inagaki’s Unified Combination Rule . . . . . . . . . . . . . . . . .

41 41 41 44 45 47 47 49 52 54 54 55

Part II The Problems 3

Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Nuclear Deterrence Subgoals . . . . . . . . . . . . . . . . . . . . . . . 3.2 Smart Power and Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Subgoals of Smart Power . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 66 67 70

4

Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Factors in Successful Democratization . . . . . . . . . . . . . . . . . . . . . 4.2 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Description of Political Stability Data . . . . . . . . . . . . . . 4.4.2 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Subfactors of Political Stability . . . . . . . . . . . . . . . . . . . . 4.5 Failed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 75 77 78 79 81 83 84

5

Issues of Hearing Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Testing Instruments for Hearing Impaired Children . . . . . . . . . 5.1.1 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Closing the Language Gap for Deaf and Hard of Hearing Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 88

Part III Applications 6

The Analytic Hierarchy Process . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Analytic Hierarchy Process . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Katy Goes to College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Attribute Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 College Comparison by Attribute . . . . . . . . . . . . . . . . . . . 6.2 Aggregating Multiple Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 95 96 98 100

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6.3 Analysis of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Smart Power and Deterrence . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . 6.5 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Failed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Issues of Hearing Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Hearing Impaired Children . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Deaf and Hard of Hearing Children . . . . . . . . . . . . . . . . .

102 104 104 104 105 107 108 108 110 110 115 116 116 117

7

The Guiasu Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Reaching a Verdict by Weighting Evidence . . . . . . . . . . . . . . . . . 7.1.1 Fuzzy Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Dempster’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Jeffrey’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 The Guiasu Model with Probabilistic Evidence . . . . . . 7.1.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 AHP and Guiasu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Smart Power and Deterrence . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . 7.4 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Factors in Successful Democratization . . . . . . . . . . . . . . . 7.4.3 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Failed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Issues of Hearing Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Hearing Impaired Children . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Deaf and Hard of Hearing Children . . . . . . . . . . . . . . . . .

119 119 122 123 123 125 126 127 128 128 129 131 132 132 133 134 135 137 138 138 139

8

Jeffrey’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Jeffrey’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Dependent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Estimating Causality in the Social Sciences . . . . . . . . . . . . . . . .

145 145 145 147

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Yen’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . 9.2 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Factors in Successful Democratization . . . . . . . . . . . . . . . 9.2.2 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 157 157 158 158 159 160

10 Methods Based on Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . 10.1 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Factors in Successful Democratization . . . . . . . . . . . . . . . 10.3 Issues of Hearing Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Hearing Impaired Children . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Plausibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 164 169 169 173 173 175

11 Set–Valued Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Smart Power and Deterrence . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . 11.3 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Factors in Successful Democratization . . . . . . . . . . . . . . . 11.3.2 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Failed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Issues of Hearing Impairment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Hearing Impaired Children . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Deaf and Hard of Hearing Children . . . . . . . . . . . . . . . . .

177 177 180 180 180 181 181 181 182 183 184 184 184 185

12 Intuitionistic Fuzzy Sets and Political Stability . . . . . . . . . . . 12.1 Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Atanassov’s Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 12.2.1 Ranking Methods of IFSs . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Weighted Average Method for MADM Problems . . . . . . . . . . . . 12.4 The Extended GOWA Operator Based MADM Method . . . . . 12.4.1 Representation of MADM Problems Using IFSs . . . . . . 12.4.2 The Extended GOWA Operator Using IFSs . . . . . . . . . . 12.4.3 MADM Method and Procedure . . . . . . . . . . . . . . . . . . . . 12.5 Combining Expert Opinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Fuzzy Preference Relations . . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 189 189 191 191 193 194 195 199

9

Contents

XVII

Part IV Analysis of Results 13 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Conditions for a Uniform Model . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Part I Rank Change from the QLI Method . . . . . . . . . . 13.3.2 Part II Rank Correlation (Spearman’s rho): . . . . . . . . . . 13.3.3 Part III Analysis of Rank Variation . . . . . . . . . . . . . . . . .

203 203 204 205 205 206 206

14 Failed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Part I Number of Rank Changes . . . . . . . . . . . . . . . . . . . 14.2.2 Part II Rank Correlation (Spearman’s rho) . . . . . . . . . . 14.2.3 Part III One-Way Analysis of Variance (ANOVA) . . . . 14.2.4 Part IV Two-Way Analysis of Variance . . . . . . . . . . . . . .

209 209 212 213 213 214 217

15 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Issues in International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Nuclear Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Smart Power and Deterrence . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Cooperative Threat Reduction . . . . . . . . . . . . . . . . . . . . . 15.2 Issues in Comparative Politics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Successful Democratization . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Economic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Political Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Economic Freedom Scores . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Deaf and Hard of Hearing Children . . . . . . . . . . . . . . . . . . . . . . .

221 221 221 221 224 226 226 229 230 235 235

Reaching a Verdict by Weighting Evidence . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

List of Figures

F.1 Factors affecting a college decision . . . . . . . . . . . . . . . . . . . . . . . . . VIII

1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

A Fuzzy Set D on a Discrete Universe X = {x1 , x2 , ..., xn } . . . . A Fuzzy Set C on a Continuous Domain X = [1, 10] . . . . . . . . . The Membership Grade of the x Values 0, 1, 2, . . . , 30 in the Fuzzy Set B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Sets A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A ∩ B, the Intersection of Fuzzy Sets A and B . . . . . . . . . . . . . . A ∪ B, the Union of Fuzzy Sets A and B . . . . . . . . . . . . . . . . . . . Ac , the Complement of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sugeno Complement Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yager Complement Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphs of the max and min Functions. . . . . . . . . . . . . . . . . . . . . . Schweizer and Sklar t-norm and t-conorm Functions for p = 1 . A Graph of the Relations R and S . . . . . . . . . . . . . . . . . . . . . . . . . A Fuzzy Order Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Katy goes to college . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

1.1 1.2 1.3

6 8 9 14 14 15 16 17 18 20 20 34 39

List of Tables

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Common Universal Sets of Numbers in Mathematics . . . . . . . . . Measures of Subsethood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common t-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common t-conorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Truth Table for Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Implication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Averaging Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Binary Relation R Presented as a Table . . . . . . . . . . . . . . . . . . Types of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 13 21 23 23 26 28 30 32

2.1

An Example of a Basic Probability Assignment and the Associated Belief and Plausibility Measures . . . . . . . . . . . . . . . . . 44

3.1 3.2 3.3

Nuclear Deterrence Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Smart Power Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Cooperative Threat Reduction Data . . . . . . . . . . . . . . . . . . . . . . . 71

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Democratization Experts Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . Democratizaton Variable Data . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality of Life Data Set One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality of Life Data Set Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Freedom Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Political Stability Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Political Stability Subfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failed States Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 5.2 5.3

Expert Ranking of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Hearing Impaired Children Data . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Weight of expert opinion on test scores . . . . . . . . . . . . . . . . . . . . . 90

6.1

The Saaty Ranking Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

74 74 76 76 78 81 83 84

XXII

6.2

List of Tables

6.14 6.15 6.16 6.17

The High School Student Ranks the Attributes Against Each Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Attributes; Academics, Cost, Location, and Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Academic Stature: (a) Comparison and (b) Resultant Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Cost: (a) Comparison and (b) Resultant Rating . . . Analysis of Desirable Location: (a) Comparison and (b) Resultant Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Program Strength: (a) Comparison and (b) Resultant Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weighted Rank of Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katy’s Comparison of Cost (Affordability) . . . . . . . . . . . . . . . . . . Mom’s Comparison of Cost (Affordability) . . . . . . . . . . . . . . . . . . Dad’s Comparison of Cost (Affordability) . . . . . . . . . . . . . . . . . . Katy, Mom, and Dad’s Amalgamated Comparison of Cost (Affordability) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katy, Mom, and Dad’s Ranking of Schools by Cost (Affordability) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katy, Mom, and Dad’s Weighted Rank of Schools . . . . . . . . . . . Index of C.I. Values for Random Comparison Matrices . . . . . . . Failed States Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight of expert opinion on test scores . . . . . . . . . . . . . . . . . . . . .

7.1 7.2

Political Stability Guiasu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Normalized weight of expert opinion on test scores . . . . . . . . . . . 140

8.1

8.2 8.3 8.4 8.5 8.6

Democracy Values for Post-Communist Countries FH Freedom House Index VH - Vanhanen Index P IV - Polity IV Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Democracy Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Democracy values using Jeffrey’s Rule (Sum) . . . . . . . . . . . . . . . Democracy values using Jeffrey’s Rule (D-S Combo) . . . . . . . . . H values for 27 Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined H values for 27 Countries . . . . . . . . . . . . . . . . . . . . . .

9.1

Political Stability Yen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3 6.4 6.5 6.6

6.7

6.8 6.9 6.10 6.11 6.12 6.13

97 97 98 99

99

99 100 101 101 101 102 102 102 103 115 117

146 147 149 151 152 153

12.1 Political Stability Expert Opinions . . . . . . . . . . . . . . . . . . . . . . . . 195 12.2 Semigroup for IFS selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 15.1 Democracy Scores Predicted by the Guiasu Model and Dempster-Shafer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 15.2 G+ and G− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

List of Tables

XXIII

15.3 G Rankings and Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 15.4 Statistics for Student Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 15.5 Multiple Regression Model of Student Scores . . . . . . . . . . . . . . . . 237

Nomenclature

⊆ χA ◦t c(a) h(a, b) h(a1 , a2 , . . . , an ) ∈ F (X) μA (x) ωt (a, b) ia igg igr ig ikd il im ir iss iy ∨ ∧ A Aα Aα > Im(A) OW Aw m i aRb {a, b, c} t(a, b) s(a, b)

Subset Characteristic function of A sup–t composition of fuzzy relations Complement function Averaging function, binary Averaging function, general Element of Fuzzy power set The membership function of a fuzzy set Residuum function Lukasiewicz Goguen Gaines–Rescher Gödel Kleene-Dienes Larsen Early Zadeh Reichenbach strict Standard strict Yager Fuzzy union, ususally max Fuzzy intersection, usually min The labels of fuzzy sets Alpha-cut of A Strong alpha-cut of A Image of A Ordered weighted average Basic probability assignment Fuzzy implication operator Relation A set of objects t-norm intersection function t-conorm union function

Introduction

In Montreal it is 10º. Of course that is in Centigrade degrees. To convert this to Fahrenheit degrees we divide by five multiply by nine and add thirty–two. Mathematically the formula is F =

9 C + 32 5

which is a linear equations. Linear equations involve multiplying the input variable(s) by constants and then summing the terms, including a constant term. In science linear equations may be based theory or on empirical data. A theory based equation for force is that it is equal to mass times acceleration: F = ma . An empirically based equation is that women live about five years longer then men on the average: W = M + 5. Empirical equations are derived from measurements. The most common way to derive a linear model from data is to use least square fitting. That is, suppose we have N of data points xi , yi , where i = 1, 2. . . . N , and you want to predict the value of y that would be associated with a new value of x. We would use a linear model y = mx + b where the slope m and the intercept b are derived from the data. The values of m and b are chosen so that for the data points the actual value of yi is a close to the predicted value yˆi = mxi + b. That is m and b are chosen so that SS =

n  i=1

2

(yi − (mxi + b))

XXVIII

Introduction

is as small as possible. This value, the sum of the squares, or SS, can be minimized by taking the derivatives of SS with respect to m and b and setting them equal to zero. n Assume that the average value of the xi s is x ¯ = n1 i=1 xi and that the  average value of the yi s is y¯ = n1 ni=1 yi . The result of setting the derivatives of SS with respect to m and b to zero and solving for m and b is m=

n  (yi − y¯) (xi − x ¯) i=1

(xi − x ¯)

2

b = y¯ − m¯ x.

This result is easily generalizable to multiple dimensional input variables X. It is important to note that regression simply minimizes the sum of the square of the error, the error being the difference between the actual value yi and predicted value yˆi . Regression is a standard topic in statistics books because: if we assume that x and or y is random and has a statistical distribution, such as the normal distribution, then we can make statements on the statistical likelihood of our accuracy. This books examines the case where this assumption is unwarranted. That is, we examine cases where the number of data points is small (effects of nuclear warfare), where the experiment is not repeatable (the break–up of the former Soviet Union), and where the data is derived from expert opinion (how conservative is a given political party) which is difficult to measure. In all these cases that assumptions of randomness and/or statistical validity are questionable.

Part I

Mathematics of Uncertainty

Chapter 1

Fuzzy Set Theory

This chapter presents to the reader the fundamental concepts of fuzzy set theory applied in the later chapters of this book. Zadeh’s Zadeh (1965) seminal work is a lucid and very readable introduction to the subject. From the viewpoint of the social sciences, probably the best overview is provided by Smithson and Verkuilen (2006). Set theory provides simple classification of objects as possessing an attribute or not. Fuzzy set theory allows on object to have an attribute to some degree. The set of red object divides the world into objects that are red or objects that are not red. A fuzzy set of redness assigns each object a value indicating its relative degree of redness. Life is simpler in a world of absolutes but this is a poor model of reality. Life is more difficult in the real world, and in the real world things are often hard to classify.

1.1

Set Theory

Set theory is the foundation of all branches of modern mathematics. Even numbers and geometric objects are seldom considered as basic or primitive concepts. In formal mathematics they are defined as constructions of set theory. Set theory is based on the notion of a class or collection of objects. The fundamental concept of set theory is; given an object in the universe of discourse (the general assemblage of things that a discussion is about), a set is well defined if one can decide whether or not the given object is contained in the set. This simple notion was one of the most powerful ideas ever conceived in the field of mathematics. A set is a collection of objects called elements. Typographically the brackets “{” and “}” are used to denote the beginning and ending of the list of elements that are in the set. J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 3–40. DOI: 10.1007/978-3-642-35224-9_1 © Springer-Verlag Berlin Heidelberg 2013

4

1 Fuzzy Set Theory

A set is defined using one of three methodologies. In the first method the elements of the set are explicitly listed, as in A = {1, 3, 5, 7, 9}

(1.1)

Here we have a set, tagged with the name or label A, and containing as elements the objects one, three, five, seven, and nine. Symbolically the statement “5 is an element of set A”is written 5 ∈ A. We can also say that 6 is not an element of A, or 6 ∈ / A. Conventionally capital letters represent sets and small letters represent elements. The second method for defining a set is implemented by giving a rule or property that a potential element must obey or posses to be included in the set. An example of this is the set A = {odd numbers between zero and ten}

(1.2)

This is the same set A that was defined explicitly by listing its elements in Eq. (1.1) above. The universe or universal set is usually labeled X or U , although any symbol is allowable. Some sets have a standard label so that N is the natural numbers, Z is the integers and R is the real numbers. The third way to determine a set is through a characteristic function. If χA is the characteristic function of a set A then χA is a function from the universe of discourse X to the set {0, 1}, where χA (x) =



1 x∈A 0 x∈ /A

(1.3)

so that the value 1 indicates membership and the value 0 indicates non– membership. In the examples above where the set A is the set of odd natural numbers less than ten then the characteristic function for this set is  1 x = 1, 3, 5, 7, 9 χA (x) = 0 otherwise The traditional notation for a characteristic function uses the Greek letter χ or chi and the set A is indicated as the subscript. However it is desirable for the purposes of this text to introduce a notation that consider A as both the label of a set and as the label of its characteristic function. Thus if A is a set, then its characteristic function is indicated by A(x) =



1 x∈A 0 x∈ /A

(1.4)

1.2 Fuzzy Sets

5

The set B = {0, 1} is so important that we give it a special name, the Boole set, named in honor of Georg Boole who was one of the most important figures in the history of set theory. Table 1.1 Common Universal Sets of Numbers in Mathematics Name

Symbol

Set

Natural numbers

N

{1, 2, 3, . . .}

Bounded natural numbers

Nn

{1, 2, 3, . . . , n − 1, n}

Non-negative integers

N0

{0, 1, 2, 3, . . .}

Integers

Z

Rational numbers

Q

Real numbers

R

{. . . , −3, −2, −1, 0, 1, 2, 3, . . .} a  | a, b ∈ Z b

Positive real numbers

R+

1.2

any sequence of digits – possibly signed, possibly containing a decimal point, and possibly infinite {x | x ∈ R and x > 0}

Fuzzy Sets

Ambiguity pervades human thinking, and it is reflected in human language. For example, the response from most people to the question, “How old is Lee?” is not likely to be a numeric answer. Only his friends and relatives usually know his exact calendar age. An acquaintance will answer that he is a “teenager” or “young” or “adolescent.” Furthermore, the age of the respondent will greatly influence the answer given. A person of seventy might say Juan is a “boy” whereas a contemporary might say, “He is the same age I am, ” a response that provides no direct numerical information. Lotfi Zadeh Zadeh (1965) created fuzzy set theory in order to mathematically represent and process such ambiguity. While the term “fuzzy” might carry the connotation to some that it is associated with fuzzy thinking, one might as well criticize probability theory for leading to random actions. If probability theory is a precise mathematical device for processing data whose source is a random event, fuzzy set theory is a precise mathematical tool for processing data that is derived from vague sources. While probability deals with uncertainty associated with randomness, fuzzy set logic is concerned with uncertainty associated with imprecision rooted in vagueness and ambiguity that has nothing to do with randomness. Rather it has to do with ambiguity concerning the exact characteristic, be it the quality, location, preference, etc., of a thing.

6

1 Fuzzy Set Theory

1.0 µD(x4)

...

0.5

0.0

X x1

x2

x3

x4

...

xn-1

xn

Fig. 1.1 A Fuzzy Set D on a Discrete Universe X = {x1 , x2 , ..., xn }

1.3

Membership Functions

The fundamental idea of fuzzy set theory is that real world phenomena cannot be divided cleanly into black and white divisions. In fuzzy set theory we extend the image set of the characteristic function from the binary set B = {0, 1} which contains only two alternatives, to the unit interval U = [0, 1] which has an infinite number of alternatives. To reflect this definition, we also relabel the characteristic function as the membership function, denoted by the symbol μ, instead of χ. This introduces a rich approach for measuring the world in shades of grey. For instance, using fuzzy set theory we can characterize democracy as more than “democracy” or “not democracy.” We can choose gradations such as “very high,” “high” “uncertain,” “low,” or “very low” and assign membership values accordingly. These are not evaluations of the probability that a country is a democracy, rather they are measures of the degree to which it is a democracy. A fuzzy set is essentially a function whose domain is some universal set X. Its range is the unit interval U =[0, 1]. The notation, μA : X → [0, 1], which specifies the membership function, defines the fuzzy set. The image of μA will be a set of membership values bounded by 0 and 1 indicating the degree of membership in A of each object x ∈ X. Hence, a fuzzy set consists of a set of objects and their membership values. (We could think of a crisp set in the same way, but the membership values for all elements of the set would be 1.) Consider the following example.

1.3 Membership Functions

7

Example 1.1. Let X = {a, b, c} and set μA (a) = 1.0, μA (b) = 0.7, and μA (c) = 0.4. The example defines fuzzy set A with three objects {a, b, c}, and their respective membership values, 1, .7, and .4. Here μA tells us that a is fully a member in the set A, b is .7 a member, and c is .4 a member. The tag of the fuzzy set, A, is difficult to read as a subscript, and the membership function of a fuzzy set with a subscript, such as Ai with membership function μAi , compounds the subscript problem. Therefore, it is often desirable to use the same tag for a set and its characteristic function. We will use the tag A to represent both the fuzzy set and its membership function. Thus, if A is a fuzzy set then we will also use A as the label of a function from a universe of discourse X into the unit interval U = [0, 1] A : X → [0, 1].

(1.5)

The membership function of the fuzzy set in Example 1.1 can be rewritten as A(a) = 1.0, A(b) = 0.7, and A(c) = 0.4. Alternatively, we can list the elements of X and their membership grades as set of ordered pairs, so that A = { a, 1.0 , b, 0.7 , c, 0.4 }

(1.6)

The notation used in books about fuzzy set theory is not yet standardized. A fuzzy set was initially indicated by the tilde above the character, such ˜ Much of the early literature includes phrases such as “the fuzzy set as A. ˜ We will use the notation A in this book to denote a fuzzy set and μA to E”. denote a fuzzy set membership function only if there is a chance of confusion. Synonyms for membership function are membership grade and characteristic function. Example 1.2. Let X = {x1 , x2 , x3 , xx , ...xn } and set μD (x1 ) = 0.5, μD (x2 ) = 0.4, μD (x3 ) = 0.8, μD (x4 ) = 0.7,. . . , μD (xn−1 ) = 0.5 and μD (xn ) = 0.7. Then the discrete fuzzy set D is depicted at Figure 2.1. Let X = [1, 10] and μC (xi ) = [0, 1]. Then one example of a continuous fuzzy set C is depicted at Figure 2.2. Note that μC : [1, 10] −→ [0, 1], but the mapping is not fully specified here. Figure 2.2 is one of an infinite number of fuzzy sets that use a membership value map like μC . The only difference between a traditional set and a fuzzy set is the image of their membership functions. A traditional set has its membership grades taking values in the set {0, 1} while a fuzzy set has its membership grades in the unit interval [0, 1]. A fuzzy set can include elements that are not fully included in the set, but that are not fully excluded. A standard set will be called a crisp set whenever it is necessary to distinguish it from a fuzzy set. The universal set X is always understood to be a crisp set. A few more definitions are necessary to our discussion. Crisp set theory defines the class of all sets defined on a universe as the power set of the universe X, P(X). Fuzzy set theory makes use of an analogous concept.

8

1 Fuzzy Set Theory

Fig. 1.2 A Fuzzy Set C on a Continuous Domain X = [1, 10]

The set of all fuzzy sets defined on the universal set X is called the fuzzy power set. Definition 1.3. Suppose that X is a crisp universal set. Let the class of all fuzzy sets defined on X be denoted by F (X). F (X) is called the fuzzy power set of X. The scalar cardinality of a fuzzy set A is a count of the number of elements in A. Of course, some elements are not completely in fuzzy set A so the scalar cardinality of A is not necessarily an integer (as is the case for a discrete crisp set.) Definition 1.4. The scalar cardinality of A is the sum of the degree of membership of every element in A and is denoted |A|,  |A| = A (x) . (1.7) x∈X

Example 1.5. The scalar cardinality of A, whose membership function is given in Example (1.1) is  A (x) (1.8) |A| = x∈X

=



A (x)

(1.9)

x∈{a,b,c}

= A(a) + A(b) + A(c) = 1.0 + 0.7 + 0.4

(1.10) (1.11)

= 2.1 .

(1.12)

1.4 Alpha-cut or α-cut

9

1.0

0.8

0.6

0.4

0.2

0 0

10

20

30

Fig. 1.3 The Membership Grade of the x Values 0, 1, 2, . . . , 30 in the Fuzzy Set B

1.4

Alpha-cut or α-cut

Sometimes it is useful to discuss a subset associated with a fuzzy set. One particularly useful class of subsets comprises the elements of a fuzzy set with membership values larger than a given cutoff α. Definition 1.6. For every α ∈ [0, 1], a given fuzzy set A yields a crisp set Aα = {x ∈ X | A(x) ≥ α} which is called an α-cut of A. Since α1 < α2 implies Aα1 ⊇ Aα2 the set of all distinct α-cuts of any fuzzy set forms a nested sequence of crisp sets. Definition 1.7. We define the image set of A, Im(A) as the image of the membership function μA . It consists of all values α in the unit interval such that A(x) = α for some x ∈ X. Im(A) = {A(x) | x ∈ X} .

(1.13)

The image set can simply be thought of as the set of membership values of the objects in X. Example 1.8. The image set of A is Im(A) = {0.4, 0.7, 1.0}. Consider once more A = { a, 1.0 , b, 0.7 , c, 0.4 } . (1.14)

10

1 Fuzzy Set Theory

The α–cut at α = 0.5 is A0.5 = {a, b} since μA (a) = 1.0 ≥ 0.5 and μA (b) = 0.7 ≥ 0.5, but μA (c) = 0.4  0.5. The image set of A, Im(a) = {1.0, 0.7, 0.4} , gives all the membership values that generate distinct α-cuts. In fact, an arbitrary fuzzy set A, it is uniquely represented by the associated sequence of its distinct α-cuts via the formula A(x) = sup α · χAα (x).

(1.15)

α∈[0,1]

where χAα denotes the characteristic function of the crisp set Aα and sup designates the supremum (or maximum, when the image set Im(A) is finite) which is the largest value the expression attains as α ranges over the unit interval. Equation (1.15), usually referred to as a decomposition theorem of fuzzy sets Zadeh (1971), establishes an important connection between fuzzy sets and crisp sets. This connection provides us with a criterion for generalizing properties of crisp sets into their fuzzy counterparts: when a fuzzy set satisfies a property that is claimed to be a generalization of a property established for crisp sets, this property usually should be preserved (in the crisp sense) in all α-cuts of the fuzzy set. For example, all α-cuts of a convex fuzzy set should be convex crisp sets; the cardinality of a fuzzy set should yield for each α the cardinality of its α-cut; each α-cut of a properly defined fuzzy equivalence relation (or fuzzy compatibility relation, fuzzy ordering relation, etc.) should be an equivalence relation (or compatibility relation, ordering relation, etc., respectively) in the classical sense. Every property of a fuzzy set that is preserved in all its α-cuts is called a cutworthy property. Example 1.9. Let X = {0, 1, 2, . . . , 29, 30}. Define the fuzzy set B by ∀x ∈ X, 30x − x2 . (1.16) B(x) = 225 Then B(0) = 0, B(5) = 0.56, B(15) = 1, B(20) = 0.89, etc. This fuzzy set is depicted at Figure 1.3. The α−cut of B at α = 0.7 is a crisp set containing all those elements x ∈ X whose membership grade is greater than or equal to 0.7, that is B α = {7, 8, 9, . . . , 22, 23}. Definition 1.10. The strong alpha cut, Aα > , is defined to be all those elements that have membership strictly greater than alpha, Aα > = {x | A(x) > α}.

(1.17)

Definition 1.11. The support of a fuzzy set is the strong alpha-cut at zero:

1.5 Standard Fuzzy Set Operations

S(A) = A0> = {x | A(x) > 0} .

11

(1.18) (1.19)

Hence it is that subset of the domain which has positive membership grade in a fuzzy set. Definition 1.12. The core, peak or mode of a fuzzy set is the alpha-cut at one: C(A) = A1 = {x | A(x) = 1} .

1.5

(1.20) (1.21)

Standard Fuzzy Set Operations

The mathematics of fuzzy set theory is often more difficult than that of the traditional set theory because the continuous interval U = [0, 1] is inherently more complex than the binary set B = {0, 1}. Furthermore, if we work from the premise that both crisp sets and fuzzy sets can be defined using membership functions, crisp sets are, in a sense, a special case of fuzzy sets. The crisp set membership values zero and one are contained in the unit interval and crisp sets can be thought of as fuzzy sets with a restricted image set. In this section we define what it means for a fuzzy set to be contained in another fuzzy set. We also provide operators for fuzzy sets that correspond to “and” “or”, and “not” in human logic. We define a fuzzy intersection to represent “and”, a fuzzy union to represent “or”, and a fuzzy complement to represent “not”. All the definitions for complements, unions, and intersections are given in terms of membership functions. We use the same notation for fuzzy intersections, unions, and complements as those for crisp intersections, unions, and complements. The definition of a fuzzy intersection is crafted so that if it has {0, 1} as its image set then the fuzzy intersection behaves like the intersection of crisp sets. This will also be true for the fuzzy union, fuzzy complement, and the fuzzy subset relation. Hence, we need not add the term “fuzzy” to these operations. The fuzzy intersection is the crisp intersection whenever the sets involved correspond to crisp sets.

1.5.1

Subsets

A fuzzy set A is a subset of a fuzzy set B if A(x) is less than or equal to B(x) for all x in X. Thus A ⊆ B if and only if ∀x A(x) ≤ B(x).

(1.22)

12

1 Fuzzy Set Theory

For A to be equal to B the membership values of A(x) and B(x) must be equal for all x. Therefore A = B if and only if ∀x A(x) = B(x).

(1.23)

It is simple to show that A ⊆ B and A ⊇ B imply that A = B. The class of all fuzzy subsets of the universe X is the fuzzy power set F (X). There are an infinite number of fuzzy subsets of any non–empty universe.

1.5.2

Subsethood

Zadeh’s original definition of A ⊆ B, when he created fuzzy sets, simply insisted that ∀x μA (x) ≤ μB (x) where μA is the membership function of fuzzy set A and μB is the membership function of fuzzy set B. It is interesting to note that the obvious definition of A → B in fuzzy logic is that A ⊆ B when A and B are viewed as fuzzy sets. However, Zadeh’s definition only tells us whether or not A is a fuzzy subset of B or not, and is in a sense a crisp decision. Because of this, many authors have tried to introduce measures of subsethood, Wierman et al. (2007), that determine to what degree A is a subset of B. The most famous such measure is the one introduced by Sanchez (1977) and popularized by (Kosko, 1990): |A ∩ B| |A|  A(x) ∧ B(x)  = x∈X . x∈X A(x)

S(A, B) =

(1.24)

Here |A| is the scalar cardinality, i.e., the sum of the membership function of A over X. The obvious similarities between the fuzzy subsethood measure, Eq. (1.24), and the conditional probability, Eq. (1.25), P (B|A) =

P (A ∩ B) . P (A)

(1.25)

have been explored deeply by Kosko Kosko (1990, 1993, 2004) and others. The next to be considered was proposed by Bandler and Kohout (1980). Bandler and Kohout’s measure determines the extent to which a fuzzy set A is a fuzzy subset of the fuzzy set B as follows: SY (A, B) = min[1 − A(x) ∨ B(x)] x∈X

(1.26)

1.5 Standard Fuzzy Set Operations

13

Ogawa et al. (1985) offers another subsethood method:

So (A, B) =



x∈X



min[μA (x), μB (x)]  μB (x)

(1.27)

x∈X

[A(x) ∨ B(x)] x∈X  = B(x) x∈X

The requirement that the subsethood measure be cutworthy is examined in Wierman (1997), he proposed 

1

χ⊆ (Aα , B α ) dα 0          ˜ ˜  = 1− [ B(x), A(x)]   x:A(x)>B(x)   

SW (A, B) =

(1.28)

(1.29)

where Aα is the α–cut of A, χ⊆ is the characteristic function of the crisp subsethood relation, and ||E|| represents the measure of the set E. We have already mentioned Sanchez’s measure, S(A, B) given by Eq. (1.24) and Ishizuka simply divides by the fuzzy cardinality of B rather than A. |A ∩ B| (1.30) SI (A, B) = |B| Table 1.2 Measures of Subsethood Name

Definition

Ishizuka

|A∩B| |A| |A∩B| |B|  [A(x)∨B(x)] x∈X  B(x)

Kosko Ogawa

 1 x∈X α α Wierman χ (A , B ) dα 0 ⊆ Bandler and Kohout x∈X [(1 − A(x)) ∨ B(x)]

1.5.3

Intersection

Suppose we have two continuous fuzzy sets, A and B, as depicted at Figure 1.4.

14

1 Fuzzy Set Theory

1.0

0.5

A

B

0.0

X 1

2

3

4

5

6

7

8

9

10

Fig. 1.4 Fuzzy Sets A and B

1.0

A

0.5

0.0

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 0000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

B

X 1

2

3

4

5

6

7

8

9

10

Fig. 1.5 A ∩ B, the Intersection of Fuzzy Sets A and B

The intersection of two fuzzy sets is the fuzzy set C defined by C(x) = (A ∩ B) (x), where (A ∩ B) (x) = min{A(x), B(x)}

(1.31)

for all x in X. The intersection of fuzzy sets A and B is the hatched area in Figure 1.5.

1.5 Standard Fuzzy Set Operations

15

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00B 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

1.0

0.5

0.0

X 1

2

3

4

5

6

7

8

9

10

Fig. 1.6 A ∪ B, the Union of Fuzzy Sets A and B

The minimum operator is represented by the symbol “∧”so that min{a, b} can be written a∧b and the intersection membership function is often written (A ∩ B) (x) = A(x) ∧ B(x) .

(1.32)

The intersection of two fuzzy sets contains all the elements of minimum membership grade in A and B.

1.5.4

Union

The union of two fuzzy sets is the fuzzy set C defined by C(x) = (A ∪ B) (x), where (A ∪ B) (x) = max{A(x), B(x)} (1.33) for all xin X. The union of fuzzy sets A and B is the hatched area in Figure 1.6. The maximum operator is represented by the symbol “∨”so that max{a, b} can be written a ∨ b and the membership function for union can be written (A ∪ B) (x) = A(x) ∨ B(x) .

(1.34)

Thus the union of two fuzzy sets contains all the elements of the maximum degree consistent with A or B.

16

1 Fuzzy Set Theory

1.0

0.5

0.0

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00A 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

X 1

2

3

4

5

6

7

8

9

10

Fig. 1.7 Ac , the Complement of A

1.5.5

Complement

The complement represents the notion of “not”in human language. The complement of a set A is another set Ac that contains each element x in the universe X to the opposite degree that the original set contained x. The complement of fuzzy set A (in Figure 1.4) is depicted in Figure 1.7. The standard complement operator is Ac (x) = 1 − A(x) .

(1.35)

This formula represents one of the most important differences between fuzzy set theory and standard set theory. In set theory it is always true that a set and its complement have nothing in common. In fuzzy set theory a set and its complement can be identical. Consider the fuzzy set A of Eq. (1.6). Then its complement is A = { a, 0.0 , b, 0.3 , c, 0.6 } . (1.36) and the membership grade of the intersection of A and Ac is A ∩ Ac = { a, 0.0 , b, 0.3 , c, 0.4 } .

(1.37)

This is a better picture of reality since, for instance, we could go to a movie and like and dislike it at the same time. People often hold contradictory feelings, opinions and evaluations about the same exact thing. We often prevaricate in our evaluations, for instance of an unusual dish ordered at a restaurant,

1.6 Generalized Fuzzy Set Operations

17

jumping from one side of a decision to the other with monotonous regularity (or with each new bite).

1.6

Generalized Fuzzy Set Operations

The standard fuzzy set operations generalize the operations of crisp set theory. However, they are not the only way this can be done. The following sections briefly cover the theory of generalized fuzzy set complement, intersection, and union operators.

1.6.1

Complement

An arbitrary complement operators, c : [0, 1] → [0, 1], must satisfy the following three axioms:

Fig. 1.8 Sugeno Complement Functions

(c1) Membership dependency — The membership grade of x in the complement of A depends only on the membership grade of x in A. (c2) Boundary condition — c(0) = 1 and c(1) = 0, that is c behaves as the ordinary complement for crisp sets.

18

1 Fuzzy Set Theory

(c3) Monotonicity — For all a, b ∈ [0, 1], if a < b, then c(a) ≥ c(b), that is c is monotonic nonincreasing. Two additional axioms, which are usually considered desirable, constrain the large family of functions that would be permitted by the above three axioms; they are: (c4) Continuity — c is continuous. (c5) Involution — c is involutive, that is c(c(a)) = a.

Fig. 1.9 Yager Complement Functions

The standard fuzzy complement function is: c(a) = 1 − a Some of the functions that conform to these five axioms besides the standard fuzzy complement are in the Sugeno class of fuzzy complements defined for all a ∈ [0, 1] by 1−a c[λ](a) = , (1.38) 1 + λa with λ ∈ (−1, ∞). The curves generated by the Sugeno complement for various values of λ are illustrated in Fig (1.8). The Yager class of fuzzy complements defined for all a ∈ [0, 1] by

1.6 Generalized Fuzzy Set Operations

c[w](a) = (1 − aw )1/w ,

19

(1.39)

with w ∈ (0, ∞). . The curves generated by the Yager complement for various values of w are illustrated in Fig (1.9). Observe that the standard fuzzy complement, c(a) = 1 − a, is obtained as the Sugeno complement at zero, c[λ = 0] and as the Yager complement at one, c[w = 1]. An example of fuzzy complements that conforms to (c1)–(c3) but not to (c4) and (c5) are the threshold fuzzy complements  1 when a ∈ [0, t] c[t](a) = , (1.40) 0 when a ∈ (t, 1] with t ∈ [0, 1]. Subsequently, we shall write Ac for an arbitrary complement of the fuzzy set A; its membership function is Ac (x) = c(A(x)). An equilibrium, ec , of a fuzzy complement c, if it exists, is a number in [0, 1] for which c(ec ) = ec . Every fuzzy complement has at most one fuzzy equilibrium and if a fuzzy complement is continuous (i.e., if it satisfies axioms (c1)–(c4)), the existence of its equilibrium is guaranteed Klir and Yuan (1996). For example, the equilibrium of fuzzy complements in the Yager class (1.39) are 1 ec [w] = 0.5 w (1.41) for each w ∈ (0, ∞).

1.6.2

Fuzzy Set Intersections

In the previous section we saw that complementation can be based on a function c that manipulates membership values. Using the same line of reasoning, intersection of fuzzy sets A and B can be based upon a function t that that takes two arguments. The function t takes the membership grade of x in the fuzzy set A and the membership grade of x in the fuzzy set B and returns the membership grade of x in the fuzzy set A ∩ B. The intersection of two fuzzy sets must be a function that maps pairs of numbers in the unit interval into the unit interval [0, 1], t : [0, 1] × [0, 1] → [0, 1]. It is now well established that triangular norms or t-norms do possess all properties that are intuitively associated with fuzzy intersections. These functions are, for all a, b, c, d ∈ [0, 1], characterized by he following axioms: (i1) (i2) (i3) (i4)

Boundary condition — t(1, a) = a. Monotonicity — t(a, b) ≤ t(c, d) whenever a ≤ c and b ≤ d. Commutativity — t(a, b) = t(b, a). Associativity — t(a, t(b, c)) = t(t(a, b), c).

20

1 Fuzzy Set Theory

Fig. 1.10 Graphs of the max and min Functions.

Fig. 1.11 Schweizer and Sklar t-norm and t-conorm Functions for p = 1

It turns out that functions that obey these four rules had been extensively studied in the literature of probability, long before the creation of fuzzy set theory. Functions that possess properties (i1) through (i4) were given the name t-norm Menger (1942), short for triangular-norm. Theorem 1.13. t(0, 0) = t(0, 1) = t(1, 0) = 0 and t(1, 1) = 1. Proof. By axiom (i1) t(1, a) = a so set a = 1 and we have that t(1, 1) = 1. By axiom (i1) t(1, a) = a so set a = 0 and we have that t(1, 0) = 0. Commutativity (i3) says that if t(1, 0) = 0 then t(0, 1) = 0.

1.6 Generalized Fuzzy Set Operations

21

Finally 0 ≤ 1 and 1 ≤ 1 so axiom (i2) says that t(0, 0) ≤ t(1, 0) = 0 but since t(a, b) cannot be negative the only value less than or equal to zero is zero we conclude that t(0, 0) = 0. ⊔ ⊓ The largest t-norm is the minimum function and the smallest is the drastic product ⎧ ⎨ a when b = 1 tmin (a, b) = b when a = 1 , (1.42) ⎩ 0 otherwise in the sense that if t is any t-norm then for any a, b ∈ [0, 1] tmin (a, b) ≤ t(a, b) ≤ min(a, b).

(1.43)

One of the most commonly applied alternative t-norms is the algebraic product: tp (a, b) = a · b, (1.44) which is calculated by simply multiplying a and b. Since multiplication is commutative and associative, axioms i3 and i4 are satisfied by tp . Multiplication of nonnegative numbers is monotonic so axiom i2 is satisfied. Finally tp (1, a) = 1 · a = a so that axiom i1 is satisfied. Thus tp is a t − norm. Another common alternative to the standard intersection operator min is the bounded difference: tb (a, b) = max(0, a + b − 1) .

(1.45)

It can be shown that the basic t-norms have the following order: tmin (a, b) ≤ max (0, a + b − 1) ≤ ab ≤ min(a, b) The most common t-norms are given in Table (1.3). Table 1.3 Common t-norms Name

t-norm

standard intersection t(a, b) = min(a, b) algebraic product

tp (a, b) = ab

bounded difference

tb (a, b) = max(0, a + b − 1) ⎧ ⎨ a when b = 1 tmin (a, b) = b when a = 1 ⎩ 0 otherwise

drastic product

(1.46)

22

1 Fuzzy Set Theory

It is sometimes convenient to think of a t-norm as binary operator, and use an in-line notation when applying it to a specific problem Thus instead of writing t(a, b) we would write a ∧t b, or a t b, or just a ∧ b, with the understanding that ∧ is some intersection operator modeled by a t-norm t.

1.6.3

Fuzzy Set Unions

Similar ideas from the section on fuzzy set intersections lead to the following axiomatic skeleton for a function s to model the union operator. The union of two fuzzy sets must be a function that maps pairs of numbers in the unit interval into the unit interval, s: [0, 1] × [0, 1] → [0, 1]. As is well known, functions known as triangular conorms or t-conorms, possess all the properties that are intuitively associated with fuzzy unions. They are characterized for all a, b, c, d ∈ [0, 1] by the following axioms: (u1) (u2) (u3) (u4)

Boundary condition — s(0, a) = a. Monotonicity — s(a, b) ≤ s(c, d) whenever a ≤ c and b ≤ d. Commutativity — s(a, b) = s(b, a). Associativity — s(a, s(b, c)) = s(s(a, b), c).

The smallest t-conorm is the maximum function and the largest is the drastic sum sometimes called the drastic union: ⎧ ⎨ a when b = 0 smax (a, b) = b when a = 0 , (1.47) ⎩ 1 otherwise in the sense that if s is any t-conorm then for any a, b ∈ [0, 1] max(a, b) ≤ s(a, b) ≤ smax (a, b).

