Fractal Functions, Dimensions and Signal Analysis [1st ed.] 9783030626716, 9783030626723

Citation preview

Understanding Complex Systems

Santo Banerjee D. Easwaramoorthy A. Gowrisankar

Fractal Functions, Dimensions and Signal Analysis

Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems— cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations, and the “Springer Briefs in Complexity” which are concise and topical working reports, case studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Indexed by SCOPUS, INSPEC, zbMATH, SCImago. Series Editors Henry D. I. Abarbanel, Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA, USA Dan Braha, New England Complex Systems Institute, University of Massachusetts, Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, USA; Hungarian Academy of Sciences, Budapest, Hungary Karl J. Friston, Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Jürgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Ronaldo Menezes, Department of Computer Science, University of Exeter, UK Andrzej Nowak, Department of Psychology, Warsaw University, Warszawa, Poland Hassan Qudrat-Ullah, School of Administrative Studies, York University, Toronto, Canada Linda Reichl, Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster, Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer, System Design, ETH Zürich, Zürich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zürich, Zürich, Switzerland Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria

Understanding Complex Systems More information about this series at http://www.springer.com/series/5394

Santo Banerjee D. Easwaramoorthy A. Gowrisankar •

•

Fractal Functions, Dimensions and Signal Analysis

123

Santo Banerjee Department of Mathematical Sciences Politecnico di Torino Turin, Italy

D. Easwaramoorthy Department of Mathematics School of Advanced Sciences Vellore Institute of Technology Vellore, Tamil Nadu, India

A. Gowrisankar Department of Mathematics School of Advanced Sciences Vellore Institute of Technology Vellore, Tamil Nadu, India

ISSN 1860-0832 ISSN 1860-0840 (electronic) Understanding Complex Systems ISBN 978-3-030-62671-6 ISBN 978-3-030-62672-3 (eBook) https://doi.org/10.1007/978-3-030-62672-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The traditional interpolation techniques generate smooth or piecewise differentiable interpolation function, despite the data points being irregular. Nevertheless, most of the natural objects such as lightning, clouds, mountain ranges, and wall cracks have irregular and complex structures in which Euclidean geometry cannot be applied successfully to describe them, since Euclidean geometry which deals with regular objects and functions is used to approximate the data obtained from the realistic environment. Hence, there is a need of a nonlinear tool in the approximation theory to resolve these problems. Barnsley invented the fractal interpolation function based on the theory of iterated function systems, which precisely suited for the approximation of naturally occurring functions which possess some kind of self-similarity under magnification. In the approximation theory, Fractal Interpolation Functions (FIF) play a vital role. It is important and necessary to discuss the variance ratio of such functions. But they are often nowhere differentiable, everywhere continuous. Hence, FIFs are sophisticated in approximating the rough curve and precisely reconstructing the naturally occurring functions when compared with classical interpolants. Nevertheless, a smooth curve is needed to approximate some kind of functions which include self-similarity under magnification. Hence, Barnsley presented the indefinitely integrated fractal interpolation function generated from some special type of the iterated function system which interpolates a certain set of data, and the integral of the fractal interpolation function retains its properties. Further, they have explored the construction of n-times frequently differentiable FIF when the derivative values reaching up to the nth order are available at the initial endpoint of the interval. The examples of differentiable functions cannot actually be considered as fractals, but they retain the name fractal interpolation function because of the flavor of the scaling in the equation and the Hausdorff–Besicovitch dimension of their graphs are non-integer. Due to the sophisticated usage of such fractal interpolants in nonlinear approximation, there are continuous efforts have extended on fractal interpolation functions.

v

vi

Preface

As mentioned, the literature and natural appearance of fractal functions motivated us to write the book titled Fractal Functions, Dimensions and Signal Analysis with the following facts. Initially, this book focuses on the construction of fractals in a metric space through various iterated function systems. Chapter two discusses the mathematical background behind the fractal interpolation functions and presents its graphical representations. In chapters three, we appeal fractional integrals and fractional derivatives on a linear fractal interpolation function. Further, the existence of a fractal interpolation function with a countable iterated function system is demonstrated while taking xn as a monotone and bounded sequence, and yn as a bounded sequence. Further, fractional order integral and integer-order integral of an FIF for a sequence of data are described when the value of the integral of an FIF is predefined at the initial endpoint or the final endpoint. Finally, the fractional derivatives of an FIF and its fractional integrals are given due importance as they are more precise and suitable for FIFs which are nowhere differentiable but continuous at all points. Hence, fractional calculus is a mathematical operator which best suits for analyzing such an FIF and which may also change the fractal dimension, when applied to fractal objects or functions. As an application part, this book discusses the biomedical signal analysis in two chapters. Chapter 5 concisely presents the overview of signal processing and its mathematical background. Chapter 6 presents a wavelet-based denoising method for the recovery of the Electroencephalogram (EEG) signal contaminated by non-stationary noises and investigates the recognition of healthy and epileptic EEG signals by using multifractal measures such as generalized fractal dimensions, since the identification of abnormality in EEG signals is the vast area of research in neuroscience. Especially, the classification of healthy and epileptic subjects through EEG signals is a crucial problem in the biomedical sciences. Denoising of EEG signals is another important task in signal processing. The noises must be corrected or reduced before the subsequent decision analysis. Moreover, this chapter explores the three different methods to explicitly recognize the healthy and epileptic EEG signals: Modified, Improved, and Advanced forms of generalized fractal dimensions. The newly proposed scheme is based on generalized fractal dimensions and the discrete wavelet transform for analyzing the EEG signals. Fractal functions have been covered with wavelet transformation and signals in almost all the books published so far. This book for the first time emphasizes the fractional calculus of fractal functions in various settings with applications of fractal dimensions in biomedical signal analysis. We are delighted to welcome our readers for a walkway in the domain of fractal functions, fractal dimensions, and their manifestations in biomedical signals. This book has been composed with six sections which are organized as follows. Chapter 1 starts with a broad outline of the iterated function system of contraction mappings. The Barnsley framework of IFS has been extended to the local iterated function system and the countable iterated function system for constructing the deterministic fractals, which is offered in Chap. 1. Consequently, the existence of an attractor of the local countable iterated function system is investigated. Moreover, it is proved that the local attractor of the local CIFS is expressed as the

Preface

vii

limit of a convergence sequence of attractors of the local IFS. At the end of Chap. 1, various notions of fractal dimensions are succinctly presented which will be applied in the forthcoming chapters. In Chap. 2, the concepts of fractal interpolation functions and their generalizations such as hidden variable fractal interpolation and a-fractal function are described. The last part of the chapter is devoted to the traditional calculus theory, an experienced tool that supports to implement the concept of spline approximation to fractal functions. In Chap. 3, the Riemann–Liouvllie fractional calculus of different types of fractal interpolation functions is explored. The Riemann–Liouville fractional integral of order b [ 0 of a quadratic fractal interpolation function with both constant and variable scaling parameter is examined. In addition to that, the prominent influence of free parameters in the shape of a fractal interpolation function is illustrated with suitable examples. In the traditional method of fractal interpolation and in practically the entirety of its expansions referenced beforehand, a fractal interpolation function is constructed for a finite data set. That is, the interpolation theory deals with the reconstruction of a continuous function associated with the finite set of data fðxn ; yn Þ : n ¼ 1; 2; . . .; N g. Nevertheless, there are practical phenomena where infinite data points might be justified, for example, in the theory of sampling and reconstruction. Recently, the standard development of the univariate fractal interpolation function is reached out from the finite case to the instance of a countable set of data. This development of a fractal interpolation function for a prescribed countable set of data frames the reason for the discussion of fractional calculus of a fractal interpolation function for countable data in Chap. 4. However, Chap. 4 introduces and describes the sequence of data and the corresponding interpolation function which is a generalization of the Secelean framework. The existence of the continuous function f which interpolates the sequence of prescribed data fðxn ; yn Þ : n 2 Ng is discussed, where ðxn Þ1 n¼1 is a is a bounded sequence of real numbers. monotonic real sequence and ðyn Þ1 n¼1 Besides, the existence of a countable iterated function system is investigated when the fractal interpolation function for a sequence of data is given. Further, the existence of the Riemann–Liouville fractional integral and the derivative of fractal interpolation function is established. In Chaps. 5 and 6, the designed multifractal methods were performed significantly in the detection of epileptic seizures in EEG signals and ECG signals through the cardiac inter-heartbeat time interval dynamics. The multifractal measures have shown significant differences among normal, interictal, and ictal EEGs and discriminate the young and elderly subjects by ECG inter-heartbeat signals. The fuzzy-based multifractal theory for signals is established in order to define the fuzzy generalized fractal dimensions by introducing the fuzzy membership function in the classical generalized fractal dimensions method, and it is used for the classification

viii

Preface

of chaotic behaviors in the fractal waveforms. In Chap. 6, a fuzzy multifractal measure for biomedical signals to identify the age-group of subjects is presented. Turin, Italy Vellore, India Vellore, India