(1.48)

One of the most commonly applied alternative s-norms is the algebraic sum: sp (a, b) = a + b − ab,

(1.49)

which is also called the probabilistic sum. Another common alternative to the standard union operator max is the bounded sum: (1.50) sb (a, b) = min(1, a + b) . It can also be shown that the basic t-conorms have the following order: smax (a, b) ≥ min (1, a + b) ≥ a + b − ab ≥ max(a, b)

(1.51)

The most common t-norms are given in Table (1.4). It is sometimes convenient to think of a t-conorm (sometimes called an s-norm) as binary operator, and use an in-line notation when applying it to

1.7 Fuzzy Implication

23 Table 1.4 Common t-conorms

Name

t-conorm

standard union s(a, b) = max(a, b) algebraic sum sp (a, b) = a + b − ab bounded sum drastic sum

sb (a, b) = min(1, a + b) ⎧ ⎨ a when b = 0 smax (a, b) = b when a = 0 ⎩ 1 otherwise

a specific problem Thus instead of writing s(a, b) we would write a ∨s b, or a s b, or just a ∨ b, with the understanding that ∨ is some union operator modeled by a t-conorm s.

1.6.4

Residuum Operator – Omega Operator

Let t be a continuous t-norm. Define the residuum operator, also called the ω (omega) operator generated by t, ωt , for every a, b ∈ [0, 1] by the following definition ωt (a, b) = sup {x ∈ [0, 1] | t(a, x) ≤ b} . (1.52) The residuum operator operator is in one sense a model of material implication. Basically, we ask how much evidence (x) can we add to a before we break the threshold of belief b.

1.7

Fuzzy Implication

Material implication, if A then B, is modeled in classical logic with the right arrow operator A → B. The result of implication for every possible truth assignment is given in the Truth Table (1.5), where 0 represents false and 1 represents true. Table 1.5 The Truth Table for Implication AB A→B 0 0 1 1

0 1 0 1

1 1 0 1

24

1.7.1

1 Fuzzy Set Theory

Implication

Fuzzy logic expands classical logic by simple expanding the role of the characteristic or membership function of A → B. A fuzzy implication operator i : X × Y → [0, 1] allows one to calculate the value of (A → B)(u, v) given the values of of A(u) and B(v), (A → B)(u, v) = i(A(u), B(v)).

(1.53)

We shall also use the notation (A → B)(u, v) = A(u) → B(v)

(1.54)

to represent the fuzzy set that corresponds to the implication if u is A then v is B, where A and B are fuzzy sets on X and Y respectively. One possible extension of material implication to implications with intermediate truth values is  1 if A(u) ≤ B(v) (1.55) A(u) → B(v) = 0 otherwise This implication operator is called Standard Strict . A smoother extension of material implication operator can be derived from a trivial consequence of the isomorphism between classical logic and classical set theory. When A, B and C are classical sets in a universe X, then the following equivalence can be demonstrated A → B ≡ sup {C | A ∩ B ⊂ C} .

(1.56)

where sup is by set inclusion. Using the above equivalence one can define the following fuzzy implication operator, where w ∈ [0, 1] A(u) → B(v) = sup {w | min{A(u), w} ≤ B(v)} that is, A(u) → B(v) =



1 if A(u) ≤ B(v) B(v) otherwise

(1.57)

(1.58)

This operator is called Gödel implication. Using the definitions of negation and union of fuzzy subsets the material implication operator p → q = ¬p ∨ q can be extended to fuzzy sets with the following definition A(u) → B(v) = max{1 − A(u), B(v)}

(1.59)

1.7 Fuzzy Implication

25

This operator is called Kleene-Dienes implication. In many practical applications one uses Mamdani’s implication operator to model causal relationship between fuzzy variables. This operator simply takes the minimum of truth values of fuzzy predicates A(u) → B(v) = min{A(u), B(v)}

(1.60)

It is easy to see this is not a correct extension of material implications, because 0 → 0 yields zero while the truth table for classical logic gives the truth value of 0 → 0 as 1. However, in the knowledge-based systems that will be demonstrated in the following chapters, rules for which the antecedent, the A part, are false are simply ignore. This is like a rule that says “If it is raining, then open your umbrella.”If it is not raining we don’t even think about this rule and could care less what it says to do. That is because fuzzy logic is for application not for symbolic manipulation. Symbolic manipulation is important, without it we would not have digital computers, however, symbolic manipulation is not the whole of thinking, though some have claimed that it is. There are three important classes of fuzzy implication operators: • s-implications: defined by a → b = s(c(a), b)

(1.61)

where s is a t-conorm (or s-norm, hence the name, s-implication) and c is a negation operator on [0, 1]. These implications arise from the Boolean formalism a → b ≡ ¬a ∨ b . (1.62) Typical examples of s-implications are the Lukasiewicz and Kleene-Dienes implications. • R-implications: obtained by residuation of continuous t-norm t, i.e., a → b = sup {c ∈ [0, 1] | t(a, b) ≤ c}

(1.63)

These implications arise from the Intutionistic Logic formalism. of Eq. (1.56) Typical examples of R–implications are the Gödel and Gaines implications. • t-norm implications: if t is a t-norm then a → b = t(a, b) .

(1.64)

t-norm implications are used as model of implication in many applications of fuzzy logic. These implication operators do not verify the properties of material implication, specifically, for any t-norm t(0, 0) = 0 but in logic

26

1 Fuzzy Set Theory

0 → 0 = 1. Typical examples of t-norm implications are the Mamdani (a → b = min{a, b}) and Larsen (x → y = xy) implications. Remark 1.14. Note that R–implication is identical to the omega operator introduced earlier. The papers Young (1996) and Fan et al. (1999) show that any fuzzy implication operator i(a, b) can be used to generate a subsethood measure, either by averaging i(A(x), B(x)) over X or by taking the inf over X. In addition, Baets et al. (2002) provide a slew of candidates under the name inclusion operators. The most often used fuzzy implication operators are listed in the following table. Table 1.6 Fuzzy Implication Operators Name Early Zadeh Lukasiewicz Mamdani* Larsen

Label im ia imm il

Standard Strict iss Gödel

ig

Gaines–Rescher igr

1.8

Goguen

igg

Kleene-Dienes Reichenbach

ikd ir

Yager

iy

Definition of a → b a → b = max [1 − a, min [a, b]] a → b = min [1, 1 − a + b] a → b = min [a, b] a → b = ab 1 if a ≤ b a→b= 0 otherwise 1 if a ≤ b a→b= b otherwise 1 if a ≤ b a→b= 0 a>b 1 if a ≤ b a→b= b otherwise a a → b = max [1 − a, b] a→b = 1−a+ab 1 if a = b = 0 a→b= ba otherwise

Averaging Operator

The averaging operators are a third class of binary operators used to average its arguments a and b. Since all intersection operators produce results that are below the minimum of a and b and all union operators produce results that are greater than the maximum of a and b, there is a large range of values that are excluded by these two classes of operators. Into this gap we now introduce averaging operators h(a, b). These operators do not correspond exactly to any logical connective, the way intersection operators model and and union operators model or. These averaging operators take two arguments and produce a result that greater than or equal to the min(a, b) and less than

1.9 Aggregation Operations

27

or equal to max(a, b). An averaging operator is a function h : [0, 1] × [0, 1] → [0, 1] such that following axioms hold. (h1) (h2) (h3) (h4)

Idempotency — h(a, a) = a. Monotonicity — h(a, b) ≤ h(c, d) whenever a ≤ c and b ≤ d. Commutativity — h(a, b) = h(b, a). Continuity — h is a continuos function.

The following properties could have been listed as a condition but since they are consequences of the previous axioms it could also be stated as a lemma. This is another example of keeping an axiomatic skeleton sparse. If we added the following two conditions as axioms student would have to show six things in a homework problem to show that an operator was an averaging operator. Instead they only have to show four things, and that is certainly easier. (g5) Extremes — h(0, 0) = 0 and h(1, 1) = 1. (g6) Boundary conditions — min(a, b) ≤ h(a, b) ≤ max(a, b). Averaging operators allow for an interaction between the values of two fuzzy sets. It allows the resultant averaged value to be better than the worst case but less than the best case. In fact the average value is often right in the middle, which should come as no surprise. However there are other averaging operators beside the geometric mean, such as the harmonic mean. Assume f is any continuous strictly monotone function (this means that it is always increasing or always decreasing). It is a fact that all continuous strictly monotone functions have inverses so we know that f −1 exists. Then  −1 f (a) + f (b) h(a, b) = f 2 is called a quasi-arithmetic means. Such a function h is always an averaging operator Aczél (1966). Let α ∈ [0, 1] then h(a, b) = f −1 [αf (a) + (1 − α)f (b)] is a more general form of quasi-arithmetic operator Aczél (1966). Some of the more important averaging operators are given in Table (1.7).

1.9

Aggregation Operations

The idea of an averaging operator can be extended to m-ary aggregation operators. Since averaging operations are in general not associative, they must be defined as functions of m arguments (m ≥ 2). That is, an averaging operation h is a function of the form h : [0, 1]m → [0, 1].

(1.65)

28

1 Fuzzy Set Theory Table 1.7 Averaging Operators Name

Operator

Generator

arithmetic mean

a+b 2

x

generalized p-mean



xp

p

ap +bp 2

2ab a+b

harmonic mean

1 x

√ geometric mean xy log x  dual of geometric mean 1 − (1 − x) (1 − y) log (1 − x)

Averaging operations are characterized by the following set of axioms: (h1) Idempotency — for all a ∈ [0, 1], (1.66)

h(a, a, a, . . . , a) = a.

(h2) Monotonicity — for any pair of m–tuples of real numbers in [0, 1], a1 , a2 , a3 , . . . , am and b1 , b2 , b3 , . . . , bm , if ak ≤ bk for all k ∈ Nm then h(a1 , a2 , a3 , . . . , am ) ≤ h(b1 , b2 , b3 , . . . , bm ). (1.67) It is significant that any function h that satisfies these axioms produces numbers that, for any m–tuple a1 , a2 , a3 , . . . , am ∈ [0, 1]m , lie in the closed interval defined by the inequalities min(a1 , a2 , a3 , . . . , am ) ≤ h(a1 , a2 , a3 , . . . , am ) ≤ max(a1 , a2 , a3 , . . . , am ). (1.68) The min and max operations qualify, due to their idempotency, not only as fuzzy counterparts of classical set intersection and union, respectively, but also as extreme averaging operations. An example of a class of symmetric averaging operations are generalized means, which are defined for all m–tuples a1 , a2 , a3 , . . . , am in [0, 1]m by the formula hp (a1 , a2 , a3 , . . . , am ) =

1 1 p (a1 + ap2 + ap3 + · · · + apm ) p , m

(1.69)

where p is a parameter whose range is the set of all real numbers excluding 0. For p = 0, hp is not defined; however for p → 0, hp converges to the well known geometric mean. That is, we take 1

h0 (a1 , a2 , a3 , . . . , am ) = (a1 a2 a3 . . . am ) m .

(1.70)

For p → −∞ and p → ∞, hp converges to the min and max operations, respectively.

1.9 Aggregation Operations

29

Assume again that f is any continuous strictly monotone function, then 

m  1 f (ai ) h(a1 , a2 , .., am ) = f −1 m i=1 is still called a quasi-arithmetic means . Let w1 , w2 , ..., wm be weights with wi ∈ [0, 1] then

m   h(a1 , a2 , .., am ) = f −1 wi f (ai ) i=1

is a more general form of quasi-arithmetic operator Aczél (1966).

1.9.1

OWA Operators

Yager introduced ordered weighted averaging(OWA) operators in Yager (1988). They are by nature averaging operators that treat a fuzzy set in its possibility theory interpretation. Let a = a1 , a2 , a3 , . . . , am be an mdimensional vector of values and let w = w1 , w2 , w3 , . . . , wm be an mdimensional vector of weights, with both ai ∈ [0, 1] and wi ∈ [0, 1] , for 1 ≤ i ≤ m. Define the vector b = b1 , b2 , b3 , . . . , bm to be the vector a sorted in decreasing order of magnitude, so that bi ≥ bi+1 then the OWA average of a is m  OW Aw (a) = wi bi . i=1

At first it would seem that OWA operators are very artificial. However. let us examine three special cases of OWA operators Let w∗ = 1, 0, 0, ..., 0 then OW Aw∗ (a) = b1 = max [a1 , a2 , a3 , . . . , am ] . Let w∗ = 0, 0, 0, ..., 1 then OW Aw∗ (a) = bm = min [a1 , a2 , a3 , . . . , am ] .  1 1 1 Let w∗ = m , m , m , ..., m then 1

m

OW Aw∗ (a) =

m

1  1  bi = ai . m i=1 m i=1

Thus OWA operators allow us to perform a delicate mix of values emphasizing either large values in a by making wi big for low values of i and tiny for higher values of i or vice versa.

30

1 Fuzzy Set Theory Table 1.8 A Binary Relation R Presented as a Table R 123 a 101 b 010 c 011

1.10

Classical Relations

A classical binary relation is a set of ordered pairs derived from a product space X × Y .

1.10.1

Types of Classical Relations

Definition 1.15 (Binary Relation). A binary relation between the sets X and Y is a subset of X × Y . If R is a subset of X × Y and if x, y ∈ R then we say that x is related to y. If the sets X and Y are the same set, X = Y , then a subset R of X × X is called a relation on X. We can list the ordered pairs that are in the relation using set notation; R = { a, 1 , a, 3 , b, 2 , c, 2 , c, 3 }

(1.71)

or use an in-line notation; a R 1,

a R 3,

b R 2,

c R 2,

cR3

(1.72)

We can also use a characteristic function to denote a relation, since a relation is just a set (of ordered pairs);  1 x, y ∈ R χR (x, y) = (1.73) 0 otherwise By far the simplest way to present a binary relation R is in a table. In Table (1.8) a 1 in row a column 3 indicates that a is related to 3, or a R 3, and the 0 in row b column 1 indicates that b is not related to 1, or b  R 1. Let R be a binary relation on a classical set X . For all of the following definitions, we assume that x, y, z ∈ X. Definition 1.16 (reflexive). R is reflexive if R(x, x), ∀x ∈ X. Definition 1.17 (irreflexive). R is irreflexive if it is not reflexive. Definition 1.18 (antireflexive). R is antireflexive if ¬R(x, x), ∀x ∈ X.

1.10 Classical Relations

31

Definition 1.19 (symmetric). R is symmetric when R(x, y) if and only if R(y, x). Definition 1.20 (asymmetric). R is asymmetric if it is not symmetric. Definition 1.21 (antisymmetric). R is antisymmetric if R(x, y) and R(y, x) imply that x = y. Definition 1.22 (transitive). R is transitive if R(x, y) and R(y, z) imply that R(x, z). Definition 1.23 (nontransitive). R is nontransitive if it is not transitive. Definition 1.24 (antitransitive). R is antitransitive if R(x, y) and R(y, z) imply that ¬R(x, z). Example 1.25. Consider the classical identity, inequality and order relations on the real line. It is clear that equality, =, is reflexive, symmetric and transitive. Inequality, =, is antireflexive and symmetric. Less than, 0 and R(y, x) > 0 imply that x = y. Finally, transitivity, has a host of variations in the fuzzy world, as described in the next section.

1.13.0.1

Transitivity

As is typical in fuzzy set theory, fuzzification introduces some new concepts and new words into the lexicon. For example, suppose the relation R on X between elements a and b is stronger than any two step connection from a to b through some c. By this we mean that that the membership grade in the relation for the pair a, b is greater than the membership grade of both a, c

and c, b for any all c ∈ X . This property of a relation is called sup-min transitivity . Definition 1.45 (max-min transitivity). If R, R : X × X → [0, 1], is a fuzzy relation then it is sup-min transitive if, for all ∀x, y ∈ X,

1.13 Properties of Fuzzy Relations

37

R(x, y) ≥ sup min [R(x, z), R(z, y)]

(1.81)

z∈Z

The standard min operator of Zadeh’s original fuzzy set theory is sometimes replaced by an alternate intersection operator, a t-norm. If we use this substitution in the definition of sup-min transitivity we get sup-t transitivity. Definition 1.46 (max-t transitivity). If R, R : X × X → [0, 1], is a fuzzy relation and t is a t-norm then the relation R is sup-t transitive if R(x, y) ≥ sup t [R(x, z), R(z, y)]

(1.82)

z∈Z

for all ∀x, y ∈ X. Example 1.47. Let C be a fuzzy set in the universe of discourse {1 , 2 , 3}and let R be a binary fuzzy relation on {1, 2, 3}. Assume that C = 0.2 + 0.4 + 0.5 1 2 3 and R is giving in the following table R 1

2

3

1 1 0.8 0.3 . 2 0.8 1 0.8 3 0.3 0.8 1

(1.83)

Suppose that we use the product t–norm. Using the definition of sup −t composition we get ⎡ ⎤ 1 0.8 0.3  t   C ◦ R = 0.2 0.4 0.5 ◦ ⎣ 0.8 1 0.8 ⎦ = 0.32 0.4 0.5 0.3 0.8 1 so

C ◦R=

0.32 0.4 0.5 + + . 1 2 3

The terms max-min transitivity and max-t transitivity are often used when sup-min transitivity and sup-t transitivity are technically correct.

1.13.1

Types of Fuzzy Relations

Definition 1.48 (fuzzy equivalence relation). If a fuzzy relation R : X×X → [0, 1], is reflexive, symmetric, and sup–min transitive then it is called a fuzzy equivalence relation, or a similarity relation Zadeh (1971).

38

1 Fuzzy Set Theory

Example 1.49. The table following presents a fuzzy equivalence relation S S a b c d a on X = {a, b, c, d} b c d

1.0 0.8 0.7 1.0

0.8 1.0 0.7 0.8

0.7 0.7 1.0 0.7

1.0 0.8 0.7 1.0

The α-cut of a fuzzy equivalence relation R are crisp equivalence relations. Example 1.50. The α-cut of S given in Example (1.49) at α = 0.75 is an equivalence relation as illustrated in the following table S 0.75 a b c d

a 1 1 0 1

b 1 1 0 1

c 0 0 1 0

d 1 1 0 1

(1.84)

Example 1.51. The α-cut of S given in Example (1.49) at α = 0.85 is an equivalence relation as illustrated in the following table S0.85 a b c d

a 1 0 0 1

b 0 1 0 0

c 0 0 1 0

d 1 0 0 1

(1.85)

An equivalence relation always produces a partition. In the example above (Eq. (1.85)) S 0.85 has partition classes {a, d}, {b}, and {c}. The partition corresponding to S 0.75 (Eq. (1.84)) has classes {a, b, d} and {c}. Note that as α increased in the two examples above the partition got finer. It is not hard to see, or to prove, that the family of partitions of the alphacuts of a fuzzy equivalence relation forms a nested sequence of partitions as α increases from zero to one. This means each element of the α = 0.75 partition {{a, d} , {b} , {c}} is a subset of the α = 0.85 partition {{a, b, d} , {c}}. Definition 1.52 (fuzzy preorder relationship). A fuzzy relation that is reflexive, and sup-min transitive is called a fuzzy preorder relationship on X. Example 1.53. The fuzzy relation R on X = {a, b, c} specified in the following equation is a fuzzy preorder. R a b c

a 0.2 0.0 0.0

b 1.0 0.6 1.0

c 0.4 0.3 0.3

(1.86)

1.13 Properties of Fuzzy Relations

39

Fig. 1.13 A Fuzzy Order Relation

Definition 1.54 (fuzzy order relationship). A fuzzy relation that is reflexive, antisymmetric, and sup-min transitive is called a fuzzy order relationship on X. Example 1.55. Fig. (1.13) gives a graphical version of a fuzzy order relation on X = {a, b, c, d}. Definition 1.56 (fuzzy partial order). A fuzzy relation that is reflexive, perfectly antisymmetric, and sup-min transitive is called a perfect fuzzy order relationship on X or a fuzzy partial order relationship on X. Definition 1.57 (fuzzy linear order). A fuzzy order such that for all x, y ∈ X; x = y implies that either R(x, y) = 0 or equivalently R(y, x) = 0 is called a fuzzy linear order on X (also called a total fuzzy order relation). Definition 1.58 (compatibility relationship). A fuzzy relation that is reflexive and symmetric is called a compatibility relationship on X.

1.13.2

Transitive Closure

Let R be a binary relation on a set X. It is always possible to construct a relation that is sup-min transitive, based upon R, by forming the transitive closure of R. To form the transitive closure of R, set C = R and then repeatedly form D = (C ◦ C) ∪ C until D = C, which it eventually must. This relation, C, is then the transitive closure of R.

40

1 Fuzzy Set Theory

Definition 1.59. Let R be a binary relation on a finite set X. Set C = R and the repeatedly form D = (C ◦ C)∪C until C = D. The transitive closure of R is then C. Example 1.60. Let R be the fuzzy relation: ⎤ ⎡ 0.70 0.50 0.00 0.00 ⎢ 0.00 0.00 0.00 1.0 ⎥ ⎥ R=⎢ ⎣ 0.00 0.40 0.00 0.00 ⎦ 0.00 0.00 0.80 0.00

Set C = R and and form D = (C ◦ C) ∪ C: ⎤ ⎡ 0.70 0.50 0.00 0.50 ⎢ 0.00 0.00 0.80 1.0 ⎥ ⎥ D=⎢ ⎣ 0.00 0.40 0.00 0.40 ⎦ 0.00 0.40 0.80 0.00

Since D = C replace C with D and calculate the new D = (C ◦ C) ∪ C: ⎤ ⎡ 0.70 0.50 0.50 0.50 ⎢ 0.00 0.40 0.80 1.0 ⎥ ⎥ D=⎢ ⎣ 0.00 0.40 0.40 0.40 ⎦ 0.00 0.40 0.80 0.40

Again D = C so we replace C with D and calculate the next D = (C ◦ C)∪C: ⎤ ⎡ 0.70 0.50 0.50 0.50 ⎢ 0.00 0.40 0.80 1.0 ⎥ ⎥ D=⎢ ⎣ 0.00 0.40 0.40 0.40 ⎦ 0.00 0.40 0.80 0.40

This is the same result as before so D is the transitive closure of R.

The same process can be applied using any sup-t composition to form the sup-t transitive closure.

Chapter 2

Evidence Theory

2.1

Introduction

In classical mathematics measures are additive. If Ann brings in 5 pounds of apples and Bob brings in 6 pounds of bananas we have 5 + 6 = 11 pounds of fruit. Suppose on the other hand that Ann and Bob are witnesses to a crime. Take individually, Ann or Bob’s testimony would convince a juror that there is a 50% likelihood that Carla is guilty. Yet taken together, their testimony will not convince a juror that Carla is guilty without a doubt. That is because evidence, in general, is non–additive.

2.2

Evidence Theory

Evidence theory (ET) is one of the broadest frameworks for the representation of uncertainty. Its origins lie in the works of Dempster (1967b,a) and Shafer (1976) are heavily influenced by probability theory, one of the oldest uncertainty frameworks. Evidence theory is especially important because it is a kind of Swiss army knife in the field of uncertainty. ET encompasses belief, plausibility, necessity, possibility and probability among a host of other measures. Here we present Evidence Theory as it was originally characterized by Shafer. Evidence theory is based on two fuzzy measures: belief measures and plausibility measures. Belief and plausibility measures can be conveniently characterized by a function m from the power set of the universal set X into the unit interval. In this chapter we will assume at all times that X is finite. Let P(X) be the power set of X, that is, the set of all subsets of X. The function m, where m : P(X) → [0, 1] , is required to satisfy two conditions: J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 41–55. DOI: 10.1007/978-3-642-35224-9_2 © Springer-Verlag Berlin Heidelberg 2013

42

2 Evidence Theory

(1) m(∅) = 0  (2) m(A) = 1.

(2.1)

A∈P(X)

The function m is called a basic probability assignment (bpa). For each set A ∈ P(X), the value m(A) expresses the proportion to which all available and relevant evidence supports the claim that a particular element of X belongs to the set A. This value, m(A), pertains solely to one set, set A; it does not imply any additional claims regarding subsets or supersets of A. If there is some additional evidence supporting the claim that the element belongs to a subset of A, say B ⊆ A, it must be expressed by another value m(B). Given a basic probability assignment, m, every set A ∈ P(X) for which m(A) = 0 is called a focal element. The pair F, m , where F denotes the set of all focal elements induced by m is called a body of evidence and we may denote it by B = F , m . From a basic probability assignment m, the corresponding belief measure and plausibility measure are determined for all sets A ∈ P(X) by the formulas  m(B), (2.2) Bel(A) = B⊆A

and Pl(A) =



m(B)

(2.3)

B∩A=∅

Thus the belief in a set A is the sum of all the evidence (basic probability) assigned to A or to any subset of A. On the other hand, the plausibility of A is the sum of all the evidence (basic probability) that overlaps with A. It can be shown that the plausibility of an event is one minus the belief of the complement of that event, and vice verse. That is Bel(A) = 1 − Pl(Ac ) Pl(A) = 1 − Bel(Ac ) Since we can calculate the belief from the plausibility, and the plausibility from the belief, and both belief and plausibility can be derived from the basic probability assignment, we only need one more formula to show that all three measures provide the same information. Given a belief measure Bel, the corresponding basic probability assignment m is determined for all A ∈ P(X) by the formula  m(A) = (−1)|A−B| Bel(B), (2.4) B⊆A

2.2 Evidence Theory

43

where |A − B| is the cardinality of the set difference of A and B, as proven by Shafer (1976). Thus each of the three function, m, Bel and Pl, is sufficient to determine the other two. Total ignorance is expressed in evidence theory by m(X) = 1 and m(A) = 0 for all A = X. Full certainty is expressed by m({x}) = 1 for one particular element of x and m(A) = 0 for all A = {x}. Example 2.1. As an example, let X = {x1 , x2 , x3 } and let m({x1 , x2 }) = 0.3 m({x3 }) = 0.1

(2.5)

m({x2 , x3 }) = 0.2 m(X) = 0.4 .

be a given basic probability assignment on P(X). The focal set of this basic probability assignment is the set F = {{x1 , x2 }, {x3 }, {x2 , x3 }, {x1 , x2 , x3 }};

(2.6)

and we always assume that m(A) = 0 for all A ∈ / F , that is, m is zero for any sets that are not listed or mentioned. Using the given basic probability assignment we can calculate the belief and plausibility of any subset of X. For example, our belief in {x2 , x3 } is Bel({x2 , x3 }) = m({x2 , x3 }) + m({x3 }) = 0.2 + 0.1

(2.7)

= 0.3. since {x2 , x3 } and {x3 } are the only subsets of {x2 , x3 } in the focal set. The plausibility of {x3 } is Pl({x3 }) = m(X) + m({x2 , x3 }) + m({x3 }) = 0.4 + 0.2 + 0.1

(2.8)

= 0.7, since X, {x2 , x3 }, and {x3 } are in the focal set and their intersection with {x3 } is non-empty. Table 2.1 is complete listing of the basic probability assignment, belief, and plausibility of all subsets of X for this example. Two special cases of evidence are important.

44

2 Evidence Theory

Table 2.1 An Example of a Basic Probability Assignment and the Associated Belief and Plausibility Measures Set ∅ {x1 } {x2 } {x1 , x2 } {x3 } {x1 , x3 } {x2 , x3 } X

2.2.1

m Bel Pl 0.0 0.0 0.0 0.3 0.1 0.0 0.2 0.4

0.0 0.0 0.0 0.3 0.1 0.1 0.3 1.0

0.0 0.7 0.9 0.9 0.7 1.0 1.0 1.0

Probability Theory

In the first special case, suppose that each element of the focal set has size one. Thus each set Ai ∈ F contains a single element of the universe X. In this case, since the sum of the weights of the focal elements is one, and each focal element element contains one object from the universe. Define p : X → [0, 1] by p(x) = m({x}) then p is always positive, and sums to one. Since we assume X is finite p is a probability distribution, and the belief, plausibility, and possibility of a subset A of X are all identical, Bel(A) = P l(A) = P (A). Example 2.2. Let us examine the following body of evidence defined on X = {x1 , x2 , x3 }. m ({x1 }) = 0.3 m ({x2 }) = 0.2 m ({x3 }) = 0.5 The focal set is F = {{x1 } , {x2 } , {x3 }}. We note that each focal object contains a single element. Define p1 = p(x1 ) = m ({x1 }) = 0.3 p2 = p(x2 ) = m ({x2 }) = 0.2 p3 = p(x3 ) = m ({x3 }) = 0.5 and we claim that p is a probability distribution on X. It satisfies all the requirement of a discrete probability distribution, all the probabilities are non–negative and the probabilities sum to one. The following Table shows the bpa, belief, plausibility, and probability of all the subsets of X.

2.2 Evidence Theory

45

Set ∅ {x1 } {x2 } {x1 , x2 } {x3 } {x1 , x3 } {x2 , x3 } X

2.2.2

m Bel 0 0 0.3 0.3 0.2 0.2 0 0.5 0.5 0.5 0 0.8 0 0.7 0 1

Pl P 0 0 0.3 0.3 0.2 0.2 0.5 0.5 0.5 0.5 0.8 0.8 0.7 0.7 1 1

Possibility Theory

In the second special case, assume that the elements of the focal set are consonant. This means that there is an order of the focal sets such that each set is nested in its successors. That is, if Ai and Aj are elements of the focal set F then, if i < j then Ai ⊂ Aj . The special branch of evidence theory that deals only with bodies of evidence whose focal elements are nested is referred to as possibility theory [Dubois and Prade, 1988]. Special counterparts of belief measures and plausibility measures in possibility theory are called necessity measures and possibility measures, respectively. Thus P os(A) = P l(A) and N ec(A) = Bel(A) It is not hard to show that in the case of consonant focal sets that: i. N ec(A ∩ B) = min[N ec(A), Nec(B)] for all A, B ∈ P(X); ii. P os(A ∪ B) = max[Pos(A), Pos(B)] for all A, B ∈ P(X). Define r(x) = max [P os(A)] x∈A

then r is called a possibility distribution. We now that the values of r must be between 0 and 1 so that r:X → [0, 1]. It also turns out that P os(A) = max r(x) x∈A

(2.9)

for each A ∈ P(X). Example 2.3. Let us examine the following body of evidence defined on X = {x1 , x2 , x3 }. m ({x1 }) = 0.3 m ({x1 , x2 }) = 0.2 m ({x1 , x2 , x3 }) = 0.5 The focal set is F = {{x1 } , {x1 , x2 } , {x1 , x2 , x3 }}. We note that the focal object can be ordered by containment, {x1 } ⊆ {x1 , x2 } ⊆ {x1 , x2 , x3 }.

46

2 Evidence Theory

The following Table shows the bpa, possibility, and necessity of all the subsets of X. Set m Nec Pos ∅ 0 0 0 {x1 } 0.3 0.3 1 {x2 } 0 0.0 0.7 {x1 , x2 } 0.2 0.5 1 {x3 } 0 0 0.5 {x1 , x3 } 0 0.3 1 {x2 , x3 } 0 0 0.7 X 0.5 1 1 Define a possibility distribution on X r1 = r(x1 ) = max [Pos(A)] = 0.3 x1 ∈A

r2 = r(x2 ) = max [Pos(A)] = 0.5 x2 ∈A

r3 = r(x3 ) = max [Pos(A)] = 1 x3 ∈A

then for each subset A of X we have that the possibility of A is just the largest distributional value assigned to an element of A.

2.2.2.1

Possibility Theory as Fuzzy Set Theory

The most visible interpretation of possibility theory, is connected with fuzzy sets. This interpretation was introduced by Zadeh (1978). To explain the fuzzy set interpretation of possibility theory, let X denote a variable that takes values in a given set X, and let information about the actual value of the variable be expressed by a fuzzy proposition “X is F ”, where F is a standard fuzzy subset of X (i.e., F (x) ∈ [0, 1] for all x ∈ X). To express information in measure-theoretic terms, it is natural to interpret the membership degree F (x) for each x ∈ X as the degree of possibility that X = x. This interpretation induces a possibility distribution, rF , on X that is defined by the equation (2.10) rF (x) = F (x) for all x ∈ X. If F is normal then a fuzzy set is, in a sense, equivalent to a possibility distribution. Example 2.4. Let F be the normal fuzzy set X = {x1 , x2 , x3 } defined by F (x1 ) = 0.3 F (x2 ) = 0.5 F (x3 ) = 1.0

2.4 Dempster’s Rule

47

Define a possibility distribution rF on X rF (x1 ) = 0.3 rF (x2 ) = 0.5 rF (x3 ) = 1 . We note that the normal fuzzy set F , the possibility distribution rF and the possibility distribution of Example 2.3 are all identical.

2.3

Data Fusion

The question of how to combine data received from different sources and, especially, from different methodologies, is the subject of the following sections. Since a body of evidence can be possibilistic (fuzzy), probabilistic, or neither, evidence theory provides an important framework for data fusion.

2.4

Dempster’s Rule

Suppose we have two bodies of evidence, one from Ann and one from Bob. Suppose we want to fuse this data into a single body of evidence. We assume that both bodies of evidence concern the same universe X but that Ann and Bob view the situation differently. Let Ann’s evidence be B1 = F1 , m1 and Bob’s evidence be B2 = F2 , m2 . Let K be the conflict among the bodies evidence. Conflict occurs when Ann and Bob focus evidence on focal sets that have nothing in common. Ann’s evidence for A ∈ F1 and Bob’s evidence for B ∈ F2 conflict whenever A ∩ B = ∅. Define the total conflict as  K= m1 (A) m2 (B) (2.11) A∩B=∅

where A ∈ F1 and B ∈ F2 . Then we can create a fused body of evidence B = F , m from Ann’s evidence B1 = F1 , m1 and Bob’s evidence be B2 = F2 , m2 by setting the focal set of B = F , m to F = {A ∩ B | A ∩ B = ∅ and A ∈ F1 and B ∈ F2 } and defining the basic probability assignment of B = F , m for C ∈ F as  m1 (A) m2 (B) m(C) = A∩B=C (2.12) 1−K If we do not divide by the conflict term 1 − K then m will not sum to one. The normalization factor above, 1 − K, has the effect of completely ignoring

48

2 Evidence Theory

conflict and attributing any mass associated with conflict to the null set. This combination rule for evidence can therefore produce counterintuitive results when there is significant conflict or not. Example 2.5. Let us examine the following bodies of evidence defined on three suspects of a crime x1 = Ralph, x2 = Sue, and x3 = T om, so that X = {x1 , x2 , x3 }. Ann is an investigative reporter and determines that there are four critical pieces of evidence, that assign guilt thusly, m1 ({x1 , x2 }) = 0.3

(2.13)

m1 ({x3 }) = 0.1 m1 ({x2 , x3 }) = 0.2

m1 (X) = 0.4 .

We interpret m({x1 , x2 }) = 0.3 as one piece of evidence, say eye–witness testimony, that places Ralph and Sue in proximity of the crime scene. The reliability of the eye–witness causes us to weight this evidence as having 30% of the available credibility. The focal set for this body of evidence is: F1 = {{x1 , x2 }, {x3 }, {x2 , x3 }, {x1 , x2 , x3 }};

(2.14)

Bob is NCIS and arrives at the following body of evidence as a result of his investigation. m2 ({x1 }) = 0.3 m2 ({x1 , x2 }) = 0.2 m2 ({x1 , x2 , x3 }) = 0.5

(2.15)

The focal set for this body of evidence is F2 = {{x1 } , {x1 , x2 } , {x1 , x2 , x3 }} .

(2.16)

Conflict between bodies of evidence occurs when focal sets have an empty intersection. For example {x3 } ∩ {x1 } = ∅ so the product of their weights m1 ({x3 }) and m2 ({x1 }) , or 0.1 × 0.3, gets added to K. K = m1 ({x3 }) × m2 ({x1 }) + m1 ({x3 }1 ) × m2 ({x1 , x2 }) + m1 ({x2 , x3 }1 ) × m2 ({x1 }) = 0.1 × 0.3 + 0.1 × 0.2 + 0.2 × 0.3 = 0.11

2.4 Dempster’s Rule

49

In the fused body of evidence m12 ({x1 }) would be the sum of all the evidence where Ann and Bob’s evidence agrees only on {x1 } divided by 1 − K. Thus: m1 ({x1 , x2 }) × m2 ({x1 }) + m1 (X) × m2 ({x1 }) 1−K 0.3 × 0.3 + 0.4 × 0.3 = 0.89 = 0.236 .

m12 ({x1 }) =

The following Table shows the bpa, possibility, and necessity of the fused body of evidence for all the subsets of X. Set ∅ {x1 } {x2 } {x1 , x2 } {x3 } {x1 , x3 } {x2 , x3 } X

2.4.1

m

Bel

Pl

0.00 0.00 0.00 0.236 0.236 0.787 0.045 0.045 0.708 0.326 0.607 0.944 0.056 0.056 0.393 0.000 0.292 0.955 0.112 0.213 0.764 0.225 1.000 1.000

Combining n Special Bodies of Evidence

We derive here a method for combining expert opinion for any number of experts in such away that the combination can occur all at once rather than in a serial fashion. Our results are for a universe X of any cardinality, but with exactly the same two focal elements, namely A and B with A ∩ B = ∅, for each basic probability assignment. We derive a formula for combining k basic probability assignments on X for arbitrary k ∈ N, the positive integers. Let n ∈ N, n ≥ 2. We use the following notation in Theorem 2.6. Let  Cjn (A) = mi1 (A)mi2 (A)...mij (A) (2.17) 1≤i1 0, the conditional probability (credibility) distribution on P (X) given B is mi|j (A | B) = mij (A, B) /mj (B). The corresponding class of focal pairs of subsets is

Fij = {(A, B) | A ⊆ X, B ⊆ X, mij (A, B) > 0} . In a natural way we can introduce the functions BelBel, BelP l, P lP l, BelAmb, etc., on P (X). Thus, for instance,

BelP lij (A, B) =





mij (C, D) .

C⊆A, C=∅D∩B=∅

Obviously, if the bodies of evidence i and j are independent, then BelP lij is equal to Beli × P lj , or BelP lij (A, B) = Beli (A) P lj (B) . Definition 7.3 (The Judge). A judge, or decision maker, or jury has to reach a verdict about the culpability or innocence of the suspects based on the available evidence and his own judgement. Mathematically, the judge is associated with a family of conditional weights which is a family of nonnegative functions on P (X) conditioned by the available evidence. For the body of evidence #i for which mi is the probability (credibility) distribution induced on P (X) the judge must determine weights wi (· | ·)1 1

In probability theory, it is common to use a dot as a placeholder for an unnamed and unnumbered variable.

7.1 Reaching a Verdict by Weighting Evidence

121

where wi (· | ·) : P (X) × Fi −→ [0, ∞) . The judge assignees weight wi (C | A) which represents the culpability of the subset C ∈ P (X) if the i-th body of evidence focuses on the culpability of the subset A ∈ Fi . Here both C and A are a collections of suspects; A must be a focal set of body of evidence #i but C is arbitrary. The larger the weight the larger the potential culpability of the corresponding subset of suspects. From mathematical point of view, except nonnegativity, the only condition imposed on the family of weights is 



C∈P (X)A∈Fi

wi (C | A) mi (A) = 1 .

(7.3)

The judge may assign positive weights to many wi (C | ·), since, in his view, the credibility in many focal sets of of the body of evidence #i may transfer to the same C ∈ P (X). When we multiply the weight wi (C | ·) by mi (·), and sum over focal sets of body of evidence #i we get a new credibility distribution on P (X) given by  μi (C) = wi (C | A) mi (A) , (7.4) A∈F (X;mi )

abbreviated by μi = wi ⋆ mi . The larger the weight the larger the potential culpability of the corresponding subset of suspects. From mathematical point of view, except nonnegativity, the only condition imposed on the family of weights is   wi (C | A) mi (A) = 1, (7.5) C∈P (X)A∈F (X;mi )

The conditions on the weights of positivity and satisfying Eq. 7.5 guarantee that that μi given by Eq. 7.4 is a probability (credibility) distribution on P (X). Definition 7.4. A family of weights is probabilistic if they satisfy the equalities  wi (C | A) = 1 for every A ∈ F ( X; mi) . (7.6) C∈P (X)

Obviously, (7.6) implies (7.5) but the converse is not necessarily true. If the family of weights is probabilistic and objective, based exclusively on relative frequencies, then wi (C | A) may be calculated using the standard formula for conditional probabilities. If, however, the family of weights is both nonprobabilistic and subjective, then wi (C | A) simply reflects what the judge believes about the culpability of C if the direct evidence focuses on the subset A and no special rules is necessarily used for getting it.

122

7 The Guiasu Method

Some important kinds of judge weighting are: Reliance. If the judge fully relies on the i-th body of evidence, then wi (A | A) = 1 and wi (C | A) = 0 if C is different from A, for every A ∈ F (X; mi ), which implies μi (A) = mi (A). Indifference. If the judge focuses on B ∈ P (X) regardless of what the i-th body of evidence says, then wi (B | A) = 1 for every A ∈ F (X; mi ), which implies μi (B) = 1.

Generalization The formulas above can be generalize easily to weighting joint credibility distributions. They can also be generalized to cases where the universes of the experts do not match. For example, Ann and Bob may arrive at different lists of subsets, Ann may focus on subsets of X = {Ralph, Sue, T om} while Bob focuses on subsets of Y = {Ralph, Sue, U rsula}. In addition, it may be that the Judge has his own list of suspects Z. We assign weights wij ( | , ) to a mixed evidence inducting the joint credibility distribution mij on P (X) × P (Y ) . Thus the credibility distribution induced on P (X) by the weighted mixed (i, j)-the body of evidence is  μij (C) = wij (C | A, B) mij (A, B) with C ∈ P (Z) , (7.7) (A,B)∈Fij

where wi,j (C | A, B) is the judge’s weight of the subset C ∈ P (Z) given the mixed evidence (A, B) ∈ Fij .

7.1.1

Fuzzy Evidence

Let F : X −→ [0, 1] be a fuzzy set. Then, the number F (x) is the degree of membership of the element x ∈ X to the fuzzy set F . It is not necessary that {F (x) | x ∈ X} is a probability distribution on X but, as shown in Guiasu (1993), F does induce a probability distribution mF on P (X), defined by   mF (A) = F (x) [1 − F (y)] with A ∈ P (X) . (7.8) x∈A

y∈A

We note that if 0 < XF (x) < 1, for every x ∈ X, then F = P (X). All the consideration made above could be applied to the case when the available evidence is provided by fuzzy sets defined on X.

7.1 Reaching a Verdict by Weighting Evidence

123

Special Cases 7.1.2

Dempster’s Rule

Suppose that Ann and Bob have arbitrary bodies of evidence. The judge adds up all the weights of evidence where Ann and Bob have some agreement, i.e., where A is a focal element for Ann and B is a focal element for Bob and A ∩ B = ∅. The judge uses this weight as a normalizing factor so that ⎡ ⎤−1 C∩D=∅  w1,2 (A ∩ B | A, B) = ⎣1 − m1 (C) m2 (D)⎦ , C∈F1 ,D∈F2

for all A ∈ F1 , B ∈ F2 , A ∩ B = ∅. With these weights Eq. 7.4, becomes  m1 (A) m2 (B) C=A∩B  μ1,2 (C) = , 1− m1 (A) m2 (B) A∩B=∅

which is Dempster’s rule Dempster (1967b) of combining two independent bodies of evidence. It gives equal credit to the common evidence and discards any other evidence. According to Shafer (1976), in the special case of a universe containing only two elements, this rule was used by J.H. Lambert in his Neues Organon published in 1764.

7.1.3 7.1.3.1

Jeffrey’s Rule First Interpretation

Suppose that Ann’s information is a probability distribution pY on Y and that Bob’s evidence is a probability distribution qZ on Z. The judge will fuse evidence onto X = Y × Z. We will use the shorthand notation of y × Z for the set of ordered pairs {y} × Z = { y, z1 , y, z2 , · · · , y, zn }, and, similarly, Y ×z is shorthand for the set of pairs Y ×{z} = { y1 , z , y2 , z , · · · , ym , z } Assume that Ann’s evidence is of the form F1 = {y × Z | y ∈ Y } , m1 (y × Z) = pY (y) , and Bob’s evidence is of the form F2 = {Y × z | z ∈ Z} , m2 (Y × z) = qZ (z) .

124

7 The Guiasu Method

Finally we will in addition assume that there exists a conditional probability distribution pZ (· | y) on Z given y ∈ Y , and pZ , the prediction probability distribution on Z is defined by  pZ (z) = pZ (z | y) pY (y) , y∈Y

where pY is interpreted as being the prior probability distribution on Y . The judge fuses the evidence with weights w ({ y, z } | y × Z, z × Y ) = pZ (z | y) /pZ (z) .

(7.9)

giving

μ1,2 (C) =

 

A∈F1 B∈F2

w (C | A, B) m1 (A) m2 (B) .

Then, when C contains a single ordered pair, we conclude μ1,2 ({ y, z }) =

pY (y) pZ (z | y) qz (z) , pZ (z)

which is a probability distribution on Y × Z. Its marginal probability distribution, namely pY (y | qZ ) = Bel12 (y × Z)  = μ1,2 ({ y, z })

(7.10)

z∈Z

=

 pY (y) pZ (z | y) qz (z) , pZ (z)

z∈Z

which is Jeffrey’s rule (Jeffrey, 1983) for calculating the posterior probability distribution on Y given the actual probability distribution qZ on Z.

7.1.3.2

Second Interpretation

Jeffrey’s rule may be obtained more directly from Eq. 7.4, as a weighting with indirect evidence. Indeed, let X and Y be two finite crisp (Cantor) sets and m a probability distribution on P (Y ) such that F = {{y} | y ∈ Y }, m ({y}) = q(y),

7.1 Reaching a Verdict by Weighting Evidence

125

where q is the actual probability distribution on Y . Taking the only positive weights on P (X) to be w ({x} | {y}) = 

p (y | x) p (x) x∈X p (y | x) p (x)

where p is a prior probability distribution on X and p (· | x) is a conditional probability distribution on Y given x ∈ X, the weighted probability distribution becomes p(y | qZ ) = μ ({x})  = w ({x} | {y}) q (y)

(7.11) (7.12)

y∈Y

=



y∈Y

p (y | x) p (x) q (y) , x∈X p (y | x) p (x)



which is Jeffrey’s rule for calculating the posterior probability distribution on X.

7.1.4

The Guiasu Model with Probabilistic Evidence

If  the probability (credibility) distribution m on X is such that A∈F(X;m) m(A) = 1 and ∀A ∈ F (X; m), |A| = 1, then we call m probabilistic. Given n experts and m goals. Assume the experts assign numbers to each goal as to their importance with respect to the overarching goal to form an m × n-matrix W = [wij ]. When the columns of the matrix W are normalized, we can consider the each column of the resulting matrix N to be a probability (credibility) distributions for each expert. These probability (credibility) distributions are probabilistic with the focal elements being the singleton sets consisting of a goal. We now show that the row averages provide the Guiasu weights, one for each goal. k j+1 Lemma 7.5. jCjk = 0, where Cjk denotes the number of comj=1 (−1) binations of k things taken j at a time. Proof. The proof is by induction on k. Suppose k = 2. Then 2 j+1 (−1) jCj2 = C12 − 2C22 = 0. Suppose the result is true for k = i. j=1 Then 0=

i 

(−1)j+1 jCji =

j=1

=

i+1  j=1

i  j=1

j+1

(−1)

(−1)j+1 j

i! i+1 × j!(i − j)! i + 1

i+1

 (i + 1)! j = (−1)j+1 jCji+1 . j!(i + 1 − j)! j=1

⊔ ⊓

126

7 The Guiasu Method

Let Ij = {(i1 , ..., ij )|ir ∈ {1, ..., n}, r = 1..., n, i1 < ... < in }, j = 1, ..., n. In the following theorem, we use the notation, m1 = m11 + m12 + ... + m1n , m2 = m12 m13 + .. + m1i m1k + ... + m1,n−1 m1n , i < k,  mj = m1i1 ...m1ij , j = 1, ..., n. (i1 ,...,ij )∈Ij

Theorem 7.6. The row averages of W give the Guiasu weights, wi , i = 1, ..., m. Proof. It suffices to prove the result for the first row. By Lemma 2.1, we have n  j=1

w1j =

n 

(−1)j+1 jmj + 2

j=1

+t

n  j=2

n  j=t

(−1)j (j − 1)mj + ...

(−1)(j − t + 1)mj + ... + n

n 

j=n

(−1)(j − n + 1)mj

= m1 − [2m2 + 2m2 ] + [3m3 + 2(−3m3 ) + 3m3 ] + ... +

k 

(−1)j+1 jCjk mk + ... + (−1)n+1 nmn

j=1

= m1 = m11 + m12 + ... + m1n . The desired result now follows by dividing each side of the equation by n. ⊔ ⊓

7.1.5

Comments

In Shafer’s approach to evidence m (∅) has to be always equal to zero. This is an unnecessary restriction because m is not obtained by extending a probability distribution on X to a probability measure on P (X), but is directly defined as a probability distribution on P (X), in which case m (∅) could be positive, corresponding to the frequent case when there is a positive probability of having nobody guilty in the universe X. “The process of reaching a verdict essentially depends on how the available evidence is used by the judge or jury. The evidence may be significant, partially relevant, or misleading and the judge may use it in an objective or subjective way.”

7.2 AHP and Guiasu

7.2

127

AHP and Guiasu

In this section, we give a necessary and sufficient condition for the weights of the AHP and the Guiasu Method to coincide. We also develop a new method to determine the weights using Dempster-Shafer theory. Let A = [aij ] denote the m × n-matrix whose entries are the weights of the m factors as to their importance determined by the n experts. That is, aij is the weight of the i-th factor given by expert j with i = 1, ..., m; j = 1, ..., n. Let N denote the m × n-matrix determined from A by normalizing the columns of A, i.e.,  aij N = m . k=1 akj Then the weights of the analytic hierarchy process are given by n 1 j=1 aij n Ai = m 1 n i=1 n j=1 aij n j=1 aij = m n i=1 j=1 aij

for i = 1, ..., m. The weights Gi of the Guiasu method are given by n

for i = 1, ..., m. Theorem 7.7. Let Cj = and only if

1 a  ij Gi = n j=1 m k=1 akj m

k=1

akj , j = 1, ..., n. For i = 1, ..., m, A = Gi if

ai1 ain ai1 ain + ... + = + ... + . C1 + ... + Cn C1 + ... + Cn nC1 nCn Proof. We have that n

j=1 aij Ai = Gi , i = 1, ..., m ⇔ m n i=1

j=1

n

aij

=

1 a m ij , i = 1, ..., m n j=1 k=1 akj

ai1 + ... + ain ai1 ain = + ... + , i = 1., , , .m C1 + ... + Cn nC1 nCn ai1 ain ai1 ain ⇔ + ... + = + ... + , i = 1, ..., m. C1 + ... + Cn C1 + ... + Cn nC1 nCn

⇔

⊔ ⊓ Corollary 7.8. If Ci = Cj , i, j = 1, ..., n, then Ai = Gi for i = 1, ..., m.

128

7 The Guiasu Method

Proposition 7.9. Ci = Cj , i, j = 1, ...., n if and only if for all j = 1, ..., n we have that C1 + ... + Cn = nCj . Proof. The proof follows from a routine solution of the linear system of equations, C1 + ... + Cn = nCj , j = 1, ..., n. ⊔ ⊓ We assume throughout that we have m factors, Fi , i = 1, ..., m, and n experts Ej , j = 1, ..., n. The experts have weighted the factors as to their importance in achieving the overarching goal. The weights are from the closed interval [0, 1] with 1 considered as giving the highest importance and 0 the lowest. These weights are placed in an m×n-matrix, W. The row averages are used in the analytic hierarchy process. The columns of the matrix W are normalized to obtain a matrix N = [bij ]. The row averages of N are used in the Guiasu method. We consider each expert as a fuzzy focal element Aj with each Aj probabilistic, j = 1, ..., n. The values of the Aj are given by N, i.e., Aj (Fi ) = m bij , i = 1, ..., m; j = 1, ...., n and i=1 Aj (Fi ) = 1, j = 1, ..., n.

7.3 7.3.1

Issues in International Relations Nuclear Deterrence

The factors involved in Nuclear Deterrence are discussed in Section 3.1 on page 61. The matrix R comes from the Table 3.1 on page 62. We next consider the weights of the experts in order to obtain the linear equations. We normalize the columns of Table 3.1 to obtain the following Table. G1 G2 G3 G4 G5 G6

E1 E2 E3 E4 E5 Row Averages .25 .26 .24 .21 .29 .25 .15 .17 .05 .17 .10 .13 .15 .22 .14 .17 .14 .16 .15 .22 .29 .21 .24 .22 .20 .04 .19 .14 .19 .15 .10 .09 .10 .10 .05 .09

The row averages of the previous table yield the weights of the Gj by the Guiasu method since the basic probability assignments are probabilistic. We have P = .25G1 + .13G2 + .16G3 + .22G4 + .15G5 + .09G6 ,

(7.13)

where P denotes the predictive value determined by substituting in for the Gj the coded results.

7.3 Issues in International Relations

129

Expert opinion as to the extent to which the U.S. was achieving goals G1 , ...G6 yielded the following numbers: G1 G2 G3 G4 G5 G6 Expert 1 .7 0 .5 .9 .7 .7 Expert 2 1 .25 .75 .75 .5 .25 Substituting these numbers into the two linear equations above yielded the following values for Expert 1: .62 and for Expert 2: .66.

7.3.2

Smart Power and Deterrence

The factors for Smart Power and Deterrence is described in Section 3.2 on page 66 and the data is presented in Table 3.2 on page 68. The following Table presents a column normalized matrix derived from the matrix in Section 6.4.2 on page 105. Columns Normalized G1 G2 G3 G4 G5 G6

E1 E2 E3 E4 E5 Row Avgs Row Products .29 .20 .23 .24 .23 .24 .0007363 .29 .16 .20 .14 .18 .19 .0002338 .14 .22 .14 .21 .20 .18 .0001811 .14 .18 .20 .05 .05 .12 .0000126 .11 .13 .20 .19 .18 .16 .0000978 .03 .11 .02 .17 .16 .10 .0000017

The row averages of the previous matrix give the Guiasu weights of the Gj since the basic probability assignments are probabilistic. We have G = .24G1 + .19G2 + .18G3 + .12G4 + .16G5 + .10G6 .

(7.14)

We note that the equations are nearly identical.

7.3.2.1

Coefficients for the Gij : The Guiasu Method

The following matrices are determined by normalizing the columns of the corresponding matrices of the AHP section ( 6.4.2.1 on page 105). G1,1 G1,2 G1,3 G1,4

E1 E2 E3 E4 E5 Row Avgs. .27 .26 .05 .18 .14 .18 .24 .24 .48 .27 .31 .31 .27 .29 .29 .30 .31 .29 .21 .21 .19 .24 .24 .22

130

7 The Guiasu Method

The row averages of the previous matrix give the Guiasu weights of the G1 j since the basic probability assignments are probabilistic. We have G1 = .18G1,1 + .31G1,2 + .29G1,3 + .22G1,4 .

G2,1 G2,2 G2,3 G2,4 G2,5 G2,6 G2,7 G2,8 G2,9 G2,10 G2,11

(7.15)

E1 E2 E3 E4 E5 Row Avgs. .05 .14 .12 .16 .10 .11 .03 .05 .05 .08 .06 .05 .05 .11 .05 .02 .09 .06 .05 .09 .09 .15 .11 .10 .09 .04 .07 .02 .07 .06 .09 .02 .14 .02 .06 .07 .14 .07 .08 .02 .07 .08 .14 .18 .14 .16 .14 .15 .11 .12 .08 .16 .11 .12 .14 .16 .12 .16 .13 .14 .12 .02 .05 .06 .06 .06

The row averages of the previous matrix give the Guiasu weights of the G2 j since the basic probability assignments are probabilistic. We have G2 = .11G2,1 + .05G2,2 + .06G2,3 + .10G2,4 + .06G2,5 + .07G2,6 (7.16) +.08G2,7 + .15G2,8 + .12G2,9 + .14G2,10 + .06G2,11 .

G3,1 G3,2 G3,3 G3,4

E1 E2 E3 E4 E5 Row Avgs. Row Products .21 .24 .21 .39 .25 .26 .0010319 .14 .21 .28 .26 .21 .22 .0004494 .31 .29 .34 .30 .32 .31 .0029343 .34 .26 .17 .04 .21 .20 .0001262

The row averages of the previous matrix give the Guiasu weights of the G3 j since the basic probability assignments are probabilistic. We have G3 = .26G31 + .22G32 + .31G33 + .20G34 .

G4,1 G4,2 G4,3 G4,4 G4,5

(7.17)

E1 E2 E3 E4 E5 Row Avgs .12 .21 .25 .29 .21 .22 .17 .15 .15 .18 .15 .16 .33 .18 .15 .03 .15 .17 .21 .21 .20 .26 .24 .23 .17 .24 .25 .24 .24 .23

The row averages of the previous matrix give the Guiasu weights of the G4 j since the basic probability assignments are probabilistic. We have

7.3 Issues in International Relations

G4 = .22G41 + .16G42 + .17G43 + .23G44 + .23G45 .

G5,1 G5,2 G5,3 G5,4

131

(7.18)

E1 E2 E3 E4 E5 Row Avgs. .26 .29 .67 .56 .40 .44 .29 .21 .07 .33 .24 .23 .26 .26 .20 .06 .20 .20 .20 .24 .07 .06 .16 .15

The row averages of the previous matrix give the Guiasu weights of the G5 j since the basic probability assignments are probabilistic. We have G5 = .44G5,1 + .23G5,2 + .20G5,3 + .15G5,4 .

G6,1 G6,2 G6,3 G6,4 G6,5 G6,6 G6,7 G6,8

(7.19)

E1 E2 E3 E4 E5 Row Avgs. .09 .13 .19 .03 .12 .11 .16 .12 .11 .08 .12 .12 .14 .06 .15 .18 .14 .13 .09 .08 .13 .08 .10 .10 .14 .17 .02 .21 .12 .13 .12 .10 .02 .13 .08 .09 .16 .19 .19 .21 .18 .19 .12 .15 .19 .08 .14 .14

The row averages of the previous matrix give the Guiasu weights of the G6 j since the basic probability assignments are probabilistic. We have G6 = .11G6,1 +.12G6,2 +.13G6,3 +.10G6,4 +.13G6,5 +.09G6,6 +.19G6,7 +.14G6,8 . (7.20) The equations for the two methods are not identical, but they are quite similar. We now turn to a third method which as we will demonstrate returns significantly different results.

7.3.3

Cooperative Threat Reduction

If  the probability (credibility) distribution m on X is such that A∈F(X;m) m(A) = 1 and ∀A ∈ F(X; m), |A| = 1, then we call m probabilistic. Given n experts and m goals. Assume the experts assign numbers to each goal as to their importance with respect to the overarching goal to form an m × n-matrix W = [wij ]. When the columns of the matrix W are normalized, we can consider each column of the resulting matrix N to be a probability (credibility) distribution for each expert. These probability (credibility) distributions are probabilistic with the focal elements being the singleton sets consisting of a goal.

132

7 The Guiasu Method

The following Table presents a column normalized matrix derived from the matrix in Section 6.4.3 on page 107. Cols. Normalized O1 O2 O3 O4

E1 E2 E3 E4 E5 E6 Row Avg = Gi .25 .25 .206 .229 .357 .25 .26 .25 .25 .206 .257 .286 .313 .26 .25 .25 .294 .257 .214 .219 .25 .25 .25 .294 .257 .143 .219 .24

We obtain the following linear equation: G = .26O1 + .26O2 + .25O3 + .24O4

7.4

(7.21)

Issues in Comparative Politics

7.4.1 7.4.1.1

Quality of Life The First Set of Experts

For Expert Group One’s data concerning Quality of Life, Guiasu produces the same results as the one produced by the AHP method, see Sec. 6.5.1. Q = .094F1 + .128F2 + .128F3 + .106F4 + .167F5

(7.22)

+.189F6 + .142F7 + .100F8 + .047F9

7.4.1.2

Second Group of Experts

We normalize the columns of the matrix given in Subsection 6.5.1.2. Cols. Normalized F1 F2 F3 F4 F5 F6 F7 F8 F9

E1 E2 E3 E4 E5 E6 E7 E8 Row Avg .111 .125 .044 .121 .130 .156 .178 .137 .125 .044 .109 .133 .106 .152 .044 .156 .098 .105 .133 .094 .156 .106 .130 .178 .133 .157 .133 .067 .094 .111 .091 .087 .133 .089 .078 .094 .178 .125 .178 .121 .174 .200 .111 .176 .158 .200 .141 .200 .136 .196 .111 .200 .157 .168 .089 .109 .022 .091 .043 .067 .044 .118 .073 .156 .094 .067 .121 .065 .089 .067 .159 .090 .022 .109 .089 .106 .022 .033 .022 .039 .054

The Row Avg column of the previous matrix sums to 1.

7.4 Issues in Comparative Politics

133

Q=.125F1 +.105F2 +.133F3 +.094F4 +.158F5+.168F6 +.073F7 +.090F8 +.054F9 (7.23) Again, this is the same result as given by AHP.

7.4.2

Factors in Successful Democratization

Guiasu’s method describes the process of reaching a verdict by a probabilistic weighting of the expert opinions, which permits us to determine a numerical relationship between the Hj . The classical rules from decision theory proposed by Hooper, Dempster, Bayes, and Jeffrey are special cases of Guiasu’s weighting process. Guiasu Guiasu (1994) addresses differences in expert opinion over the likelihood (probability) of something being true as well as the credibility of the claims (our belief in its truth). Since we are dealing with theoretical perspectives on democratic consolidation in this paper, we will consider the credibility of the five respective claims that we have identified. The factors necessary for successful democratization are discussed in 4.1. The data is given in Table 4.1. We consider our five previously identified expert theoretical perspectives (opinions) as five bodies of evidence. Treating each of them as equally credible, we will determine the weights attached to the each of the causes that they collectively identify in order to arrive at a fully specified model of consolidated democracy. We begin by defining the corresponding basic credibility assignments on the universe of causal factors X as follows: m1 ({H8 }) = 1,

m2 ({H1 }) = m2 ({H4 }) = m2 ({H5 }) = 1/3 , m3 ({H4 }) = m3 ({H6 }) = m3 ({H7 }) = m3 ({H8 }) = 1/4 ,

m4 ({H1 }) = m4 ({H3 }) = m4 ({H4 }) = m4 ({H5 }) = m4 ({H7 }) = 1/5 , m5 ({H2 }) = m5 ({H6 }) = 1/2 .

In words, none of our experts indicates the relative credibility of any factor as greater or lesser than any other identified. For instance, our first expert identifies only one factor (H8 ), that one factor accounts for all (1.00) of the variance in democratic consolidation. On the other hand expert four identifies five causes, each one of which accounts for one-fifth (.20) of the variance. Since we have no reason to give greater or lesser importance to any of the five experts, we proceed as follows. We calculate, for all C ∈ P(X), the weight w12345 (C| | A1 , A2 , A3 , A4 , A5 ) where

134

7 The Guiasu Method

A1 = {{H8 }} , A2 ∈ {{H1 }, {H4 }, {H5 }} ,

A3 ∈ {{H4 }, {H6 }, {H7 }, {H8 }} , A4 ∈ {{H1 }, {H3 }, {H4 }, {H5 }, {H7 }} , A5 ∈ {{H2 }, {H6 }} .

under the assumptions that each body of evidence (expert opinion) is independent and of equal importance. Thus we obtain the following matrix. The row averages give the coefficients for the linear equation by the Guiasu method. Since each column sums to 1, the row averages also give the coefficients of the linear equation by the Analytic Hierarchy Process. H1 H2 H3 H4 H5 H6 H7 H8

E1 0 0 0 0 0 0 0 1

E2 E3 E4 E5 Row Avg 1 0 15 0 .11 3 0 0 0 12 .10 0 0 15 0 .04 1 1 1 0 .16 3 4 5 1 1 0 0 .11 3 5 0 14 0 12 .15 0 14 15 0 .09 0 14 0 0 .25

(7.24)

Thus, given the μ(C) calculated for each causal factor (Hj) using equation (1), the resulting fully specified model for democratic consolidation (D) is: D = .11H1 + .10H2 + .04H3 + .16H4 + .11H5 + .15H6 + .09H7 + .25H8 (7.25) Based on the factors identified by our expert (theorists) at Equation (7.25) the most important factor effecting democratic consolidation (D) is the choice of regime type (H8 ); followed by the emergence of a multi-party system (H4 ), and the decentralization of the state (H6 ); then stateness (H1 ) and a reformed public administration (H5 ); then low levels of corruption (H2 ); an autonomous judiciary (H7 ); and finally privatization (H3 ) in that order.

7.4.3

Economic Freedom

The factors involved in Economic Freedom are discussed in Section 4.3 on page 77. We now determine the weights of a linear model of Economic Freedom based on Expert opinion using the Guiasu method. The following matrix is determined from the matrix in Section 6.5.2 on page 110 by normalizing its columns.

7.4 Issues in Comparative Politics

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

135

E1 E2 E3 E4 E5 E6 E7 E8 E9 Row Avgs. .12 .10 .11 .08 .11 .13 .10 .10 .06 .10 .09 .11 .11 .11 .09 .07 .06 .12 .06 .09 .09 .11 .09 .05 .09 .12 .26 .10 .06 .11 .08 .06 .09 .05 .09 .04 .06 .10 .06 .07 .11 .10 .10 .14 .09 .13 .03 .10 .15 .11 .07 .09 .10 .11 .09 .06 .03 .10 .15 .09 .09 .10 .10 .14 .11 .07 .03 .09 .19 .08 .11 .13 .11 .15 .11 .15 .32 .12 .19 .15 .14 .11 .11 .11 .11 .10 .06 .12 .06 .10 .09 .08 .10 .08 .11 .10 .03 .05 .06 .08

The row averages of the previous matrix give the Guiasu weights, Theorem 7.6. Thus the following equation gives a measure of the economic freedom of a country by the Guiasu method: EF = .1F1 + .09F2 + .11F3 + .07F4 + .11F5 + .09F6 + .08F7 + .15F8 + .1F9 + .08F10 .

7.4.4

(7.26)

Political Stability

In this section, we analyze the independent variables F1 , F2 , F3 , F4 concerning political stability that were derived in Sec 4.4. Let Cj denote the sum of the entries in the j-th column of W = [wij ]. Let N ′ denote the matrix [wij /Cj ]. Each column of N ′ to be the probability (credibility) distribution for each expert. These probability (credibility ) distributions are probabilistic with the focal elements being singleton sets. It is shown in Theorem 7.6 that the row averages of N ′ provide the Guiasu weights Gi for i = 1 . . . m. We next normalize the columns of the μ and ν tables, the results are in Table (7.1). The entries in the Row Avg column yield the coefficients for G+ and G− . G+ = .26F1 + .24F2 + .29F3 + .19F4 . G− = .16F1 + .15F2 + .13F3 + .56F4 .

136

7 The Guiasu Method Table 7.1 Political Stability Guiasu (a) Guiasu G+

(b) Guiasu G−

µ E1 E2 E3 Row Avg

ν E1 E2 E3 Row Avg

F1 F2 F3 F4

7.4.4.1

.26 .26 .26 .21

.23 .27 .30 .20

.30 .22 .30 .17

.26 .24 .29 .19

F1 F2 F3 F4

.2 0 .2 .6

0 0 0 1

.27 .45 .18 .09

.16 .15 .13 .56

Subfactors of Political Stability

Subsection 7.4.4.1 presents the experts estimates of the importance of the subfactors that influence political stability. In Section 6.5.3.1 on page 111 the row and column averages are given, and we do not repeat them here. We use the Guiasu method to derive to determine linear equations of the dependent variables Fi+ and Fi− , i = 1, 2, 3, 4, in terms of the variables Fi , i = 1, ..., 19. The same procedure is used as previously described. Due to space restrictions, we do not present the appropriate tables here. They can be obtained upon request. F1+ = 0F1,1 + .08F1,2 + .06F1,3 + .03F1,4 + .09F1,5 +.09F1,6 + .01F1,7 + .07F1,8 + .10F1,9 + .10F1,10 +.03F1,11 + .02F1,12 + .03F1,13 + .03F1,14 + .01F1,15 +.05F1,16 + .04F1,17 + .08F1,18 + .09F1,19

F+ 2 = 0F2,1 + .09F2,2 + .01F2,3 + .05F2,4 + .06F2,5 +.07F2,6 + .04F2,7 + .08F2,8 + .07F2,9 + .07F2,10 +.03F2,11 + .02F2,12 + .05F2,13 + .07F2,14 + .03F2,15 +.04F2,16 + .07F2,17 + .08F2,18 + .06F2,19

F+ 3 = .01F3,1 + .08F3,2 + .04F3,3 + .04F3,4 + .07F3,5 +.07F3,6 + .02F3,7 + .08F3,6 + .07F3,9 + .07F3,10 +.03F3,11 + .02F3,12 + .03F3,13 + .05F3,14 + .02F3,15 +.04F3,16 + .08F3,17 + .10F3,18 + .08F3,19

7.4 Issues in Comparative Politics

137

F+ 4 = 0F4,1 + .06F4,2 + .02F4,3 + .04F4,4 + .09F4,5 +.09F4,6 + .02F4,7 + .06F4,8 + .05F4,9 + .07F4,10 +.07F4,11 + .03F4,12 + .07F4,13 + .01F4,14 + .04F1,15 +.09F1,16 + .05F1,17 + .07F4,18 + .07F4,19

F1− = .26F1,1 + .06F1,2 + .05F1,3 + 0F1,4 + .01F1,5 +.05F1,6 + .07F1,7 + .09F1,8 + 0F1,9 + 0F1,10 +.07F1,11 + .06F1,12 + 0F1,13 + .09F1,14 + .06F1,15 +.07F1,16 + .01F1,17 + .03F1,18 + .02F1,19

F− 2 = .30F2,1 + 0F1,2 + .07F2,3 + .06F2,4 + .05F2,5 +.05F2,6 + .05F2,7 + .01F1,8 + .01F2,9 + .01F2,10 +.06F2,11 + .03F2,12 + .05F2,13 + .05F2,14 + .08F2,15 +.07F2,16 + .02F2,17 + 0F2,18 + .06F2,19

F− 3 = .23F3,1 + .06F3,2 + .05F3,3 + .05F3,4 + .06F3,5 +.05F3,6 + .02F3,7 + .05F3,8 + .05F3,9 + .05F3,10 +.06F3,11 + .05F3,12 + 0F3,13 + .04F3,14 + .05F3,15 +.04F3,16 + .05F3,17 + .01F3,18 + .05F3,19

F− 4 = .19F4,1 + 0F4,2 + .06F4,3 + .06F4,4 + 0F4,5 +.04F4,6 + .08F4,7 + .02F4,8 + .04F2,9 + .04F4,10 +0F4,11 + .09F4,12 + .05F4,13 + .06F4,14 + .12F4,15 +0F4,16 + .05F4,17 + .06F4,18 + .05F4,19

7.4.5

Failed States

The factors involved in Failed States are discussed in Section 4.5 on page 84. We now determine the weights of a linear model of Failed States based on Expert opinion using the Guiasu method. The following matrix is determined from the matrix in Section 6.5.4 on page 115 by normalizing its columns.

138

7 The Guiasu Method

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 Total

E1 E2 E3 E4 Row Avg Row Product .02 .04 .06 .09 .05 .0000043 .03 .08 .06 .03 .05 .0000043 .12 .04 .06 .13 .09 .0000374 .03 .12 .06 .03 .06 .0000065 .07 .02 .07 .11 .07 .0000108 .10 .12 .10 .11 .11 .0001320 .14 .20 .11 .14 .15 .0004312 .10 .02 .10 .11 .08 .0000220 .09 .06 .11 .08 .08 .0000475 .10 .10 .10 .05 .09 .0000500 .14 .02 .10 .05 .08 .0000140 .05 .18 .10 .08 .10 .0000720 1.01 .0008320

The row averages in the above table give the coefficients by the Guiasu method, see Section 7.1.4 and especially Theorem 7.6. F SG = .05F1 + .05F2 + .09F3 + .06F4 + .07F5 + .11F6

(7.27)

+ .15F7 + .08F8 + .08F9 + .09F10 + .08F11 + .10F12 If we normalize the Row Product column of the above table, we obtain the coefficients using the Dempster-Shafer method (see Section 2.4.2). F SDS = .005F1 + .005F2 + .045F3 + .008F4 + .013F5 + .159F6

(7.28)

+ .518F7 + .026F8 + .057F9 + .060F10 + .017F11 + .087F12

7.5 7.5.1

Issues of Hearing Impairment Hearing Impaired Children

In this section, we use the rankings of the experts to determine the most predictive tests and also to determine linear equations which give a measure of the predictive success of a hearing impaired child when mainstreamed. See Section 5.1. In the following matrix, Ej denotes an expert for j = 1, ..., 6 and the entries denote their weights of tests Ti , i = 1, . . . , 10.

7.5 Issues of Hearing Impairment

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

E1 .5 .6 .5 .5 .4 0 0 0 0 0

E2 .8 .9 .6 .7 .3 0 0 0 0 0

E3 .7 .8 0 0 .7 .7 .5 0 0 0

139

E4 .6 .9 .7 0 0 .6 0 .5 0 0

E5 .8 0 0 .4 .4 0 0 0 .4 .3

E6 Row Average .7 .683 = w1 .8 .667 = w2 0 .300 = w3 0 .267 = w4 .4 .367 = w5 .5 .300 = w6 .3 .133 = w7 0 .083 = w8 0 .067 = w9 0 .05 = w10

We now normalize the columns of the previous matrix. Normalized T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

E1 E2 E3 E4 E5 E6 Row Average .2 .24 .21 .18 .35 .26 .240 .24 .27 .24 .27 0 .30 .220 .2 .18 0 .21 0 0 .098 .2 .21 0 0 .17 0 .097 .16 .09 .21 0 .17 .15 .130 0 0 .21 .18 0 .19 .097 0 0 .15 0 0 .11 .043 0 0 0 .15 0 0 .025 0 0 0 0 .17 0 .028 0 0 0 0 .13 0 .022

The row averages of the previous table yield the weights of the Tj by the Guiasu method since the basic probability assignments are probabilistic. We have P = .240T1 + .220T2 + .098T3 + .097T4 + .130T5 + .097T6 + .043T7 + .025T8 + .028T9 + .022T10 ,

(7.29)

where P denotes the predictive value determined by substituting in for the Tj the coded test results for an individual child.

7.5.2

Deaf and Hard of Hearing Children

The tests used in evaluating deaf and hard of hearing children are described in Section 5.2 on page 88. The data is presented in Tables 5.2 on page 88 and 5.3 on page 90.

140

7 The Guiasu Method Table 7.2 Normalized weight of expert opinion on test scores Normalized T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

E1 .28 .28 .2 .12 .12

E2 E3 E4 E 5 E6 .25 .37 .19 .23 .23 .225 .30 .24 .23 .26 .2 .21 .18 .15 .175 .19 .21 .18 .18 .15 .18

E7 .14 .28 .17 .22 .19

E8 .24 .27 .24

E9 .14 .25 .22 .19

.15 .19 .09

.05 .14 .18 .11

The row averages of the previous matrix give the Guiasu weights. Thus CG = .23T1 + .26T2 + .17T3 + .03T4 + .14T5 +.06T6 + .04T7 + .02T8 + .04T9 + .01T10 . Due to the lack of data for T3 , T4 , T6 , T7 , T8 , T9 , and T10 , we use only T1 , T2 and T5 . We normalize their coefficients to obtain CG = .35T1 + .41T2 + .24T5 for the analytic hierarchy method and CG = .37T1 + .41T2 + .22T5 for the Guiasu method. Due to the nearly identical equations for the Guiasu method and the analytic hierarchy method, we use the equation CG = .36T1 + .41T2 + .23T5 obtained by averaging the coefficients. The scores for tests T 2 and T 5 are determined by the following tests: MLU PLAI1 PLAI2 PLAI3 PLAI4 E

R

Mean Length of Utterance (the number of words used in each utterance) PLAI level 1 PLAI level 2 PLAI level 3 PLAI group 4 Expressive (the language the student uses to express himself/herself. Can the student produce the language spontaneously to answer the question?) Receptive (Can the student recognize a word or phrase? If the student understands the language can he/she pick out an appropriate picture?)

7.5 Issues of Hearing Impairment

141

PLAI is an assessment that tests both expressive and receptive language. Can the student understand the question and answer appropriately. Level 1 is the easiest while level 4 is the most abstract. Using expert opinion, we obtain the following equations. T2 = P LAI = .1 P LAI1 + .2 P LAI2 + .3 P LAI3 + .4 P LAI4, T5 = P LS = .4 E + .6 R. Making the appropriate substitutions, we obtain the following equations: CG = .135 M LU + .135 T T R + .032 P LAI1 + .064 P LAI2 + .096 P LAI3 +.128 P LAI4 + .088 EL + .132 RL + .076 E + .114 R. for the analytic hierarchy method and CG = .145 M LU + .145 T T R + .032 P LAI1 + .064 P LAI2 + .096 P LAI3 +.128 P LAI4 + .084 EL + .126 RL + .072 E + .108 R. for the Guiasu method. Using the equation CG = .36 T1 + .41 T2 + .23 T5 , we obtain the equation CG = .36 T1 + .42(.1 P LAI1 + .2 P LAI2 + .3 P LAI3 + .4 P LAI4) + .23(.4 E + .6 R).

The closing the gap scores, CG, are determined by substituting the coded test scores into the previous equation. The latter equation was used to determine the closing the gap values for children of various ages. We next present those values for ages 3 - 5. Values for other ages can be found in Mordeson et al. (2011b). This table is reproduced with the permissions of the managing editors from the article “Closing The Language Gap For Deaf And Hard Of Hearing Children,” New Mathematics and Natural Computation 7(1) 2011, 51-62. 7.5.2.1

3-5 Year Age Group

The following tables show the linear models for Students 6–20. Student 6 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.5) + .41(.1) + .23(.18) = .2624 CG = .36(.5) + .41(.16) + .23(.5) = .3606 CG = .36(.7) + .41(.58) + .23(.5) = .6048

Student 7 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.3) + .41(.1) + .23(.1) = .172 CG = .36(.3) + .41(.1) + .23(.3) = .218 CG = .36(.5) + .41(.32) + .23(.38) = .3986

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7 The Guiasu Method

Student 8 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.3) + .41(.1) + .23(.3) = .219 CG = .36(.3) + .41(.1) + .23(.3) = .218 CG = .36(.3) + .41(.34) + .23(.34) = .3256

Student 9 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.4) + .41(.1) + .23(.3) = .254 CG = .36(.5) + .41(.66) + .23(.5) = .5656 CG = .36(.9) + .41(.68) + .23(.58) = .7362

Student 10

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.5) + .41(.12) + .23(.3) = .2982 CG = .36(.7) + .41(.64) + .23(.5) = .6294 CG = .36(.6) + .41(.9) + .23(.7) = .746

Student 11

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.3) + .41(.1) + .23(.18) = .1904 CG = .36(.5) + .41(.1) + .23(.18) = .2624 CG = .36(.7) + .41(.34) + .23(.3) = .4604

Student 12

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.5) + .41(.1) + .23(.18) = .2624 CG = .36(.8) + .41(.1) + .23(.1) = .352 CG = .36(.4) + .41(.2) + .23(.22) = .2766

Student 13

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.1) + .41(.1) + .23(.3) = .146 CG = .36(.3) + .41(.22) + .23(.22) = .2488 CG = .36(.3) + .41(.28) + .23(.22) = .2734

Student 14 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.3) + .41(.1) + .23(.22) = .2456 CG = .36(.3) + .41(.16) + .23(.3) = .2702 CG = .36(.5) + .41(.38) + .23(.5) = .4508

Student 15 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.5) + .41(.1) + .23(.18) = .2624 CG = .36(.5) + .41(.1) + .23(.42) = .3176 CG = .36(.5) + .41(.36) + .23(.42) = .4242

Student 16 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.6) + .41(.7) + .23(.5) = .618 CG = .36(.9) + .41(.52) + .23(.5) = .6522 CG = .36(.7) + .41(.8) + .23(.58) = .7134

7.5 Issues of Hearing Impairment

Student 17 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.2) + .41(.1) + .23(.3) = .182 CG = .36(.5) + .41(.2) + .23(.42) = .3586 CG = .36(.6) + .41(.62) + .23(.42) = .5668

Student 18

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.7) + .41(.1) + .23(.5) = .408 CG = .36(.5) + .41(.1) + .23(.38) = .3084 CG = .36(.5) + .41(.4) + .23(.5) = .459

Student 19

CG = .36 LSA + .41 P LAI + .23 P LS

Age 3 Age 4 Age 5

CG = .36(.7) + .41(.22) + .23(.1) = .367 CG = .36(.9) + .41(.22) + .23(.5) = .5292 CG = .36(.5) + .41(.53) + .23(.5) = .5123

Student 20 Age 3 Age 4 Age 5

CG = .36 LSA + .41 P LAI + .23 P LS CG = .36(.3) + .41(.44) + .23(.3) = .3574 CG = .36(.3) + .41(.42) + .23(.5) = .3052 CG = .36(.5) + .41(.72) + .23(.58) = .6086

143

Student 6 7 8 9 10 11 12 % increase 130 132 49 190 150 142 5 Student 13 14 15 16 17 18 19 20 % increase 87 84 62 15 211 12 40 70 Scores for students in other age groups can be found in Mordeson et al. (2011b).

Chapter 8

Jeffrey’s Rule

8.1

Jeffrey’s Rule

Section 7.1.3 introduced Jeffrey’s Rule as a special case of the Guiasu method. In this Chapter we will use Jeffrey’s Rule to produce a prediction of how democratic a country is based on the eight hypothetical causes introduced in Section 4.1. This chapter also introduces three popular indices of democratic consolidation; the Freedom House, Vanhanen, and Polity IV indexes. We contrast our results of applying Jeffrey’s Rule with the values produced by fuzzifying the indices of democratic consolidation.