Santo Banerjee D. Easwaramoorthy A. Gowrisankar

Contents

1 Mathematical Background of Deterministic Fractals . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Iterated Function System . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Countable Iterated Function System . . . . . . . . . . . . . . . . . 1.4 Local Countable Iterated Function System . . . . . . . . . . . . 1.4.1 Local Iterated Function System . . . . . . . . . . . . . . 1.4.2 Existence and Analytical Properties of LCIFS . . . . 1.5 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Generalized Fractal Dimensions . . . . . . . . . . . . . . . . . . . . 1.6.1 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Limiting Cases of Generalized Fractal Dimensions 1.6.3 Range of Generalized Fractal Dimensions . . . . . . . 1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

1 1 5 9 11 11 12 15 17 18 19 19 19

2 Fractal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . 2.3 Fractal Interpolation Function . . . . . . . . . . . . . . . . . 2.4 Hidden Variable Fractal Interpolation Function . . . . . 2.5 Classical Calculus on Fractal Interpolation Functions 2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

21 21 22 23 29 32 35

3 Fractional Calculus on Fractal Functions . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Fractal Interpolation Function . . . . . . . . . . . . . . . . . 3.3 Riemann–Liouville Fractional Calculus Quadratic FIF . . . . 3.4 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

37 37 39 49 53

..... .....

54 59

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

ix

x

Contents

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

61 61 62

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

68 77

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

79 79 79 79 81 83 83 83 83

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

83 84 85 87 88 89

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. 95 . 96 . 99 . 100 . 117

4 Fractal Interpolation Function for Countable Data . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of FIF for Countable Data Set . . . . . . . . . . . . . 4.3 Fractional Calculus on Interpolation Function of Sequence of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimental Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Synthetic Weierstrass Signals . . . . . . . . . . . . . . . 5.2.2 Clinical EEG Signals . . . . . . . . . . . . . . . . . . . . . 5.3 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 ANOVA Test . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Box Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Normal Probability Plot . . . . . . . . . . . . . . . . . . . 5.4 Development of Multifractal Analysis in EEG Signal Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Modified Generalized Fractal Dimensions . . . . . . 5.4.2 Improved Generalized Fractal Dimensions . . . . . . 5.4.3 Advanced Generalized Fractal Dimensions . . . . . 5.4.4 Methods to Analyze the Fractal Time Signals . . . 5.4.5 Results and Discussions . . . . . . . . . . . . . . . . . . . 5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Discrete Wavelet Transform . . . . . . . . . . . . . . . . 5.5.2 Wavelet Denoising of Signals . . . . . . . . . . . . . . . 5.5.3 Results and Discussions . . . . . . . . . . . . . . . . . . . 5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Fuzzy Multifractal Analysis in ECG Signal Classification . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fuzzy Multifractal Analysis for Fractal Signals . . . . . . . . . . . . . 6.2.1 Fuzzy Renyi Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Fuzzy Generalized Fractal Dimensions for Signals . . . . . . 6.3 Fuzzy Generalized Fractal Dimensions for Deterministic Fractal Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Experimental ECG Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Fuzzy Generalized Fractal Dimensions for Clinical ECG Signals . 6.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

119 119 119 119 120

. . . .

121 124 125 127

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 1

Mathematical Background of Deterministic Fractals

1.1 Introduction The historical backdrop of describing natural objects by mathematics, with respect to Euclidean geometry, is as old as the advent of science itself. In our intuitive understanding, traditionally lines, squares, rectangles, circles, spheres, and so forth have been the fundamental shapes of geometry. Mathematics is mostly concerned with these fundamental shapes to model or approximate or often analyze the natural phenomena. Indeed, nature is not confined to such Euclidean entities thatare typically characterized only by integer dimensions, but we restricted ourselves to integer dimensions only. Actually, even now in the beginning phase of our teaching, we discover that objects which have just length are one dimensional, objects with length and width are two dimensional, and those that have length, width, and breadth are three dimensional. Did we ever raise the question for what reason we are always moving toward integer dimension? The response as far as we could possibly know is Never. We never addressed such a question, if there exist objects with non-integer dimensions. However, one person did and he was none other than Benoit B. Mandelbrot. He included another word in the scientific jargon through his inventive work which he termed as a fractal. He was bewildered with such thoughts for many years and realized that nature is not restricted to Euclidean or integer-dimensional space. Rather, a large portion of the natural objects which we see around us are complex in shape, and traditional Euclidean geometry is not adequate to depict them. The notion of fractal geometry has all the earmarks of being irreplaceable for portraying such complex structure at least quantitatively. Mandelbrot has revolutionized Euclidean geometry with the notion of a fractal that has generated extensive attention in almost every branch of science. He presented his new concept through his amazing book “The fractal geometry of nature ”in a striking manner and since then it stayed as the standard reference book for both amateurs and scientists [1]. In a sense, the idea of a fractal has brought numerous apparently disconnected subjects under one umbrella.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_1

1

2

1 Mathematical Background of Deterministic Fractals

There has been a surge of research activities in applying the influential fractal idea in pretty much every part of scientific disciplines to increase profound bits of knowledge into numerous unresolved problems. However, there is no flawless and complete definition of the fractal. The simplest way to define a fractal is as an object which appears self-similar under varying degrees of magnification. A ‘Fractal’ is generally a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity. The word Fractal is derived from the Latin word, fractus meaning broken or fractured to describe objects that were too irregular to fit into a traditional geometrical setting. In Mandelbrot’s original article, the Fractal is mathematically defined as a set with the Hausdorff dimension strictly greater than its topological dimension. Roughly speaking, a fractal set is a set that is more ‘irregular’ than the sets considered in classical geometry. Fractal sets have the additional property of being in some sense either strictly or statistically self-similar; this property has been widely applied to model the numerous natural phenomena by Mandelbrot and others. However, the notion of the strict self-similar property has been theoretically framed by Hutchinson and popularized by Barnsley. The iterated function system is a convenient and powerful way tool for generating fractals in a metric space with specified self-similarity properties. Hutchinson introduced the conventional explanation of deterministic fractals through the theory of Iterated Function System (IFS). Meanwhile, Barnsley formulated the theory of IFS called the Hutchinson–Barnsley (HB) theory in order to define and construct the fractals as a non-empty compact invariant subset of a complete metric space which is generated by the Banach fixed point theorem, known as IFS theory [2, 3]. In view of the applications of fractals to comprehend the natural phenomena, every area of science has been concerned about the fractal analysis. Hence, fractal analysis plays a central role in mathematics as a tool for nonlinear applications and as a theory of interest. Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable, called Weierstrass Function, whose graph would today be considered as a fractal. The complexity and irregularity can be found in many physical and biological nonlinear systems naturally and have been analyzed by the tools of fractal theory and computed by the non-integer or fractional measure called Fractal Dimension. In approximation theory, the traditional interpolation techniques generate a smooth or piecewise differentiable interpolation function, despite the data points being irregular. Nevertheless, most of the natural objects such as lightning, clouds, mountain ranges, and wall cracks have irregular and complex structures in which Euclidean geometry cannot be applied successfully to describe them. Hence, there is a need for a nonlinear tool in the approximation theory to resolve these problems. M. F. Barnsley invented the Fractal Interpolation Function (FIF) based on the theory of IFS, which precisely suited for the approximation of naturally occurring functions which possess some kind of self-similarity under magnification [4]. Fractal interpolation functions are not necessarily differentiable even though they are continuous everywhere. Hence, FIFs are sophisticated in approximating the rough curve which precisely reconstructs the naturally occurring functions when compared to the classical interpolants. Nevertheless, a smooth curve