8.1.1

The Dependent Variable

We calculate a variable for democratic consolidation using three indexes of democracy, the Freedom House, Vanhanen, and Polity IV indexes. The indexes and their fuzzy equivalents are at Table 8.1. These indexes compare the post-soviet countries along with the other countries of the world. Hence, a value of 1.0 on the fuzzy Vanhanen index does not mean that a country is the most consolidated democracy in the set of post-soviet states, but means that the country is the most consolidated democracy that can exist anywhere in the world. The first index, the Freedom House measure of democracy, considers electoral processes, civil society, independent media, national and local democratic governance, and corruption to calculate a rating of democracy on a scale from 1 to 7, 1 being the highest level of democratic governance. We rescaled these data by subtracting the Free House value from seven so that high values indicate a positive correlation with democracy. The second measure, the Vanhanen index, considers political competitiveness and participation to create a scale with a maximum value of 49 corresponding to the J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 145–153. DOI: 10.1007/978-3-642-35224-9_8 © Springer-Verlag Berlin Heidelberg 2013

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8 Jeffrey’s Rule

Table 8.1 Democracy Values for Post-Communist Countries FH - Freedom House Index VH - Vanhanen Index P IV - Polity IV Index

Country

Fuzzy Fuzzy Fuzzy FH VH P IV FH VH P IV

Albania Armenia Azerbaijan Belarus Bosnia & Herz. Bulgaria Croatia Czech Rep. Estonia Georgia Hungary Kazakhstan Kyrgyzstan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia & Mont. Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

4.04 5.18 5.86 6.64 4.18 3.18 3.75 2.29 1.96 4.96 1.96 6.29 5.64 2.14 2.21 3.89 5.07 2 3.39 5.61 3.75 2 1.68 5.79 6.93 4.5 6.43

12 17.7 11.8 7.2 24.8 24.1 27.5 39.3 24.1 14.8 25.4 8.2 11 27.7 28.2 20.2 22 22.3 20.7 29.3 * 36.8 29.1 9.2 0 32.7 3.7

7 5 0 0 0.40 9 7 10 7 5 10 0 1 8 10 9 8 10 8 7 6 9 10 1 0 7 0

0.42 0.26 0.16 0.05 0.51 0.55 0.46 0.67 0.72 0.29 0.72 0.10 0.19 0.69 0.68 0.44 0.28 0.71 0.52 0.20 0.46 0.71 0.76 0.17 0.01 0.36 0.08

0.24 0.36 0.24 0.15 0 0.49 0.56 0.80 0.49 0.30 0.52 0.17 0.22 0.57 0.58 0.41 0.45 0.46 0.42 0.60 0.00 0.75 0.59 0.19 0.00 0.67 0.08

0.7 0.5 0 0 0.3 0.9 0.7 1 0.7 0.5 1 0 0.1 0.8 1 0.9 0.8 1 0.8 0.7 0.6 0.9 1 0.1 0 0.7 0

highest level of democratic governance. The third index, Polity IV, offers an 11 point scale, ranging from 0 to 10, an additive scale totaling weights in the following categories: competitiveness of political participation, openness and competition of executive recruitment, and constraints on the chief executive. We divide the values in each of these three scales by the scale’s maximum value to fuzzify the data (7 for Freedom House, 49 for Vanhanen, and 10 for Polity IV). Table 8.1 is reproduced from the article “Specifying Theories in Comparative Politics: Toward a More Thoroughly Deductive Approach,” published in New Mathematics and Natural Computation 3(2) 2007, 165-189 with permission of the managing editor.

8.2 Estimating Causality in the Social Sciences

147

Table 8.2 Combined Democracy Values

8.2

Country

Weight

Albania Armenia Azerbaijan Belarus Bosnia Bulgaria Croatia Czech Rep Estonia Georgia Hungary Kazakhstan Kyrgyzstan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

m123 ({D}) = .347929 m123 ({D}) = .165021 m123 ({D}) = 0 m123 ({D}) = 0 m123 ({D}) = 0 m123 ({D}) = .913559 m123 ({D}) = .716693 m123 ({D}) = 1 m123 ({D}) = .852174 m123 ({D}) = .148973 m123 ({D}) = 1 m123 ({D}) = 0 m123 ({D}) = .007298 m123 ({D}) = .921887 m123 ({D}) = 1 m123 ({D}) = .830911 m123 ({D}) = .56 m123 ({D}) = 1 m123 ({D}) = .758333 m123 ({D}) = .466667 m123 ({D}) = 0 m123 ({D}) = .985098 m123 ({D}) = 1 m123 ({D}) = .005310 m123 ({D}) = 0 m123 ({D}) = .727132 m123 ({D}) = 0

Estimating Causality in the Social Sciences: A Fuzzy Set Approach Based on Jeffrey’s Rule

We present a method based on Jeffrey’s Rule (Guiasu, 1994) that permits us to estimate an outcome variable given prior probability assignments over a set of input factors estimating the degree of correlation between fuzzy variables. We also derive a method for combining expert opinion for any number of experts in such away that the combination can occur all at once rather than in sequential order. We demonstrate the approach’s utility with a case study of democratic consolidation in post-communist states. The approach commends itself for use in the social sciences when data are inherently ambiguous or vague. Using a data set of factors hypothesized to be important in successful democratic consolidation, we calculate the posterior probability distribution on C, the set of countries, given an actual distribution qH on H, the set of

148

8 Jeffrey’s Rule

factors in our data set. Our results are for a universe X of any cardinality, but with exactly the same two focal elements, namely A and B with A ∩ B = ∅, for each basic probability assignment. We derive a formula for combining n basic probability assignments on X for arbitrary n ∈ N. The factors in the data set represent fuzzy set inclusion values for the following factors that are hypothesized to be important to democratic consolidation, namely H1 , . . . , H8 . Let A = {D}, where D stands for democracy. We now combine the three basic probability assignments of Freedom House, Vanhanen, and Polity by using the formula,

m123 ({D}) =

1−

3

m1 ({D})m2 ({D})m3 ({D}) 2 j 3 i=1 mi ({D}) + j=2 (−1) Cj ({D})

3i=1

= m1 ({D})m2 ({D})m3 ({D})/ [1 − m1 ({D}) − m2 ({D}) − m3 ({D})

+ m1 ({D})m2 ({D}) + m1 ({D})m3 ({D}) + m2 ({D})m3 ({D})]

Table 8.2 details the combined democracy values for the 27 countries being examined. Bunce (1995) demonstrates that the predicted values for democracy based on the model derived from Guiasu’s method (also applied to the values for the hypothesized variables in the data set) closely approximate those DempsterShafer values for democracy. Mordeson et al. (2007) demonstrated a deductive approach for more full specifying theories. It did not have as its primary purpose testing correlations between variables. Jeffrey’s Rule provides us with a tool for doing so when the data are fuzzy. We now turn to the primary purpose of this paper, a demonstration and evaluation of Jeffrey’s Rule. We use the first interpretation of Jeffrey’s Rule, Guiasu (1994). The second can also be found in Guiasu (1994). Let C = {C1 , . . . , C27 }. Take a 1×27 row vector, D = [di ], where di denotes the democracy score of country Ci , i = 1, . . . , 27, determined by any method such as the median of the three expert’s 27scores or their combined score by the Dempster-Shafer method. Let S = i=1 di . Let pC be a probability distribution on C defined by pC (Ci ) = di /S, i = 1, . . . , 27. Let H = {H1 , . . . , H8 }. Let qH be any probability (credibility) distribution defined on H. Here we define qH by qH (Hj) = μ(Hj), where μ(Hj) is the value of Hj determined by the Guiasu’s method, j = 1, ..., 8. Let X = C × H. Let m1 , m2 be the basic probability assignments on X defined as follows: m1 ({(C, H)|H ∈ H}) = pC (C)∀C ∈ C, m2 ({(C, H)|C ∈ C}) = qH (H)∀H ∈ H.

8.2 Estimating Causality in the Social Sciences

149

Table 8.3 Democracy values using Jeffrey’s Rule (Sum) Country Albania Armenia Azerbaijan Belarus Bosnia Bulgaria Croatia Czech Repub. Estonia Georgia Hungary Kazakhstan Kyrgystan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

Sum 8 H’s Median 4.36 2.97 1.89 0.53 3.14 6.70 4.92 6.82 6.77 2.58 7.34 1.71 2.25 6.18 6.37 5.59 3.24 5.61 5.85 2.31 3.31 6.96 7.33 1.31 0.20 3.19 1.03

0.42 0.36 0.16 0.05 0.40 0.55 0.56 0.80 0.70 0.30 0.72 0.10 0.19 0.69 0.68 0.44 0.45 0.71 0.52 0.60 0.46 0.75 0.76 0.17 0.00 0.67 0.08

di Si

Jeffrey’s

0.096330 0.121212 0.084656 0.094340 0.127389 0.082090 0.113821 0.117302 0.103397 0.116279 0.098093 0.058480 0.084444 0.111650 0.106750 0.078712 0.138889 0.126560 0.088889 0.259740 0.139183 0.107759 0.103683 0.130268 0.000000 0.210031 0.077670

0.036659 0.022953 0.008850 0.003794 0.032851 0.044225 0.044163 0.065539 0.055205 0.018534 0.056922 0.004913 0.012166 0.054649 0.053644 0.037008 0.038144 0.054399 0.041732 0.035971 0.040660 0.060678 0.062938 0.009451 0.000000 0.049234 0.004718

Assume F (X; m1 ) = {{(C, H)|H ∈ H}|C ∈ C}, F (X; m2 ) = {{(C, H)|C ∈ C}|H ∈ H}. Let pH (·|C) be a conditional probability distribution on H given C ∈ C, and pH the probability distribution on H defined by  pH (H) = pH (H|C)pC (C), C∈C

where pC is interpreted as being the prior probability distribution on C. Define m12 : P(X) → [0, 1] by ∀W ∈ P(X),   m12 (W ) = w(W | Y, Z) m1 (Y ) m2 (Z), Y ∈F (X;m1 ) Z∈F (X;m2 )

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8 Jeffrey’s Rule

where F (X; m12 ) = {{(C, H)} | C ∈ C, H ∈ H}

= {Y ∩ Z | Y ∈ F(X, m1 ), Z ∈ F(X; m2 )}

and w({(C, H)} | {(C, H) | H ∈ H}, {(C, H)} | C ∈ C}) =

pH (H | C) . pH (H)

Then we have ∀(C, H) ∈ C × H, m12 ({(C, H)}) =

pC (C) pH (H | C) qH (H). pH (H)

Now m12 is a probability distribution on C × H and its marginal probability distribution, pC (C|qH ) = Bel({(C, H)} | H ∈ H}; m2 )}  = m12 ({(C, H)})

(8.1)

H∈H

=



[pC (C)pH (H|C)/pH (H)]qH (H),

H∈H

is Jeffrey’s Rule for calculating the posterior probability distribution on C given the actual distribution qH on H. Let CH denote the 27 × 8 matrix whose entries are the H values for each country. Let CH = [hij ]. Let Si = hi1 + ... + hi8 for i = 1, ..., 27. Let pH (Hj|Ci ) = hij /Si for j = 1, ..., 8; i = 1, ..., 27. Then for Hj ∈ H, j = 1, ..., 8, pH (Hj ) =

27  i=1

=

27 

pH (Hj | Ci )pC (Ci ) (hij/Si ) (di/S )

i=1

Also, pC (C | qH ) = Thus for i = 1, ..., 27,

8  pC (C)pH (Hj | C) qH (Hj ). pH (Hj ) j=1

8.2 Estimating Causality in the Social Sciences

151

8 

(di/S ) (hij/Si ) qH (Hj ) 27 hij di i=1 ( /Si ) ( /S ) j=1

 8  hij d = ( i/S) di qH (Hj ). 27 hij i=1 ( /Si ) j=1

pC (Ci |qH ) =

Let IC denote the 27 × 27 matrix whose i, i-th element is di /Si , i = 1, ..., 27, and is 0 elsewhere. Let G denote the 1 × 8 matrix whose elements are the Guiasu scores, and let C denote the 8 × 27 matrix

 hij . 27 di i=1 hij Si

Then G · C · IC is the 1 × 27 matrix [pC (Ci | qH )]. The results can be found in Tables 8.3 and 8.4. Table 8.4 Democracy values using Jeffrey’s Rule (D-S Combo) Country

D-S Comb.

di/S

Jeffrey’s

Albania Armenia Azerbaijan Belarus Bosnia Bulgaria Croatia Czech Repub. Estonia Georgia Hungary Kazakhstan Kyrgystan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

0.347929 0.165021 0.000000 0.000000 0.000000 0.913559 0.716693 1.000000 0.852174 0.148973 1.000000 0.000000 0.007298 0.921887 1.000000 0.830911 0.560000 1.000000 0.758333 0.466667 0.000000 0.985098 1.000000 0.005310 0.000000 0.727132 0.000000

0.079800 0.055563 0.000000 0.000000 0.000000 0.136352 0.145669 0.146628 0.125875 0.057741 0.136240 0.000000 0.003244 0.149173 0.156986 0.148642 0.172840 0.178253 0.129630 0.202020 0.000000 0.141537 0.136426 0.004069 0.000000 0.227941 0.000000

0.027165 0.009515 0.000000 0.000000 0.000000 0.065816 0.050594 0.073340 0.060275 0.008353 0.070879 0.000000 0,000424 0.065433 0.070682 0.062559 0.042448 0.068607 0.054515 0.025383 0.000000 0.071424 0.074112 0.000271 0.000000 0.048204 0.000000

i

152

8 Jeffrey’s Rule

In Mordeson et al. (2007), we substituted the H values from the matrix CH for each country into the equation D = .09H1 + .10H2 + .04H3 + .14H4 + .09H5 + .15H6 + .09H7 + .25H8 to obtain a democracy value for each country. These scores can be found in Table 8.6. We ran a sample rank correlation between the predicted values of democracy in Mordeson et al. (2007) based on the Guiasu model and the ones in the above table obtained by Jeffrey’s Rule with the use of the median democracy values. We obtained a sample rank correlation of 0.94. We also ran a sample rank correlation between the democracy values determined by the combination of the three experts using Dempster-Shafer theory and the democracy values found by Jeffrey’s method with the use of Dempster-Shafer democracy values. We obtained a rank correlation of 0.99. Table 8.5 gives the H values for each country. Table 8.6 gives the combined H values for each country. Table 8.5 H values for 27 Countries Country Albania Armenia Azerbaijan Belarus Bosnia Bulgaria Croatia Czech.Rep. Estonia Georgia Hungary Kazakhstan Kyrgystan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

H1 H2 H3 H4 H5 H6 H7 H8 .37 .12 .23 .04 .55 .85 .55 .82 .90 .42 .95 .12 .25 .72 .80 .48 .25 .75 .78 .38 .25 .82 1 .125 0 .40 .10

.20 .30 .10 0 .42 .76 .40 .75 .90 .23 .85 .12 .11 .78 .70 .45 .10 .48 .80 .21 .375 .72 .85 .03 .05 .32 .12

.65 .83 .78 .05 .40 .88 .80 .95 1 .75 1 .77 .80 .90 .92 .85 .48 .90 .78 .75 .48 1 .75 .65 .10 .65 .33

.75 .40 .10 .05 .37 .90 .80 1 .82 .15 .95 .20 .30 .80 1 .85 .72 .87 .68 .20 .38 .95 .95 .10 0 0 .08

.38 .47 .12 0 .40 .85 .49 .70 .80 .45 .87 .35 .30 .75 .72 .48 .19 .78 .82 .20 .35 .82 .98 0 0 .40 .10

.81 .28 .10 .15 .35 .83 .40 .85 .80 .08 .95 .03 .20 .65 .78 .90 .70 .18 .47 .05 .62 .95 .90 .15 0 .32 .05

.45 .45 .33 22 .20 .80 .78 .80 .75 .30 .95 .10 .10 .80 .75 .72 .40 .85 .68 .35 .25 .80 .90 .15 .05 .65 .20

.75 .12 .13 .02 .45 .83 .7 .95 .80 .20 .82 .02 .19 .78 .70 .86 .40 .80 .84 .17 .60 .90 1 .10 0 .45 .05

8.2 Estimating Causality in the Social Sciences

153

Table 8.6 Combined H values for 27 Countries Country

D-S Comb.

di/S

Jeffrey’s

Albania Armenia Azerbaijan Belarus Bosnia Bulgaria Croatia Czech.Rep. Estonia Georgia Hungary Kazakhstan Kyrgystan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

0.347929 0.165021 0.000000 0.000000 0.000000 0.913559 0.716693 1.000000 0.852174 0.148973 1.000000 0.000000 0.007298 0.921887 1.000000 0.830911 0.560000 1.000000 0.758333 0.466667 0.000000 0.985098 1.000000 0.005310 0.000000 0.727132 0.000000

0.079800 0.055563 0.000000 0.000000 0.000000 0.136352 0.145669 0.146628 0.125875 0.057741 0.136240 0.000000 0.003244 0.149173 0.156986 0.148642 0.172840 0.178253 0.129630 0.202020 0.000000 0.141537 0.136426 0.004069 0.000000 0.227941 0.000000

0.027165 0.009515 0.000000 0.000000 0.000000 0.065816 0.050594 0.073340 0.060275 0.008353 0.070879 0.000000 0,000424 0.065433 0.070682 0.062559 0.042448 0.068607 0.054515 0.025383 0.000000 0.071424 0.074112 0.000271 0.000000 0.048204 0.000000

i

Tables 8.3–8.6 are reproduced with the permission of the managing editor from the article “An Inductive Approach to Determining Causality in Comparative Politics: A Fuzzy Set Alternative,” New Mathematics and Natural Computation 3(2) 2007, 191-207.

Chapter 9

Yen’s Method

Mixing Evidence Theory and Fuzzy Set Theory In Yen (1990), Yen developed an approach that addressed the issue of managing imprecise and vague information in evidential reasoning by combining Dempster-Shafer theory with fuzzy set theory. Later we will apply Yen’s method to arrive at the degree of belief of certain subsets of a set {x1 , ..., xk }. We assume we have n subsets of X, Aj , j = 1, . . . , n, which are the focal elements of a function m of the power set of X into the closed interval [0, 1], i.e., m(Aj ) > 0, j = 1, . . . , n. We first formulate the linear programming problems solved by the belief function in order to lay the foundation from which various components of the optimization problem can be generalized. P l(B) and Bel(B) are upper and lower probabilities of a subset B of X. The belief function can be obtained by solving the following optimization problem: min

n  

m(xi : Aj )

xi ∈B j=1

subject to



m(xi : Aj ) ≥ 0, i = 1, ..., k; j = 1, ..., n , m(xi : Aj ) = 0 ∀xi ∈ / Aj , m(xi : Aj ) = m(Aj ), j = 1, ..., n .

i

The variable m(xi : Aj ) denotes the probability mass allocated to xi from the basic probability of a focal element Aj . The objective function computes the total probability of the set B, where the inner summation gives the probability of an element xi . The distributions of focals’ masses do not interact J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 155–162. DOI: 10.1007/978-3-642-35224-9_9 © Springer-Verlag Berlin Heidelberg 2013

156

9 Yen’s Method

with one another. Hence they can be optimized individually to reach a global optimal solution. The optimal value is the sum of the following subproblems j = 1, . . . , n :  min m(xi : Aj ) xi ∈B

subject to



m(xi : AJ ) ≥ 0, / Aj , m(xi : Aj ) = 0 ∀xi ∈ m(xi : Aj ) = m(Aj ).

i

The linear programming problem for computing the plausibility of a set B follows in a similar manner as that for the belief function except that min is replaced with max. Then the optimal solutions to the subproblems are denoted by m∗ (B : Aj ) and m∗ (B : Aj ), respectively. By the adding the solutions of subproblems we obtain  Bel(B) = m∗ (B : Aj ), Aj ⊆T

P l(B) =



m∗ (B : Aj ).

Aj ⊆T

In the above discussion, B was a crisp set. We now consider the case,where B is a fuzzy subset of X. Theorem 9.1. Yen (1990) Suppose A is a nonfuzzy focal element. The minimum and maximum probability masses that can be allocated to a fuzzy subset B from A are m∗ (B : A) = m(A) × inf{B(x)|x ∈ A},

m∗ (B : A) = m(A) × sup{B(x)|x ∈ A}.

We now consider the situation for fuzzy focal elements. That is, when A is a fuzzy subset of X. Definition 9.2. Yen (1990) The decomposition of a fuzzy focal element A is a collection of nonfuzzy subsets such that they are A’s t-level sets that form a resolution identity and their basic probabilities are m(Ati ) = (ti − ti−1 ) × m(A), i = 1, . . . , q, where Im(A) = {t0 , t1 , . . . , tq } with t0 = 0 and tq = 1.

In Yen (1990), the following formulas are derived for computing the belief function and the plausibility function for a fuzzy subset B, where the sums are taken over the fuzzy focal elements A :

9.1 Issues in International Relations

Bel(B) =



m(A)

P l(B) =

 ti

A



157

m(A)

A

Theorem 9.3. P l(χ{xi } ) =

[ti − ti−1 ] × inft B(x), x∈A

i



[ti − ti−1 ] × sup B(x).

j=1

m(Aj )

ti

n

x∈Ati

Im(Aj ) = {t0j , . . . , tqj }, , j = 1, . . . , n, and

qj

i=1 tij , t xi ∈ / Ajij .

i = 1, . . . , k, where

n qj t Proof. P l(χ{xi } ) = j=1 m(Aj ) i=1 [tij − ti−1,j ] sup{χ{xi } (y) | y ∈ Ajij }. Since t0j = 0, we have that qj  i=1

t

[tij − ti−1,j ] sup{χ{xi } (y) | y ∈ Ajij } = [t1j − 0] × 1 + [t2j − t1j ] × 1 + . . . + [tij − ti−1j ] × 1 = tij t

t

since xi ∈ / Ajh,j and so χ{xi } (y) = 0 ∀y ∈ Ajhj , h = i + 1, . . . , qj .

⊔ ⊓

Corollary 9.4. Suppose m(Ai ) = m(Aj ) for all i, j = 1, . . . , n. Then n

P l(χ{xi } ) =

1 tij n j=1

for i = 1, . . . , k. Proof. m(Aj ) =

1 n

for j = 1, . . . , n.

⊔ ⊓

Yen’s method is developed under the assumption that the fuzzy focal elements are normal. If the fuzzy focal elements are not normal, he normalizes them as follows.  : X → [0, 1] Let A be a fuzzy focal element that is not normal. Define A A(x)  by ∀x ∈ X, A(x) = Amax , where Amax = max{A(x) | x ∈ X}. Then define  by m(  = Amax m(A). the basic probability assignment m  on A  A)

9.1 9.1.1

Issues in International Relations Cooperative Threat Reduction

The factors necessary for Cooperative Threat Reduction are discussed in Section 3.3. The data is given in Table 3.3. In the expert’s table of weights, we divide each element of the column by the column’s maximal entry. We thus obtain the following matrix from which we derive a linear equation for G by Yen’s method.

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9 Yen’s Method

If m(Aj ) = n1 for j = 1, ..., n, then P l(1{xi } ) is the average of the entries in the i-th row. This average gives the coefficients Yi , i = 1, ..., m, for the linear equation. O1 O2 O3 O4 Sum

E1 1 1 1 1

E2 1 1 1 1

E3 .7 .7 1 1

E4 .9 1 1 1

E5 1 .8 .6 .4

E6 Row Avg .8 .90 1 .92 .7 .88 .7 .85 3.55

Yi .25 .26 .25 .24

G = .25O1 + .26O1 + .25O3 + .24O4 .

9.2 9.2.1

Issues in Comparative Politics Factors in Successful Democratization

The factors necessary for Successful Democratization are discussed in 4.1. The data is given in Table 4.1. We consider Yen’s method applied to Democracy Data. We start by transforming each column into a normal fuzzy set. This is done by dividing each column by the maximum value in that column. We apply this to the columns of the matrix given in 7.24 on page 134 to produce the following matrix. H1 H2 H3 H4 H5 H6 H7 H8

E1 0 0 0 0 0 0 0 1

E2 1 0 0 1 1 0 0 0

E3 0 0 0 1 0 1 1 1

E4 1 0 1 1 1 0 1 0

E5 Row Avg 2/5 0 1/5 1 1/5 0 3/5 0 2/5 0 2/5 1 2/5 0 2/5 0

The row average column sums to 3. Normalizing the row column, i. e., dividing each entry by 3 yields the coefficients of the linear equation. We have D = .13H1 + .07H2 + .07H3 + .20H4 + .13H5 + .13H6 + .13H7 + .13H8 . (9.1)

9.2 Issues in Comparative Politics

9.2.2

159

Economic Freedom

Define the fuzzy focal elements as follows in terms of the normalized matrix, e. g., Aj ({Fi }) = ij-th element of the normalized matrix, where j = 1, ..., 9 and i = 1, ..., 10. For example, A3 ({F4 }) = .09. We normalize these fuzzy focal elements to obtain the following matrix.

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A .86 .77 1 .53 1 .87 .31 .83 .32 .64 .85 1 .73 .82 .47 .19 .1 .32 .64 .85 .82 .33 .82 .80 .81 .83 .32 .57 .46 .82 .33 .82 .27 .19 .83 .32 .79 .77 .91 .93 .82 .87 .09 .83 .79 .50 .64 .91 .73 .82 .40 .09 .83 .79 .64 .77 .91 .93 1 .47 .09 .75 1 .79 1 1 1 1 1 1 1 1 1 .85 1 .73 1 .67 .19 1 .32 .64 .62 .91 .53 1 .67 .09 .42 .32

Corollary 9.4 says that if m(Aj ) = n1 for j = 1, . . . , n then P l(1{xj } ) is the average in the i–th row, i = 1, . . . , k. By Corollary 9.4, we thus have 1 [.79 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1] = .98, 9 1 P l(χX\{F 8} ) = [1 + .85 + 1 + .93 + 1 + .87 + .75 + 1 + 1] = .93. 9 P l(χ{F 8} ) =

In fact, P l(χ{F1 } ) = .72 P l(χ{F2 } ) = .57 P l(χ{F3 } ) = .86 P l(χ{F4 } ) = 51 P l(χ{F5 } ) = .76 P l(χ{F6 } ) = .63 P l(χ{F7 } ) = .73 P l(χ{F8 } ) = .98 P l(χ{F9 } ) = .75 P l(χ{F10 } ) = .58  Now 10 i=1 P l(χ{Fi } ) = 7.07. Thus if we let wi = P l(χ{Fi } )/7.07, i = 1, ..., 10, we have w1 = .1, w2 = .08, w3 = .12, w4 = .07, w5 = .11, w6 = .09, w7 = .1, w8 = .14, w9 = .11, w10 = .08 .

160

9 Yen’s Method

Hence if we use these weights as coefficients for the ten economic freedoms, we have EF = .1F1 + .08F2 + .12F3 + .07F4 + .11F5 + .09F6 + .1F7 + .14F8 + .11F9 + .08F10 . The coefficients of the F i are very close to those obtained by the Guiasu method and the analytic hierarchy process.

9.2.3

Political Stability

In this section, we analyze the independent variables F1 , F2 , F3 , F4 concerning political stability that were derived in Sec 4.4. We assume we have n subsets of X, Aj , j = 1, ..., n, which are focal elements of a function m of the power set of X into the closed interval [0, 1], i.e., m(Aj ) > 0, j = 1, ..., n. Let X = {F1 , ..., F4 }. Let N ′′ be determined from N ′ by dividing each entry of N ′ by the maximal element of the column. The following result is shown in Corollary 9.4. Theorem 9.5. If m(Aj ) = 1/n for j = 1, ..., n, then P l(χ{Fj } ) is the average of the entries in the i-th row of the matrix N ′′ . We normalize the columns of the previous tables in the sense of Yen. Table 9.1 Political Stability Yen (a) Yen G+ µ E1 E2 E3 Row Avg F1 F2 F3 F4

1 1 1 .81

.77 .90 1 .67

1 .73 1 .57

.92 .88 1 .68

(b) Yen G− ν E1 E2 E3 Row Avg F1 F2 F3 F4

.33 0 .33 1

0 .6 0 1 0 .4 1 .2

.31 .33 .24 .73

The sum of the entries in the Row Avg column of the μ table is 3.48 and that of the ν table is 1.61. Normalizing these columns yields the coefficients for the linear equations of G+ and G− . G+ = .26F1 + .25F2 + .29F3 + .20F4 . G− = .19F1 + .20F2 + .15F3 + .45F4 .

9.2 Issues in Comparative Politics

9.2.3.1

161

Analysis of Subfactors of Political Stability

Subsection 4.4.3 presents the experts estimates of the importance of the subfactors that influence political stability. We use the Yen method to derive to determine linear equations of the dependent variables Fi+ and Fi− , i = 1, 2, 3, 4, in terms of the variables Fi , i = 1, ..., 19. The same procedure is used as previously described. Once again we do not provide the pertinent tables in order to conserve space. They can be obtained upon request. F1+ = 0F1,1 + .07F1,2 + .05F1,3 + .04F1,4 + .08F1,5 +.09F1,6 + .02F1,7 + .06F1,8 + .09F1,9 + .09F1,10 +.03F1,11 + .02F1,12 + .04F1,13 + .04F1,14 + .02F1,15 +.05F1,16 + .04F1,17 + .08F1,18 + .09F1,19 F+ 2 = 0F2,1 + .08F2,2 + .01F2,3 + .05F2,4 + .06F2,5 +.07F2,6 + .04F2,7 + .08F2,8 + .07F2,9 + .07F2,10 +.04F2,11 + .02F2,12 + .05F2,13 + .07F2,14 + .02F1,15 +.05F2,16 + .07F2,17 + .08F2,18 + .06F2,19 F+ 3 = 0F3,1 + .07F3,2 + .04F3,3 + .04F3,4 + .06F3,5 +.07F3,6 + .02F3,7 + .08F3,8 + .06F3,9 + .06F3,10 +.04F3,11 + .02F3,12 + .04F3,13 + .05F3,14 + .02F1,15 +.05F1,16 + .07F1,17 + .09F1,18 + .08F1,19 F+ 4 = 0F4,1 + .06F4,2 + .03F4,3 + .04F4,4 + .08F4,5 +.08F4,6 + .02F4,7 + .06F4,8 + .05F4,9 + .07F4,10 +.06F4,11 + .03F4,12 + .06F4,13 + .01F4,14 + .04F4,15 +.08F4,16 + .06F4,17 + .07F4,18 + .07F4,19 F1− = .19F1,1 + .07F1,2 + .06F1,3 + 0F1,4 + .02F1,5 +.06F1,6 + .06F1,7 + .10F1,8 + 0F1,9 + 0F1,10 +.06F1,11 + .05F1,12 + 0F1,13 + .08F1,14 + .05F1,15 +.09F1,16 + .01F1,17 + .05F1,18 + .04F1,19

162

9 Yen’s Method

F− 2 = .25F2,1 + 0F2,2 + .06F2,3 + .06F2,4 + .06F2,5 +.06F2,6 + .06F2,7 + .01F2,8 + .01F2,9 + .01F2,10 +.05F2,11 + .02F2,12 + .06F2,13 + .05F2,14 + .09F2,15 +.08F2,16 + .02F2,17 + 0F2,18 + .03F2,19 F− 3 = .23F3,1 + .11F3,2 + .09F3,3 + .06F3,4 + .11F3,5 +.03F3,6 + .02F3,7 + .03F3,8 + .03F3,9 + .03F3,10 +.06F3,11 + .05F3,12 + 0F3,13 + .02F3,14 + .05F3,15 +.02F3,16 + .03F3,17 + .02F3,18 + .03F3,19 F− 4 = .14F4,1 + 0F4,2 + .07F4,3 + .07F4,4 + 0F4,5 +.04F4,6 + .04F4,7 + .03F4,8 + .05F3,9 + .05F4,10 +0F4,11 + .07F4,12 + .06F4,13 + .05F4,14 + .11F4,15 +0F4,16 + .06F4,17 + .08F4,18 + .06F4,19

Chapter 10

Methods Based on Fuzzy Set Theory

In this chapter, we apply another method developed by Yen (1992). This method involves various measures of subsethood and extends DempsterShafer theory by including a measure of subsethood S(A, B), the degree to which the fuzzy subset A is included in the fuzzy subset B. Let F (X) be the fuzzy power set of X, F (X) thus denotes the set of all fuzzy subsets of X. In Section 1.7.1 we noted that an implication operator I on [0, 1] that models ⇒ induces a measure of subsethood as follows: ∀A, B ∈ F (X),  I(A, B) = {A(x) ⇒ B(x) | x ∈ X}  = {I(A(x), B(x)) | x ∈ X} . Let

I = {I | I : F (X) → F (X)} be the set of all inclusion (implication and subsethood) operators. Yen’s extended belief measure is given by the formula,  I(A, B) m(A), Bel(B) = A∈F (X)

where m : F (X) → [0, 1], the basic probability assignment, is such that  A∈F (X) m(A) = 1. We recall here the essential definitions of fuzzy set theory pertinent to our discussion. Let X be a nonempty set and A and B be fuzzy subsets of X. Let α ∈ [0, 1] and Aα = {x ∈ X | A(x) ≥ α}. Then Aα is called an α-cut of A. Define Ac : X → [0, 1] by ∀x ∈ X, Ac (x) = 1 − A(x). Then Ac is known as the standard complement of A. The following are several measures of inclusion (the degree to which a set is a subset of another) (see Bandler and Kohout (1980); Sanchez (1977); Yager (1982); Ogawa et al. (1985)): ∀A, B ∈ F (X) : J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 163–175. DOI: 10.1007/978-3-642-35224-9_10 © Springer-Verlag Berlin Heidelberg 2013

164

10 Methods Based on Fuzzy Set Theory

Bandler and Kohout: IBK (A, B) = ∧{A(x) ∨ B(x) | x ∈ X}, Sanchez: IS (A, B) =



∧ B(x)) x∈X A(x),

x∈X (A(x)

Wierman: IW (A, B) =





1

χ⊆ (Aα , B α )dα,

0

where χ⊆ (Aα , B α ) = 1 if Aα ⊆ B α and 0 otherwise.

10.1 10.1.1

Issues in International Relations Nuclear Deterrence

We produce normal fuzzy sets by taking the data in the matrix in Table 3.1 and dividing each column by the maximal element in that column (as in Ishizuka et al. (1982)) to obtain

G1 G2 G3 G4 G5 G6

1 A 1 .6 .6 .6 .8 .4

2 A 3 A 4 A 5 A 1 .83 1 1 .65 .17 .81 .34 .85 .48 .81 .48 .85 1 1 .83 .15 .66 .67 .66 .35 .34 .48 .17

(10.1)

In Yen (1989, 1990, 1992), the following formulas are derived for computing the belief function and the plausibility function for a fuzzy subsets B, where the sums are taken over the fuzzy focal elements A, and where Im(A) = {t0 , t1 , ..., tn } : Bel(B) =



m(A)

ti

A

Pl(B) =

 A



m(A)

 ti

[ti − ti−1 ] × inft B(x), x∈A

i

[ti − ti−1 ] × sup B(x). x∈Ati

i , i = 1, ..., 6, we have Thus for the normalized fuzzy focal elements, A 1 Pl(χ{G1 ,G4 } ) = [1 + 1 + 1 + 1 + 1] = 1, 5

10.1 Issues in International Relations

165

1 [.8 + .85 + .66 + .81 + .66] = .76, 5 1 Pl(χ{G1 } ) = [1 + 1 + .83 + 1 + 1] = .97, 5 1 Pl(χ{G4 } ) = [.6 + .85 + 1 + 1 + .83] = .86. 5

Pl(χ{G2 ,G3 ,G5 ,G6 } ) =

This indicates that goals G1 and G4 are believed by the experts to be the most important goals, according to Yen’s method. In fact, the degrees of belief are quite high. We note that P l(χY ) = Bel(χY ) for all Y ⊆ {Gi | i = 1, ..., 6}. 10.1.1.1

Belief Functions

We use the normalized fuzzy focal elements of the previous section for the next four belief functions since otherwise we would allow large belief for the fuzzy empty set. For N the characteristic function of the universe, we note that Bandler/Kohout’s, Sanchez’s, and Wierman’s methods yield Bel(N ) = 1 since i ⊆ N ) = IS (A i ⊆ N ) IBK (A i ⊆ N ) = IW (A =1 5

i ) = 1. for i = 1, ..., 5. Thus Bel(N ) = i=1 m(A Recall χY denotes the characteristic function of a subset Y of X. Let F = {f | f : [0, 1]2 → [0, 1], ∀(p.q) ∈ [0, 1]2 , f (p, q) = 1 if p ≤ q}. Let I = {I | I : F (X) → F (X)} . Proposition 10.1. Let f, g ∈ F and I, J ∈ I be such that ∀A, B ∈ F(X),  I(A, B) = {f (A(x), B(x)) | x ∈ X} and

J(A, B) =

 {g(A(x), B(x)) | x ∈ X}.

If f (p, 0) = g(p, 0) for ∀p ∈ [0, 1], then I(A, χY ) = J(A, χY ) for ∀Y ∈ P(X). Proof. Let Y ∈ P(X). Then

166

10 Methods Based on Fuzzy Set Theory

I(A, χY ) = = = = =

  

 

{f (A, χY (x)) | x ∈ X}  {1 | x ∈ Y } ∧ {f (A, χY (x)) | x ∈ / Y} {f (A(x), 0) | x ∈ / Y}

{g (A(x), 0) | x ∈ / Y}  {1 | x ∈ Y } ∧ {f (A, χY (x)) | x ∈ / Y}

= J(A, χY ) .

⊔ ⊓

Recall that an implication operator ⇒ on [0, 1] defines a measure of subsethood as follows: ∀A, B ∈ F(X), I(A, B) = ∧{A(x) ⇒ B(x) | x ∈ X}. Then Lukasiewicz’s measure of subsethood as induced by the implication operator, p ⇒ q = 1 ∧ (1 − p + q), is IL (A, B) = ∧{1 ∧ (1 − A(x) + B(x)) | x ∈ X}. Corollary 10.2. Let IBK , and IL be the measures of subsethood of Bandler and Kohout and Lukasiewicz, respectively. Then IBK (A, χY ) = IL (A, χY ) ∀Y ∈ P(X) Proof. For both measures I, I(A, χY ) = ∧{1 − A(x) | x ∈ / Y }.

⊔ ⊓

In the following, we use the normalized fuzzy focal elements of the previous section (10.1). We next find the belief functions of certain subsets of X = {Gj | j = 1, ..., 6} with respect to IBK = Ac (x) ∨ B(x) and IL . By Theorem 9.3 on page 157 and Corollary 9.4 on page 157, we have that Bel(χY ) =

5  i=1

5

i )I(A i ⊆ χY ) = m(A

1 i (x) | x ∈ (∧{1 − A / Y }), 5 i=1

where I stands for any of the two measures of subsethood. Thus

1 [.2 + .15 + .34 + .19 + .34] = .24, 5 1 Bel(χ{G2 ,G3 ,G5 ,G6 } ) = [0 + 0 + 0 + 0 + 0] = 0, 5 1 Bel(χ{G1 } ) = [.2 + .15 + 0 + 0 + .17] = .10, 5 1 Bel(χ{G4 } ) = [0 + 0 + .17 + 0 + 0] = .03. 5 Bel(χ{G1 ,G4 } ) =

10.1 Issues in International Relations

167

We next consider Reichenbach’s measure of subsethood as induced by his implication operator,  1 if p ≤ q, p⇒q= 1 − p + pq if p > q. Then IR (A ⊆ B) = ∧{1 − A(x) + A(x)B(x) | x ∈ X}. Hence  i ⊆ χY ) = {1 | x ∈ Y }) ∧ (∧{1 − A i (x) | x ∈ IR (A / Y },  i (x) | x ∈ = {1 − A / Y} ,

for i = 1, ..., 5. Thus it follows that BelR (χY ) for Reichenbach’s measure of subsethood is the same as for Bandler/Kohout and Lukasiewicz. We next consider Wierman’s measure of subsethood. We have Bel(χY ) =

5  j=1

j ) m(A

5

1 = 5 j=1



1

0



0

1

j )t , (χtY ))dt I((A

j )t , (χt ))dt . I((A Y

Hence it follows that BelW (χY ) for Wierman’s measure of fuzzy subsethood is the same as that for Bandler/Kohout, Lukasiewicz, and Reichenbach. Let IS denote of subsethood. Let Y ∈ P(X). Then Sanchez’s measure  IS (A ⊆ χY ) = x∈Y A(x)/ x∈X A(x). Let Ai be the fuzzy focal element defined above, i = 1, ..., 5. Then Bel(χY ) =

5  i=1

m(Ai )IS (Ai ⊆ χY )

5

=

1 (IS (Ai ⊆ χY ). 5 i=1

6 Since j=1 Ai (Gj) = 1 for i = 1, ..., 5,we have the following beliefs for Sanchez’s measure of subsethood: 1 (.4 + .47 + .53 + .42 + .53) = .47 5 1 Bel(χG2 ,G3 ,G5 ,G6 } ) = (.6 + .53 + .47 + .58 + .47) = .53 5 Bel(χ{G1 ,G4 } ) =

168

10 Methods Based on Fuzzy Set Theory

1 (.25 + .26 + .24 + .21 + .29) = .25 5 1 Bel(χ{G4 } ) = (.15 + .22 + .29 + .21 + .24) = .22. 5

Bel(χ{G1 } ) =

10.1.1.2

Plausibility

In this subsection, we consider the plausibility of certain subsets of X = {G1 , . . . , G6 }. The plausibility of a fuzzy subset B of X is a function from F (X) into [0, 1] given as follows: ∀B ∈ F(X), Pl(B) = 1 − Bel(B c ) , where B c is the standard complement of B. Thus  Pl(B) = 1 − I(A, B c )m(A),

(10.2)

A

where the sum is taken over all fuzzy focal elements A. Thus it follows that for a subset Y of X Pl(χY ) = 1 − Bel(χY c ) . In Dempster-Shafer theory, the plausibility of a set represents not only the total evidence or belief that the element in question belongs to the set or any of its subsets, but also the additional evidence or belief associated with sets that overlap with it. It follows that for a subset Y of X, Pl(χY ) = 1−Bel(χY c ) for the measures of subsethood of Bandler/Kohout, Lukasiewicz, and Reichenbach, Pl(χ{G1 ,G4 } ) = 1 − Bel(χ{G2 ,G3 ,G5 ,G6 } ) = 1.

For Sanchez’s measure of subsethood, we have Pl(χ{G1 ,G4 } ) = 1 − Bel(χ{G2 ,G3 ,G5 ,G6 } ) = 1 − .53 = 0.47 . We used techniques from fuzzy mathematics to develop metrics for measuring how well the U. S. is doing in protecting the country, its allies and its friends from nuclear attack and from coercive pressures by states possessing nuclear weapons. The metrics presented are linear equations that weight the six components of the overarching goal on the basis of expert opinion.

10.2 Issues in Comparative Politics

169

A future research problem is the determination of how well the U.S. policies are in effect. Possible methods one might employ can be found in Beers and Vachtsevanos (1997). Other possible methods are the minimal disagreement approach Lau and Lam (2002) and the data envelopment analysis method Charnes (1994). The study of global nuclear stability in its entirety is an important undertaking for future examination. Beginning reading can be found in Cleary and Delaney (2003)Finkleman (2001)O’Neil (2005)Roberts (2000). In Charnes (1994), insights to strategic stability environment and presents issues worthy of future research are given. It is demonstrated in Finkleman (2001) that ballistic missile defense and space system can diminish the prospects of nuclear stability and contribute to international stability if pursued properly. It is argued in O’Neil (2005) that the current non-proliferation regime is very inadequate. Roberts (2000) addresses stability issues associated with a multipolar world. A new approach to the study of the spread of nuclear weapons can be found in Kroenig (2009). The paper focuses on the supply side of nuclear proliferation.

10.2 10.2.1

Issues in Comparative Politics Factors in Successful Democratization

Aside from the empirical validity of our results, there are also some open theoretical questions. Among the most important of them are the degree to which we have properly specified the model. The salience of the question is further underlined by the apparent disagreement among our experts on the causal factors of democratic consolidation. But just how divided are our experts? In what follows, we arrive at an estimate of the disagreement among the expert theoretical perspectives used to build our theoretical model. Let X = {H1 , H2 , H3 , H4 , H5 , H6 , H7 , H8 }. Then X is considered to be our universe. Once again, we do not make any judgments of which factors are more important to Democratic Consolidation in the absence of the experts themselves doing so (See Section 4.1). We define the following fuzzy focal elements on X: A1 (H8 ) = 1, A2 (H1 ) = A2 (H4 ) = A2 (H5 ) = 1/3 , A3 (H4 ) = A3 (H6 ) = A3 (H7 ) = A3 (H8 ) = 1/4 ,

170

10 Methods Based on Fuzzy Set Theory

A4 (H1 ) = A4 (H3 ) = A4 (H4 ) = A4 (H5 ) = A4 (H7 ) = 1/5 , A5 (H2 ) = A5 (H6 ) = 1/2 . There are a number of alternatives to Yen’s method Yen (1992). These approaches combine Dempster-Shafer theory and fuzzy set theory. We obtain similar results, demonstrating the robustness of conclusions reached with Yen’s approach assuming normalized data. Let Y denote a subset of X. We define the characteristic function χY of Y by ∀x ∈ X, ψY (x) = 1 if x ∈ Y and χY (x) = 0 if x ∈ / Y.It is clear that χY is a fuzzy subset of X and that χY represents Y . For N the characteristic function of the universe, we note that Bandler and Kohout’s, Sanchez’s, and Wierman’s methods yield Bel(N ) = 1 since IBK (Ai ⊆ N ) = IS (Ai ⊆ N )

= IW (Ai ⊆ N ) =1

5 for i = 1, 2, 3, 4, 5. Thus Bel(N ) = Σi=1 m(Ai ) = 1. Since N is the universe essentially, this is as it should be in most cases. We next find the belief functions of certain subsets of X = {H1 , H2 , H3 , H4 , H5 , H6 , H7 , H8 } with respect to IBK and IL . By Proposition 9.3 on page 157 and Corollary 9.4 on page 157 we have that

Bel(χY ) =

5  i=1

=

m(Ai ) I(Ai ⊆ χY )

5 1  {1 − Ai (x)} , 5 i=1 x∈Y /

where I stands for either of the two measures of subsethood IBK or IL .

Bel(χ{H8 } ) =

2 3 4 1 1 (1 + + + + ) = .7433, 5 3 4 5 2

Bel(χ{H1 ,H4 ,H5 } ) =

3 4 1 1 (0 + 1 + + + ) = .6100, 5 4 5 2

Bel(χ{H4 ,H6 ,H7 ,H8 } ) =

1 2 4 1 (1 + + 1 + + ) = .7933, 5 3 5 2

10.2 Issues in Comparative Politics

171

Bel(χ{H1 ,H3 ,H4 ,H5 ,H7 } ) =

1 2 3 1 (0 + + + 1 + ) = .5833, 5 3 4 2

2 3 4 1 (0 + + + + 1) = .6433. 5 3 4 5 We next consider Reichenbach’s measure of subsethood. Then  IR (A ⊆ B) = {1 − A(x) + A(x)B(x) | x ∈ X} . Bel(χ{H2 ,H6 } ) =

Hence

IR (Ai ⊆ χY ) = =



x∈Y



x∈Y /

{1) ∧



x∈Y /

{1 − Ai (x)}

{1 − Ai (x)} ,

for i = 1, 2, 3, 4, 5. Thus it follows that BelR (χY ) for Reichenbach’s measure of subsethood is the same as Bel(χY ) for Bandler and Kohout and Lukasiewicz, where Y ∈ H, H = {{H8 }, {H1 , H4 , H5 }, {H4 , H6 , H7 , H8 }, {H1 , H3 , H4 , H5 , H7 }, {H2 , H6 }}. We next consider Wierman’s measure of subsethood. Now A1 ⊆ χ{H8 } and A1 ⊆ χ{H4 ,H6 ,H7 ,H8 } . Also (Ai )α ⊆ Supp(Aj ) if and only if (Ai )α = ∅, when i = j and i, j = 2, 3, 4, 5. Hence it follows that BelW (χY ) for Wierman’s measure of subsethood is the same as Bel(χY ) for Bandler and Kohout and Lukasiewicz.

10.2.1.1

Other Belief Measures

We next consider other belief measures. Let X1 = {H8 }, X2 = {H1 , H4 , H5 }, X3 = {H4 , H6 , H7 , H8 }, X4 = {H1 , H3 , H4 , H5 , H7 }, X5 = {H2 , H6 }. We use the following table.

172

10 Methods Based on Fuzzy Set Theory

A1 (x) A2 (x) A3 (x) 1 x ∈ X1 1 0 4 1 1 1 x ∈ X2 0, 0, 0 0, 41 , 0 3, 3, 3 1 1 1 1 1 x ∈ X3 0, 0, 0, 1 3 , 0, 0, 0 4 , 4 , 4 , 4 x ∈ X4 0, 0, 0, 0, 0 13 , 0, 13 , 13 , 0 0, 0, 14 , 0, 41 x ∈ X5 0, 0 0, 0 0, 14 Let Y ∈ P(X). Then

A4 (x) A5 (x) 0 0 1 1 1 , , 0, 0, 0 5 5 5 1 1 1 , 0, , 0 0, , 0, 0 5 5 2 1 1 1 1 1 1 , , , , 0, , 0, 0, 0 5 5 5 5 5 2 1 1 0, 0 , 2 2

 x∈Y A(x)  IS (A ⊆ χY ) = . x∈X A(x)

Let above, i = 1, 2, 3, 4, 5. Then  Ai be the fuzzy focal element defined  x∈X Ai (x) = 1. Thus IS (A ⊆ χY ) = x∈Y Ai (x) for i = 1, 2, 3, 4, 5. Using the above table, we thus obtain the following beliefs: BelS (χ{H8 } ) =

1 1 (1 + 0 + + 0 + 0) = .25, 5 4

BelS (χ{H1 ,H4 ,H5 } ) = BelS (χ{H4 ,H6 ,H7 ,H8 } ) =

3 1 3 1 (0 + + + + 0) = .37, 5 3 4 5 1 1 4 2 1 (1 + + + + ) = .6467, 5 3 4 5 2

BelS (χ{H1 ,H3 ,H4 ,H5 ,H7 } ) =

3 2 5 1 (0 + + + + 0) = .5, 5 3 4 5

1 1 2 (0 + 0 + + 0 + ) = .25. 5 4 2 Many measures of subsethood are arrived at by the use of implication operators in fuzzy set theory. There are many families of implication operators. They have been created for various reasons. Godel’s and Goguen’s are from the same family and they come from multivalued logic systems. Let f ∈ F. For Godel’s, f (p, q) = q if p > q and for Goguen’s, f (p, q) = q/p if p > q. In both cases, f (p, 0) = 0. Hence for Y ∈ P(X), we have   I(A ⊆ χY ) = {1 | x ∈ Y } ∧ {0 | x ∈ X\Y, A(x) > 0} . BelS (χ{H2 ,H6 } ) =

Thus for Ai the fuzzy focal element defined above, i = 2, 3, 4, 5, I(Ai ⊆ χY ) = 0 for Y ⊆ H\Supp(Ai) and I(A1 ⊆ χY ) = 0 for Y ⊆ H\Supp(A1 )∪Supp(A3 ). Hence it follows that for both Godel’s and Gougen’s measures of subsethood, Bel(χ{H8 } )) =

1 (1 + 0 + 0 + 0 + 0) = .20, 5

10.3 Issues of Hearing Impairment

Bel(χ{H1 ,H4 ,H5 } ) =

173

1 (0 + 1 + 0 + 0 + 0) = .20, 5

Bel(χ{H4 ,H6 ,H7 ,H8 } ) =

1 (1 + 0 + 1 + 0 + 0) = .40, 5

Bel(χ{H1 ,H3 ,H4 ,H5 ,H7 } ) =

1 (0 + 0 + 0 + 1 + 0) = .20, 5

1 (0 + 0 + 0 + 0 + 1) = .20 . 5 We have demonstrated that the methods of Bandler and Kohout, Wierman, Reichenbach, and Lukasiewicz give the same belief for χY as Yen’s method (with m(∅)  = .5433). Some other papers dealing with measures of subsethood and implication operators can be found in Bosc and Pivert (2006) Boudraa et al. (2004) Cornelis et al. (2003b) De Cock and Kerre (2003) Fan et al. (1999) Kosko (2004) Yager (2004b) Young (1996) Ishizuka et al. (1982). We demonstrated a fuzzy approach for specifying a model predicting democratic consolidation based on propositions drawn from the theoretical literature. A test of the model resulted in a relatively good fit between the model’s predictions of the degree to which the twenty-seven countries of postcommunist Europe and the former Soviet Union have achieved consolidated democracy and a fuzzy measure of consolidated democracy based on three indexes. The results indicate that institutional factors are the most salient in democratic consolidation. Among these are regime type, decentralization, and a multi-party system. Behavioral and attitudinal factors are less important, to include corruption and the legitimacy of the state. However, our confidence in these results is reduced by the substantial degree of disagreement among scholars concerning what factors are the most important to the success of democratic consolidation. Bel(χ{H2 ,H6 } ) =

10.3 10.3.1

Issues of Hearing Impairment Hearing Impaired Children

In this section, we apply Yen’s method Yen (1992) to arrive at the degree of belief of certain subsets of tests selected by the experts. Yen’s method involves various measures of subsethood and extends Dempster-Shafer theory by defining a measure of subsethood I(A, B), the degree to which the fuzzy subset A is included in the fuzzy subset B.

174

10 Methods Based on Fuzzy Set Theory

Normalized T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

E1 E2 E3 E4 E5 E6 Row Average .83 .89 .88 .67 1 .87 .85 1 1 1 1 0 1 .83 .83 .67 0 .78 0 0 .38 .83 .78 0 0 .49 0 .35 .67 .33 .88 0 .49 .50 .48 0 0 .88 .67 0 .63 .36 0 0 .62 0 0 .37 .16 0 0 0 .56 0 0 .09 0 0 0 0 .49 0 .08 0 0 0 0 .34 0 .06

The Row Average column sums to 3.64. Dividing the entries of the Row Average column by 3.64 yields the coefficient of the linear equation for P by Yen’s method. We have P = .234T1 + .228T2 + .104T3 + .096T4 + .132T5 + .099T6 + .044T7 + .025T8 + .022T9 + .016T10 . In Yager (2004b), the following formulas are derived for computing the belief function and the plausibility function for a fuzzy subsets B, where the sums are taken over the fuzzy focal elements A, and where Im(A) = {t0 , t1 , ..., tn } : Bel(B) =



m(A)

ti

A

Pl(B) =

 A



m(A)

 ti

[ti − ti−1 ] × inft B(x), x∈A

i

[ti − ti−1 ] × sup B(x). x∈Ati

i , i = 1, ..., 6, we have Thus for the normalized fuzzy focal elements, A Pl(χ{T1 ,T2 ,T3 ,T4 ,T5 } ) =

Pl(χ{T6 ,T7 ,T8 ,T9 ,T10 } ) = + = P l(χ{T1 } ) = P l(χ{T2 } ) = P l(χ{T1 ,T2 } ) = P l(χ{T10 } ) =

1 [1 + 1 + 1 + 1 + 1 + 1] = 1, 6 1 [(.625 − 0)1+(.875 − 625)1+(.556 − 0)1 + (.667 − .556)1 6 (.371 − 0)1 + (.49 − .371)1 + (.367 − 0)1 + (.633 − .367)1] 1 [.875 + .667 + .49 + .633] = .444, 6 1 [.83 + .89 + .875 + .667 + 1 + .867] = .85, 6 1 [1 + 1 + 1 + 1 + 0 + 1] = .83, 6 1, 1 [(.371 − 0)1] = .06. 6

10.3 Issues of Hearing Impairment

10.3.2

175

Plausibility

In this section, we consider the plausibility of certain subsets of X = {T 1, ..., T 10}. The plausibility of a fuzzy subset A of X is given in Equation 10.2. Hence for the measures of subsethood of Bandler/Kohout, Lukasiewicz, Reichenbach, and Wierman, P l(χ{T1 ,T2 ,T3 ,T4 ,T5 } ) = 1 − Bel(χ{T6 ,T7 ,T8 ,T9 ,T10 } ) = 1−0 = 1.

For Sanchez’s measure of subsethood, we have P l(χ{T1 ,T2 ,T3 ,T4 ,T5 } ) = 1 − .215 = .785 .

The plausibility of other fuzzy subsets are easily determined. We can conclude the plausibilities of χ{T1 ,T2 ,T3 ,T4 ,T5 } are in agreement.

Chapter 11

Set–Valued Statistical Methods

11.1

Introduction

We now present a summary of Set–Valued Statistical Methods as introduced by Li and Yen (1995). This method can be used to overcome the difficulty of determining pairwise comparisons for large universes. Let G denote a finite set. In our case, G = {G1 , ..., Gm } is a set of factors making up the overarching goal. Let E denote a set of subjects. In our case, E = {E1 , ..., En } is the set of experts. The problem is to find the degree of membership of G1 , ..., Gm , the set of factors (subgoals) making up the overarching goal S, i.e., the coefficients of the linear equation in our application. 1. First choose an integer q, 1 ≤ q ≤ m, and then an Ej in order to carry out the following steps: a. Select r1 = q elements from G such that they are the first group of  by Ej . This yields a subset elements best fit to A ! (j) (j) (j) (j) G1 = Gi1 , Gi2 , ..., Giq (11.1)

of G. b. Select r2 = 2q elements from G (which includes the q elements already selected in step (1) in such a way that all 2q elements are considered  than other elements of G by Ej . This yields the following better fit to A subset of G : ! (j) (j) (j) (j) (j) (j) G2 = Gi1 , Gi2 , ..., Giq , Giq+1 , ..., Gi2q . (11.2) J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 177–185. DOI: 10.1007/978-3-642-35224-9_11 © Springer-Verlag Berlin Heidelberg 2013

178

11 Set–Valued Statistical Methods (j)

(j)

c. Continuing this process, we construct the s-th subset G s = G1 ∪ ! (j) (j) (j) (j) (j) ... ∪ Gs−1 ∪ Gi(s−1)+1 , ..., Gisq . It follows that Gs ⊃ Gs−1 .

2. This process is continued until we obtain a positive integer t such that m = tq + v, where 1 ≤ v ≤ q. 3. We next calculate m(Ei ), the average frequency of Ei using the following formula t+1  n  1 m(Gi ) = χ (j) (Gi ) , (11.3) n(t + 1) s=1 j=1 Gs (j)

for i = 1, ..., m where χG(j) is the characteristic function of the set Gs in s G, j = 1, ..., n. 4. Finally we normalize the m(Gi ) to produce the membership grade of goal Gi in the overarching goal S. Thus m(Gi ) S(Gi ) = n , k=1 m (Gk ) for i = 1, . . . n.

Example 11.1. Suppose we have five Culinary Judges who seek to evaluate dishes served at a cooking contest. They agree that taste, creativity, presentation, and the use of the contest ingredient are important but disagree over the relative merits of the four criteria. Then E = {E1 , ..., E5 } represents the five Culinary Judges and G = {G1 , ..., G4 } represents the four goals. While it is possible to have the experts add pairs (q = 2) or triples (q = 3) of goals in each iteration, it is most common to set q = 1 so that the experts add one factor at a time. Expert One must now choose the single sub–goal (since q is one) that most influences (or fits) the overall goal. She chooses G1 = taste so that (1) (1) G1 = {G1 }. She next chooses G4 = ingredient so that G2 = {G1 , G4 }. (1) (1) Next comes G3 = presentation so that G3 = {G1 , G4, , G3 }. Finally G4 = {G1 , G4, , G3 , G2 } since only G3 = creativity is left. We note that the vector [G1 , G4, , G3 , G2 ] completely summarizes this information. It shows the Expert (j) Ones order of inclusion of goals into the various Gs . The following table summarizes the order of inclusion for each of the five experts. Expert One Two Three Four Five 4 s=1 χG(j) (Gi ) s

First Second Third Fourth G1 G4 G2 G3 G1 G2 G3 G4 G2 G1 G4 G3 G4 G1 G2 G3 G1 G2 G4 G3 4 3 2 1

11.1 Introduction

179

Equation 11.3 can now be used to calculate membership grades. The summations are over n = 5 experts and t + 1 = 4 inclusions. For G1 we get m(G1 ) =

=

t+1

n

n

t+1

 1 χ (j) (G1 ) n(t + 1) s=1 j=1 Gs

 1 χ (j) (G1 ) n(t + 1) j=1 s=1 Gs

1 (4 + 4 + 3 + 3 + 4) 4∗5 18 = . 20 =

6 = 14 , m(G3 ) = 1+2+1+1+1 = 20 , and Similarly m(G2 ) = 2+3+4+2+3 20 20 20 3+1+2+4+2 12 m(G4 ) = = . 20 20 i) Finally, norming the m(Gi ) produces S with Si = S(Gi ) = m(G 40/20 . Thus S1 = 18/40, S2 = 14/40, S3 = 6/40, and S4 = 12/40. Finally, we note that the following table presents the same information as the previous table, but in a manner that makes the calculations easier.

4

s=1

χG(j) (Gi ) One Two Three Four Five Sum s

G1 G2 G3 G4

4 2 1 3

4 3 2 1

3 4 1 2

3 2 1 4

4 3 1 2

18 14 6 12

Example 11.2. If we repeat the previous example with q = 2, and assume that the Judges add factors in the same order, we get the following table. 4

s=1

χG(j) (Gi ) One Two Three Four Five Sum s

G1 G2 G3 G4

2 1 1 2

2 2 1 1

2 2 1 1

2 1 1 2

and the resulting fuzzy membership grades are S1 = 5/30, and S = 7/30. 4

2 2 1 1

10 8 5 7

10/30,

S2 = 8/30, S3 =

180

11.2 11.2.1

11 Set–Valued Statistical Methods

Issues in International Relations Nuclear Deterrence

We start with the Nuclear Deterrence data of Table 3.1. Since Expert One grades factor G1 a 5, which is the highest grade, we assume that he would (1) (1) add G1 to G1 . Next Expert One adds G5 to produce G2 = {G1 , G5 }. The factors G2 , G3 and G4 have a tie grade and go in next in an arbitrary order. (1) Finally, Expert One adds his lowest ranked factor G6 to produce G6 . The other exerts are processed similarly. The following table summarizes the scores produced by this procedure for each factor and expert. Nuclear Deterrence Scores E1 E2 E3 E4 E5 Sum Si G1 6 6.0 5 5.5 6 28.5 0.271 G2 3 3.0 1 3.5 2 12.5 0.119 G3 3 4.5 3 3.5 3 17.0 0.162 G4 3 4.5 6 5.5 5 24.0 0.229 G5 5 1.0 4 2.0 4 16.0 0.152 G6 1 2.0 2 1.0 1 7.0 0.067 Sum 105.0 This produces the linear equation: S = 0.271 G1 + 0.119 G2 + 0.162 G3 + 0.229 G4 + 0.152 G5 + 0.067 G6 .

11.2.2

Smart Power and Deterrence

We apply the set valued statistical method to the data in Table 3.2 to produce the following table.

G1 G2 G3 G4 G5 G6

Smart Power Scores E1 E2 E3 E4 E5 Sum Si 5.5 5 6 6 6.0 28.5 0.271 5.5 3 4 2 3.5 18.0 0.171 3.5 6 2 5 5.0 21.5 0.205 3.5 4 4 1 1.0 13.5 0.129 2.0 2 4 4 3.5 15.5 0.148 1.0 1 1 3 2.0 8.0 0.067 105.0

This produces the linear equation: S = 0.271 G1 + 0.171 G2 + 0.205 G3 + 0.129 G4 + 0.148 G5 + 0.076 G6 .

11.3 Issues in Comparative Politics

11.2.3

181

Cooperative Threat Reduction

In this section, we use the set-valued statistical method of to determine the coefficients of the linear equation expressing G in terms of O1, O2, O3, and O4, (see Section 11.1 and Table 3.3). We assume that the experts would add the higher rated attributes first. If two (or more) attributes have the same rating then we average their ranks. The averages of the row sums give the coefficients Si , for i = 1, ..., m, of the linear equation by the set-valued statistical method. Cooperative Threat Reduction E1 E2 E3 E4 E5 E6 Sum O1 2.5 2.5 1.5 1 4 3.0 14.5 O2 2.5 2.5 1.5 3 3 4.0 16.5 O3 2.5 2.5 3.5 3 2 1.5 15.0 O4 2.5 2.5 3.5 3 1 1.5 14.0 60.0

Scores Si 0.242 0.275 0.25 0.233

The linear equation is thus: G = 0.242 O1 + 0.275 O2 + 0.25 O3 + 0.233 O4 .

11.3 11.3.1

Issues in Comparative Politics Factors in Successful Democratization

We next consider the set valued statistical method applied to the matrix given in 7.24 on page 134. H1 H2 H3 H4 H5 H6 H7 H8

E1 4 4 4 4 4 4 4 8

E2 7 3 3 7 7 3 3 3

E3 E4 2.5 6 2.5 2 2.5 6 6.5 6 2.5 6 6.5 2 6.5 6 6.5 2

E5 Row Total 3.5 23 7.5 19 3.5 19 3.5 27 3.5 23 7.5 23 3.5 23 3.5 23

The row total column sums to 180. Normalizing this column yields the coefficients for the linear equation. We have

182

11 Set–Valued Statistical Methods

D = .13H1 + .11H2 + .11H3 + .15H4

(11.4)

+.13H5 + .13H6 + .13H7 + .13H8 . The ranking of countries using this method can be easily obtained.

11.3.2

Quality of Life

11.3.2.1

Expert Group One

Again we are starting from the matrix given in Subsection 6.5.1.2.

F1 F2 F3 F4 F5 F6 F7 F8 F9

E1 2 8 3 5 7 9 1 6 4

Expert Group One E2 E3 E4 E5 E6 E7 E8 Sum Si 3 8 7 2 5 2 5 34 0.094 7 7 2 6 2 8 6 46 0.128 5 6 8 7 6 4 7 46 0.128 6 4 6 5 3 6 3 38 0.106 8 5 9 8 8 7 8 60 0.167 9 9 5 9 9 9 9 68 0.189 1 2 3 1 4 1 2 15 0.042 4 3 4 3 7 5 4 36 0.100 2 1 1 4 1 3 1 17 0.047

Each element in the Row Total column is divided by 72 to obtain the m(F i), i = 1, ..., 9. The m(F i) are then normalized to obtain the coefficients for Quality of Life. m(F1 ) m(F2 ) m(F3 ) m(F4 ) m(F5 ) m(F6 ) m(F7 ) m(F8 ) m(F9 ) Row Total 0.472 .639 .639 .528 .833 .944 .208 .500 .236 5 We have Q = .094F1 + .128F2 + .128F3 + .106F4 + .167F5 +.189F6 + .042F7 + .100F8 + .047F9 . The ranking of countries using this method can be found in Chapter 13.

11.3.2.2

Expert Group Two

Again we are starting from the matrix given in Subsection 6.5.1.2.

11.3 Issues in Comparative Politics

F1 F2 F3 F4 F5 F6 F7 F8 F9

E1 5 2 6 3 8 9 4 7 1

E2 7.5 5.0 2.0 2.0 7.5 9.0 5.0 2.0 5.0

183

Expert Group Two E3 E4 E5 E6 E7 E8 Sum Si 2 7.0 5.5 7 8 6.5 48.5 0.135 6 4.0 7.0 2 7 4.0 37.0 0.103 7 4.0 5.5 8 6 6.5 45.0 0.125 5 1.5 4.0 6 4 3.0 28.5 0.079 8 7.0 8.0 9 5 9.0 61.5 0.171 9 9.0 9.0 5 9 8.0 67.0 0.186 1 1.5 2.0 3 2 5.0 23.5 0.065 3 7.0 3.0 4 3 2.0 31.0 0.086 4 4.0 1.0 1 1 1.0 18.0 0.050

Each element in the Row Total column is divided by 72 to obtain the m(F i), i = 1, ..., 9. The m(F i) are then normalized to obtain the coefficients for Quality of Life. m(F 1) m(F 2) m(F 3) m(F 4) m(F 5) m(F 6) m(F 7) m(F 8) m(F 9) Row Total .674 .514 .625 .396 .854 .931 .326 .431 .250 5

We have Q = .135F 1 + .103F 2 + .125F 3 + ..080F 4 + .171F 5 +.186F 6 + .065F 7 + .086F 8 + .050F 9 . The ranking of countries using this method can be found in Chapter 13.

11.3.3

Economic Freedom

Here we start from the Table 4.5. We use this data to score the factors influencing Economic Freedom to produce the following Table:

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 Sum

Economic Freedom Scores E1 E2 E3 E4 E5 E6 E7 E8 E9 9.0 5 8.5 3.5 8 8.5 8.0 5 3.5 4.5 8 8.5 6.0 3 3.5 6.0 9 3.5 4.5 8 1.5 1.5 3 7.0 9.0 5 3.5 2.0 1 1.5 1.5 3 1.0 6.0 5 3.5 7.5 5 4.5 8.5 3 8.5 2.5 5 7.5 1.0 3 4.5 6.0 3 2.0 2.5 5 7.5 4.5 5 4.5 8.5 8 3.5 2.5 2 9.5 7.5 10 8.5 10.0 8 10.0 10.0 9 9.5 10.0 8 8.5 6.0 8 5.5 6.0 9 3.5 4.5 2 4.5 3.5 8 5.5 2.5 1 3.5

Sum Si 59.0 0.119 52.0 0.105 43.0 0.087 24.5 0.049 52.0 0.105 34.5 0.070 48.0 0.097 82.5 0.167 64.5 0.130 35.0 0.071 495.0

184

11 Set–Valued Statistical Methods

The linear equation is thus: S = 0.119F1 + 0.105F2 + 0.087F3 + 0.049F4 + 0.105F5 +0.07F6 + 0.097F7 + 0.167F8 + 0.13F9 + 0.071F10 .

11.3.4

Failed States

Here we start from the matrix 4.8. We use this data to score the factors influencing Failed States to produce the following Table: Failed States Scores E1 E2 E3 E4 Sum Si F1 1.0 4.5 2.5 7.0 15.0 0.048 F2 2.5 7.0 2.5 1.5 13.5 0.043 F3 10.0 4.5 2.5 11.0 28.0 0.090 F4 2.5 9.5 2.5 1.5 16.0 0.051 F5 5.0 2.0 5.0 9.0 21.0 0.067 F6 8.0 9.5 8.0 9.0 34.5 0.111 F7 11.5 12.0 11.5 12.0 47.0 0.151 F8 8.0 2.0 8.0 9.0 27.0 0.087 F9 6.0 6.0 11.5 5.5 29.0 0.093 F10 8.0 8.0 8.0 3.5 27.5 0.088 F11 11.5 2.0 8.0 3.5 25.0 0.080 F12 4.0 11.0 8.0 5.5 28.5 0.091 Sum 312.0 The corresponding linear equation is thus: S = 0.048F1 + 0.043F2 + 0.09F3 + 0.051F4 +0.067F5 + 0.111F6 + 0.151F7 + 0.087F8 +0.093F9 + 0.088F10 + 0.08F11 + 0.091F12 .

11.4 11.4.1

Issues of Hearing Impairment Hearing Impaired Children

We are trying to determine the best method for testing Hearing Impaired Children with the goal of prediction a successful transition to a mainstream school. The data from Table 5.2 allows to score the various Tests and produces the following Table.

11.4 Issues of Hearing Impairment

185

Hearing Impaired E1 E2 E3 E4 E5 E6 Sum Si T1 8 9 8 7.5 10 9 51.5 0.156 T2 10 10 10 10.0 3 10 53.0 0.161 T3 8 7 3 9.0 3 3 33.0 0.100 T4 8 8 3 3.0 8 3 33.0 0.100 T5 6 6 8 3.0 8 7 38.0 0.115 3 3 8 7.5 3 8 32.5 0.098 T6 T7 3 3 6 3.0 3 6 24.0 0.073 T8 3 3 3 6.0 3 3 21.0 0.064 T9 3 3 3 3.0 8 3 23.0 0.070 T10 3 3 3 3.0 6 3 21.0 0.064 Sum 330.0 The corresponding linear equation is thus: S = 0.156T1 + 0.161T2 + 0.1T3 + 0.1T4 + 0.115T5 +0.098T6 + 0.073T7 + 0.064T8 + 0.07T9 + 0.064T10 .

11.4.2

Deaf and Hard of Hearing Children

Now we are interested in metrics for measuring how well deaf and hard of hearing students of the preschool are closing the gap with respect to language growth. The data from Table 5.3 is used to produce the following Table using the set–valued statistical method. Deaf and Hard Of Hearing E1 E2 E3 E4 E5 E6 E7 E8 E9 Sum Si T1 9.5 10 9.0 5.0 6.5 8.0 3.0 8.5 6.0 65.5 0.132 T2 9.5 9 8.0 8.0 6.5 9.0 7.0 10.0 10.0 77.0 0.156 T3 8.0 8 2.5 6.5 4.0 5.0 4.0 8.5 9.0 55.5 0.112 T4 6.5 7 2.5 9.0 1.5 10.0 1.5 3.0 3.0 44.0 0.089 T5 6.5 3 7.0 6.5 4.0 6.5 6.0 3.0 7.5 50.0 0.101 T6 3.0 6 2.5 10.0 1.5 6.5 5.0 3.0 3.0 40.5 0.082 T7 3.0 3 10.0 2.0 8.0 2.5 1.5 7.0 7.5 44.5 0.090 T8 3.0 3 5.0 2.0 9.0 2.5 8.0 6.0 3.0 41.5 0.084 T9 3.0 3 2.5 4.0 4.0 2.5 9.0 3.0 3.0 34.0 0.069 T10 3.0 3 6.0 2.0 10.0 2.5 10.0 3.0 3.0 42.5 0.086 Sum 495.0 The corresponding linear equation is then: S = 0.132T1 + 0.156T2 + 0.112T3 + 0.089T4 + 0.101T5 +0.082T6 + 0.09T7 + 0.084T8 + 0.069T9 + 0.086T10 .

Chapter 12

Intuitionistic Fuzzy Sets and Political Stability

12.1

Intuitionistic Fuzzy Sets

A human being is, in general, not a computer. For a person the amount of whiteness of the sky and the amount of not whiteness of the sky are not always complementary. To model this inconsistency, Krassimir Atanassov introduced the concept of an intuitionistic fuzzy set (IFS): Atanassov (1986, 1999); Atanassov et al. (2005); Cornelis et al. (2003a). An IFS is characterized by two functions that expressing, respectively, the degree of belongingness and non-belongingness of an object to an attribute, for example the object could be the sky and the attribute could be whiteness. Furthermore, Atanassov does not rule out that an observer may object to committing themselves completely to a whiteness – not whiteness view of the sky. An observer may decide that an attribute is a poor classifier of an object and refuse to commit to either belongingness or not belongingness. Thus if an object is abstract, like love, an observer may question the amount of blueness we could attribute to love. Similarly, if the object is concrete, say that we are examining a rock, it may be hard to determine its honesty, an abstract quality.

12.2

Atanassov’s Intuitionistic Fuzzy Sets

An Atanassov IFS is comprised of a pair of membership functions. The uncertainty about an object x’s attribute value is divided into membership, or μ, non–membership, or ν, and indeterminacy, which is 1 − (μ + ν). Since all these values should be in the unit interval μ + ν must be less than or equal to one, otherwise μ and ν are more or less independent. Definition 12.1. Let X be a universal set. An Atanassov IFS A may be expressed as the set of triples: A = { x, μA (x), νA (x) | x ∈ X} , J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 187–200. DOI: 10.1007/978-3-642-35224-9_12 © Springer-Verlag Berlin Heidelberg 2013

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12 Intuitionistic Fuzzy Sets and Political Stability

where μA : X → [0, 1] and νA : X → [0, 1] are the degree of membership and the degree of non-membership of an element x of X in the IFS set A, respectively, with the requirement that they satisfy the requirement that 0 ≤ μA (x) + νA (x) ≤ 1 for all x ∈ X. Definition 12.2. For μA : X → [0, 1] and νA : X → [0, 1] we define μ + ν as shorthand for μA (x)+νA (x) with x ∈ X. Similarly, indeterminacy, 1−(μ+ν) is shorthand for 1 − (μA (x) + νA (x)) with x ∈ X. Definition 12.3. Let A be an Atanassov IFS. Define πA : X → [0, 1] by ∀x ∈ X, πA (x) = 1−μA (x)−νA (x). The value πA (x) is called the intuitionistic index of an element x in the set A. From Definitions 12.1 and 12.3, it is clear that 0 ≤ πA (x) ≤ 1. Definition 12.4. Let A and B be two IFSs on the set X and let c be a positive real number. Atanassov defined the following operations: Atanassov (1986, 1999); Li (2010a); De et al. (2000); Li and Wu (2010) 1. 2. 3. 4. 5. 6.

A ⊆ B if and only if for all x ∈ X, μA (x) ≤ μB (x) and νA (x) ≥ νB (x) A = B if and only if for all x ∈ X, μA (x) = μB (x) and νA (x) = νB (x) A + B = { x, μA (x) + μB (x) − μA (x)μB (x), νA (x)νB (x) | x ∈ X} AB = { x, μA (x)μB (x), νA (x) + νB (x) − νA (x)νB (x) | x ∈ X} cA = { x, 1 − (1 − μA (x))c , (νA (x))c | x ∈ X} Ac = {< x, (μA (x))c , 1 − (1 − νA (x))c >| x ∈ X} .

It is apparent that the above equations can be unsuitable from the viewpoints of sets and algebraic operations. Furthermore, Atanassov’s IFSs are missing a definition of subtraction nor is there a division operation. This is due to the fact that the above algebraic operations were not defined on the basis of the extension principle of Atanassov’s IFSs, which should produce an extension of the algebraic operations of fuzzy sets or real numbers. In Li and Wu (2010) the extension principle is used to define an algebraic operation on IFSs when there is binary operation on X. Definition 12.5. Let (X, ∗) be a mathematical system. Define the binary operation ∗ on the set of all Atanassov IFSs, IF S(X), as follows: ∀A, B ∈ IF S(X) let " μA∗B (z) = {μA (x) ∧ μB (y) | z = x ∗ y | x, y ∈ X} , and

νA∗B (z) = then



{νA (x) ∨ νB (y) | z = x ∗ y | x, y ∈ X} ,

A ∗ B = z, μA∗B (z), νA∗B (z) .

12.3 Weighted Average Method for MADM Problems

12.2.1

189

Ranking Methods of IFSs

A few methods exist for ranking IFSs. A method based on both the score function and the accuracy function is presented here because of the conciseness and intuitive nature of these functions. Definition 12.6. Let A = μA , νA = { x, μA , νA } be an IFS. Then the score of the function ∆ of A is defined as follows: ∆(A) = μA − νA . The score represents the net degree of membership of the element x belonging to the set A. Definition 12.7. Let A = μA , νA = { x, μA , νA } be an IFS. Then the accuracy of the function H of A is defined as follows: H(A) = μA + νA . Clearly, H(A) = μA + νA = 1 − πA . Hence, the accuracy function H(A) represents the non-hesitation degree. In order to rank two IFSs A = μA , νA and B = μB , νB , their score and accuracy functions must be respectively compared. The score and accuracy function can be explained in terms of the mean and variance (respectively) in statistics. Hence, the score function is generally more important than the accuracy function. Thus, according to lexicographic method, an order relation is defined as follows. Definition 12.8. Let A = μA , νA and B = μB , νB be two IFSs. Then, 1. If ∆(A) > ∆(B) then A is bigger than B. 2. If ∆(A) = ∆(B) then a. If H(A) = H(B) then A is equal to B; b. If H(A) < H(B) then A is smaller than B; c. If H(A) > H(B) then A is larger than B.

12.3

Weighted Average Method for MADM Problems

The later chapters of this book will look at applications of IFSs to MultiAttribute Decision Making (MADM). The weighted average method is the simplest and most commonly used method in MADM problems. A decision matrix which which represents a MADM problem is given in the following definition.

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12 Intuitionistic Fuzzy Sets and Political Stability

Definition 12.9. Suppose that a MADM problem has m alternatives Xi (i = 1, 2, ..., m) evaluated with respect to n attributes Aj (j = 1, 2, ..., n), both qualitatively and quantitatively. Denote X = {X1 , X2 , ..., Xm } and A = {A1 , A2 , ..., An }. Let μij be the value of each alternative Xi (i = 1, 2, ..., m) on any attribute Aj (j = 1, 2, ..., n). All these values may be expressed concisely in the decision matrix format: F = (μij )m×n . Each attribute may have different importance. Theorem 12.10. Assume the relative weight of each attribute Aj be ωj (j = 1, 2, ..., n) where ωj satisfies the normalization conditions, i.e., ωj ∈ [0, 1] n ωj = 1. Let W = (ω1 , ω2 , ..., ωn )τ be the relative weight vector of all and Σj=1 attributes. Using the weighted average method (Chen & Hwang), the weighted comprehensive value of each alternative Xi (i = 1, 2, ..., m) is calculated as follows: n Vi = Σj=1 ωj μij .

In the maximization situation, the best alternative is the one with the largest weighted comprehensive value.

12.3.0.1

GOWA Operator

Let f : Rn → R be a mapping such that f (a1 , a2 , · · · , an ) = T

n 

ωj b j ,

j=1

where ω = (ω, ω2 , · · · , ωn ) is a weight vector associated with f , satisfying n  ωj = 1 ; bj is the j−th largest of all ωj ∈ [0, 1] (j = 1, 2, · · · , n ) and j=1

numerical values (a1 , a2 , · · · , an ). The function is called an OWA operator of n–dimension (Yager, 1988) . Definition 12.11. (Yager, 2004a) Let M : Rn → R be a mapping such n  T that M (a1 , a2 , · · · , an ) = ( ωj bλj )1/λ , where ω = (ω1 , ω2 , · · · , ωn ) is a j=1

weight vector associated with M , satisfying ωj ∈ [0, 1] (j = 1, 2, · · · , n ) and n  ωj = 1 ; bj is the j−th largest of all numerical values ak (k = 1, 2, · · · , n); j=1

λ ∈ (−∞, +∞) is a parameter. The function is called a GOWA operator of n-dimension. We list some properties for the GOWA operator M . 1. For all weights ωj = 0 (j = 1, 2, · · · , n), M (a1 , a2 , · · · , an ) = min{aj |j = 1, 2, · · · , n} = bn if λ → −∞ . That is, the GOWA operator M reduces to the min operator.

12.4 The Extended GOWA Operator Based MADM Method

2. M (a1 , a2 , · · · , an ) =

n 

j=1

191

ω

bj j if λ → 0. That is, the GOWA operator M

reduces to the ordered weighted geometric (OWG) operator. n  3. M (a1 , a2 , · · · , an ) = ωj bj if λ = 1. That is, the GOWA operator rej=1

duces to the ordered weighted average (OWA) operator. 4. For all weights ωj = 0 (j = 1, 2, · · · , n), M (a1 , a2 , · · · , an ) = max{aj | j = 1, 2, · · · , n} = b1 if λ → −∞. That is, the GOWA operator M reduces to the max operator. As stated above, when the parameter takes different values in the interval , the GOWA operator has different specific forms.