1.1 Introduction

3

is needed to approximate some kind of functions which include the self-similarity under magnification. Barnsley and Harrington explored the construction of p-times frequently differentiable FIF when the derivative values reach up to pth order at the initial endpoint of the interval [5]. During the past few years, numerous mathematical techniques were broadly correlated with fractal analysis. Considering the importance of fractional calculus for discussing the variance ratio of fractal functions, the general relationship between the fractional calculus and fractals has been investigated. Based on these studies, there are continuous efforts connecting the fractional calculus, and the graphs of the fractal function are available in the literature. Generally, the interpolation and approximation of functions have been constructed by means of smooth functions, often piecewise differentiable. However, the waves and signals from the real world do not share this aspect. Hence, the main goal of the current book is to provide the idea to approximate the rough (nowhere differentiable) functions through the fractal interpolation function and its fractional calculus. Against this background, it is proposed to generalize any interpolation function, smooth or rough, by means of a family of fractal functions. It is also proposed to construct the various fractal interpolation functions for a sequence of data and to investigate their shape-preserving properties. Fractional calculus of such functions will be explored for approximating the rough functions, additionally to correlate the fractional order and fractal dimension of the fractal interpolation function. The brain cells of humans start in between the 120th day and the 160th day of pregnancy. It is accepted that from this beginning phase and throughout life, electrical signals produced by the brain which speak to the cerebrum work as well as the status of the entire body. In this direction, there are numerous advanced digital signal processing methods applied on the biomedical signals, and thereby understand the brain activity of a human. Signals from the human body play a vital role for the early diagnosis of a variety of diseases. Such electrobiological signals can be in the form of an electrocardiogram (ECG) taken from the heart, electromyogram (EMG) measured in muscles, electroencephalogram (EEG) from the brain, magnetoencephalogram (MEG) given by the brain, electrogastrogram (EGG) from the stomach, and electrooculogram (EOG) generated by eye nerves. As an application part, this book discusses the signal analysis with respect to a multifractal perspective. Particularly, fractal methods are applied on a medical signal, such as an Electroencephalogram and Electrocardiogram, to help in the diagnosis. An EEG signal is a capacity of currents that flow during synaptic excitations of the dendrites of many pyramidal neurons in the cerebral cortex. The synaptic currents are formed within the dendrite when neurons are activated. This current produces a magnetic field measurable by electromyogram machines and a secondary electrical field over the scalp measurable by EEG systems. There is ever-expanding worldwide interest for more moderate and powerful clinical and medicinal services. New strategies and apparatus should be created along these lines to help in the diagnosis, monitoring, and treatment of abnormalities and diseases of the human body. Biomedical signs (biosignals) in their complex structures are rich data sources, which when properly prepared can possibly encourage such developments. In the present revolution, such preparing is probably

4

1 Mathematical Background of Deterministic Fractals

going to be digital, as affirmed by the incorporation of computerized signal handling ideas as center preparing in biomedical science degrees. Ongoing developments in digital signal preparing are relied upon to support key parts of things to come progress in biomedical exploration and innovation, and the reason for this book is to feature this pattern for the handling of estimations of brain activity, essentially electroencephalograms. Similarly, Electrocardiogram is the vital sign most commonly used in the clinical environment. It provides an insight into the understanding of many agerelated cardiac disorders. The vast existing literature on automatic ECG classification comprises lot more methods based on pattern recognition methodologies, artificial neural networks, support vector machines, linear discriminant analysis, clustering techniques, and other soft computing techniques that have been analyzed by various researchers. In all physical diagnosis, the physicians are recognizing the diseases based on the age-group of the patients. At the outset of the diagnosing process, age can be discriminated through the ECG by inter-heartbeat interval dynamics. Fractal Analysis provides a powerful mathematical tool which is applicable for modeling many natural phenomena with high complexity and irregularity, and cannot be well treated by Euclidean geometry. This chaotic behavior can also be observed in biological time series representing dynamics of complex processes. Therefore, multifractal theory, based on Generalized Fractal Dimensions (GFD), could be a useful technique to compute the degree of disorders in the Human Brain, especially Epilepsy disease in physiological and pathological conditions by using the time signal called Electroencephalogram (EEG). GFD is the measure to compute the complexity, irregularity, and the chaotic nature of the EEG signals. The identification of an abnormality in EEG signals is the vast area of research in neuroscience. In this chapter, the three forms of GFD namely, Modified GFD, Improved GFD, and Advanced GFD are newly framed in order to discriminate the healthy and epileptic EEG time signals. Also, a novel method based on wavelet denoising is proposed to improve the preprocessing analysis of EEG signals and the identification of healthy and epileptic EEG signals is investigated by using multifractal measures. Among all the nonlinear techniques, the correlation dimension measurement is more accessible in dealing with experimental systems. The absolute value of the estimated correlation dimension does not represent the complexity of the data especially since it is only one scalar value from the system of the fractal dimension spectrum. The single-valued dimensional quantity is insufficient to characterize the non-uniformity or inhomogeneity of the chaotic waveforms. Generally, chaotic attractors are inhomogeneous. Such an inhomogeneous set is called a Multifractal and is characterized by Generalized Fractal Dimensions or Renyi Fractal Dimensions. The usage of the whole family of fractal dimensions should be very useful in comparison with using only some of the dimensions. So far, the chaotic nature of nonlinear signals has been analyzed under various settings by the multifractal measure called Generalized Fractal Dimensions. Later, GFD has been defined for noisy images to analyze the rate of complexity. Also, Fuzzy Generalized Fractal Dimensions (F-GFD) has been generalized from the classical GFD for estimating the chaotic nature of the mathematical waveforms generated by Weierstrass Functions.

1.1 Introduction

5

The aim of this chapter is to describe the fundamental concepts of deterministic fractals in view of their use in fractal interpolation functions. This chapter will thus present a modest introduction to the concepts of fixed point theorem, iterated function systems, and fractal dimensions. Since we are interested in fractal interpolation, most theoretical results are given without proofs which are available in the references mentioned where appropriate. This chapter first describes the deterministic fractal through the several notions of iterated function systems which provide a global characterization of a fractal. All the concepts presented here are more fully established in [4]. The idea of dimension applies to sets in a metric space more broad than signals. Although the notion of dimension makes sense for more complex entities, for example, measures or classes of functions, several interesting concepts of dimension exist. Hence, the concept of fractal dimensions is concisely discussed at the end of this chapter.

1.2 Iterated Function System Hutchinson introduced the conventional explanation of deterministic fractals through the theory of the iterated function system. Meanwhile, Barnsley formulated the theory of the iterated function system called the Hutchinson–Barnsley theory in order to define and construct the fractals as a non-empty compact invariant subset of a complete metric space generated by the Banach fixed point theorem (for further reading, [2, 3, 7–9]). This section concisely discusses the construction of deterministic fractals (or metric fractals) in the complete metric space generated by the IFS of Banach contractions. Definition 1.1 A self-mapping f on the metric space (X, d) is called a contraction mapping (simply contraction) if there exists a constant α ∈ [0, 1) such that d( f (x), f (y)) ≤ αd(x, y) for all x, y ∈ X.

(1.1)

The constant α is called the contraction factor or contraction ratio. Definition 1.2 (fixed point) A point x in a metric space (X, d) is said to be a fixed point of the mapping f : X → X , if x satisfies the equation f (x) − x = 0. The fixed point of the given function f is invariant under the mapping f , hence it is also known as an invariant point. Fixed points of the function f defined on R are the points of intersection of the curve y = f (x) and the line y = x, as shown in Fig. 1.1. An identity function that is defined on R has all the points in R as fixed points, whereas in the same domain f (x) = x 2 has two fixed points namely, 1, 0. Consider that the function f (x) = x 2 − 1 on R does not have a fixed point in R. These examples show that all the functions not necessarily have a unique fixed point on their domain. The notable result which gives the uniqueness of a fixed point in a complete metric space was explored by Stefan Banach in 1922.

6

1 Mathematical Background of Deterministic Fractals 1.6 1.4

1.5

1.2

1

1

0.5

0.8

0

0.6

-0.5

0.4

-1

0.2 0 -0.2 -0.2

-1.5 0

0.2

0.4

0.6

0.8

1

1.2

-1

-0.5

0

0.5

1

Fig. 1.1 Graphical representation of a fixed point

Let f : X → X be a contraction mapping with the contraction factor α. Let x0 ∈ X be an arbitrary point on the metric space (X, d) such that x0 does not lie on the set {x ∈ X : f (x) = x}. Inductively define xn+1 = f (xn ) for n ≥ 0. This iteration sequence is constructively used to find the fixed point of f in X , which is also know as Picard’s successive approximation process. The distance between any two points in the sequence (xn ) is estimated as follows: d(xn , xn+1 ) = d( f (xn−1 ), f (xn )) ≤ αd(xn−1 , xn ) ≤ α 2 d(xn−2 , xn−2 ) .. . ≤ α n d(x1 , x0 ). For any n, m ∈ N with n < m, d(xm , xn ) ≤

m−1 

d(xi , xi+1 )

i=n

≤

m−1  i=n

α i d(x1 , x0 ) ≤

αn d(x1 , x0 ). 1−α

0 ,x 1 )α Hence, for given  > 0, choose n 0 large enough that d(x1−α < . Then, for n, m ≥ n 0 , we have d(xm , xn ) < . It shows that (xn ) is Cauchy. If X is complete, then every Cauchy sequence is convergent. Assume X is a complete space, then there exists a point x  in X such that xn → x  as n → ∞. Observe that the limit as m → ∞, one can get αn d(xn , x  ) ≤ d(x1 , x0 ). 1−α n