12.4

12.4.1

The Extended GOWA Operator Based MADM Method Representation of MADM Problems Using IFSs

Suppose that there exists an alternative set A = {A1 , A2 , · · · , Am } which consists of m non-inferior alternatives from which the best alternative is to be selected. Each alternative is assessed on n attributes, both quantitatively and qualitatively. Denote the set of all attributes by X = {x1 , x2 , · · · , xn } . Assume that aij are ratings of alternatives Ai ∈ A on attributes xj ∈ X, respectively. Let αj ∈ [0, 1] , βj ∈ [0, 1], δj ∈ [0, 1], and γj ∈ [0, 1] satisfy the following conditions: 0  αj + βj  1 and 0  δj + γj  1 . These parameters are chosen by the decision maker a a priori according to needs in real-life situations. If ratings aij are crisp values, they need to be normalized since the physical dimensions and measurements of different quantitative attributes are different. The formulae for relative degrees of membership and relative degrees of non-membership are chosen as follows:

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12 Intuitionistic Fuzzy Sets and Political Stability

μij = and

⎧ ⎨ αj

aij amax j min ⎩δj aj aij

if j ∈ F 1 if j ∈ F 2

$ ⎧ # ij ⎨βj 1 − amax if j ∈ F 1 aj # $ υij = min ⎩γj 1 − aj if j ∈ F 2 aij

respectively, where F 1 and F 2 are subscript sets of benefit quantitative attributes and cost quantitative attributes, respectively, with F 1 ∪ F 2 = {1, 2, . . . , n}, F 1 ∩ F 2 = ∅ and = max {aij } amax j 1im

amin = min {aij } j 1im

It follows that 0

aij 1 amax j

0

amin j 1. aij

and

Hence, 01−

aij 1 amax j

01−

amin j 1. aij

and

Clearly 0  μij  αj 0  νij  βj for (j ∈ F 1 ) and 0  μij  δj 0  νij  γj for (j ∈ F 2 ). If ratings, rij , are linguistic variables, their semantics can be expressed with IFSs designated a priori. The corresponding relations between linguistic

12.4 The Extended GOWA Operator Based MADM Method

193

variables and IFSs are given by the decision maker a priori according to characteristics and needs in real-life situations. Thus, ratings rij of alternatives Ai ∈ A on qualitative and quantitative attributes xj ∈ X may be transformed into IFSs. Let the vector of IFSs of all n attributes for an alternative Ai ∈ A be (ri1 , ri2 , · · · , rin ) = (< μi1 , υi1 >, < μi2 , υi2 >, · · · , < μin , υin >) . Thus an MADM problem under the IFS environment can be expressed concisely in the matrix format as follows: (12.1)

F = (< μij , υij >)m×n or, specifically, F A1 A2 .. . Am

⎛ ⎜ ⎜ ⎜ ⎝

x1 μ11 , υ11

μ21 , υ21

.. .

x2 μ12 , υ12

μ22 , υ22

.. .

... ... ... .. .

xn μ1n , υ1n

μ2n , υ2n

.. .

μm1 , υm1 μm2 , υm2 . . . μmn , υmn

⎞ ⎟ ⎟ ⎟ ⎠

which is referred to as an IFS decision matrix used to represent the MADM problem under the IFS environment.

12.4.2

The Extended GOWA Operator Using IFSs

Let M be a mapping such that ⎛ ⎞ λ1 n  M (c1 , c2 , · · · , cn ) = ⎝ ωj dλj ⎠

(12.2)

j=1

where

T

1. ω = (ω1 , ω2 , · · · , ωn )

is a weight vector associated with M , satisfying n  ωj ∈ [0, 1] (j = 1, 2, · · · , n) and ωj = 1 ; j=1

2. dj = μj , υj is the j–th largest of all IFSs ck (k = 1, 2, · · · , n ); 3. λ ∈ (0, +∞) is a parameter, which may be chosen by the decision maker a priori.

The mapping of Eq. (12.2) is called an extended GOWA operator using IFSs. Sometimes Eq. (12.2) is called an the extended GOWA operator for short.

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12 Intuitionistic Fuzzy Sets and Political Stability

According to Definition (12.4) and Eq. (12.2), it follows that ⎛ ⎞ λ1 n  M (c1 , c2 , · · · , cn ) = ⎝ ωj dλj ⎠ j=1

⎞1 ⎛ n + , λ  =⎝ ωj μλj , 1 − (1 − νj )λ ⎠ j=1

⎛ ⎞ λ1 n + / 0 ,  ω  .ωj λ j ⎠ =⎝ 1 − 1 − μλj , 1 − (1 − νj ) (12.3) j=1

=

=

⎧1 ⎨ ⎩

1−

1⎡

⎣1 − ⎛

n  -

j=1

n  -

j=1

2⎫ λ1 n / ⎬ 0  ω . j ωj λ 1 − μλj , 1 − (1 − νj ) ⎭

1 − μλj

j=1

.ωj

⎤ λ1

⎦ ,

⎞1 2 n / 0ωj λ  λ ⎠ 1 − ⎝1 − 1 − (1 − νj ) j=1

The following conclusions are easily derived from Eq. ((12.3)). 1. M (c1 , c2 , · · · , cn ) =

n 

j=1

ω

dj j if λ → 0. That is, the extended GOWA oper-

ator M reduces to theOWG operator using IFSs. n 2. M (c1 , c2 , · · · , cn ) = j=1 ωj dj if λ = 1. That is, the extended GOWA operator M reduces to the OWA operator using IFSs. 3. For all weights ωj = 0 (j = 1, 2, · · · , n ), M (c1 , c2 , · · · , cn ) = max{cj | j = 1, 2, · · · , n} = b1 if λ → ∞. The extended GOWA operator M reduces to the Max operator using IFSs. We see that, when the parameter λ takes different values in the interval (0, +∞), the extended GOWA operator may have different specific forms.

12.4.3

The GOWA Operator Based MADM Method and Procedure

Suppose that an MADM problem under the IFS environment is expressed as the IFS decision matrix in Eq. ((12.1)). Then an algorithm and decision process of the extended GOWA operator based methodology is summarized as follows:

12.5 Combining Expert Opinion

195

Step 1: Obtain the IFS decision matrix F. Applying the method represented in Section (12.4.1), the IFS decision matrix F can be obtained. Step 2: Compute the overall assessments of alternatives. Using Eq. (12.3) and the rows of the IFS decision matrix F in Eq. (12.1), the overall assessments of alternatives ri = μi , νi are computed as ri = M (ri1 , ri2 , · · · , rin ) , respectively. Step 3: Rank ordering of all alternatives. The ranking order of the alternative set X is generated according to the scores ∆(ri ) and the accuracies H(ri ) (i = 1, 2, · · · m) using the method represented in Section (12.2.1).

12.5

Combining Expert Opinion

In this section, we use expert opinion to define a binary operation on F , the set of components of political stability, (see Section 4.4 and Table 4.6 on page 81). This binary operation gives a method to combine the experts’ opinions. The approach is in keeping with that in Mordeson et al. (2012, 2011d, 2010a, 2011c,a) and is similar to that in George et al. (2007). Table 12.1 Political Stability Expert Opinions (a) Values for µ.

(b) Values for ν.

µ E1 E2 E3 Row Avg

ν E1 E2 E3 Row Avg

F1 F2 F3 F4

.9 .9 .9 .7

.7 .8 .9 .6

.7 .5 .7 .4

.77 .73 .83 .57

Let F = {F1 , F2 , F3 , F4 }. Define σ : F → [0, 1] by σ(F1 ) = .77, σ(F2 ) = .73, σ(F3 ) = .83, σ(F4 ) = .57 . Define τ : F → [0, 1] by τ (F1 ) = .13, τ (F2 ) = .17, τ (F3 ) = .10, τ (F4 ) = .20 .

F1 F2 F3 F4

.1 0 .1 .3

0 0 0 .2

.3 .5 .2 .1

.13 .17 .10 .20

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12 Intuitionistic Fuzzy Sets and Political Stability

Definition 12.12. Define the binary operations ∗σ and ∗τ as follows: ∀Fi , Fj ∈ F,  Fi if σ(Fi ) ≥ σ(Fj ), Fi ∗σ Fj = Fj otherwise.  Fi if τ (Fi ) ≤ τ (Fj ), Fi ∗τ Fj = Fj otherwise. The next result is immediate from the definitions. Theorem 12.13. ∗σ = ∗τ if and only if ∀Fi , Fj , σ(Fi ) ≥ σ(Fj ) ⇔ τ (Fi ) ≤ τ (Fj ). Thus in our application, we have ∗σ = ∗τ and we obtain the following operation table for ∗ = ∗σ = ∗τ : Table 12.2 Semigroup for IFS selection ∗ F1 F2 F3 F4

F1 F1 F1 F3 F1

F2 F1 F2 F3 F2

F3 F3 F3 F3 F3

F4 F1 F2 F3 F4

We note that the (F, ∗) is a commutative semigroup with identity F4 and that every element is idempotent. We also note that this method gives a measure of combined dominance. Before continuing we also note that the operation ∗ is associative. Proposition 12.14. Associative (Fi ∗ Fj ) ∗ Fk = Fi ∗ (Fj ∗ Fk ). Proof. Suppose σ(Fi ) ≥ σ(Fj ) ∧ σ(Fk ). Then both sides equal Fi . Suppose σ(Fi ) < σ(Fj ) ≥ σ(Fk ). Then both sides equal Fj . Suppose σ(Fi ) ≤ σ(Fj ) < σ(Fk ). Then both sides equal Fk . Finally, suppose σ(Fi ) < σ(Fj ) = σ(Fk ). Then both sides equal Fj . ⊔ ⊓ Definition 12.15. Li (2010b) Let (X, ∗) be a semigroup. Let μ1 , ν1 and μ2 , ν2 be intuitionistic fuzzy sets on X. Define the intuitionistic fuzzy set μ1 ∗ μ2 , ν1 ∗ ν2 on X as follows: ∀z ∈ X, z, μ1 ∗ μ2 , ν1 ∗ ν2 = z, μ1 ∗ μ2 (z), ν1 ∗ ν2 (z)

where μ1 ∗ μ2 (z) = ∨ {μ1 (x) ∧ μ2 (y) | z = x ∗ y, x, y ∈ X} , and ν1 ∗ ν2 (z) = ∧ {ν1 (x) ∨ ν2 (y) | z = x ∗ y, x, y ∈ X} .

12.5 Combining Expert Opinion

197

Let μi , νi denote the intuitionistic sets associated with the i-th expert, i = 1, 2, 3. We denote μ1 ∗ μ2 by μ12 , ν1 ∗ ν2 by ν12 , , (μ1 ∗ μ2 ) ∗ μ3 by μ123 and (ν1 ∗ ν2 ) ∗ ν3 by ν123 . For the above semigroup, we have (using Definition 12.15) μ12 (F1 ) = ∨{μ1 (F1 ) ∧ μ2 (F1 ), μ1 (F1 ) ∧ μ2 (F2 ), μ1 (F1 ) ∧ μ2 (F3 ), μ1 (F2 ) ∧ μ2 (F1 ), μ1 (F3 ) ∧ μ2 (F1 )} = ∨{.9 ∧ .7, .9 ∧ .8, .9 ∧ .6, .9 ∧ .7, .7 ∧ .7} = .8,

μ12 (F2 ) = ∨{μ1 (F2 ) ∧ μ2 (F2 ), μ1 (F2 ) ∧ μ2 (F3 ), μ1 (F3 ) ∧ μ2 (F2 )} = ∨{.9 ∧ .8, .9 ∧ .6, .7 ∧ .8} = .8,

μ12 (F3 ) = ∨{μ1 (F1 ) ∧ μ2 (F3 ), μ1 (F2 ) ∧ μ2 (F3 ), μ1 (F3 ) ∧ μ2 (F3 ), μ1 (F3 ) ∧ μ2 (F3 ), μ1 (F3 ) ∧ μ2 (F1 ), μ1 (F3 ) ∧ μ2 (F2 ),

μ1 (F3 ) ∧ μ2 (F3 )} = ∨{.9 ∧ .9, .9 ∧ .9.9 ∧ .9, .7 ∧ .9, .9 ∧ .7, .9 ∧ .8, .9 ∧ .6} = .9,

μ12 (F4 ) = μ1 (F4 ) ∧ μ2 (F4 ) = .7 ∧ .6 = .6,

μ123 (F1 ) = ∨{μ12 (F1 ) ∧ μ3 (F1 ), μ12 (F1 ) ∧ μ3 (F2 ), μ12 (F1 ) ∧ μ3 (F4 ), μ12 (F2 ) ∧ μ3 (F1 ), μ12 (F4 ) ∧ μ3 (F1 )} = ∨{.8 ∧ .7, .8 ∧ .5, .8 ∧ .4, .8 ∧ .7, .6 ∧ .7} = .7,

μ123 (F2 ) = ∨{μ12 (F2 ) ∧ μ3 (F2 ), μ12 (F2 ) ∧ μ3 (F4 ), μ12 (F4 ) ∧ μ3 (F2 )} = ∨{.8 ∧ .5, .8 ∧ .4, .6 ∧ .5} = .5, μ123 (F3 ) = ∨{μ12 (F1 ) ∧ μ3 (F3 ), μ12 (F2 ) ∧ μ3 (F3 ), μ12 (F3 ) ∧ μ3 (F3 ),

μ12 (F4 ) ∧ μ3 (F3 ), μ12 (F3 ) ∧ μ3 (F1 ), μ12 (F3 ) ∧ μ3 (F2 ), μ12 (F3 ) ∧ μ3 (F4 )} = ∨{.8 ∧ .7, .8 ∧ .7., 9 ∧ .7, .6 ∧ .7, .9 ∧ .7, .9 ∧ .5, .9 ∧ .4} = .7,

μ123 (F4 ) = μ12 (F4 ) ∧ μ3 (F4 ) = .6 ∧ .4 = .4,

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12 Intuitionistic Fuzzy Sets and Political Stability

ν12 (F1 ) = ∧{ν1 (F1 ) ∨ ν2 (F1 ), ν1 (F1 ) ∨ ν2 (F2 ), ν1 (F1 ) ∨ ν2 (F4 ), ν1 (F2 ) ∨ ν2 (F1 ), ν1 (F4 ) ∨ ν2 (F1 )} = ∧{.1 ∨ 0, .1 ∨ 0, .1 ∨ .2, 0 ∨ 0, .3 ∨ 0} = 0,

ν12 (F2 ) = ∧{ν1 (F2 ) ∨ ν2 (F2 ), ν1 (F2 ) ∨ ν2 (F4 ), ν1 (F4 ) ∨ ν2 (F2 )} = ∧{0 ∨ 0, 0 ∨ .2, .3 ∨ 0} = 0, ν12 (F3 ) = ∧{ν1 (F1 ) ∨ ν2 (F3 ), ν1 (F2 ) ∨ ν2 (F3 ), ν1 (F3 ) ∨ ν2 (F3 ), ν1 (F4 ) ∨ ν2 (F3 ), ν1 (F3 ) ∨ ν2 (F1 ), ν1 (F3 ) ∨ ν2 (F2 ),

ν1 (F3 ) ∨ ν2 (F4 )} = ∧{.1 ∨ 0, 0 ∨ 0, 1 ∨ 0, .3 ∨ 0, .1 ∨ 0, .1 ∨ 0, .1 ∨ .2} = 0,

ν12 (F4 ) = ν1 (F4 ) ∨ ν2 (F4 ) = .3 ∨ .2 = .3

ν123 (F1 ) = ∧{ν12 (F1 ) ∨ ν3 (F1 ), ν12 (F1 ) ∨ ν3 (F2 ), ν12 (F1 ) ∨ ν3 (F4 ), ν12 (F2 ) ∨ ν3 (F1 ), ν12 (F4 ) ∨ ν3 (F1 )}

= ∧{0 ∨ .3, 0 ∨ .5, 0 ∨ .1, 0 ∨ .3, .3 ∨ .3} = .1, ν123 (F2 ) = ∧{ν12 (F2 ) ∨ ν3 (F2 ), ν12 (F2 ) ∨ ν3 (F4 ), ν12 (F4 ) ∨ ν3 (F2 )} = ∧{0 ∨ .5, 0 ∨ .1, .3 ∨ .5} = .1, ν123 (F3 ) = ∧{ν12 (F1 ) ∨ ν3 (F3 ), ν12 (F2 ) ∨ ν3 (F3 ), ν12 (F4 ) ∨ ν3 (F3 ), ν12 (F3 ) ∨ ν3 (F1 ), ν12 (F3 ) ∨ ν3 (F2 ), ν12 (F3 ) ∨ ν3 (F4 )}

= ∧{0 ∨ .2, 0 ∨ .2, 0 ∨ .3, .3 ∨ .3, 0 ∨ .3, 0 ∨ .5, 0 ∨ .1} = .1,

ν123 (F4 ) = ν12 (F4 ) ∨ μ3 (F4 ) = .3 ∨ .1 = .3 . Definition 12.16. Li and Nan (2011)Let A = μA , νA and B = μB , νB be intuitionistic fuzzy sets in the set X and let c be a nonnegative real number. Define the fuzzy intuitionistic sets as follows: ∀x ∈ X, (1 ) (2 ) (3 ) (4 )

A + B = x, μA (x) + μB (x) − μA (x)μB (x), νA (x)νB (x) ; AB = x, μA (x)μB (x), νA (x) + νB (x) − νA (x)νB (x) ; cA = x, 1 − (1 − μA (x))c , (νA (x))c ; Ac = x, (μA (x))c , 1 − (1 − νA (x))c .

Using this definition, we obtain the following equation for G.

12.5 Combining Expert Opinion

199

1 1 1 1 .7, .1 + .5, .1 + .7, .1 + .4, .3

4 4 4 4 + , 6 6 4 4 = 1 − (.3)(.5)(.3)(.6), (.1)(.1)(.1)(.3)

G=

= 1 − .40536, .13160

= .59, .13 .

This equation is used in Section 15.2.3 to obtain an equation that yields the final political stability score for each country.

12.5.1

Fuzzy Preference Relations

In this section, we give a necessary and sufficient condition that the semigroup (Table 12.2 on page 196) derived from the row average column of the Political Stability data (Table 12.1 on page 195) is the same as the one that would be arrived at using the method in George et al. (2007). Let X = {Fi | i = 1, . . . . , m}. Let E denote a finite set of experts and let |E| = n. We use the following fuzzy binary relations ρk , k = 1, ..., n on X to give the experts’ preference relations. Let a ∈ (0, 1). Define ρk : X×X → [0, 1] by ∀Fi , Fj ∈ X,  (eik − ejk + a) ∧ 1 if eik ≥ ejk , ρk (Fi , Fj ) = 1 − [(ejk − eik + 1 − a) ∧ 1] if ejk > eik , n k k . where i, j = 1, ..., m; k = 1, ..., n. Let rij = ρk (Fi , Fj ). Let rij = n1 k=1 rij Let ρ be the preference relation on X given by the matrix R = [rij ]. Define the binary operation ∗ρ on X by ∀Fi , Fj ∈ X,  Fi if ρ(Fi , Fj ) ≥ ρ(Fj , Fi ), Fi ∗ρ Fj = Fj otherwise.

This equation is used in Section 15.2.3 on page 230 to obtain an equation that yields the final political stability score for each country. Definition 12.17. Let E be a finite set of experts. We say the E is a−agreeable if ∀k ∈ E, ∀Fi , Fj ∈ X, |eik − ejk | ≤ a. Lemma 12.18. Let E be a−agreeable. Let Fi , Fj ∈ X. Then σ(Fi ) ≥ σ(Fj ) if and only if ρ(Fi , Fj ) ≥ ρ(Fj , Fi ).

k k Proof. Since E is a−agreeable, rij = eik − ejk + a if eik ≥ ejk and rij = k 1 − [ejk − eik + 1 − a] = eik − ejk + a if ejk > eik . Thus rij = eik − ejk + a. Hence n 1 rij = (eik − ejk + a). n k=1

200

12 Intuitionistic Fuzzy Sets and Political Stability

Thus σ(Fi ) ≥ σ(F j) ⇔ ⇔

n 

k=1

n

n

n

k=1

k=1 n 

k=1

 1 1 eik ≥ ejk ⇔ (eik − ejk ) ≥ 0 n n

(2eik − 2ejk ) ≥ 0 ⇔ n

k=1 n 

1 1 ⇔ (eik − ejk + a) ≥ n n k=1

(eik − ejk ) ≥

k=1

n 

k=1

(ejk − eik )

(ejk − eik + a)

⇔ rij ≥ rji ⇔ ρ(Fi , F j) ≥ ρ(F j, Fi ).  Theorem 12.19. Let E be a−agreeable. Then ∗σ = ∗ρ . The linear equations representing political stability have been derived in Sections 6.5.3 on page 110, 7.4.4 on page 135, and 9.2.3 on page 160. The rankings of the countries can be found in 15.2.3 on page 230.

Part IV

Analysis of Results

Chapter 13

Quality of Life

13.1

Conditions for a Uniform Model

Chapter 6 applied the Analytic Hierarchy Method to various data sets. Chapter 7 applied Guiasu’s method to the data. Chapter 9 applied Yen’s method to the data. In Chapter 10 on page 163 we applied set valued statistical methods. We now show that all four method yield the same equation for the Quality of Life – Expert Group One data. Let A = [aij ] denote the original m × n-matrix giving the weights of m factors by n experts. Suppose that the weights come from the set {1, ..., m}. Suppose further that {aij | i = 1, ..., m} = {1, ..., m}, j = 1, ..., n. That is, suppose every Ej has no ties for the weights aij , i = 1, ..., m. Then the coefficients for the factors for the four methods are given as follows: AHP coefficients

n aij m j=1 n i=1

Guiasu coefficients

Yen coefficients

j=1

aij

,

n n ( j=1 aij )/n j=1 aij m n = m n , i=1 ( j=1 aij )/n i=1 j=1 aij n n ( j=1 aij /m)/n j=1 aij m n = m n . i=1 ( j=1 aij /m)/n i=1 j=1 aij

J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 203–207. DOI: 10.1007/978-3-642-35224-9_13 © Springer-Verlag Berlin Heidelberg 2013

204

13 Quality of Life

For the set-valued statistical method we have that for i = 1, ..., m, t+1 n s=1 j=1 χF s(j) (F i) m(F i) = mn n a j=1 ij = . mn Thus we have shown that the set valued statistical method coefficients are given by the following formula. Set-valued statistical coefficients m(F i) m = i=1 m(F i)

n

j=1 aij m nm n i=1

j=1

mn n

aij

j=1 aij = m n i=1

j=1

aij

.

We note that if A is a Latin square, then all four methods are simply a nonweighted average of the factors. For Quality of Life – Expert Group Two the Yen method and the AHP are equivalent since the maximum values of each column are equal, Mordeson et al. (2011d). In this study, we try to compare five methods of ranking the quality of life of 194 countries in Year 2010. The five sets (methods) of rankings are: QLI Exp AHP Guiasu SSM

13.2

The International Living’s Quality of Life Index 2010 (www.internationalliving.com/gofl 2010/) First Set of Experts’ Weighting Second Set of experts’ Weighting using the Analytic hierarchy Process (AHP) The Guiasu Method The Set-valued Statistical Method

Results

We now present the rankings of the countries obtained by our methods. We use the following abbreviations, Luxemb. for Luxembourg and U.K. for United Kingdom.

13.3 Statistical Analysis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

205

Country

QLI

Country

Exp

Country

AHP

Country

GM

Country

SSM

France Australia Switzerland Germany N. Zealand Luxemb. U.S. Belgium Canada Italy Netherlands Norway Austria Lichtenstein Malta Denmark Spain Finland Uruguay Hungary Portugal Lithuania Andorra Czech. Rep. U.K. Argentina Slovenia Greece Monaco

82 81 81 81 79 78 78 78 77 77 77 77 77 76 76 76 76 75 75 74 73 73 73 73 73 72 72 72 72

France Switzerland Germany Australia N. Zealand Luxemb. Austria Denmark Lichtenstein Italy Belgium Norway Malta Canada Spain Netherlands Finland U.S. U.K. Hungary Portugal Sweden Czech. Rep. Andora Uruguay Lithuania Japan Monaco Ireland

85.6 85.5 84.5 84.0 83.0 82.7 82.0 81.7 81.4 81.1 81.1 81.1 80.7 80.3 80.1 80.1 80.0 78.8 78.3 78.2 78.1 77.9 77.6 77.6 77.3 77.2 76.9 76.4 75.8

France Switzerland Germany Australia N. Zealand Luxemb. Belgium Norway Canada Austria Italy Netherlands Denmark Malta Lichtenstein U.S. Spain Finland Hungary Portugal Uruguay U.K. Czech. Rep. Andora Lithuania Sewden Greece Monaco Estonia

84.4 83.9 83.2 83.1 81.4 80.6 79.9 79.6 79.5 79.4 79.1 79.1 79.1 78.8 78.8 78.6 78.4 77.9 76.9 76.1 76.1 76.0 75.9 75.7 75.6 74.9 74.1 74.0 74.0

France Switzerland Germany Australia N. Zealand Luxemb. Belgium Norway Canada Austria Denmark Italy Netherlands Lichtenstein Malta U.S. Spain Finland Hungary Portugal U.K. Uruguay Czech. Rep. Andora Lithuania Sweden Greece Monaco Estonia

84.3 83.9 83.2 83.0 81.5 80.7 79.9 79.8 79.6 79.5 79.2 79.1 79.1 79.0 78.8 78.5 78.5 77.9 76.9 76.1 76.1 76.0 76.0 75.8 75.6 74.9 74.1 74.0 74.0

France Switzerland Germany Australia N. Zealand Luxemb. Belgium Canada Malta Italy Austria Norway Netherlands Spain Denmark U.S. Lichtenstein Finland Hungary Portugal Uruguay Czech Rep. Andora. Lithuania U.K. Estonia Costa Rica Greece Chile

85.1 84.0 83.5 83.4 82.2 80.7 80.3 80.3 80.0 79.9 79.8 79.6 79.6 79.3 79.3 78.8 78.7 78.2 77.9 77.2 76.9 76.8 76.7 76.7 76.4 75.1 74.9 74.8 74.8

13.3

Statistical Analysis

We now proceed with a statistical analysis of our results concerning Quality of Life.

13.3.1

Part I Number of Countries with Rank Change from the QLI Method # of changes (%) 1st Quarter (n=48) 2nd Quarter (n=48) 3rd Quarter (n=49) 4th Quarter (n=49)

Exp

39 (81%) 43 (90%) 46 (94%) 45 (92%) 173 Column Total (89%)

AHP Guiasu SSM 41 (85%) 43 (90%) 47 (98%) 46 (94%) 178 (92%)

39 (81%) 46 (96%) 46 (94%) 47 (96%) 178 (92%)

38 (79%) 44 (92%) 49 (100%) 45 (92%) 176 (91%)

Row Total 158 (83%) 176 (92%) 188 (96%) 183 (93%)

206

13 Quality of Life

1. Observations: a. Most of the ranks in the QLI were changed when the other four methods were used (89-92%). b. Countries in the third quarter of the QLI (rank 97 to 145) have the highest number of rank changes (96%). c. Countries in the first quarter of the QLI (rank 1 to 48) have the lowest number of rank changes (83%), indicating the higher consistency of the rankings among all five sets of rankings in regard to the countries with the highest quality of life. d. The percentages of rank changes from the QLI rankings to the other four methods (Exp, AHP, Guiasu, and SSM) are significantly different among quarters (83% to 96%). e. The percentages of rank changes from the QLI rankings to the other four methods are not significantly different among the four methods (89% to 92%). f. In the following parts (II and III) we shall take the magnitude of the rank changes into consideration.

13.3.2

Part II Rank Correlation (Spearman’s rho): Rank Correlation Exp AHP QLI .977 .976 Exp .990 AHP Guiasu

Guiasu SSM .980 .960 .999 .976 .990 .984 .977

2. Observations: a. Although the number of rank changes in each method is high as shown in Part I, the magnitudes of most changes are relatively small. The rankings of the 194 countries follow a general pattern in all five sets with high correlation between any two methods.

13.3.3

Part III Analysis of Rank Variation

Part III Analysis of Rank Variation (magnitude of rank change) - Using absolute values of rank difference of each country between two methods:

13.3 Statistical Analysis

207

Mean Std. Deviation QLI-Exp QLI-AHP QLI-Guiasu QLI-SSM Exp-AHP Exp-Guiasu Exp-SSM AHP-Guiasu AHP-SSM Guiasu-SSM

8.23 7.53 7.49 9.48 2.86 2.23 4.23 1.31 3.17 3.36

8.89 9.70 8.23 12.28 7.58 2.06 11.50 7.68 9.64 11.43

One-tailed t–statistic 12.89 10.81 12.67 10.34 5.25 5.07 7.33 2.37 4.58 4.09

p–value .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0093 .0000 .0000

3. Observations: a. The rankings from the four methods (Exp, AHP, Guiasu, and SSM) are clearly different from the original QLI method with average rank variation from 7.53 to 9.48 (p = .0000). b. The rankings from the first set of experts (Exp) are also significantly different from the other three methods (AHP, Guiasu, SSM) with average rank variation from 2.33 to 4.23 (p = .0000). c. Among the three methods of AHP, Guiasu, and SSM, AHP and Guiasu are the closest in ranking counties (average rank variation = 1.31). However, AHP and SSM, as well as Guiasu and SSM, are significantly different. d. All five sets of rankings showed a general consistency in ranking 194 countries by their quality of life with a high correlation between any two sets. e. The ranking among the five sets of ranking methods are significantly different from one to another, showing a difference in the scoring functions between any two methods.

Chapter 14

Failed States

14.1

Results

The values determined by the Fund for Peace (available online from Fund for Peace, The (2009)) are substituted into the Fi , i = 1, . . . , 12, to obtain a failed state index by each of the three methods. These results together with those of the Fund for Peace are compared next. We present the rankings and scores of the countries by the methods used above. We use several abbreviations: Dem. Rep. C. stands for Democratic Republic of Congo, C. Af. Rep. stands for Central African Republic, P. N. Guinea stands for Papua New Guinea, Brunei Dar. stands for Brunei Darussalam, and Ant. & Bar stands for Antigua and Barbuda. The Table is reproduced with the permission of the managing editor from the article “Failed state ranking of 177 countries: A weighted average approach,” Journal of Fuzzy Mathematics, 19(4) 2011, 975-987. Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Rank

Country Avg Country Somalia 9.56 Somalia Zimbabwe 9.50 Zimbabwe Sudan 9.37 Chad Chad 9.35 Sudan Dem. Rep. C. 9.06 Afghanistan Iraq 9.05 Iraq Afghanistan 9.02 Dem. Rep. C C. Af. Rep. 8.78 C. Af. Rep. Guinea 8.72 Guinea Pakistan 8.68 Pakistan Ivory Coast 8.54 Ivory Coast Haiti 8.48 Haiti Burma 8.46 Burma Kenya 8.45 N. Korea Nigeria 8.32 Kenya Ethiopia 8.24 Nigeria N. Korea 8.19 East Timor Bangladesh 8.18 Ethiopia Yemen 8.18 Niger East Timor 8.10 Bangladesh Uganda 8.08 Yemen Sri Lanka 8.06 Sri Lanka Niger 8.04 Uganda Country Avg Country

AHP 9.62 9.49 9.40 9.35 9.11 9.05 9.01 8.87 8.83 8.68 8.62 8.62 8.51 8.46 8.42 8.38 8.23 8.19 8.18 8.17 8.17 8.04 8.03 AHP

Country Somalia Zimbabwe Chad Sudan Afghanistan Dem. Rep. C. Iraq C. Af. Rep. Guinea Pakistan Ivory Coast Haiti N. Korea Burma Kenya Nigeria East Timor Niger Yemen Bangladesh Ethiopia Sri Lanka Guinea-Bissau Country

GM 9.62 9.50 9.40 9.36 9.18 9.08 9.06 8.87 8.84 8.67 8.63 8.63 8.48 8.47 8.36 8.35 8.25 8.19 8.18 8.09 8.08 8.03 8.03 GM

Country Somalia Zimbabwe Chad Afghanistan Sudan N. Korea Guinea C. Af. Rep. Haiti Iraq Ivory Coast Burma East Timor Dem. Rep. C. Pakistan Niger Kenya Nigeria Cameroon Guinea-Bissau Uzbekistan Georgia Comoros Country

D-S 9.83 9.60 9.51 9.44 9.33 9.29 9.26 9.07 8.99 8.94 8.88 8.85 8.84 8.77 8.69 8.56 8.53 8.49 8.34 8.34 8.33 8.26 8.24 D-S

J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 209–219. DOI: 10.1007/978-3-642-35224-9_14 © Springer-Verlag Berlin Heidelberg 2013

210 Rank 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 Rank

14 Failed States Country Burundi Nepal Cameroon Guinea-Bissau Malawi Lebanon Rep. Congo Uzbekistan Sierra Leone Georgia Liberia Burkina Faso Eritrea Tajikistan Iran Syria Solomon Is. Columbia Kirgizstan Egypt Laos Rwanda Mauritania Eq. Guinea Bhutan Cambodia Togo Bolivia Comoros Phillipines Moldova Angola Azerbaijan China Isreal/WB Turkmenistan Zambia Indonesia P. N. Guinea Bosnia Nicaraga Swaziland Belarus Lesotho Madagascar Ecuador Tanzania Russia Mozambique Algeria Cuba Djibouti Guatemala Venezuela Sebia Thailand Gambia Fiji Maldives Mali Cape Verde Turkey Jordan India Dom. Rep. Saudi Arabia Country

Avg 7.98 7.95 7.94 7.90 7.82 7.79 7.76 7.73 7.68 7.65 7.65 7.61 7.53 7.53 7.50 7.48 7.47 7.43 7.43 7.42 7.42 7.42 7.39 7.36 7.28 7.28 7.27 7.19 7.19 7.15 7.09 7.08 7.05 7.05 7.05 7.03 7.02 7.01 7.01 6.94 6.88 6.87 6.86 6.82 6.80 6.77 6.76 6.73 6.73 6.72 6.72 6.72 6.72 6.63 6.60 6.60 6.58 6.57 6.57 6.56 6.54 6.52 6.49 6.48 6.48 6.46 Avg

Country AHP Country GM Nepal 8.01 Uganda 8.02 Guinea-Bissau 7.99 Nepal 7.98 Cameroon 7.97 Burundi 7.91 Burundi 7.95 Cameroon 7.90 Malawi 7.86 Rep. Congo 7.86 Uzbekistan 7.85 Uzbekistan 7.85 Rep. Congo 7.82 Lebanon 7.77 Georgia 7.77 Georgia 7.71 Lebanon 7.76 Malawi 7.69 Solomon Is. 7.67 Burkina Faso 7.65 Burkina Faso 7.66 Eritrea 7.62 Eritrea 7.64 Solomon Is. 7.61 Sierra Leone 7.63 Tajikistan 7.56 Tajikistan 7.63 Kirgizstan 7.56 Liberia 7.60 Liberia 7.52 Laos 7.53 Syria 7.52 Syria 7.52 Sierra Leone 7.50 Kirgizstan 7.52 Eq. Guinea 7.50 Egypt 7.52 Laos 7.47 Eq. Guinea 7.52 Iran 7.46 Iran 7.50 Comoros 7.45 Comoros 7.47 Mauritania 7.42 Mauritania 7.45 Egypt 7.38 Rwanda 7.40 Togo 7.32 Cambodia 7.37 Cambodia 7.30 Togo 7.32 Colombia 7.28 Colombia 7.31 Moldova 7.24 Bhutan 7.28 Phillipines 7.20 Bolivia 7.24 Turkmenistan 7.18 Moldova 7.23 Rwanda 7.14 Phillipines 7.21 Bhutan 7.10 Turkmenistan 7.20 Bolivia 7.10 Azerbaijan 7.10 Azerbaijan 7.06 Belarus 7.09 Swaziland 7.06 P. N. Guinea 7.07 P. N. Guinea 7.03 Swaziland 7.07 Zambia 7.01 Israel/WB 7.02 Bosnia 7.06 Angola 7.05 Belarus 6.97 Bosnia 7.05 Indonesia 6.96 Zambia 7.04 Angola 6.94 China 6.97 Israel/WB 6.90 Indonesia 6.97 Nicaraga 6.87 Nicaraga 6.93 China 6.86 Lesotho 6.90 Mozambique 6.83 Madagascar 6.84 Cuba 6.81 Mozambique 6.81 Lesotho 6.79 Djibouti 6.79 Guatemala 6.78 Cuba 6.77 Madagascar 6.74 Ecuador 6.75 Algeria 6.74 Tanzania 6.74 Djibouti 6.73 Russia 6.74 Mali 6.71 Guatemala 6.73 Ecuador 6.67 Fiji 6.73 Tanzania 6.67 Algeria 6.72 Gambia 6.67 Gambia 6.68 Russia 6.64 Thailand 6.65 Fiji 6.63 Serbia 6.64 Thailand 6.60 Venezuela 6.63 Cape Verde 6.60 Cape Verde 6.62 Saudi Arabia 6.58 Maldives 6.60 Venezuela 6.56 Saudi Arabia 6.57 El Salvador 6.56 Mali 6.52 Serbia 6.52 Honduras 6.49 Maldives 6.52 Vietnam 6.49 Honduras 6.47 Sao Tome 6.49 Mexico 6.47 Turkey 6.48 Peru 6.45 Country AHP Country GM

Country Sri Lanka Bangladesh Eritrea Rep. Congo Tajikistan Yemen Malawi Solomon Is. Nepal Eq. Guinea Ethiopia Syria Egypt Swaziland Uganda Belarus Kirgizstan Laos Cambodia Turkmenistan Burkina Faso Burundi Lerbanon Iran Phillipines Moldova Rwanda Fiji Sierra Leone Azerbaijan Togo Bosnia Zambia Liberia Bolivia Mauritania Bhutan Gambia P. N. Guinea Lesotho Colombia China Angola Cuba Saudi Arabia Mozambique Djibouti Thailand Cape Verde Nicaraga Kazakhstan Russia Israel/WB Sao Tome Algeria Paraguay Vietnam Serbia Maldives Macedonia Guatemala Indonesia Madagascar Honduras Ecuador Tanzania Country

D-S 8.22 8.20 8.20 8.18 8.18 8.16 8.12 8.12 8.07 8.03 8.00 8.00 8.00 8.00 7.94 7.94 7.93 7.87 7.86 7.80 7.79 7.75 7.67 7.67 7.66 7.66 7.54 7.53 7.51 7.51 7.48 7.48 7.46 7.36 7.32 7.31 7.29 7.27 7.25 7.25 7.21 7.19 7.12 7.12 7.12 7.10 7.09 7.04 7.04 7.03 7.02 7.00 6.99 6.96 6.92 6.89 6.88 6.86 6.86 6.85 6.84 6.80 6.77 6.74 6.73 6.73 D-S

14.1 Results Rank 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 Rank

Country El Salvador Honduras Morocco Peru Vietnam Sao Tome Namiba Benin Mexico Gabon Macedonia Armenia Senegal Suriname Guyana Kazakhstan Paraguay Micronesia Samoa Albania Ukraine Belize Libya Brazil Cyprus Malaysis Botswana Jamaica Brunei Dar. Grenada Seychelles Tunisia S. Africa Trinidad Ghana Kuwait Ant. & Bar. Mongolia Bulgaria Romania Bahamas Croatia Panama Bahrain Montenegro Barbados Latvia Costa Rica Qatar U. Arab Em. Estonia Hungary Poland Malta Slovakia Lithuania Oman Greece Argentina Mauritius Italy Spain Czech Rep. S. Korea Uruguay Chile Country

211 Avg 6.43 6.43 6.43 6.43 6.41 6.39 6.30 6.29 6.28 6.20 6.20 6.19 6.18 6.10 6.08 6.04 6.00 5.99 5.95 5.83 5.81 5.79 5.78 5.76 5.74 5.74 5.73 5.72 5.68 5.66 5.64 5.63 5.62 5.56 5.52 5.28 5.23 5.16 5.13 5.11 5.08 5.01 4.98 4.92 4.83 4.77 4.55 4.38 4.33 4.32 4.27 4.23 4.13 4.07 4.05 4.00 3.93 3.84 3.73 3.73 3.66 3.61 3.55 3.47 3.43 3.13 Avg

Country Jordan Peru El Salvador Morocco Dom. Rep. India Macedonia Mexico Benin Kazakhstan Gabon Paraguay Armenia Suriname Namibia Senegal Guyana Micronesia Samoa Albania Ukraine Libya Jamaica Brunei Dar. Belize Brazil Malaysia Cyprus Seychelles Grenada Tunisia Botswana Trinidad Mongolia S. Africa Ghana Kuwait Ant. & Bar. Romania Bulgaria Bahrain Bahamas Croatia Panama Montenegro Barbados Latvia Qatar U. Arab Em. Hungary Costa Rica Estonia Oman Poland Slovakia Malta Lithuania Mauritius Greece Argentina Italy Czech Rep. Spain S. Korea Uruguay Chile Country

AHP 6.44 6.43 6.42 6.41 6.37 6.34 6.33 6.32 6.27 6.26 6.20 6.19 6.17 6.16 6.15 6.15 6.13 6.05 6.02 5.98 5.92 5.91 5.88 5.79 5.77 5.77 5.77 5.76 5.74 5.73 5.70 5.59 5.57 5.46 5.40 5.39 5.34 5.25 5.21 5.20 5.08 5.04 4.98 4.95 4.91 4.74 4.56 4.49 4.46 4.33 4.30 4.25 4.20 4.12 4.06 4.03 3.99 3.91 3.87 3.74 3.69 3.56 3.51 3.47 3.38 3.10 AHP

Country Sao Tome Turkey Vietnam Jordan India Benin Morocco Namibia Senegal Dom. Rep. Macedonia Guyana Suriname Kazakhstan Gabon Samoa Armenia Micronesia Jamaica Paraguay Albania Brazil Belize Libya Malaysia Grenada Seychelles Tunisia Brunei Dar. Botswana Ukraine Cyprus Trinidad Mongolia Ant. & Bar. Bulgari Ghana S. Africa Kuwait Romania Panama Bahamas Croatia Bahrain Montenegro Barbados Latvia U. Arab Em. Qatar Costa Rica Malta Hungary Oman Estonia Poland Greece Lithuania Mauritius Slovakia Italy Argentina Czech Rep. Spain Uruguay S. Korea Slovenia Country

GM 6.44 6.42 6.42 6.40 6.40 6.34 6.32 6.30 6.29 6.28 6.28 6.26 6.20 6.18 6.09 6.08 6.06 6.05 6.04 6.02 5.96 5.89 5.87 5.77 5.76 5.73 5.72 5.70 5.69 5.67 5.58 5.57 5.57 5.41 5.31 5.26 5.25 5.25 5.18 5.16 5.09 5.08 4.98 4.96 4.80 4.80 4.46 4.34 4.31 4.22 4.22 4.12 4.06 4.02 4.02 3.99 3.93 3.93 3.91 3.71 3.65 3.46 3.45 3.44 3.33 3.14 GM

Country Morocco Venezuela Albania El Salvador Mexico Jamaica Gabon Peru Brunel Dar. Libya Armenia Suriname Ukraine Benin Guyana Turkey Micronesia Samoa Seychelles Mongolia Jordan Mali Senegal Grenada Tunisia Belize Dom. Rep. Malaysia Brazil Bahrain India Kuwait Romania Trinidad Bulgaria Botswana Namibia Cyprus Ant. & Bar. Qatar U. Arab Em. Ghana Bahamas Oman S. Africa Montenegro Croatia Hungary Panama Latvia Barbados Maurtius Estonia Poland Costa Rica Malta Greece Lithuania Slovakia Italy S. Korea Argentina Czech Rep. Singapore Uruguay Slovenia Country

D-S 6.72 6.70 6.64 6.62 6.53 6.53 6.51 6.50 6.49 6.44 6.43 6.43 6.43 6.42 6.41 6.34 6.33 6.33 6.30 6.26 6.25 6.15 6.08 6.06 5.97 5.93 5.92 5.83 5.82 5.81 5.70 5.69 5.67 5.65 5.56 5.55 5.54 5.50 5.45 5.40 5.35 5.15 5.11 5.10 5.07 4.89 4.88 4.88 4.85 4.80 4.67 4.59 4.36 4.25 4.19 4.17 4.16 4.06 4.00 3.98 3.67 3.64 3.60 3.46 3.13 3.05 D-S

212 Rank 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 Rank

14.2

14 Failed States Country Slovenia Germany France U. S. Singapore U. K. Belgium Portugal Japan Iceland Canada Austria Luxembourg Netherlands Australia N. Zealand Denmark Ireland Switzerland Sweden Finland Norway Country

Avg 3.03 3.02 2.94 2.83 2.82 2.80 2.79 2.73 2.60 2.42 2.31 2.30 2.30 2.25 2.16 1.94 1.93 1.80 1.77 1.72 1.60 1.53 Avg

Country Slovenia Singapore Germany France Belgium U. S. U. K. Portugal Japan Iceland Luxembourg Austria Canada Netherlands Australia N. Zealand Denmark Ireland Switzreland Sweden Finland Norway Country

AHP 3.04 2.93 2.91 2.84 2.84 2.80 2.72 2.71 2.62 2.60 2.38 2.24 2.20 2.15 2.09 1.93 1.90 1.80 1.71 1.60 1.54 1.51 AHP

Country Chile Germany France U. S. Singapore Belgium Portugal U. K. Japan Iceland Luxembourg Austria Netherlands Canada Australia Denmark N. Zealand Ireland Switrzerland Sweden Finland Norway Country

GM 3.13 2.93 2.77 2.70 2.70 2.70 2.69 2.68 2.62 2.55 2.26 2.13 2.12 2.09 2.03 1.92 1.91 1.78 1.71 1.58 1.54 1.50 GM

Country Iceland Belgium U. S. Chile Germany Luxembourg Spain Japan Portugal France U. K. Austria Canada Ireland Netherlands Australia Denmark N. Zealand Switzerland Norway Finland Sweden Country

D-S 3.00 2.85 2.82 2.74 2.60 2.60 2.52 2.49 2.37 2.36 2.23 1.91 1.85 1.84 1.79 1.78 1.71 1.68 1.48 1.37 1.33 1.27 D-S

Statistical Analysis

In this section, we compare the four methods used to determine the score of the countries. We start by considering rank change.