1.2 Iterated Function System

7

This inequality gives explicit error estimation in xn when regarded as an estimate of the fixed point x. Initially, we take the contraction mapping f and hence f is uniformly continuous. Therefore, xn → x  implies f (xn ) → f (x  ). It follows f (x  ) = x  . Assume x  , y  are two points in X such that f (x  ) = x  and f (y  ) = y  . Since f is a contraction mapping, d(x  , y  ) = d( f (x  ), f (y  )) ≤ αd(x  , y  ) < d(x  , y  ), which implies d(x  , y  ) = 0. Hence x  = y  . The above argument summaries the following theorem namely, the Banach contraction principle. Theorem 1.1 Let (X, d) be a complete metric space and f be a contraction mapping on X . Then f has a unique fixed point x ∗ . The Banach contraction principle presented in Theorem 1.1 is applied in many fields. In particular, this section describes an application of the Banach contraction principle to the construction of fractals, widely known as the deterministic fractal, in a complete metric space through iterated function systems. The study of fractals and their properties in a complete metric space associated with the Banach contraction principle is a constructive method to understand fractal geometry which is named as the Hutchinson–Barnsley theory. For a large number of examples and details, refer to the books ([2, 8, 9]). Definition 1.3 For n ∈ N, let Nn denote the subset {1, 2, . . . , n} of N. Consider a finite family of contraction mappings f 1 , f 2 , . . . , f n on X with contraction ratios αk ∈ [0, 1), k ∈ Nn , simply written as ( f k )k∈Nn . Then the system {X ; f k : k ∈ Nn } is called an Iterated Function System (IFS) or finite iterated function system. Let (X, d) be a complete metric space and H(X ) be the class of all non-empty compact subsets of X . Usually, H(X ) is known as a hyperspace of X which includes all non-empty compact subsets of X . Define the distance between a point x in X and a compact subset A in H(X ) as follows: d(x, A) = inf{d(x, a) : a ∈ A}.

(1.2)

Now define the distance between two sets A, B ∈ H(X ) as d(A, B) = sup{d(a, B) : a ∈ A}.

(1.3)

The Hausdorff distance between A and B in H(X ) is defined as Hd (A, B) = max{d(A, B), d(B, A)}. The hyperspace H(X ) is complete with respect to the Hausdorff metric Hd . Definition 1.4 Define the self-mapping F : H(X ) −→ H(X ) by

(1.4)

8

1 Mathematical Background of Deterministic Fractals

F(A) = f 1 (A) ∪ f 2 (A) ∪ · · · ∪ f n (A)  f k (A), for all A ∈ H(X ). =

(1.5)

k∈Nn

This self-mapping F is called the Hutchinson–Barnsley mapping (HB mapping) on H(X ). Let f : H(X ) → H(X ) be defined by f (A) = { f (a) : a ∈ A} for all A ∈ H(X ). If f is a contraction on X with a contraction ratio α, then f is a contraction on H(X ) with the same contraction ratio α. That is, if f is a contraction on X , then it is also a contraction on the hyperspace of X with the same contraction ratio. The following theorem establishes the fact that the Hutchinson–Barnsley mapping is a contraction provided associated functions f k ’s are contractions. Theorem 1.2 Let (X, d) be a metric space and H(X ) be an associated hyperspace of non-empty compact subsets of X with the Hausdorff metric Hd . If f k ’s are contraction mappings on X for all k ∈ Nn , then the HB mapping F is a contraction on H(X ). Proof Let A, B ∈ H(X ) and consider the case n = 2. Then we get  Hd (F(A), F(B)) = Hd

2 

f k (A),

k=1

2 

 f k (B)

k=1

≤ max {Hd ( f 1 (A), f 1 (B)), Hd ( f 2 (A), f 2 (B))} ≤ max {α1 Hd (A, B), α2 Hd (A, B)} ≤ α Hd (A, B). Here α = max{αk : k ∈ N2 }. Theorem 1.3 Let (X, d) be a complete metric space and (H(X ), Hd ) be an associated Hausdorff metric space. If the self-mapping F, in Eq. (1.5), is defined by the IFS {X ; f k : k ∈ Nn }, then F has a unique fixed point A∗ in H(X ), that is, there exists a unique non-empty set A∗ ∈ H(X ) such that F satisfies the self-referential equation A∗ = F(A∗ ) =



f k (A∗ ).

k∈Nn

Moreover, for any B ∈ H(X ), lim F ◦ p (B) = A∗ ,

p→∞

the limit being taken with respect to the Hausdorff metric.

1.2 Iterated Function System

9

1 0.8

0

0.6

-0.2

0.4

-0.4

0.2

-0.6

0 -0.2

-0.8

-0.4

-1

-0.6

-1.2

-0.8 -1 -1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

-1.4

-1

-0.5

0

0.5

1

1.5

2

Fig. 1.2 Iteration of function

Proof The completeness of the space (X, d) gives (H(X ), Hd ) which is a complete metric space. Theorem 1.2 shows that F is a contraction mapping on H(X ). Hence, by the Banach fixed point theorem 1.1, the contraction mapping F on the complete metric space (H(X ), Hd ) has a unique fixed point. This completes the proof. In Theorem 1.3, F ◦ p denotes the pth composition of the HB mapping F, that is, ◦p ◦ · · · ◦ F . The iteration of function is described in Fig. 1.2. F = F ◦ F  p times

Definition 1.5 A non-empty compact set A∗ obtained from Theorem 1.3 is called an invariant set or self-referential set or attractor or deterministic fractal of the IFS {X ; f k : k ∈ Nn } . Due to the sophisticated usage of fractals in nonlinear approximation, there are sequel efforts to extend the Hutchinson–Barnsley classical framework for fractals to more general spaces and countable IFS or infinite IFS [28–37]. Consequent sections review some basic definitions and results on the theory of the generalized iterated function system and its attractors.

1.3 Countable Iterated Function System The Hutchinson–Barnsley theory of finite iterated function systems has been studied in the last decades with some extensions of the spaces and the contractions of various types. In this section, an extension of the Hutchinson–Barnsley theory of finite iterated function systems to a countable iterated function system is presented. N.A. Secelean has generalized the HB theory by constructing the deterministic fractals through a countable iterated function system [28]. After a brief review of the fundamental properties of a countable iterated function system (CIFS), this section describes the features and approximation of attractor of CIFS with the attractors of IFS.

10

1 Mathematical Background of Deterministic Fractals

Consider the sequence of Banach contractions ( f n )n≥1 on a compact metric space X with contraction factors rn ∈ [0, 1), n ≥ 1. Then the system {X ; f n : n ≥ 1} is called a Countable Iterated Function System (CIFS). Define a self-map F : H(X ) −→ H(X ) by F (B) =

∞ 

f n (B), for all B ∈ H(X ).

(1.6)

n=1

Then the self-map F is a Banach contraction with a contraction factor r ≤ sup rn , n ≥ 1. Theorem 1.4 The self-map F defined in Eq. (1.6) has a unique fixed point A ∗ , that is, there exists a unique non-empty set A ∗ ∈ K (X ) such that F satisfies the self-referential equation A ∗ = F (A ∗ ) =

∞ 

f n (A ∗ ).

n=1

Furthermore, for any B ∈ H(X ), limn→∞ F ◦n (B) = A ∗ , the limit being taken with respect to the Hausdorff metric. A non-empty compact invariant set A ∗ obtained from Theorem 1.4 is called an attractor of the CIFS {X ; f n : n ≥ 1}. The attractor A ∗ is known as a deterministic fractal generated by the CIFS of contractions. According to Theorem 1.3, the attractor A∗ is dependent on the corresponding IFS. Suppose A∗N is an attractor of the IFS {X ; f n : n = 1, 2, . . . , N } and FN is the associated HB operator, for N ≥ 1. Then, the following Theorem 1.5 reveals the relation between the attractors of IFS and the attractor of CIFS of Banach contractions on a compact metric space X . Theorem 1.5 If B ∈ H(X ), then F (B) = lim FN (B) = lim N →∞

N →∞

N 

f n (B).

n=1

In particular, if A is the attractor of CIFS {X ; f n : n ≥ 1}, then A = F (A ) = lim A∗N = lim lim FN◦n (B), N →∞

N →∞ n→∞

also, A = lim N →∞ limn→∞ FN◦n (B). The above theorem shows that the attractor of CIFS {X ; f n : n ≥ 1} is approximated by the attractors of IFS {X ; f n : n = 1, 2, . . . , N } , N ≥ 1, with respect to the Hausdorff metric.