Abbreviation List O: A: G: D: AG: AD: GD: AGD: ABS(d1):

ABD(d2):

Original ranking by the Fund for Peace Analytic Hierarchy Process Guiasu Method Dempster-Shafer Method Combined average using the Analytic Hierarchy Process and the Guiasu Method Combined average using the Analytic Hierarchy Process and the Dempster-Shafer Method Combined average ranking using the Guiasu Method and the Dempster-Shafer Method Combined average using all three: A, G, and D Absolute value of the first difference, X(i + 1) − X(i), in each set of rankings, where X is the rank-value and i is the rank-order of country (i) in the original set, O. Absolute value of the second difference [X(i + 2) − X(i + 1)] − [X(i + 1) − X(i)], in each set of rankings.

14.2 Statistical Analysis

14.2.1

213

Part I Number of Rank Changes from the Original Ranking #of changes A G D Row Total (%) Top Third 43 46 55 144 (n=59) (73%) (78%) (93%) (87%) Middle Third 51 53 55 154 (n=59) (86%) (90%) (93%) (87%) Bottom Third 36 37 50 123 (n=59) (61%) (63%) (85%) (69%) 130 136 160 Column Total (73%) (77%) (90%)

1. Observations: a. Most of the original ranks were changed when A,G, or D were used. b. The differences among the total number of changes from the three methods (A,G, and D) are statistically significant, with D having the highest percentage of changes (90%). c. The differences among the total number of changes from the three subgroups (Top Third, Middle Third, and Bottom Third) are statistically significant, with the Middle Third group having the highest percentage of changes (87%). d. Within each of the three subgroups, Method D constantly produced the highest numbers of rank changes. e. In the following parts of the data analysis, we will take the magnitude of the rank changes into consideration.

14.2.2

Part II Rank Correlation (Spearman’s rho)

14.2.2.1

Rank Correlation among Four Sets: O, A, G, and D r O A G D O .998 .997 .973 A .998 .982 G .980

2. Observations: a. Although the number of rank changes in each set is high, the magnitude of each change is small. The rankings among the four sets are still highly correlated.

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b. Among the three sets, A, G, and D, the set D shows the highest rank change from O, A and G are very closely correlated to O, and to each other. 14.2.2.2

Rank Correlation among Combinations of the Four Sets r AG GD Ad AGD O .998 .990 .990 .994 AG .996 .995 .998 GD 1.000 .999 AD .980

3. Observations: a. These correlations indicate that combination of methods produces largely the same result in A) above. b. When A or G, or both, are combined with D, the resulted correlation moves toward the rankings of A and G.

14.2.3

Part III One-Way Analysis of Variance (ANOVA)

In this part, we will treat the actual magnitude of the rank changes between and within each set as continuous data. We want to know if the four sets of methods (O, A, G, and D) are significantly different from one another with such data.

14.2.3.1

Analysis of Rank Variation between Sets - Using Absolute Values of Rank Difference of Each Country between Two Sets

1. Means and Standard Deviations: n = 177 O-A O-G O-D A-G A-D G-D Mean 2.03 2.76 8.54 2.03 6.92 7.27 Std. Dev. 2.08 2.58 8.12 2.02 6.80 7.17 2. ANOVA Table: Source SS df F p-value Between Group 7793 5 52.36 0.00 Error 31436 1056

14.2 Statistical Analysis

215

3. Pairwise Multiple Comparison: (Scheffe’s Method) Pair Comparison O-A vs O-G O-A vs O-D O-A vs A-G O-A vs A-D O-A vs G-D O-G vs O-D O-G vs A-G O-G vs A-D O-G vs G-D O-D vs A-G O-D vs A-D O-D vs G-D A-G vs A-D A-G vs G-D A-D vs G-D

Is the Difference Significant at the .05 Level? No Yes No Yes Yes Yes No Yes Yes Yes No No Yes Yes No

4. Observations: a. There is an “over-all” statistical difference among the six groups (OA,O-G, etc.), with the p-value near zero (Type-one error probability near zero). b. The post-hoc pairwise comparisons show that the difference between A and G is not significant - indicating that these two ranking methods yield similar results. c. Thew post-hoc pair-wise comparisons show that the difference comes from the set D — indicating that the Dempster-Shafer ranking method is different from the other methods among the four sets. d. A large number of multiple comparison contrasts (Scheffe’s method) were performed and the results also pointed the “stand-out” of set D. The details are too lengthy to include in this report.

14.2.3.2

Analysis of Rank Variation within Set - Using the Absolute Values of the First-Difference X(i + 1) − X(i), in Each Set

1. Means and Standard Deviations ABS(d1) O A G D AG GD AD AGD Mean 1.0 3.38 4.30 11.89 3.45 7.26 7.30 5.79 Std. Dev. 0 2.77 3.57 10.06 2.98 6.24 6.19 4.98

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2. ANOVA Table with Four Groups: O, A, G, and D Source Ss df F p-value Between Group 11701 3 128.35 0.000 Error 21272 700 3. Multiple Comparison and Contrasts (Scheffe’s Method) Pair Comparison O vs A O vs G O vs D O vs AG O vs AD O vs GD A vs G A vs D A vs GD G vs D G vs AD D vs AG

Is the Difference Significant at the .05 Level? Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes

4. Observations: a. With the exception of Set A and Set G, all other sets and various combinations are significantly different from one another when the first difference [X(i + 1) − X(i)] is used. b. Set D shows the largest amount of rank difference measure by the first-difference approach.

14.2.3.3

Analysis Rank Variation within Sets - Using the Absolute Values of the Second Difference, i.e., [X(i + 2) − X(i + 1)] − [X(i + 1) − X(i)]

1. Means and Standard Deviations: ABS(d2) O A G D AG GD AD AGD Mean 0 35.61 7.29 19.54 5.89 12.05 12.06 9.60 Std. Dev. 0 4.78 6.28 16.48 5.11 10.28 10.17 8.22 2. ANOVA Table with Three Groups: A, G, and D Source Ss df F p-value Between Group 20242 2 90.64 0.00 Error 58159 533

14.2 Statistical Analysis

217

3. Multiple Comparison: Pair Comparison A vs G A vs D A vs GD G vs D G vs AD D vs AG

Is the Difference Significant at the .05 Level? No Yes Yes Yes Yes Yes

4. Observations: a. The results with the second difference are similar to the results with the first difference in B) above. b. The results with the second difference confirm that D is the main source of difference among the sets A, G, and D.

14.2.4

Part IV Two-Way Analysis of Variance

In this part we add a second factor to the ANOVA model: Factor B, with three levels: Top Third, Middle Third, and Bottom Third in each set.

14.2.4.1

Analysis of Rank Variation between Sets

1. Means: Factor B (3 levels) Top Third (n=59) Middle Third (n=59) Bottom Third (n=59) Column Mean

Factor A (6 levels) O-A O-G O-D A-G A-D G-D Row Mean 2.02 2.83 8.22 1.39 6.44 6.07 4.49 3.05 4.08 12.67 3.47 10.25 11.63 7.52 1.02 1.39 4.75 1.22 4.07 4.10 2.76 2.03 2.76 8.54 2.03 6.92 7.27 4.92

2. ANOVA Table: Source Factor A Factor B A×B Error

SS df F p-value 7793 5 62.31 0.00 4113 2 82.21 0.00 1209 10 4.83 0.00 26114 1044

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3. Observations: a. The ranking change among the top third, middle third, and bottom third of the 177 countries is significantly different, with the middle third varying the most (mean = 7.52). b. Interaction between factor A and factor B (A×B) is significant since the total sum of ranks in each set is a constant. c. Rank changes are also constantly higher in all three levels of Factor B in set D.

14.2.4.2

Two-Way ANOVA of First-Difference (ABS(d1))

1. Means: Factor B (3 levels) Top Third Middle Third Bottom Third Column Mean 2. NOVA Table

Factor A (3 levels) A G D Row Mean 3.75 4.93 13.14 7.27 4.10 5.53 14.80 8.14 2.24 2.39 7.54 4.06 3.36 4.28 11.83 6.49 Source Factor A Factor B A×B Error

SS df F p-value 7637 2 103.49 0.00 1637 2 22.49 0.00 509 4 3.45 0.00 19260 522

3. Observations: a. The results are consistent with previous findings. The three levels in Factor B are different from one another, with the middle third varying the most, and the bottom third the least. b. In Factor A, rank variation in Set D is significantly different from Sets A and G.

14.2.4.3 1. Means:

C) Two-Way ANOVA of Second Difference (ABS(d2))

Factor A (3 levels) Factor B (3 levels) A G D Row Mean Top Third 6.37 8.56 22.00 12.31 Middle Third 6.88 9.42 23.47 13.26 Bottom Third 3.39 3.64 12.49 6.51 Column Mean 5.55 7.21 19.32 10.69

14.2 Statistical Analysis

219

2. ANOVA Table Source SS df F p–value Factor A 20013 2 97.96 0.00 Factor B 4729 2 23.15 0.00 A × B 1031 4 2.52 0.04 Error 53322 522 3. Observations: a. The two-way ANOVA of the second difference, ABS(d2), shows that the rankings in Set D is the major deviation in Factor A, consistent with the findings in Part III, Section C, where the one-way ANOVA shows that rankings in Set D have the highest deviation (Mean: 19.54, three to four times higher than those from Set A and G.) b. Factor B in this analysis shows that rank variation measured by the second difference remains the largest in the Middle Third of the 177 countries, but the Top Third result is very close. c. The variations among the three sets, A, G, and D, of the Bottom Third remain the smallest compared to the Top and Middle Thirds, even in this third (The Bottom Third) the rankings in D is almost four times as high as Sets A and G. Rank changes occurred from the original ranking by the Fund for Peace when the Analytic Hierarchy Process, the Guiasu Method, or the Dempster-Shafer Method was applied. While the rank correlation remained high among the four sets of rankings, the Dempster-Shafer Method displayed the greatest variation from the other three methods in all of the data analyses. We may conclude that the Dempster-Shafer Method is clearly different from the other methods.

Chapter 15

Additional Results

15.1 15.1.1

Issues in International Relations Nuclear Deterrence

Based on the expert opinions, we developed a metric for measuring US success employing two approaches in the literature on fuzzy mathematics: the Guiasu Method Guiasu (1994) and the Analytic Hierarchy Method Saaty and Vargas (2001). Each model weights the expert opinions on the relative importance of the six components G1 − G6 of the US strategic goal, and each results in a linear equation assessing the degree of US success in achieving that goal. We found that expert opinion believes the US has been most successful in achieving goals G1 and G4 to date. We also took the weights given by the experts to provide preference relations on the set of goals. We first determined the core, i.e., the set of undominated goals - those not defeated by pairwise comparison by a required majority. We found that goals G1 and G4 are not dominated by most experts. We also found that goals G1 and G4 followed by G5 are preferred (considered more important) to most goals.

15.1.2

Smart Power and Deterrence

We used two methods to determine the success the U. S. is having in achieving the goal of smart power based on the expert opinions: the Guiasu method Guiasu (1994) and the analytic hierarchy process (AHP) Saaty and Vargas (2001)Saaty (2008). The two approaches yield linear equations with G1 –G6 as the independent variables and the overarching goal as the dependent variable. They provide the coefficients for each independent variable of the linear equations. In a subsequent survey, we asked the experts to indicate the degree J.N. Mordeson et al.: Linear Models in the Mathematics of Uncertainty, SCI 463, pp. 221–237. DOI: 10.1007/978-3-642-35224-9_15 © Springer-Verlag Berlin Heidelberg 2013

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to which the U.S. is achieving each sub goal. The resulting models weight the relative importance of the six sub goals G1 –G6 of the overarching goal.

15.1.2.1

How Well the U.S. Is Doing in Achieving Goals

We asked a different set of experts to score how well they felt the U.S. is doing in achieving its goals. The following matrices provide the experts’ opinions. E1 E2 E3 E4 Row Avg. G 3 5 7.5 7 5.625 G1 G2 G3 G4 G5 G6

E1 5 5 5 3 5 3

E2 7 7 7 7 8 3

E3 8 8 8 8 8 8

E4 Row Avg 8 7 7 6.75 6 6.5 9 6.75 9 7.5 5 4.75

G1,1 G1,2 G1,3 G1,4

E1 6 6 1 2

E2 1 1 9 6

E3 9 9 8 8

E4 Row Avg 5 7.75 5 7.75 5 5.75 5 5.25

G2,1 G2,2 G2,3 G2,4 G2,5 G2,6 G2,7 G2,8 G2,9 G2,10 G2,11

E1 8 6 8 8 7 1 6 6 5 5 5

E2 9 9 8 8 6 3 1 2 6 6 7

E3 8 8 8 8 8 8 5 6 7 7 9

E4 Row Avg 10 8.75 6 7.25 6 7.5 6 7.5 6 7 5 4.25 3 3.75 7 5.25 5 7.75 5 5.75 5 6.5

G3,1 G3,2 G3,3 G3,4

E1 7 7 7 7

E2 8 8 2 8

E3 8 8 8 9

E4 Row Avg 5 7 5 7 8 6.25 9 8.25

15.1 Issues in International Relations

15.1.2.2

223

G4,1 G4,2 G4,3 G4,4 G4,5

E1 8 0 0 0 0

E2 7 1 1 8 8

E3 8 5 5 5 5

E4 Row Avg 10 8.25 5 2.75 1.5 5 4.5 7 5

G5,1 G5,2 G5,3 G5,4

E1 1 1 0 0

E2 7 4 1 1

E3 1 2 1 1

E4 Row Avg 5 3.5 5 3 5 1.75 5 1.75

G6,1 G6,2 G6,3 G6,4 G6,5 G6,6 G6,7 G6,8

E1 0 2 2 2 7 2 0 0

E2 1 2 5 5 2 0 1 0

E3 5 5 6 6 7 7 1 5

E4 Row Avg 3 2.25 4 3.25 4 4.25 4 4.25 6 5.5 6 3.75 6 2 9 3.5

SubGoals

We next substitute the opinions of the experts concerning how well the U.S. is doing in achieving its goals into the appropriate equation. We then compare this result with their original opinions. For example, E1 ’s opinions for G1 , G2 , G3 , G4 , G5 , G6 are 5, 5, 5, 3, 5, 3, respectively. When these numbers are substituted into the equation G = .23G1 + .19G2 + .19G3 + .13G4 + .17G5 of the AHP, we obtain 4.56. This compares to the number 3 that E1 assigned to how well he felt the U.S. was achieving the overarching goal. Also, E1 ’s opinions for G11 , G12 , G13 , G14 were 6, 6, 1, 2, respectively. When substituted into the equation G1 = .19G1,1 + .29G1,2 + .29G1,3 + .22G1,4 , we obtain the value 3.61 for the value of how well the U.S. is achieving G1 according to E1 . This value is then compared to the value 5 which E1 felt how well the U.S. was doing in achieving G1 .

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G G1 G2 G3 G4 G5 G6 15.1.2.3

E1 E2 E3 E4 Row Avg 4.56 6.77 8.00 7.42 6.69 3.61 4.41 8.40 4.95 5.34 5.99 5.47 7.34 6.02 6.27 7.00 6.14 8.22 6.81 7.04 1.84 5.61 5.47 5.86 4.76 0.63 4.07 1.25 5.05 2.75 1.85 2.08 5.16 5.42 3.63

How Well the U.S. Is Doing in Achieving Goals AHP

Here we follow the same procedure as for AHP using the data from Section 15.1.2.1. G G1 G2 G3 G4 G5 G6

E1 E2 E3 E4 Row Avg 4.51 6.69 7.92 7.35 6.62 3.67 4.42 8.49 5.00 5.40 3.67 4.42 7.26 5.96 5.33 6.93 6.06 8.12 6.68 6.95 1.76 5.55 5.71 5.76 4.70 0.67 4.35 1.25 5.10 2.84 1.79 1.95 4.96 5.45 5.04

We find that generally, the experts’ opinions on how well the US is doing in achieving various goals tend to be less than the corresponding values determined by use of the equations. 15.1.2.4

Analysis

The Guiasu method and Analytic Hierarchy Process returned similar results.

15.1.3

Cooperative Threat Reduction

We present four models aggregating the expert opinions based on four different approaches in the literature on fuzzy mathematics: the Analytic Hierarchy Method Saaty and Vargas (2001)Saaty (2008), Guiasu’s Method Guiasu (1994), Yen’s Method Yen (1992), and the Set–Valued Statistical Method Li and Yen (1995). The models weight the relative importance of the four objectives and the various programs of the objectives. A linear equation for G in terms of the Oi and a linear equation for each Oi in terms of the Pij is then determined. Although the analysis is partly based on meetings with experts from US government agencies, it is neither endorsed by these bodies nor reflects an official position by them.

15.1 Issues in International Relations

225

The three matrices below give the weights of the experts for the Pij as to their importance. The linear equations for the Oi in terms of the Pij are determined in the same manner as were the linear equations for the overarching goal in terms of the Oi . Analytic Hierarchy Process: P1,1 P1,2 P1,3 P1,4 P1,5 Sum

E1 .8 1 1 1 1

E2 .5 1 1 1 1

E3 .9 .9 .9 .9 .9

E4 1 1 .9 .9 .9

E5 1 .8 .9 .7 .6

E6 Row Avg 1 .87 .5 .87 .3 .83 .3 .80 .5 .82 4.18

.21 .21 .20 .19 .20

O1 = .21P1,1 + .21P1,2 + .20P1,3 + .19P1,4 + .20P1,5 .

P2,1 P2,2 P2,3 P2,4 Sum

E1 1 1 1 1

E2 1 1 1 1

E3 1 1 1 1

E4 1 .9 1 .9

E5 .8 .8 .6 .4

E6 Row Avg 1 .97 .8 .92 .8 .90 .6 .82 3.60

.27 .25 .25 .23

O2 = .27P2,1 + .25P2,2 + .25P2,3 + 23P2,4 . E1 E2 E3 E4 E5 E6 Row Avg P3,1 1 1 1 .8 .6 .5 .82 1 Sum .82 O3 = P3,1 . E1 E2 E3 E4 E5 E6 Row Avg P4,1 1 1 1 1 .8 .7 .92 .51 P4,2 .7 1 1 1 1 .5 .87 .49 Sum 1.79 O4 = .51P4,1 + .49P4,2 . Guiasu’s Method O1 = .22P1,1 + .21P1,2 + .19P1,3 + .19P1,4 + .19P1,5 . O2 = .27P2,1 + .26P2,2 + .25P2,3 + .22P2,4 .

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15 Additional Results

O3 = P3,1 . O4 = .52P4,1 + .48P4,2 . Yen’s Method O1 = .21P11 , +.21P1,2 + .20P1,3 + .19P1,4 + .20P1,5 . O2 = .27P2,1 + .26P2,2 + .25P2,3 + 22P2,4 . O3 = P3,1 . O4 = .52P4,1 + .48P4,2 . Set-Valued Statistical Method O1 = .22P1,1 + .23P1,2 + .19P1,3 + .17P1,4 + .18P1,5 . O2 = .31P2,1 + .25P2,2 + .26P2,3 + .18P2,4. O3 = P3,1 . O4 = .53P4,1 + .47P4,2 .

15.2

Issues in Comparative Politics

15.2.1

Successful Democratization

15.2.1.1

Testing the Democracy Model

How well does our model predict democratic consolidation? To answer this question, we estimate the level of democratic consolidation in the twentyseven countries of post-communist Europe and the former Soviet Union. We use fuzzy measures of democratic consolidation derived from the three most commonly used indexes. The data transformation is described in the Appendix. Despite the debate over best measure of democratic consolidation Munck and Verkuilen (2002), there is a high degree of correlation among

15.2 Issues in Comparative Politics

227

these indexes Vanhanen (2003): Vanhanen Vanhanen (2000), Polity IV Marshall and Jagger (2004), and Freedom House House (2005). Pearson’s correlations for the transformed data in our sample range from .708 (for the fuzzy Vanhanen index) to .961 (for the Freedom House index). All three correlations are significant at the p < .000 level. Table 1 compares the results of our model’s predictions with the fuzzy score of democratic consolidation. The results indicate that our model performs quite well. However, we should note that the data set includes only twentyseven countries. That leaves open the question of how generalizable our results are. We combined the three measures of democratic consolidation in the following way. Let C be any country. Let X = {D, N D}, where D denotes democracy and N D nondemocracy. We have three basic probability assignments, one for each of Freedom House, Vanhanen, and Polity IV indexes. That is, m({D}) is the democracy score for C given by the particular index, m({N D}) = 1 − m({D}), m(X) = 0 = m(∅). The democracy scores of the three indexes are then combined using Dempster’s rule of combination. This is done 27 times, once for each country. Since the universe X consists of two elements, m({D}) = Bel({D}). Table 15.1 reports the Dempster Shafer values and the predicted values for the countries in our analysis. We obtained the latter using equation (2) and fuzzy values for the Hj in our data base. The Pearson’s correlation between the two is .905 (significant at p < .000). The Table is reproduced with the permission of the managing editor from the article “Specifying Theories in Comparative Politics: Toward a More Thoroughly Deductive Approach,” New Mathematics and Natural Computation 3(2) 2007, 165-189.

15.2.1.2

How Well the U.S. Is Doing in Achieving Goals

The following matrices provide experts’ opinions on well they felt the U.S. is doing in achieving its goals. The experts scores were based on a scale from 1 to 10 with 1 meaning very poor and 10 meaning very well. G O1 O2 O3 O4 E1 6 5 7 5 2 E2 5 5 6 4 3 E1 E2

P1,1 P1,2 P1,3 P1,4 P1,5 P2,1 P2,2 P2,3 P2,4 P3,1 P4,1 P4,2 7 6 9 9 8 6 7 7 8 9 6 4 7 7 8 8 8 7 6 8 6 6 5 5

The equations for the Analytic Hierarchy Process, Guiasu’s method and Yen’s method are nearly identical. Thus we use only the equations for AHP and

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15 Additional Results

Table 15.1 Democracy Scores Predicted by the Guiasu Model and DempsterShafer Country Albania Armenia Azerbaijan Belarus’ Bosnia Bulgaria Croatia Czech Republic Estonia Georgia Hungary Kazakhstan Kyrgyzstan Latvia Lithuania Macedonia Moldova Poland Romania Russia Serbia Slovakia Slovenia Tajikistan Turkmenistan Ukraine Uzbekistan

Guiasu Dempster-Shafer 0.5680 0.2848 0.1639 0.0599 0.3783 0.7942 0.5828 0.8268 0.7853 0.2413 0.8548 0.1316 0.2210 0.7228 0.7431 0.6992 0.4106 0.6470 0.6921 0.2127 0.4299 0.8321 0.8922 0.1157 0.0135 0.3490 0.0924

0.3479 0.1650 0.0000 0.0000 0.0000 0.9136 0.7167 1.0000 0.8522 0.1490 1.0000 0.0000 0.0073 0.9219 1.0000 0.8309 0.5600 1.0000 0.7583 0.4667 0.0000 0.9851 1.0000 0.0053 0.0000 0.7271 0.0000

the set-valued statistical method (SSM). When the scores of the experts were substituted into the corresponding linear equations the following results were obtained: AHP G O1 O2 O3 O4 E1 4.8 7.84 6.96 9 5.02 E2 4.53 7.66 6.77 6 5 SSM G O1 O2 O3 O4 E1 5.0 7.6 6.87 9 5.06 E2 4.63 7.49 6.76 6 5 These results can then be compared with the experts’ opinions for G, O1 , O2 , O3 , and O4 .

15.2 Issues in Comparative Politics

15.2.2

229

Economic Freedom

Because we are dealing with an entire population rather than with a sample, the comparisons of statistical difference are limited. The general findings are as follows. (a) The scores of the top 15 ranked countries are higher under the Analytic Hierarchy Process (AHP) than the scores computed as the index of Economic Freedom (IEF) by the Heritage Foundation (HF). (b) For countries 16 through 145 by the IEF, scores computed with the AHP are lower for about 80% of those countries. (c) For countries as rated “repressed” by the IEF, the scores computed with the AHP are much smaller. (d ) For almost every country, the score computed with the AHP are much smaller. (e) For countries rated as “repressed” by the IEF, the scores computed with Guiasu method are much smaller. (f ) The major difference in the resultant groups under the three methods (IEF, AHP, G) lies in the number of countries classified as “repressed”. These are countries whose scores are less than 50, according to IEF. In 2009 there were 29 countries ranked as “repressed under IEF scoring, 36 under AH scoring, and 40 under Guiasu scoring. (g) Countries missing subcategory scores, but nevertheless given IEF scores by the Heritage Foundation (all “repressed”) were considered to be “repressed” under AHP and G rankings as well (in light of (c) and (e) above). (h) Correlations on ranks: The rankings of the three systems are tightly correlated. Only the countries that were not missing data were used. corr(IEF, AHP ) = .995433 corr(IEF, Guiasu) = .995002 corr(AH, Guiasu) = .999213 (i) Categories: The Heritage Foundation divides the countries into five categories based on their index of Economic Freedom (IEF). Number of countries in each category for each method: free (80+) mostly free (70+) moderately free (60+ mostly unfree (50+) repressed (< 50)

IEF 7 23 53 67 29

AHP Guiasu 10 6 21 20 50 45 62 68 36 40

(j ) Statistical Analysis: If we consider the scores for the countries as a random sample from some hypothetical population, the population means are found to be not statistically different at the 5% alpha level since the

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15 Additional Results

95% confidence levels overlap. Only those countries not missing data were used. IEF AHP Guiasu Average 60.53 59.63 58.08 St. Dev. 10.43 12.66 12.18 95% CI ±1.53 ±1.86 ±1.79 upper 95% 62.06 61.49 59.87 lower 95% 59.00 57.77 56.29 Given our results taken from the index, it becomes apparent that countries which are economically freer also display high ratings in a number of other political and social indicators. There is, for example, a very strong relationship between the level of economic freedom and the level of prosperity of a given country as measured by the purchasing power parity, given a county’s GDP per capita. Economic freedom also positively correlates with human development in a country, taking into account education, literacy, life expectancy and standard of living ( United Nations Development Index). While there has been some debate about the causality between economic freedom and democracy, it is evident that a positive relationship exists. Based on the Economist Intelligence Unit Democracy Index, measuring economic freedom with a country’s unit index of democracy displays a high positive correlation. Clearly, economic freedom works to promote not just development but development and progress for a society as a whole. More specifically, we use a weighted average approach to determine the economic freedom score for a country. Our rankings of the economic freedom of the countries are highly correlated to the ranking of the Heritage Foundation. We also find a new method to obtain the weights using a generalization of Dempster-Shafer theory. Every measure of subsethood gives a different way of determining belief functions and consequently the weights for the independent variables, Mordeson et al. (2012). Thus our finding opens a whole new research area for determining weights of the independent variables.

15.2.3

Political Stability

The μ and ν values provided by the experts result in μ values much larger than the ν values. In the normalization process of the μ and ν values for the linear equations derived from the AHP, Guiasu and Yen methods, these values become similar in size. We thus use the equation G = .59, .13 to return the μ and ν values to their original relative size. Consequently, we use 13 − the equation G = G+ − 59 G to obtain the final political stability score for each country.

15.2 Issues in Comparative Politics

231

Table 15.2 G+ and G−

Country Name Argentina Australia Austria Azerbaijan Belgium Bolivia Botswana Brazil Burkina Faso Cameroon Canada Chile China Colombia Costa Rica Cœte-d’Ivoire Croatia Czech Republic Denmark Ecuador Egypt Ethiopia Finland France Germany Ghana Greece Hungary India Ireland Israel Italy Japan Jordan Kazakhstan Kenya Latvia Lithuania Malawi Malaysia Mauritius

AHP AHP G+ G− 0.5304 0.4838 0.6442 0.4780 0.6533 0.5241 0.3123 0.2552 0.6113 0.4830 0.5000 0.3961 0.5541 0.4191 0.4786 0.5995 0.3010 0.2193 0.2806 0.4247 0.6505 0.4624 0.6640 0.5417 0.3641 0.5070 0.4908 0.4464 0.5728 0.4868 0.2907 0.2210 0.4519 0.5999 0.4787 0.3806 0.6882 0.5612 0.3837 0.5641 0.3834 0.3690 0.2990 0.2050 0.6371 0.5080 0.6123 0.7256 0.6628 0.5096 0.3325 0.2456 0.6009 0.7344 0.5902 0.4769 0.4578 0.6093 0.7024 0.5708 0.5921 0.6760 0.6060 0.5152 0.6700 0.7449 0.3923 0.5912 0.2667 0.2164 0.2816 0.4276 0.4264 0.3215 0.5236 0.4294 0.4062 0.2793 0.4677 0.3729 0.5536 0.4059

Guiasu G+ 0.4978 0.6197 0.6250 0.2938 0.5870 0.4782 0.5319 0.4487 0.2855 0.2642 0.6283 0.6313 0.3434 0.4603 0.5428 0.2727 0.4330 0.4619 0.6581 0.3578 0.3532 0.2820 0.6051 0.5832 0.6345 0.3130 0.5725 0.5634 0.4337 0.6711 0.5717 0.5743 0.6416 0.3641 0.2513 0.2651 0.4046 0.5002 0.3956 0.4456 0.5337

Guiasu G− 0.4841 0.4639 0.5157 0.2735 0.4607 0.3770 0.4020 0.6224 0.2085 0.4246 0.4326 0.5292 0.5381 0.4436 0.4932 0.2146 0.6135 0.3660 0.5532 0.5857 0.3874 0.1971 0.5140 0.7267 0.4941 0.2335 0.7458 0.4741 0.6279 0.5653 0.6732 0.5113 0.7516 0.6146 0.2167 0.4250 0.3239 0.4285 0.2545 0.3427 0.3863

Yen Yen G+ G− 0.5077 0.4779 0.6223 0.4793 0.6289 0.5216 0.2906 0.2627 0.5898 0.4837 0.4829 0.3963 0.5349 0.4136 0.4485 0.5823 0.2834 0.2099 0.2606 0.3997 0.6287 0.4653 0.6365 0.5387 0.3392 0.4975 0.4694 0.4429 0.5503 0.4901 0.2728 0.2261 0.4352 0.5747 0.4642 0.3820 0.6617 0.5588 0.3586 0.5344 0.3555 0.3697 0.2816 0.2021 0.6026 0.5154 0.5873 0.6902 0.6385 0.5048 0.3122 0.2368 0.5761 0.7048 0.5680 0.4769 0.4297 0.5882 0.6731 0.5715 0.5705 0.6520 0.5815 0.5118 0.6428 0.7132 0.3637 0.5565 0.2467 0.2282 0.2633 0.3915 0.4022 0.3337 0.5044 0.4361 0.3962 0.2668 0.4486 0.3650 0.5359 0.4028

232

15 Additional Results

Mexico 0.5331 0.4658 Moldova 0.3877 0.3104 Morocco 0.3757 0.3419 Mozambique 0.3947 0.3047 Namibia 0.4648 0.3540 Netherlands 0.6651 0.4949 New Zealand 0.6517 0.4747 Nigeria 0.3192 0.4221 Norway 0.6781 0.5599 Philippines 0.4911 0.4184 Poland 0.5728 0.5102 Portugal 0.6333 0.5456 Romania 0.4392 0.4051 Russia 0.4438 0.6177 Senegal 0.4980 0.5985 Singapore 0.5487 0.4171 Slovak Republic 0.4741 0.3932 Slovenia 0.5674 0.4723 South Africa 0.5669 0.4550 Spain 0.6268 0.4885 Sweden 0.6564 0.5348 Switzerland 0.6557 0.6919 Tanzania 0.2808 0.1984 Thailand 0.4926 0.4105 Tunisia 0.4071 0.4002 Turkey 0.5071 0.6576 Ukraine 0.2971 0.2552 United Kingdom 0.6922 0.5269 USA 0.7395 0.5180 Uzbekistan 0.2629 0.2411 Venezuela 0.4103 0.3855 Vietnam 0.3134 0.2780 Zambia 0.3490 0.2375 Zimbabwe 0.2838 0.2701

0.5027 0.3737 0.3503 0.3777 0.4375 0.6395 0.6276 0.3153 0.6478 0.4677 0.5414 0.6003 0.4148 0.4200 0.4695 0.5268 0.4532 0.5429 0.5410 0.5995 0.6283 0.6340 0.2677 0.4635 0.3750 0.4785 0.2771 0.6622 0.7099 0.2463 0.3782 0.2997 0.3327 0.2657

0.4595 0.2924 0.3533 0.2838 0.3486 0.4712 0.4625 0.3999 0.5550 0.3968 0.5091 0.5473 0.4090 0.6161 0.6084 0.3955 0.3892 0.4678 0.4380 0.4723 0.5294 0.6791 0.1822 0.4131 0.4208 0.6773 0.2624 0.5160 0.4982 0.2645 0.4007 0.2854 0.2199 0.2628

0.5093 0.4653 0.3775 0.3025 0.3509 0.3460 0.3801 0.2987 0.4358 0.3637 0.6420 0.4930 0.6297 0.4797 0.3132 0.3834 0.6520 0.5570 0.4721 0.4119 0.5485 0.5042 0.6071 0.5449 0.4215 0.4056 0.4237 0.5807 0.4698 0.5771 0.5264 0.4161 0.4580 0.3946 0.5463 0.4724 0.5440 0.4656 0.6036 0.4904 0.6314 0.5364 0.6335 0.6648 0.2640 0.1903 0.4632 0.4215 0.3774 0.3965 0.4748 0.6332 0.2765 0.2652 0.6664 0.5254 0.7099 0.5187 0.2443 0.2604 0.3822 0.3862 0.3023 0.2888 0.3289 0.2302 0.2743 0.2606

15.2 Issues in Comparative Politics

233

Table 15.3 G Rankings and Orders

Country Name Argentina Australia Austria Azerbaijan Belgium Bolivia Botswana Brazil Burkina Faso Cameroon Canada Chile China Colombia Costa Rica Cœte-d’Ivoire Croatia Czech Republic Denmark Ecuador Egypt Ethiopia Finland France Germany Ghana Greece Hungary India Ireland Israel Italy Japan Jordan Kazakhstan Kenya Latvia Lithuania Malawi Malaysia Mauritius

AHP G AHP G Guiasu G Guiasu G Rank Order Rank Order 0.4238 34 0.3911 35 0.5389 11 0.5174 10 0.5378 13 0.5114 13 0.2561 62 0.2335 63 0.5049 18 0.4855 16 0.4128 35 0.3951 34 0.4618 26 0.4433 24 0.3465 47 0.3116 49 0.2526 64 0.2396 60 0.1870 75 0.1707 75 0.5486 8 0.5330 6 0.5446 10 0.5147 11 0.2524 65 0.2249 69 0.3924 39 0.3625 41 0.4655 23 0.4341 27 0.2420 67 0.2254 68 0.3197 52 0.2978 51 0.3949 38 0.3813 36 0.5646 4 0.5362 4 0.2594 61 0.2288 64 0.3021 56 0.2679 58 0.2538 63 0.2386 61 0.5252 14 0.4919 15 0.4525 29 0.4230 30 0.5505 7 0.5256 8 0.2784 59 0.2616 59 0.4391 31 0.4082 31 0.4851 21 0.4590 21 0.3235 51 0.2953 52 0.5766 2 0.5466 3 0.4431 30 0.4234 29 0.4925 20 0.4616 20 0.5058 17 0.4760 19 0.2620 60 0.2287 65 0.2190 72 0.2035 72 0.1874 74 0.1715 74 0.3556 45 0.3332 45 0.4290 33 0.4058 32 0.3447 48 0.3395 43 0.3855 42 0.3701 39 0.4642 24 0.4485 22

Yen G Yen G Rank Order 0.4024 34 0.5167 11 0.5140 12 0.2327 65 0.4832 19 0.3955 35 0.4438 23 0.3202 48 0.2372 63 0.1726 75 0.5262 8 0.5178 10 0.2296 66 0.3718 38 0.4424 24 0.2230 68 0.3086 51 0.3801 37 0.5386 4 0.2409 61 0.2740 58 0.2371 64 0.4891 15 0.4352 28 0.5273 7 0.2600 59 0.4208 31 0.4630 21 0.3001 52 0.5472 3 0.4268 30 0.4687 20 0.4857 18 0.2410 60 0.1964 72 0.1770 74 0.3287 47 0.4083 32 0.3374 44 0.3681 41 0.4471 22

234

Mexico Moldova Morocco Mozambique Namibia Netherlands New Zealand Nigeria Norway Philippines Poland Portugal Romania Russia Senegal Singapore Slovak Republic Slovenia South Africa Spain Sweden Switzerland Tanzania Thailand Tunisia Turkey Ukraine United Kingdom USA Uzbekistan Venezuela Vietnam Zambia Zimbabwe

15 Additional Results

0.4305 0.3193 0.3004 0.3276 0.3868 0.5560 0.5471 0.2262 0.5548 0.3989 0.4603 0.5131 0.3500 0.3077 0.3661 0.4568 0.3875 0.4634 0.4667 0.5191 0.5386 0.5033 0.2371 0.4021 0.3189 0.3622 0.2409 0.5761 0.6254 0.2098 0.3254 0.2521 0.2966 0.2243

32 53 57 49 41 5 9 70 6 37 27 16 46 55 43 28 40 25 22 15 12 19 69 36 54 44 68 3 1 73 50 66 58 71

0.4015 0.3093 0.2724 0.3151 0.3607 0.5357 0.5257 0.2272 0.5255 0.3803 0.4293 0.4797 0.3247 0.2843 0.3355 0.4396 0.3675 0.4398 0.4445 0.4954 0.5117 0.4844 0.2276 0.3725 0.2822 0.3293 0.2193 0.5486 0.6002 0.1880 0.2900 0.2368 0.2843 0.2078

33 50 57 48 42 5 7 67 9 37 28 18 47 55 44 26 40 25 23 14 12 17 66 38 56 46 70 2 1 73 53 62 54 71

0.4068 0.3109 0.2747 0.3143 0.3556 0.5334 0.5240 0.2287 0.5293 0.3813 0.4375 0.4870 0.3321 0.2957 0.3426 0.4347 0.3710 0.4422 0.4414 0.4955 0.5132 0.4870 0.2221 0.3703 0.2900 0.3353 0.2181 0.5506 0.5956 0.1869 0.2971 0.2386 0.2782 0.2169

33 50 57 49 42 5 9 67 6 36 27 16 46 54 43 29 39 25 26 14 13 17 69 40 55 45 70 2 1 73 53 62 56 71

15.3 Deaf and Hard of Hearing Children

15.2.4

235

Economic Freedom Scores

In the following table, U. K. stands for the United Kingdom. Scores for other countries can be found in Mordeson et al. (2012). The Table is reproduced with the permission of the managing editor from the article “Economic freedom ranking of 171 countries in year 2009: A weighted average approach,” J. Fuzzy Math 2011, 895–906. Country

Rank

Hong Kong Singapore Australia Ireland New Zealand United States Canada Denmark Switzerland U. K. Chile Netherlands Estonia Iceland Luxembourg Bahrain Finland Mauritius Japan Belgium Macau Barbados Austria Cyprus Germany Sweden Bahamas Norway

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

15.3

I.E.F. Score

Country

90.0 Hong Kong 87.1 Singapore 82.6 New Zealand 82.2 Australia 82.0 Ireland 80.7 Denmark 80.5 Canada 79.6 United States 79.4 U. K. 79.0 Switzerland 78.3 Netherlands 77.0 Chile 76.4 Sweden 75.9 Iceland 75.2 Finland 74.8 Estonia 74.5 Luxembourg 74.3 Bahrain 72.8 Belgium 72.1 Austria 72.0 Japan 71.5 Mauritius 71.2 Germany 70.8 Barbados 70.5 Cyprus 70.5 Norway 70.3 Macau 70.2 Spain

Anal. Hier. weight

Country

Guiasu weight

90.71 87.91 84.38 84.27 83.73 82.75 82.62 82.31 81.33 81.31 79.72 79.09 78.84 78.55 78.11 78.01 77.46 74.61 74.40 74.12 73.98 73.91 73.31 73.28 73.15 72.83 71.66 71.39

Hong Kong Singapore New Zealand Australia Ireland Canada United States Denmark Switzerland U. K. Chile Netherlands Iceland Sweden Estonia Finland Luxembourg Bahrain Mauritius Japan Belgium Austria Barbados Germany Cyprus Norway Macau Bahamas

88.10 85.91 81.68 81.33 81.13 80.11 79.75 79.46 78.74 78.56 77.06 76.94 76.42 76.14 76.13 75.86 75.27 72.46 71.98 71.96 71.74 71.66 71.55 71.52 71.38 70.78 69.59 69.22

Deaf and Hard of Hearing Children

The statistical analysis in this section indicate that the students are very successful in closing the language gap, especially those students who completes the sixth year.