1.3 Countable Iterated Function System

11

Example 1.1 Consider the space X = [0, 1] with the Euclidean metric. Let q ∈ (0, 21 ], define f n : [0, 1] −→ [0, 1] by f n (x) = q n x + αn , for all n ∈ N, where n−1

α1 = 0 and αn = q n−1 + 1−2q + αn−1 , n ≥ 2. Then the attractor of the CIFS 2−3q {X ; f n : n ≥ 1} gives the infinite type Cantor set. n−1 If q = 1/3 and αn = 1 − 13 , n ≥ 1, then the sequence of Banach contractions becomes f n = 3xn + 1 − 3n1−1 , n ≥ 1. Example 1.2 Let X = [0, 1] × [0, 1] ⊂ R2 , for every n = 4 p + k, consider the following contractions: ⎧ 1 1

x + 2 p+1 − 2, 13 y , ⎪ p+1 2 3 ⎪

 √ √ ⎪ ⎪ 1 1 3 3 1 p+1 ⎨ p+1 x − y + 2 − 5, x + y , 2 6 6 6

6 √ √ f n (x, y) = 1 1 3 p+1 ⎪ − 23 , −6 3 x + 16 y + ⎪ ⎪ 2 p+1 6 x + 6 y + 2 ⎪

⎩ 1 1 4 1 x + 2 p+1 − 3 , 3 y , 2 p+1 3

if k = 1, √

3 6



if k = 2, , if k = 3, if k = 4.

Then the attractor of the CIFS {X ; f n : n ≥ 1} gives the infinite type von Koch curve.

1.4 Local Countable Iterated Function System This section concisely provides the essential material of a local iterated function system (LIFS) and describes the approximation process of the attractor of a local countable iterated function system with the attractors of the iterated function system.

1.4.1 Local Iterated Function System If {X i : i ∈ Nn } is n number of non-empty subsets of X and for each X i , there exists a continuous mapping f i on X i to X . Then the system {X ; (X i , f i ) : i ∈ N} is called a local iterated function system (local IFS) [6] whereas, an iterated function system (IFS) is a complete metric space X together with a finite set of contraction mappings, denoted by {X ; f k : k ∈ Nn }, with contraction factors ck , k ∈ Nn . If X i = X, then local IFS becomes (global) IFS. The operator Floc,n on H(X ) is defined by Floc,n (B) =

 i∈Nn

Here f i (S ∩ X i ) = { f i (x) : x ∈ S ∩ X i }.

f i (B ∩ X i ).

12

1 Mathematical Background of Deterministic Fractals

Theorem 1.6 Let X be a complete metric space with the Hausdorff metric.  Let (E n )n be a sequence of compact subsets of X such that E n ⊂ E n+1 and E = ∞ n=1 E n . Then E¯ = limn→∞ E n .

1.4.2 Existence and Analytical Properties of LCIFS Suppose {X i : i ∈ N} is a sequence of non-empty subsets of X . Further assume that for each X i , there exists a continuous mapping f i : X i −→ X, i ∈ N. Then {X ; (X i , f i ) : i ∈ N} is called a local countable iterated function system (local CIFS). If X i = X, then local CIFS becomes global CIFS. A local CIFS is called contractive or hyperbolic if each f i is a contraction on their respective domains. Let P(X ) be the power set of X , i.e., P(X ) = {S : S ⊂ X }. Define Wloc : P(X ) −→ P(X ) by  f i (B ∩ X i ). Wloc (B) = i∈N

Here f i (S ∩ X i ) = { f i (x) : x ∈ S ∩ X i }. Every local CIFS has at least one local attractor (fixed point of Wloc ) namely, A = ∅, empty set. The largest local attractor, union of all distinct local attractors, is called the local attractor of local CIFS, {X ; (X i , f i ) : i ∈ Nn }. If X is compact and X i , i ∈ N, is closed, the compact in X and where local CIFS {X ; (X i , f i ) : i ∈ Nn } is contractive, the local attractor may be computed as follow. Let K 0 = X and set K n = Wloc (K n−1 ) =



f i (K i−1 ∩ X i ), n ∈ N.

i∈N

Then {K n : n ∈ N} is a decreasing nested sequence of compact sets. If each K n is non-empty, then by the cantor intersection theorem, K =



K n = ∅

n∈N

K = lim K n , n→∞

limit taken with respect to the Hausdorff metric Hd on K (X ).

1.4 Local Countable Iterated Function System

13

K = lim K n = lim n→∞



n→∞

=



f i (K n−1 ∩ X i )

i∈N

f i (K ∩ X i )

i∈N

= Wloc (K ). Thus, K = Aloc . A condition guaranteeing that each K n is non-empty is that f i (X i ) ⊂ X i , i ∈ N. Theorem 1.7 Let X be a compact metric space and let X i , i ∈ N, be a closed subset of X . If A is an attractor of CIFS and A∗ is a local attractor of local CIFS, then A∗ is a subset of A. Proof Consider the sequence {K n : n ∈ N} such that K 0 = X and K n = Wloc  (K n−1 ) = i∈N f i (K i−1 ∩ X i ), n ∈ N. The unique attractor A is obtained as the CIFS assolimit of the sequence {K n : n ∈ N}. Let {X ; f i : i ∈ Nn } be the contractive  ciated with the set-valued map W on H(X ) defined by W(B) = i∈N f i (B). Then, the unique attractor A of the CIFS is obtained as the limit of the sequence {An : n ∈ N} such that A0 = X and An = W(An−1 ), n ∈ N. Assume K n−1 ⊆ An−1 , n ∈ N, we have A∗ = lim K n = lim n→∞

n→∞

⊆ lim

n→∞

⊆ lim

n→∞



f i (K n−1 ∩ X i )

i∈N



f i (K n−1 )

i∈N

 i∈N

f i (An−1 ) = lim An = A. n→∞

Theorem1.7 reveals the relation between the attractor of a countable iterated function system and a local countable iterated function system. Theorem 1.8 Let X be a compact metric space. Let {X ; (X i , f i ) : i ∈ N} be a local CIFS and {X ; (X i , f i ) : i ∈ Nn } be a local IFS. Suppose limn→∞ E n = E = ∅, where each n, E n ⊆ X. Then the local attractor A∗ of CIFS is approximated by the local attractors A’s of local IFSs. k (E n ) = A∗ . lim lim Floc,n n

Proof Let Floc,n (B) =

 i∈Nn

k

f i (B ∩ X i ), for n ∈ N. Then it’s enough to prove that

k k lim Floc,n (E n ) = Wloc (E),

n→∞

(1.7)

where the limit is taken with respect to the Hausdorff metric h. As {X ; (X i , f i ) : i ∈ Nn } is a local IFS, for each X i there exists a contraction mapping f i : X i →

14

1 Mathematical Background of Deterministic Fractals

X with contraction factors ci , i ∈ Nn . Denote f i1 ...ik = f i1 ◦ · · · ◦ f ik , for each k ≥ 1, i 1 , i 2 , . . . , i k are positive integers. Clearly, f i1 ...ik is a contraction mapping with contraction factor ci1 ci2 · · · cik . k k k k k k (E n ), Wloc (E)) ≤ h(Floc,n (E n ), Floc,n (E)) + h(Floc,n (E), Wloc (E)). h(Floc,n (1.8)

The first term of the left-hand side of Eq. (1.8) can be expressed as ⎛



k k h(Floc,n (E n ), Floc,n (E)) = h ⎝



f i1 ...ik (E n ∩ X n ),

i 1 ,...,i k ∈Nn

⎞ f i1 ...ik (E ∩ X n )⎠

i 1 ,...,i k ∈Nn

(1.9) ≤

sup

i 1 ,...,i k ∈Nn

h( f i1 ...ik (E n ∩ X n ), f i1 ...ik (E ∩ X n ))

≤ ci1 · · · cik h(E n ∩ X n , E ∩ X n ) ≤ h(E n , E). Since limn→∞ E n = E, h(E n , E) → 0 as n → ∞. Now, let us consider the second term in Eq. (1.8).  k (B) = f i1 ...ik (B). (1.10) Wloc i 1 ,...,i k ∈N

Continuity of f i ’s and the basic topological result give the following: ⎛



k+1 Wloc (E) = Wloc ⎝

⎞ f i1 ...ik (E)⎠

i 1 ,...,i k ∈N

∞ 

=

⎛ fi ⎝

⊂

i=1 ∞ 

=

⎛ fi ⎝

f i1 ...ik (E ∩ X i )⎠



⎞ f i1 ...ik (E ∩ X i )⎠

i 1 ,...,i k ∈N

⎛ fi ⎝

i=1

The sequence of sets (1.6)

⎞

i 1 ,...,i k ∈N

i=1 ∞ 



 i 1 ,...,i k ∈Nn



⎞ k+1 f i1 ...ik (E ∩ X i )⎠ = Wloc (E).

i 1 ,...,i k ∈N

f i1 ...ik (E ∩ X n )

n∈N

is increasing and by Theorem

1.4 Local Countable Iterated Function System

15



k lim Floc,n (E) = lim

n→∞

n→∞

=

f i1 ...ik (E ∩ X n )

(1.11)

i 1 ,...,i k ∈Nn

∞ 



f i1 ...ik (E ∩ X n )

n=1 i 1 ,...,i k ∈Nn



=

k f i1 ...ik (E) = Wloc (E).

i 1 ,...,i k ∈N k k (E n ), Wloc (E)) = 0 as n, k → 0. Applying Eqs. (1.9), (1.11) in (1.8) gives h(Floc,n Thus we conclude, k k (E n ) = lim Wloc (E) = A∗ . lim lim Floc,n

n→∞ k→∞

k→∞

Theorem1.8 establishes the approximation process of the attractor of the local iterated function system in terms of attractors of a local countable iterated function system.

1.5 Fractal Dimension Measuring the size of a set or an object helps to understand its fundamental nature. In mathematics, objects are constituted by set of points, lines, square or cubes. How do we measure the size of a set? For instance, consider a straight line with length L. The approximate length L() is estimated by the product of the number N () of line segment of length  (yardstick) required to cover the straight line, i.e.,  × N () = L().

(1.12)

As yardstick  tends to zero, it makes L() approach the length L of the straight line, i.e., (1.13) lim  × N () = L . →0

There exists a positive integer n such that  ≥ /n and if the length L is measured by a smaller size of the yardstick /n, then /n × N (/n) = L .

(1.14)

N (/n) = n N ().

(1.15)

It gives that

Extending this idea to a higher dimensional space gives

16

1 Mathematical Background of Deterministic Fractals

N (/n) = n d N (),

(1.16)

where d = 1, 2, 3, . . .. It implies that if the size of the yardstick is decreased by a factor of n, then the numerical value of the number N is increased by a factor of n d . It can be rigorously proved that the solution of Eq. (1.16) is given by the inverse power law 1 (1.17) N () ∼ d ,  where d = 1, 2, 3. The integer solution of Eq. (1.17) is known as the Euclidean dimension of the given object. If the object is a fractal, then lim  × N () = ∞.

→0

Since, if the size of the yardstick is decreased, then one can get a finer and finer fractal structure, hence N () is large. However, in this case, Eq. (1.17) has a non-integer solution which is known as the Hausdorff dimension. The Hausdorff dimension is defined as follows. If U is any non-empty subset of n-dimensional Euclidean space Rn , the diameter of U is defined as |U | = sup{|x − y| : x, y ∈ U }, i.e., the greatest distance apart from any pair of points in U . If {Ui } is a countable (or finite) collection of sets of diameter at most δ that cover F, i.e., K ⊂

∞ 

Ui

i=1

with 0 < |Ui | ≤ δ for each i, we say that {Ui } is a δ-cover of K . Suppose that K is a subset of Rn and s is a non-negative number. For any δ > 0, we define Hδs (K )

= inf

 ∞

 |Ui | : {Ui } is a δ-cover of K . s

i=1

As δ decreases, the class of permissible covers of K is reduced. Therefore, the infimum Hδs (K ) increases, and so approaches a limit as δ → 0. Thus, H s (K ) = lim Hδs (K ). δ→0

This limit exists for any subset K of Rn , though the limiting value can be 0 or ∞. We call H s (K ) as the s-dimensional Hausdorff measure of K . Then, the Hausdorff Dimension or Hausdorff–Besicovitch Dimension of K is defined as     dim H (K ) = inf s : H s (K ) = 0 = sup s : H s (K ) = ∞ ,

1.5 Fractal Dimension

17

so that  H (K ) = s

∞, if s < dim H (K ), 0, if s > dim H (K ).

If s = dim H (K ), then H s (K ) may be zero or infinite, or may be 0 < H s (K ) < ∞. The Hausdorff dimension has the advantage of being defined for any set, and is mathematically convenient, as it is based on measures, which are relatively easy to manipulate. The main disadvantage is that the explicit computation of the Hausdorff dimension of a given set K is rather difficult since it involves taking the infimum over covers consisting of balls of radius less than or equal to a given  > 0. A slight simplification is obtained by considering only covers by the balls of radius equal to . This gives rise to the concepts of the box dimension [8]. Let K ∈ H(X ) and N () denote the smallest number of closed balls of radius  > 0 required to cover K . If  dim B = lim

→0

log N () log(1/)

 (1.18)

exists, then dim B is called the box dimension or fractal dimension of K . Suppose that K is a subset in Rn . The topological dimension of K , denoted as dim T M, is inductively defined as follows: 1. dim T ∅ = −1, 2. The topological dimension of K at a point p ∈ K is less than or equal to n, p written as dim T K ≤ n, if there exist arbitrarily small neighborhood of p whose boundaries have a topological dimension at most n − 1, 3. K has a topological dimension at most n if it has a topological dimension at most n at each of its points p: p

dim T K ≤ n ⇐⇒ dim T K ≤ n for all p ∈ K . p

In addition, dim T K = ∞, if condition (2) does not hold for any n ∈ N, and dim T K = ∞ if condition (3) does not hold for any n ∈ N.

1.6 Generalized Fractal Dimensions Fractal dimension is insufficient to characterize the object of interest having complex and inhomogeneous scaling properties, since different irregular structures may have the same fractal dimension [70, 71]. Thus, generalized fractal dimensions provide more information about the space filling properties than the fractal dimension alone (see, for instance, [1]). This section describes the generalized fractal dimensions through Renyi entropy as follows.

18

1 Mathematical Background of Deterministic Fractals

Alfred Renyi introduced the measure to quantify the uncertainty or randomness of a given system. It plays  a vital role in the information theory [75]. N pi = 1, the Renyi entropy of order q is given by Given probabilities pi , i=1  q 1 log pi , 1−q i=1 N

R Eq =

where q ≥ 0 and q = 1. At q = 1, the value of R E q is potentially undefined as it generates the indeterminate form, otherwise R E q values are decreasing as a function of q. If q → 1, then R E q → R E 1 which is defined by R E 1 = −ln

N 

pi log pi .

i=1

R E 1 is called Shannon entropy. Renyi Fractal Dimensions or Generalized Fractal Dimensions (GFD) of order q ∈ (−∞, ∞) are defined, in terms of generalized Renyi Entropy, as

  q N i=1 pi 1 log2 (1.19) Dq = lim r →0 q − 1 log2 r where pi is the probability distribution. For all q, we have Dq > 0 and Dq is a monotonically decreasing function of q such that D0 ≥ D1 ≥ D2 . Also, observe that Dq = D0 = 0, for all q for a constant signal because all probabilities except one equal to zero, whereas the exceptional probability value is one.

1.6.1 Some Special Cases • If q = 0, then D0 =

log2 N , log2 r

(1.20)

which is known as the Fractal Dimension. • As q −→ 1, Dq converges to D1 , which is given by N D1 = lim

r →0

pi log2 pi , log2 r

i=1

where D1 is the Information Dimension • If q = 2, then Dq is called the Correlation Dimension.

(1.21)

1.6 Generalized Fractal Dimensions

19

1.6.2 Limiting Cases of Generalized Fractal Dimensions The two limit cases in the Generalized Fractal Dimensions Method, when q = −∞ and q = ∞, are log2 ( pmin ) log2 r log2 ( pmax ) = lim r →0 log2 r

D−∞ = lim

r →0

D∞ where

pmin = min{ p1 , p2 , . . . , p NV }, pmax = max{ p1 , p2 , . . . , p NV }.

1.6.3 Range of Generalized Fractal Dimensions The two limit cases, D−∞ and D∞ , define the Range of Generalized Fractal Dimensions of a given Fractal Time Series as RG F D = D−∞ − D∞ .

(1.22)

1.7 Concluding Remarks In this chapter, a broad outline of an iterated function system of contraction mappings is provided. The Barnsley framework of IFS has been extended to the local iterated function system and a countable iterated function system for constructing the deterministic fractals which are offered in this chapter. Consequently, the existence of an attractor of a local countable iterated function system is investigated. Moreover, the unique expression of the local attractor of the local countable iterated function system by the limit of a convergence sequence of attractors of the local iterated function system is presented. This chapter ends with the various notions of fractal dimensions which will be applied in the forthcoming chapters.