236

15 Additional Results Table 15.4 Statistics for Student Scores

n sum avg std z at 99% upper 99 lower 99

age 4 − age 3 age 5 − age 4 age 5 − age 3 age 6 − age 3 19 21 20 11 1.6503 2.2353 4.4645 3.5042 0.086858 0.111765 0.234974 0.318564 0.107249 0.084945 0.135101 0.144662 2.576 2.576 2.576 2.576 0.150239 0.159515 0.312793 0.430922 0.023476 0.064015 0.157154 0.206205

Table 15.4 summarizes the Student Score statistics by age. Each of the above 27 students have either an age 3 and/or and age 4 score. Each of these 27 have either an age 5 and/or age 6 score. All 27 have the lowest of the age 3 or age 4 scores less than .5. 14 of these have the lowest of the age 5 or age 6 scores above .5. 14 = .52. Sample Proportion of success is 27 Sample ’Success’ rate is 52%. Two students whose lowest scores were already above .5 were excluded from the data for analysis purposes. Of those students who were tested at age 6, 9 of 11, or 82% had scores above .5. The other two had scores that were close: .454 and .4326. Of those students who were tested at age 5, only 5 of 16, or 31% had scores above .5. However, 4 of the students with both age 5 and age 6 scores and all 4 of these students had closed the gap by age 5. Hence 9 of 20 students or 45% had scores above .5. It appears that the 6-th year is very valuable in “closing the gap.” There are 16 students with scores for ages 3, 4, and 5. They were used to create a multiple regression model to estimate the average score given the scores at age 3 and 4 : avg yr(5) = .921208 · yr(4) − 0.080761 · yr(3) + 0.16802. Confidence interval for average yr(5) is:

√ (avg yr(5) from model with the specified age 3/4 scores) ±z(99)·se(y). 2 n√ −3 (avg yr(5) from model with the specified age 3/4 scores) ±2.576·0.09141/ 2 13 (avg yr(5) from model with the specified age 3/4 scores) ±0.065308 Example: If yr(3) = .2 and yr(4) = .4, then avg yr(5) = 0.921208 · .4 − 0.08076 · .2 + 0.16802 = 0.552656. The 99% confidence interval on (average yr(5), .2, .4) is 0.487348 to 0.617963.

15.3 Deaf and Hard of Hearing Children

237

Table 15.5 Multiple Regression Model of Student Scores Model Values Parameters m4 m3 b 0.921208 −0.080761 0.16802 se(m4) se(m3) se(b) 0.219476 0.297544 0.086695 n = 16 r2 se(yr) 0.633072 0.09141 #N/A v1 = n − df − 1 = 2 F df 11.21463 13 #N/A v2 = df = 13 0.1877412 0.108624 #N/A F (11.21463, 2, 13) = 0.001478 ss(reg) ss(res)

The schema in Table 15.5 shows labels for parameter estimates in the regression model. The average effect age 3 to age 4 is between +0.0235 and +0.15 with 99% confidence. The average effect age 4 to age 5 is between +0.064 and +0.16 with 99% confidence. The average effect age 3 to age 5 is between +0.157 and +0.313 with 99% confidence. The average effect age 3 to age 6 is between +0.206 and +0.431 with 99% confidence.

 

 

 

A universe is a finite set X of potential suspects in a criminal investigation. Independent or dependent bodies of evidence are provided by testimonies made by witnesses or/and by facts collected by investigators. They induce basic probability assignments on the class P(X) of all subsets of X. The judge or jury reaches a verdict by weighting the evidence at hand in an objective or subjective manner. The paper deals with a mathematical model describing the process of reaching a verdict by probabilistically weighting the available evidence. The classical rules from decision theory proposed by Hooper, Dempster, Bayes, and Jeffrey prove to be special cases of such a weighting process. The evidence induced by a fuzzy set is also discussed.



Subjective logic is very often used in everyday life. When the available evidence is only partial or even contradictory and is provided by not entirely reliable sources, making up our minds about what is going on involves an inevitable weighting of this evidence. The weighting process could be based either on additional information about the reliability of the available sources of evidence or could rely on similar cases from past experiences. Being often subjective, the weighting process of evidence can yield either correct or incorrect conclusions. Detective stories are so popular just because they aim at finding the truth behind a web of partial, incomplete, and often contradictory facts, alibis and testimonies. An investigator relies on the evidence at hand but is free to make correlations and suppositions that could reveal a new interpretation of the existing evidence or show a new direction where to search for new evidence before reaching a final conclusion. In doing this, the detective uses not only the available evidence but also his intuition, feelings, and analogies from his or other people’s past experience. In his more serious pursuit, a scientific researcher is not so far Reaching a Verdict by Weighting Evidence Silviu Guiasu

Department of Mathematics and Statistics, York University, North York, Ontario, Canada

.

from what an intelligent detective is doing.

 

The objective of this paper is to discuss a mathematical model dealing with a set X of potential suspects, one or several bodies of evidence provided by not entirely relevant facts collected by investigators or by testimonies made by not always entirely reliable witnesses, and a jury or a judge trying to reach a verdict by weighting the available evidence. A body of evidence induces a probability (or credibility) distribution on the class P(X) of all possible subsets of X. A judge is associated to a family of conditional weights that are nonnegative functions defined on P(X) conditioned by the given evidence. The weights and bodies of evidence are combined for getting a weighted probability (credibility) distribution on P(X) which is used for reaching a verdict about the culpability or innocence of the subsets of the universe X. An example from an Agatha Christie’s mystery short story is given both for illustrating the significance of the mathematical symbols introduced and for underlying the difference between an objective and subjective weightings of the same body of evidence. The main part of the paper shows how the classical decision making rules formulated by Hooper, Dempster, Bayes, and Jeffrey are special cases of weighting bodies of evidence. The case of a body of evidence induced by a fuzzy set is also discussed.



1. The Model Involving Direct Evidence. The present mathematical model contains a finite set of suspects called universe, bodies of evidence inducing probability (credibility) distributions on the class of all subsets of the universe and a judge associated with a family of conditional weights which is a family of nonnegative functions on P(X) conditioned by the available evidence. The bodies of evidence and the weights are combined in order to get a weighted probability (credibility) distribution on P(X) which may be used for calculating the belief and plausibility of the subsets of the universe X and reach a verdict about the culpability or innocence of the potential suspects. A. The niverse. Let X be a crisp (Cantor) finite set called universe or frame. Let P(X) be the class of all subsets of X. Intuitively, we may think of X as being a set of potential suspects in a criminal investigation in which case P(X) is the class of all possible subsets of suspects, while the empty set ∅ means that there is no suspect in X. B. Bodies of Evidence. made by witnesses or by facts collected by investigators on the culpability or innocence of subsets of the universe X. 

The evidence is provided either by testimonies

U

240



Reaching a Verdict by Weighting Evidence

Reaching a Verdict by Weighting Evidence

241

(a) Simple Evidence. Simple evidence refers to the case when the bodies of evidence are mutually independent. A body of evidence induces a probability (credibility) distribution on P(X). Thus mi : P(X) −→ [0, 1] and  mi (A) = 1. mi (A) ≥ 0, (A ∈ P(X)), A⊆X

The number mi (A) denotes the probability (or credibility) that the suspects belong to the subset A but not to a subsubset of it. The class of focal subsets of X corresponding to mi is F(X; mi ) = {A; A ⊆ X, mi (A) > 0}. The belief, plausibility and ambiguity of A induced by mi are defined by   Bel(A; mi ) = B⊆A,B=∅ mi (B), P l(A; mi ) = B∩A=∅ mi (B), Amb(A; mi ) =



B∩A=∅,B⊆A

mi (B).

If C¯ = X − C is the complementary subset of C, then we obviously have ¯ mi ) = 1, mi (∅) + Bel(C; mi ) + P l(C;

Bel(C; mi ) ≤ P l(C; mi ).

The belief and plausibility functions are well-known in the literature whereas the ambiguity function was defined only in [6]. The ambiguity of A induced by mi , namely Amb(A; mi ), takes into account only the subsets B of the universe X that make both A and its complement A¯ plausible, i.e. it is obtained by summing up the values of mi for all the subsets of X except ¯ the proper subsets of A and the proper subsets of A. (b) Mixed Evidence. A pair of dependent bodies of evidence, let us say witness i and witness j testifying dependently, induce a joint probability (credibility) distribution, namely mij : P(X) × P(X) −→ [0, 1],   mij (A, B) ≥ 0, A⊆X B⊆X mij (A, B) = 1,

where mij (A, B) is the probability (credibility) that witness i focuses on subset A and witness j focuses on subset B. If the bodies of evidence are independent, then mij (A, B) = mi (A) mj (B). If mj (B) > 0, the conditional probability (credibility) distribution on P(X) given B is mi|j (A | B) = mij (A, B)/mj (B). The corresponding class of focal pairs of subsets is F(X, X; mij ) = {(A, B); A ⊆ X, B ⊆ X, mij (A, B) > 0}. In a natural way we can introduce the functions BelBel, BelP l, P lP l, BelAmb, etc., on P(X) × P(X). Thus, for instance,   mij (C, D). BelP l(A, B; mij ) = C⊆A,C=∅ D∩B=∅

242

Reaching a Verdict by Weighting Evidence

Obviously, if the bodies of evidence i and j are independent, then BelP l is equal to Bel × P l, or BelP l(A, B; mij ) = Bel(A; mi ) P l(B; mj ). C. The Judge. A judge, or decision maker, or jury, has to reach a verdict about the culpability or innocence of the suspects based on the available evidence and his own judgement. Mathematically, the judge is associated with a family of conditional weights which is a family of nonnegative functions on P(X) conditioned by the available evidence. Thus, the weights corresponding to the body of evidence #i for which mi is the probability (credibility) distribution induced on P(X) are wi (· | ·) : P(X) × F(X; mi ) −→ [0, ∞), where wi (C | A) represents judge’s weight assigned to the culpability of the subset C ∈ P(X) if the i-th body of evidence focuses on the culpability of the subset A ∈ F(X; mi ). Now, the i-th body of evidence weighted by the judge will provide the new probability (credibility) distribution on P(X) given by  wi (C | A) mi (A), (1) μi (C) = A∈F (X;mi )

abbreviated by μi = wi ⋆ mi . The larger the weight the larger the potential culpability of the corresponding subset of suspects. From mathematical point of view, except nonnegativity, the only condition imposed on the family of weights is   wi (C | A) mi (A) = 1, (2) C∈P(X) A∈F (X;mi )

which implies that μi given by (1) is a probability (credibility) distribution on P(X). The corresponding class of focal subsets F(X; μi ), belief function Bel(·; μi ), and plausibility function P l(·; μi ) may be defined in the usual way. A family of weights is probabilistic if they satisfy the equalities  wi (C | A) = 1, for every A ∈ F(X; mi ). (3) C∈P(X)

In such a case, very often the weights represent the judge’s credibility in the culpability of the subsets of suspects conditioned by the evidence available. Obviously, (3) implies (2) but the converse is not necessarily true. If the family of weights is probabilistic and objective, based exclusively on relative frequencies, then wi (C | A) may be calculated using the standard formula for conditional probabilities. If, however, the family of weights is both nonprobabilistic and subjective, then wi (C | A) simply reflects what the judge believes about the culpability of C if the direct evidence focuses on the subset A and no special rule is necessarily used for getting it.

Reaching a Verdict by Weighting Evidence

243

Particular Cases: (a) If the judge fully relies on the i-th body of evidence, then wi (A | A) = 1 and wi (C | A) = 0 if C is different from A, for every A ∈ F(X; mi ), which implies μi (A) = mi (A). (b) If the judge focuses on B ∈ P(X) regardless of what the i-th body of evidence says, then wi (B | A) = 1 for every A ∈ F(X; mi ), which implies μi (B) = 1. In general, a weight w is fully compatible with the probability (credibility) distribution m induced by a body of evidence if w(A | A) = 1 and w(C | A) = 0 if C is different from A, for every A ∈ F(X; m). Also, w is compatible with m if w(B | A) = 0 for every B ∈ / F(X; m). A weight is not fully compatible with m if either the corresponding body of evidence is not reliable or the judge is biased against or in favour of that evidence. A weight is fully compatible with m if the judge gives full credit to the body of evidence that induces m on P(X). The weighted body of evidence is conclusive only if there is a subset A ∈ P(X) such that μ(A) = 1. Let us notice that if F(X; μi ) = F(X; mi ) and [wi (· | ·)] is a doubly stochastic matrix, then the entropy of μi is larger than or equal to the entropy of mi , which shows that there is more confusion about what subset of X to focus on after weighting the available evidence than before. Maximum confusion is attained when wi (C | A) = 1/si for every A ∈ F(X; mi ), where si is the number of subsets of the class F(X; mi ), in which case the entropy of μi is equal to ln si . More often than not, however, the weighting process diminishes the uncertainty on the class of possible subsets of X, and sometimes even drastically. All the considerations made above may be formulated in a straightforward way when we assign weights wij (· | ·, ·) to a mixed evidence inducing the joint probability (credibility) distribution mij on P(X) × P(X). Thus, for instance, the probability (credibility) distribution induced on P(X) by the weighted mixed (i, j)-th body of evidence is  wij (C | A, B) mij (A, B), (C ∈ P(X)), (4) μij (C) = (A,B)∈F (X,X;mij )

where wi,j (C | A, B) is the judge’s weight of the subset C ∈ P(X) given the mixed evidence (A, B) ∈ F(X, X; mi,j ). Remarks: (a) The idea of associating a body of evidence to a probability distribution m on P(X) is due to Dempster [4] and extensively discussed in Shafer [9]. In their approach, however, m(∅) has to be always equal to zero, an unnecessary restriction because m is not obtained by extending a probability distribution on X to a probability measure on P(X), but is directly defined as a probability distribution on P(X), in which case m(∅) could be positive, corresponding to the frequent case when there is a positive probability of having nobody guilty in the universe X.

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Reaching a Verdict by Weighting Evidence

(b) Let χF : X −→ [0, 1] be a fuzzy set in Zadeh’s sense [12]. Then, the number χF (x) is the degree of membership of the element x ∈ X to the fuzzy set F . Obviously, {χF (x); x ∈ X} is not a probability distribution on X but, as shown in [6] and [7], it induces a probability distribution mF on P(X), defined by   mF (A) = χF (x) [1 − χF (y)], A ∈ P(X), ¯ y∈A

x∈A

describing the body of evidence induced by a fuzzy set F . Let us notice that if 0 < χF (x) < 1, for every x ∈ X, then F(X; m) = P(X). All the considerations made above could be applied to the case when the available evidence is provided by fuzzy sets defined on X. 2. The Model Involving Indirect Evidence. There are cases when the judge has to reach a verdict using indirect evidence. This more general case may be treated like the previous one with some minor changes. Thus, let X, Y , and Z be three finite crisp (Cantor) sets, where X is the judge’s universe and Y, Z are universes of two bodies of evidence. (a) Simple Evidence. If a body of evidence induces a probability (credibility) distribution m on P(Y ) and the judge’s weights are w(· | ·) : P(X) × F(Y ; m) −→ [0, ∞ ], then the weighted probability (credibility) distribution on P(X) will be  μ(C) = w(C | A) m(A), A∈F (Y ;m)

for every C ∈ P(X), where 



w(C | A) m(A) = 1.

C∈P(X) A∈F (Y ;m)

(b) Mixed Evidence. If two dependent bodies of evidence, say #1 and #2, induce a joint probability (credibility) distribution m1,2 on P(Y ) × P(Z), and w1,2 (· | ·, ·) is the family of judge’s weights on P(X) given the mixed evidence from F(Y, Z; m1,2 ), then the weighted probability (credibility) distribution on P(X) is  w1,2 (C | A, B) m1,2 (A, B), (5) μ1,2 (C) = (A,B)∈F (Y,Z;m1,2 )

where 



C∈P(X) (A,B)∈F (Y,Z;m1,2 )

for every (A, B) ∈ F(Y, Z; m1,2 ).

w1,2 (C | A, B) m1,2 (A, B) = 1,



 



All probability distributions involved in the above analysis could be either objective, i.e. based on relative frequencies of the outcomes of probabilistic experiments repeated independently, or subjective, in which case they are rather called credibility distributions. The weights themselves can be either objective or subjective. The subjective weights can be based on judge’s past experience, logic, feelings, and/or imagination. The result of the weighting process could be a right or wrong verdict, expressed in a sharp or vague form. An Agatha Christie’s mystery short story, namely ”The Tuesday Night Club” ([3], pp.1-14), offers a very good example of different kinds of evidence and weights that could be used in reaching correct, partially correct, or incorrect verdicts. Briefly, Mrs.Jones has died and the characters are: J (Mr.Jones, i.e. Mrs.Jones’s husband), C (Miss Clark, Mrs.Jones’s companion), Dr (the doctor), D (the doctor’s daughter), and G (Gladys Linch, the maid). Thus, the universe is X = {J, C, Dr, D, G}. In what follows, the body of evidence #i implies (⇒) the credibility distribution mi : P(X) −→ [0, 1], where mi ({C, D}), for instance, means the credibility that C and D, together, have murdered Mrs. Jones, as a possible result of the body of evidence #i. Also, mi (∅) means the credibility induced by the body of evidence #i that nobody is guilty and the death has occurred as an unfortunate accident. Thus, mi (∅) = 1 means that the body of evidence #i does not incriminate anybody. Also, only the positive weights are going to be explicitely mentioned. Body of Evidence #1: J, C, and Mrs.Jones sat down to a supper consisting of tinned lobster and salad, trifle, and bread and cheese. Later in the night all three were taken ill, and a doctor was hastily summoned. Two people recovered, Mrs.Jones died. Death was considered to be due to ptomaine poisoning, a certificate was given to that effect and the victim was duly buried. ⇒ m1 (∅) = 1. Body of Evidence #2: J had been staying the previous night at a small hotel in Birmingham and the chambermaid there found on the blotting paper the text: ‘Entirely dependent on my wife . . . when she is dead I will . . . hundreads and thousands . . . ’ Also, J had been very attentive to D. He also benefited by his wife’s death to the amount of £8000. ⇒ m2 ({J}) = 1. Body of Evidence #3: An exhumation was ordered. The result of the autopsy was that the deceased lady had died of arsenical poisoning. ⇒ m3 ({J}) = m3 ({C}) = m3 ({Dr}) = m3 ({D}) = m3 ({G}) = 1/5. Reaching a Verdict by Weighting Evidence



245

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. Body of Evidence #4: J’s testimony (The friendship with D had been over two months before the death; in Birmingham he wrote in fact an innocent letter to his brother; he returned from Birmingham just as supper was being served.) ⇒ m4 (∅) = 1. Body of Evidence #5: Dr’s testimony and the investigation on him made by Scotland Yard. ⇒ m5 (∅) = 1. Body of Evidence #6: After supper, J had gone down to the kitchen and had demanded a bowl of corn-flour for his wife prepared by G. He had waited in the kitchen and then carried it up to his wife’s room himself. J had motive and opportunity to kill his wife. ⇒ m6 ({J}) = 1. Body of Evidence #7: C’s testimony. (The whole of the bowl of cornflour was drunk by her. As she was banting at the time, she was always hungry and Mrs.Jones had changed her mind about tasting the corn-flour.) ⇒ m7 (∅) = 1. At this moment, Sir Henry, a commissioner of Scotland Yard, asked Joyce, Mr.Petherick, Raymond, and Miss Marple to reach individually a verdict based on the seven independent bodies of evidence. In computation, we are using here the formula (4) with m1 2, 3, 4, 5, 6, 7, = m1 m2 m3 m4 m5 m6 m7 . Mr.Petherick (a solicitor, relying on facts and money): J was guilty and C sheltered him for money, lying about drinking the corn- flour. ⇒ w({J, C} | ∅, {J}, A3 , ∅, ∅, {J}, ∅) = 1, where, here and subsequently, A3 has to be successively replaced by the possible subsets induced by the body of evidence #3, namely {J}, {C}, {Dr}, {D}, and {G}. Applying (4), we get the verdict μ({J, C}) = 1. Joyce (a young artist, relying on intuition): C was guilty because probably she was in love with J and hated his wife. ⇒ w({C} | ∅, {J}, A3 , ∅, ∅, {J}, ∅) = 1,

⇒ μ({C}) = 1.

Raymond (a young writer relying on imagination ): D was guilty. After noticing the poisoning symptoms, Dr sent a messenger home for some opium pills for Mrs.Jones to relieve her acute pain. D, who was in love with J, had motive and opportunity. Consequently, she sent back pills containing pure white arsenic. ⇒ w({D} | ∅, {J}, A3 , ∅, ∅, {J}, ∅) = 1,

⇒ μ({D}) = 1.

Miss Marple (an old lady relying on life experience and analogy): A similar case happened in the village St.Mary Mead. G murdered Mrs.Jones, pushed by J who made her his murder instrument. ⇒ w({G, J} | ∅, {J}, A3 , ∅, ∅, {J}, ∅) = 1,

⇒ μ({G, J}) = 1.

Reaching a Verdict by Weighting Evidence

247

Scotland Yard (objective weighting giving credit only to the available evidence): Taking into account only the seven bodies of evidence and calculating how many times each suspect has been involved, we get w(∅ | ∅, {J}, {J}, ∅, ∅, {J}, ∅)

=

w({J} | ∅, {J}, {J}, ∅, ∅, {J}, ∅)

=

w(∅ | ∅, {J}, {C}, ∅, ∅, {J}, ∅)

=

w({J} | ∅, {J}, {C}, ∅, ∅, {J}, ∅)

=

w({C} | ∅, {J}, {C}, ∅, ∅, {J}, ∅)

=

w(∅ | ∅, {J}, {Dr}, ∅, ∅, {J}, ∅)

=

w({J} | ∅, {J}, {Dr}, ∅, ∅, {J}, ∅)

=

w({Dr} | ∅, {J}, {Dr}, ∅, ∅, {J}, ∅)

=

w(∅ | ∅, {J}, {D}, ∅, ∅, {J}, ∅)

=

w({J} | ∅, {J}, {D}, ∅, ∅, {J}, ∅)

=

w({D} | ∅, {J}, {D}, ∅, ∅, {J}, ∅)

=

w(∅ | ∅, {J}, {G}, ∅, ∅, {J}, ∅)

=

w({J} | ∅, {J}, {G}, ∅, ∅, {J}, ∅)

=

w({G} | ∅, {J}, {G}, ∅, ∅, {J}, ∅)

=

4 , 7 3 , 7 4 , 7 2 , 7 1 , 7 4 , 7 2 , 7 1 , 7 4 , 7 2 , 7 1 , 7 4 , 7 2 , 7 1 . 7

No evidence pointed at subsets with more than one suspect. From (4), we get the following values 1 μ({C}) = 35 ; μ(∅) = 74 ; μ({J}) = 11 35 ; 1 1 1 . μ({Dr}) = 35 ; μ({D}) = 35 ; μ({G}) = 35

Consequently, although there was a high credibility (11/35) in J’s culpability, the evidence was not considered conclusive and no arrest was made. Body of Evidence #8: One year later, G’s testimony before dying. (J promissed to marry her when his wife was dead. Following J’s instructions,

248

Reaching a Verdict by Weighting Evidence

she put arsenic in trifle. Only Mrs.Jones ate it because C was on diet and J knew about the poison. She had a child with J. The child died at birth and J deserted her for another woman.) ⇒ m8 ({G, J}) = 1. Scotland Yard (based on this final evidence): ⇒ w({G, J} | {G, J}) = 1, ⇒ μ({G, J}) = 1.

(a) Hooper’s Rule. Let us take two independent bodies of evidence provided by two witnesses inducing the probability distributions m1 and ¯ where m2 on P(X), respectively. Let F(X; m1 ) = F(X; m2 ) = {A, A}, ¯ A = X − A, which shows that both witnesses focus on the same subsets A and A¯ of X. The weights are defined as ¯ = w1,2 (A | A, ¯ A) = w1,2 (A¯ | A, ¯ A) ¯ = 1, w1,2 (A | A, A) = w1,2 (A | A, A) which means that the judge gives full credit to A if at least one witness ¯ Then, focuses on A and full credit to A¯ only if both witnesses focus on A. according to (1), we have ¯ = m1 (A) ¯ m2 (A), ¯ μ1,2 (A)

μ1,2 (A) = 1 − [1 − m1 (A)][1 − m2 (A)].

According to Lindley [9], this rule for combining evidence was used by G. Hooper in 1685. (b) Dempster’s Rule. Let us take two independent bodies of evidence provided by two witnesses inducing the probability (credibility) distributions m1 and m2 on P(X). The judge takes into account only the common part of the focal sets of the two witnesses with the only positive weights ⎡ ⎤−1  m1 (C) m2 (D)⎦ , w1,2 (A ∩ B | A, B) = ⎣1 − C∈F (X;m1 ),D∈F (X;m2 ),C∩D=∅

for all A ∈ F(X; m1 ), B ∈ F(X; m2 ), A∩B = ∅. In such a case (4) becomes



   μ1,2 (C) = m1 (A) m2 (B) , m1 (A) m2 (B) 1− C=A∩B

A∩B=∅

which is Dempster’s rule [4] of combining two independent bodies of evidence. It gives equal credit to the common evidence and discards any other evidence. According to [10], in the special case of a universe containing only two elements, this rule was used by J.H. Lambert in his Neues Organon published in 1764.

Reaching a Verdict by Weighting Evidence

249

(c) Jeffrey’s Rule. First Interpretation: Let us take two bodies of evidence inducing the probability (credibility) distributions m1 and m2 on P(X), respectively, where X = Y × Z is the Cartesian product of two finite crisp (Cantor) sets. Assume F(X; m1 ) = {{(y, z); z ∈ Z}; y ∈ Y },

m1 ({(y, z); z ∈ Z}) = pY (y);

F(X; m2 ) = {{(y, z); y ∈ Y }; z ∈ Z},

m2 ({(y, z); y ∈ Y }) = qZ (z),

where pY and qZ are probability distributions on Y and Z, respectively. Let pZ (· | y) be a conditional probability distribution on Z given y ∈ Y , and pZ the prediction probability distribution on Z defined by  pZ (z) = pZ (z | y)pY (y), y∈Y

where pY is interpreted as being the prior probability distribution on Y . Let   w(C | A, B) m1 (A) m2 (B), μ1,2 (C) = A∈F (X;m1 ) B∈F (X;m2 )

where F(X; μ1,2 ) = {{(y, z)}; y ∈ Y, z ∈ Z} = {A∩B; A ∈ F(X; m1 ), B ∈ F(X; m2 )} and w({(y, z)} | {(y, z); z ∈ Z}, {(y, z); y ∈ Y }) = pZ (z | y)/pZ (z). Then, we have μ1,2 ({(y, z)}) =

pY (y) pZ (z | y) qZ (z), pZ (z)

which is a probability distribution on Y × Z. Its marginal probability distribution, namely  μ1,2 ({(y, z)}) pY (y | qZ ) = Bel({(y, z); z ∈ Z}; μ1,2 ) = z∈Z

=

 pY (y) pZ (z | y) qZ (z), pZ (z)

(6)

z∈Z

is Jeffrey’s rule ([8], [7]) for calculating the posterior probability distribution on Y given the actual probability distribution qZ on Z.

 

Second Interpretation: Jeffrey’s rule may be obtained more directly from (5), as a weighting with indirect evidence. Indeed, let X and Y be two finite crisp (Cantor) sets and m a probability distribution on P(Y ) such that F (Y ; m) = {{y}; y ∈ Y },

m({y}) = q(y),

where q is the actual probability distribution on Y . Taking the only positive weights on P(X) to be p(y | x) p(x) x∈X p(y | x) p(x)

w({x} | {y}) = 

where p is a prior probability distribution on X and p(· | x) is a conditional probability distribution on Y given x ∈ X, the weighted probability distribution (5) becomes  p(x | q) = μ({x}) = w({x} | {y}) q(y) y∈Y

=



y∈Y

p(y | x) p(x) q(y), x∈X p(y | x) p(x)



(7)

which is Jeffrey’s rule for calculating the posterior probability distribution on X. In this case, F(X; μ) = {{x}; x ∈ X}. (d) Bayes’ Rule. Taking 1, if z = z0 ; qZ (z) = 0, if z = z0 , the formula (6) becomes Bayes’ rule for calculating the posterior probability distribution, namely pY (y | z0 ) = p( y | qZ ) =

pZ (z0 | y) pY (y) . pZ (z0 )

Bayes’ rule may also be obtained from (7) by taking q to be a degenerate probability distribution focussed on a single element {y0 }, i.e. q({y0 }) = 1.

The process of reaching a verdict essentially depends on how the available evidence is used by the judge or jury. The evidence may be significant, partially relevant, or misleading and the judge may use it in an objective or subjective way. The paper discusses a general mathematical model of 250

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251

weighting the available evidence. The classical rules from decision theory proposed by Hooper, Dempster, Bayes, and Jeffreys prove to be special cases of such a weighting of evidence process. One of the eight referees of this paper wrote: ’The framework proposed is shown to generalize some known rules but nothing is said about its rationality.’ It is often said that only Bayes’ rule is rational and the other alternatives are not. The present paper pleads in favour of a relaxed or relative rationality. Instead of the drastic dichotomy ’rational or not’, a mathematical model in decision theory should be classified as ’rational with respect to a specific way of weighting the available evidence.’ It is often forgotten that Bayes’ formula is a rigorous mathematical statement only when it is applied to the events of the same probability space. If this formula is used by analogy to describe the connection between different types of entities, like the available hypotheses on one side and the outcomes of an auxiliary experiment on the other side, for instance, then it is only a viewpoint, one of the many possible ways of describing the respective interdependence. Perhaps Bayes realized this when he hesitated to publish his seminal paper [1] during his lifetime. Fortunately for us, a close friend of his found the manuscript among his papers and published it posthumously.

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Index

actual probability distribution, 124 aggregation, 100 aggregation operator, 27 AHP, see {nalytic Hierarchy Process}93 algebraic product, 21 algebraic sum, 22 alpha–cut, 9 strong, 10 amalgamate, 100 ambiguity, 5 Analytic Hierarchy Method, 221, 224 Analytic Hierarchy Process, 76, 93 antireflexive, 30 antisymmetric, 31, 36 antitransitive, 31 Arms Control and Disarmament Agency, 93 asymmetric, 31 autonomous judiciary, 134 averaging operator, 26 B-52, 65 basic probability assignment, 42 belief, 42 binary relation, 30 Biological Threat Reduction Prevention, 71, 72 Boehm Test of Basic Concepts, 86 bounded difference, 21 bounded sum, 22 Center of Strategic and International Studies, 67 characteristic function, 4

Chemical Weapons Destruction, 71 Clinical Evaluation of Language Function, 86 Cold War, 61 Common Principles Charter, 69 compatibility, 39 compatibility relation, 31 complement fuzzy set, 17 Sugeno, 18 threshold, 19 Yager, 18 conflict, 47 Consistency Index, 103 Consistency Ratio, 103 Container Security Initiative, 62 Cooperative Threat Reduction, 70, 107, 131, 157, 181 core, 11 Correlates of War, 80 Corruption Perceptions Index, 80 credibility, 119 crisp set, 7 data fusion, 47 decentralization of the state, 134 decomposition theorem, 10 Democratic Consolidation, 73, 147, 226 Dempster’s rule, 123 Dempster-Shafer theory, 85 Domestic Nuclear Detection Office, 63 drastic product, 21 drastic sum, 22 drastic union, 22

262 Early Zadeh implication, 26 Economic Freedom, 110, 134, 159, 183, 229 Economic strength, 81 eigenvalue, 94 eigenvector, 94 emocratic consolidation, 134 equilibrium, 19 equivalence, 31 equivalence relation, 10 evidence theory, 41 Failed States, 184 Fissile Material Storage Facility, 72 Freedom House, 145, 227 Fund for Peace, 84 fuzzy equivalence relation, 37 implication operator, 24 similarity relation, 37 fuzzy power set, 8, 12 Fuzzy Preference Relations, 199 fuzzy relation, 32 antisymmetric, 36 compatibility, 32, 39 linear order, 39 order, 39 partial order, 39 preorder, 38 reflexive, 36 symmetric, 36 fuzzy set, 122 complement, 17 core, 11 intersection, 19 membership, 6 subset, 11 support, 10 union, 22 fuzzy set operations, 11 intersection, 14 union, 15 fuzzy set theory, 5 fuzzy sets, 5 Gaines–Rescher implication, 26 GB-2 long range bombers, 65 Gödel implication, 24 generalized means, 28

Index Gingrich-Mitchell Task Force, 68 Goguen implication, 26 Governmental legitimacy, 81 GOWA operator, 190 Greenhouse Gas Emissions, 69 ground probabilities, 54 Guiasu, 119 Guiasu Method, 76, 119, 127, 221 Hearing Impairment, 85, 116, 138, 173, 184, 235 IAEA, 64 IEF see(Index of Economic Freedom), 77 image set fuzzy set, 9 implication Early Zadeh, 26 Gaines–Rescher, 26 Gödel, 24 Goguen, 26 Kleene-Dienes, 25 Larsen, 26 Lukasiewicz, 26 Mamdani, 25 Reichenbach, 26 standard strict, 24 Yager, 26 implication operator, 24, 163 indeterminacy, 187 Index of Economic Freedom, 77 Institutional stability, 81 Internal Leadership Program, 69 International Country Risk Guide, 80 International Living’s Quality of Life Index, 75 International Maritime Organization, 63 International Trade/World Trade Organization, 69 intersection, 14 fuzzy set, 19 intuitionistic fuzzy set, 78, 187 intuitionistic index, 188 irreflexive, 30 Jeffrey’s Rule, 123–125, 145 Joint Technology Development Center, 70

Index

263

judge, 120

OWA operator, 190

Kleene-Dienes implication, 25

Parent Infant Communication Scale, 86 partial order, 31 Peabody Picture Vocabulary Test, 86 Peace Building Commission and Support Office, 68 PEPAR, 68 plausibility, 42 Political Stability, 110, 135, 160, 195, 230 Polity IV, 146, 227 possibility measures, 45 posterior probability distribution, 124 power set fuzzy, 8 preorder relation, 31 Preschool Language Assessment Instrument, 86, 116 Preschool Language Scale, 86, 116 Privatization, 73 privatization, 134 probabilistic, 121, 125 probabilistic sum, 22

Language Sample Analysis, 86, 116 Larsen implication, 26 low levels of corruption, 134 Lukasiewicz implication, 26 MacArthur Communicative Development Inventory, 86 MADM, 189 Major Episodes of Political Violence, 80 Mamdani implication, 25 Mathematical Methods of Operations Research, 93 max–min, 34 max–t transitivity, 37 max-min transitivity, 37 membership, 187 membership function, 6 Millennium Challenge Corporation, 68 Minorities At Risk, 80 Minuteman ICBMs, 65 mixed evidence, 120 Multi-Attribute Decision Making, 189 multi-party system, 134 n-ary relation, 32 National Science Foundation, 70 National Security Education Program, 69 necessity measures, 45 non–membership, 187 nontransitive, 31 Nuclear Deterrence, 61, 104, 128, 164, 180 Nuclear Weapons Storage Security, 71 Nuclear Weapons Transportation Security, 71 omega operator, 23 ω operator, 23 operator aggregation, 27 averaging, 26 ordered weighted averaging, 29 ordered weighted averaging, 29 OWA, 29

Quality of Life, 108, 132, 182 quasi-equivalence, 31 Random Consistency Index, 103 Receptive-Expressive Emergent Language Scale, 86 reflexive, 30, 36 reformed public administration, 134 regime type, 134 Reichenbach implication, 26 relation, 30 antireflexive, 30 antisymmetric, 31 antitransitive, 31 asymmetric, 31 binary, 30 compatibility, 31, 39 equivalence, 31 fuzzy, 32 fuzzy linear order, 39 fuzzy order, 39 fuzzy partial order, 39 fuzzy preorder, 38 irreflexive, 30

264 n-ary, 32 nontransitive, 31 partial order, 31 preorder, 31 quasi-equivalence, 31 reflexive, 30 symmetric, 31 total order, 31 transitive, 31 residuum operator, 23 Reynell, 86 Rosseti Infant Toddler Language Scale, 86 s-norm, 22 scalar cardinality, 8 set theory, 3 Set–Valued Statistical Method, 76, 177, 224 similarity relation, 37 simple evidence, 120 Ski-Hi Language Scale, 86 Smart Power and Deterrence, 105, 129 standard strict implication, 24 Stateness, 73 stateness, 134 Strategic Nuclear Arms Elimination, 71 Strategic Offensive Arms Elimination, 71 strong alpha–cut, 10 Strong external (foreign) relations, 81 Structured Photographic Expressive Language Test, 86 subset, 11 subsethood, 163 Successful Democratization, 73, 133, 158, 169 sup–t transitivity, 37 sup-min transitivity, 36 support, 10 symmetric, 31, 36 t–conorm algebraic sum, 22 bounded sum, 22 drastic sum, 22 drastic union, 22 probabilistic sum, 22 t-conorm, 22 t–norm

Index algebraic product, 21 bounded difference, 21 drastic product, 21 t-norm, 19 Test of Syllable Structure and Sequencing, 86 the Defense Threat Reduction Agency, 70 The President’s Malaria Initiative, 69 Thomas Saaty, 93 total order, 31 transitive, 31 transitive closure, 39 transitivity max–t, 37 max-min, 37 sup–t, 37 sup-min, 36 triangular conorm, 22 triangular norm, 19 Trident submarines, 65 UN Department of Peacekeeping Operations, 68 uncertainty, 5 union, 15 fuzzy set, 22 United Nations Development Index, 230 United Nations Human Development Index, 78 vagueness, 5 Vanhanen, 227 Vanhanen index, 145 Weapons of Mass Destruction Infrastructure Elimination, 71 Weapons of Mass Destruction Proliferation Prevention Initiative, 72 World Bank’s World Development Indicator’s Database, 79 World Customs Organizations, 63 World Health Organization, 68 Yager implication, 26 Yen’s Method, 76, 85, 155 evidence fusion, 155 Yen’s method, 224

Index

265

Greek Alphabet Letters A α B β Γ γ ∆ δ E ε Z ζ M η Θ θ

Names Alpha Beta Gamma Delta Epsilon Zeta Eta Theta

Letters I ι K κ Λ λ M μ N ν Ξ ξ O o Π π

Names Iota Kappa Lambda Mu Nu Xi Omicron Pi

Letters P ρ Σ σ T τ Υ υ Φ φ Xχ Ψ ψ Ω ω

Names Rho Sigma Tau Upsilon Phi Chi Psi Omega

Fuzzy Sets and Fuzzy Logic English Logical Set Theory Fuzzy not A negation ¬A complement Ac 1−a A and B conjunction A ∧ B intersection A ∩ B min(a, b) A or B disjunction A ∨ B union A ∪ B 7max(a, b) 1 b≥a if A then B implication A → B subsethood A ⊆ B b a>b