Chapter 2

Fractal Functions

2.1 Introduction The interpolation theory focuses on the existence of a continuous function which reconstructs the prescribed data set thereby helping to approximate the real-valued continuous function. In the historical development of the approximation theory, beyond polynomials, splines and trigonometric functions are likewise applied in approximation methods which are directed toward the smooth approximants. Then again, numerous practical and natural phenomenal signals reveal the non-smoothness in their traces and henceforth request non-smooth functions for significant reconstruction. The notion of fractal functions is explored based on the theory of the iterated function system which was made by envisaging the universe as a fractal. Consequently, fractal functions are used for both smooth and non-smooth approximations by including various classical approximation methods, although there are differences between the fractal interpolation function and the traditional interpolation function (see for more details, [42–59]). Fractal interpolants are constructed by the theory of iterated function system which offers a self-referential functional equation for the interpolant and implies a self-similarity in magnification. Additionally, the choice of vertical scaling factors provides a flexible and optimal interpolant which also generates a specific traditional interpolant [47]. As a consequence of its fruitful success in non-smooth approximation, the theory of fractal functions has been extensively investigated and there are developments in the pipeline beyond its mathematical framework. This chapter presents the construction of fractal interpolation functions and their evolution based on the vertical scaling factors and different iterated function systems.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_2

21

22 Fig. 2.1 Interpolation functions of the data set given in Example 2.1

2 Fractal Functions 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.2 Interpolation Functions Let {x1 , x2 , . . . , xn } be a partition of the closed interval [x1 , xn ] ⊂ R such that x1 < x2 < · · · < xn , where n ∈ N. Let {(xi , yi ) ∈ [x1 , xn ] × R : i ∈ Nn } be a given data set or interpolation points and the interpolation function f is a continuous function f : [x1 , xn ] −→ R satisfying f (xi ) = yi for i ∈ Nn . The interpolation theory deals with the existence and construction of a continuous function which interpolates the given data set. Example 2.1 Consider the data {(0, 0), (1/2, 1/2), (1, 0)}. The line joining (0, 0) and (1/2, 1/2) is f 1 (x) = x. Similarly, the line joining (1/2, 1/2) and (1, 0) is f 2 (x) = −x. A simple continuous function which passes through the given data is the combination of the straight line joining the pair of points (0, 0), (1/2, 1/2) and (1/2, 1/2), (1, 0). Thus,  x, if x ∈ [0, 1/2], f (x) = −x, if x ∈ [1/2, 1] is a continuous function and connects all the points of the given data. So, f is the interpolation function of the given data, and it is known as the classical interpolation function. The graph of f is showed in Fig. 2.1. Example 2.2 Consider the data {(0, 1/5), (1/3, 1/2), (1/2, 1/3), (3/4, 3/4), (1, 1/2)}, and this interpolation data set is interpolated by the continuous function ⎧ 9 x + 15 , if x ∈ [0, 1/3], ⎪ 10 ⎪ ⎪ ⎨ 5 x − x, if x ∈ [1/3, 1/2], f (x) = 65 ⎪ 3 x − 56 , if x ∈ [1/2, 3/4], ⎪ ⎪ ⎩3 − x, if x ∈ [3/2, 1]. 2 Figure 2.2 elucidates the graphical representation of f .

2.3 Fractal Interpolation Function Fig. 2.2 Interpolation functions of the data set given in Example 2.2

23

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.3 Fractal Interpolation Function Based on the IFS theory, the fractal interpolation function is constructed as follows. Consider a data set {(xi , yi ) ∈ R2 : i ∈ Nn } with increasing abscissae. Set I = [x1 , xn ] and Ii = [xi , xi+1 ] for i ∈ Nn−1 . Let L i : I → Ii , i ∈ Nn−1 be (n − 1) contraction homeomorphisms such that L i (x1 ) = xi , L i (xn ) = xi+1 .

(2.1)

Let X := I × [a, b] for some −∞ < a < min yi ≤ max yi < b < ∞ and Ri : X → [a, b], i ∈ Nn−1 be continuous mappings satisfying Ri (x1 , y1 ) = yi , Ri (xn , yn ) = yi+1 , for x ∈ I, y, y ∗ ∈ [a, b], |Ri (x, y) − Ri (x, y ∗ )| ≤ ri |y − y ∗ |

(2.2)

where ri ∈ [0, 1), i ∈ Nn−1 . Scilicet Ri ’s are contractions with respect to the second variable. Define functions f i : X → Ii × [a, b], i ∈ Nn−1 , by f i (x, y) = (L i (x), Ri (x, y)).

(2.3)

Consider the IFS F := {X ; f i : i ∈ Nn−1 } and the set-valued map F : H(X ) → H(X ) defined by  f i (B). (2.4) F(B) = i∈Nn−1

The following theorem is narrated by the notion of the Barnsley remarkable framework for the fractal interpolation theorem [4].

24

2 Fractal Functions

Theorem 2.1 The self mapping F defined in (2.4) has a unique compact set Gg such that Gg = F(Gg ) and Gg = {(x, g(x)) : x ∈ I } is the graph of a continuous function g : I → [a, b] which obeys g(xi ) = yi for i ∈ Nn . Proof Proof of this theorem includes the following parts: (i). Uniqueness of Gg , (ii). Gg is the graph of the function g : I → [a, b], (iii). Continuity of the function g. Let us start to prove (i). Uniqueness of Gg . Let Gg be any attractor of the IFS F . Then Gg = F(Gg ) = i∈Nn−1 f i (Gg ). Let I  = {x ∈ I : (x, y) ∈ Gg for some y ∈ [a, b]}. It is clear that F(I  ) = I , but F(I ) = I because {I, L i : i ∈ Nn−1 } is hyperbolic ifs. It gives that I  = I . Now, let us to prove (ii). Gg is the graph of the function g : I → [a, b]. It is enough to prove that only one y ∈ [a, b] corresponds to each x ∈ I . Let Si = {(x, y) ∈ Gg : x = xi } for i ∈ Nn . There is no common point between S1 and f i (S1 ) except i = 1; it gives that f 1 (S1 ) = S1 . We arrive at S1 = (x1 , y1 ), since f 1 is a contraction on the compact metric space. In the same way, Sn = (xn , yn ). Note that the only points which map to Si are S1 under the mapping f i , and Sn under the mapping f i+1 for i ∈ Nn \ {1, n}; it gives that Si = f i (S1 ) ∪ f i+1 (Sn ) = (xi , yi ). It clarifies xi has only one image yi for all i ∈ Nn . Consider δ = max{|s − t| : (x, s), (x, t) ∈ Gg , some x ∈ I }. Since Gg is a compact set, maximum (δ) is achieved in Gg , say maximum attains at (x  , s), (x  , t) with δ = |s − t|. But, x  ∈ (xi , xi+1 ) for some i. Then there exist two points (L i−1 (x  ), u), (L i−1 (x  ), v) in Gg such that s = Fi ((L i−1 (x  ), u)) and t = Fi ((L i−1 (x  ), v)) because L i is a homeomorphism on I . Hence, δ = |s − t| = |Fi ((L i−1 (x  ), u)) − Fi ((L i−1 (x  ), v))| ≤ q.|u − v| ≤ q.δ. We arrived at δ = 0; it narrates that s = t, so only one y ∈ [a, b] corresponds to each x ∈ I . Hence, Gg is the graph of the function g : I → [a, b] such that g(xi ) = yi for i ∈ Nn . Define  : G → G by

(h) = Ri L i−1 (x), h ◦ L i−1 (x) , x ∈ Ii , i ∈ Nn−1 ,

(2.5)

where G = {h : I → R | h is continuous on I, h(x1 ) = y1 , h(xn ) = yn } is the complete metric space endowed with the uniform metric δ(h, h  ) = max{|h(x) − h  (x)| : x ∈ I }. Then  δ((h) − (h  )) = max |Ri (L i−1 (x), h ◦ L i−1 (x)) − Ri (L i−1 (x), h  ◦ L i−1 (x))| : x ∈ Ii , i ∈ Nn−1  ≤ max ri · |h ◦ L i−1 (x) − h  ◦ L i−1 (x)| : x ∈ Ii , i ∈ Nn−1 ≤ r · δ(h(x), h  (x)).

2.3 Fractal Interpolation Function

25

It shows that  is a contraction mapping on (G, δ) with the contraction factor r = max{ri : i ∈ Nn−1 } < 1. So,  has a unique fixed point g which satisfies the functional equation

g(x) = Ri L i−1 (x), g ◦ L i−1 (x) , x ∈ Ii , i ∈ Nn−1 .

(2.6)

Further, the graph of g is an attractor of the IFS F and g ∈ G hence g is continuous. Definition 2.1 The function g obtained in Theorem (2.1) whose graph is the attractor of the IFS F is called a Fractal Interpolation Function or Fractal function associated with the data set {(xi , yi ) ∈ R2 : i ∈ Nn }. Let a set of interpolation data {(xi , yi ) ∈ [x1 , xn ] × R : i ∈ Nn } be given. Then, the following process explains the construction of an iterated function system in R2 such that its attractor is the graph of the interpolation function of the given data. Consider the IFS {[x1 , xn ] × R; f i : i ∈ Nn−1 }, where

    x ai 0 x b fi = + i di ci αi y y

(2.7)

for i ∈ Nn−1 . By Eq. (2.1), f i maps the endpoints of the given data, i.e., (x1 , y1 ), (xn , yn ), to the endpoints of each subinterval, i.e., (xi , yi ), (xi+1 , yi+1 ), respectively. Hence, the mapping f i has constrained as

  x x f i 1 = i and y1 yi

  xn xi+1 = fi yn yi+1

(2.8) (2.9)

for all i ∈ Nn−1 . These constraints gives the following system of linear equation: ai x1 + bi = xi ai xn + bi = xi+1 ci x1 + αi y1 + bi = yi ci xn + αi yn + bi = yi+1

(2.10)

for all i ∈ Nn−1 . The transformation f i maps the line segment parallel to the y-axis to the line segment parallel to the y-axis. Moreover, if the length of the line segment L is l, then the length of the line segment f i (L) is αi l, for all i ∈ Nn−1 . In other words, the ratio between the length of the line segments L and f i (L) is |αi | (see Fig. 2.3). We say αi as the vertical scaling factor of the transformation f i for all i ∈ Nn−1 . If αi is a free parameter in the system of linear equation (2.10), then it gives a unique solution for (2.10). Hence, ai , bi , ci , di can be uniquely determined by the following equations:

26

2 Fractal Functions

Fig. 2.3 Effect of vertical scaling factor on a fractal interpolation function

xi+1 − xi xn − x1 xn xi − x1 xi+1 bi = xn − x1 (yi+1 − yi ) − αi (yn − y1 ) ci = xn − x1 (xn yi − x1 yi+1 ) − αi (xn y1 − x1 yn ) bi = . xn − x1

ai =

(2.11)

The free parameter αi determines the shape of the fractal interpolation function generated by the IFS (2.7). If αi = 0 for all i ∈ Nn−1 , one can produce the piecewise linear interpolation function which is discussed in Examples 2.1 and 2.2. We begin with N + 1 sample points. The contractive homeomorphism mappings L i , i = 1, 2, . . . , N , provides N − 1 points in each N subintervals in the first iteration. Consequently, N (N − 1) + N + 1 = N 2 + 1 distinct points are obtained at the end of the first iteration. Similarly, the second iteration yields N (N 2 − 1) + N + 1 = N 3 + 1 distinct points. In general, by induction, the FIF g interpolates N n+1 + 1 distinct points at n-iteration and the density of the interpolation function is high when the iteration successively increases on affine mappings. Example 2.3 Consider the data {(0, 0), (1/2, 1/2), (1, 0)} same as in Example 2.1 and fix the scaling factors α1 = −1/2, α2 = −1/2. Then, the fractal interpolation function f is determined from the IFS consisting of the following mappings: 1 1 x, R1 (x, y) = x − y, 2 2 1 1 1 L 2 (x) = x + , R2 (x, y) = 1 − x − y. 2 2 2 L 1 (x) =

Here, the constants ai , bi , ci , and di are estimated from Eq. (2.11). The graph of f is shown in Fig. 2.4.

2.3 Fractal Interpolation Function Fig. 2.4 Fractal interpolation functions of the data set given in Example 2.3

27 1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Example 2.4 Consider the data {(0, 1/5), (1/3, 1/2), (1/2, 1/3), (3/4, 3/4), (1, 1/2)} as given in Example 2.2 and choose the scaling factors α1 = −1/3, α2 = 1/3, α3 = −1/2, α4 = 1/2. Then, the fractal interpolation function f is generated by the IFS which includes the following mappings: 1 4 1 2 x, R1 (x, y) = x − y + , 3 5 3 15 1 1 13 1 −4 L 2 (x) = x + , R2 (x, y) = x+ y+ , 6 3 15 3 30 1 1 13 1 17 L 3 (x) = x + , R3 (x, y) = x− y+ , 4 2 30 2 30 3 1 13 1 −2 L 4 (x) = x + , R4 (x, y) = x+ y+ . 4 4 5 2 20 L 1 (x) =

By choosing the scaling factors α1 = 1/2, α2 = −1/2, α3 = 3/4, α4 = 3/4, the FIF f  is generated by the IFS which includes the mappings 1 3 x, R1 (x, y) = x 3 20 1 1 L 2 (x) = x + , R2 (x, y) = 6 3 1 1 L 3 (x) = x + , R3 (x, y) = 4 2 3 1 L 4 (x) = x + , R4 (x, y) = 4 4 L 1 (x) =

1 1 y+ , 2 10 1 3 −1 x− y+ , 60 2 5 3 11 23 x+ y+ , 120 4 60 3 3 −19 x+ y+ . 40 4 5

+

The graph of f is depicted in Fig. 2.5a and the graph of f  is given in Fig. 2.5b. It evidences that small changes in the scaling factor provide more difference in the graph of FIF. Moreover, changing scaling factors does not affect the contractive functions L i for i = 1, 2, 3, 4.

28

2 Fractal Functions

0.8

1.2

0.7

1

0.6

0.8

0.5

0.6

0.4

0.4

0.3

0.2

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a)

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Fig. 2.5 Fractal interpolation functions of the data set given in Example 2.4

Note that the fractal interpolation function g can be generated as the fixed point of the operator  : G → G defined by

(h) = Ri L i−1 (x), h ◦ L i−1 (x) , x ∈ Ii , i ∈ Nn−1 . Then,  is a contraction mapping on (G, δ) with the contraction factor r = max{ri : i ∈ Nn−1 } < 1. The fixed point of  is the FIF g corresponding to the IFS F . Therefore, g satisfies the functional equation

g(x) = Ri L i−1 (x), g ◦ L i−1 (x) , x ∈ Ii , i ∈ Nn−1 .

(2.12)

The widely studied IFS is of the form L i (x) = ai x + bi , Ri (x, y) = αi y + qi (x), i ∈ Nn−1 ,

(2.13)

where αi ∈ (−1, 1) is a parameter called as a vertical scaling factor of the transformation Ri and qi : I → R is a continuous function satisfying qi (x1 ) = yi − αi y1 ,

qi (xn ) = yi+1 − αi yn .

If αi = 0 for all i ∈ Nn−1 , the FIF g reduces to the classical interpolation function which can be described by the nature of qi ’s. By choosing the suitable continuous function qi , there are numerous works dealing with the existence of various types of FIF [48–57]. For instance, if qi (x) = ci x + di , then the corresponding fractal interpolation function is known as the linear fractal interpolation function and it satisfies g(L i (x)) = Ri (x, g(x)) = αi g(x) + ci x + di , x ∈ Ii , i ∈ Nn−1 .

(2.14)

2.3 Fractal Interpolation Function

29

If qi (x) = x 2 + ci x + di , then the associated fractal interpolation function is known as the quadratic fractal interpolation function (QFIF) and it satisfies g(L i (x)) = Ri (x, g(x)) = αi g(x) + x 2 + ci x + di , x ∈ Ii , i ∈ Nn−1 .

(2.15)

Wang et al. [47] extended the class of FIFs appearing in (2.13) by considering vertical scaling parameters as function scaling factors, i.e., αi ∈ C(I ), satisfying αi ∞ = sup{|αi | : x ∈ I } < 1 instead of constant scaling factors αi ∈ (−1, 1), i ∈ Nn−1 . The corresponding fractal interpolation satisfies g(x) = αi (L i−1 (x))g(L i−1 (x)) + qi (L i−1 (x)) ∀x ∈ Ii , i ∈ Nn−1 .

(2.16)

The linear fractal interpolation function with variable scaling factors obeys the functional equation g(L i (x)) = Ri (x, g(x)) = αi (x)g(x) + ci x + di , x ∈ Ii , i ∈ Nn−1 .

(2.17)

The quadratic fractal interpolation function with variable scaling factors abide by the functional equation g(L i (x)) = Ri (x, g(x)) = αi (x)g(x) + x 2 + ci x + di , ∀x ∈ Ii , i ∈ Nn−1 . (2.18) The graphical representation of the fractal interpolation function with constant scaling, fractal interpolation with variable scaling, quadratic fractal interpolation function with constant scaling, quadratic fractal interpolation function with variable scaling is given in Fig. 2.6 (a), (b), (c) and (d) respectively.

2.4 Hidden Variable Fractal Interpolation Function This section succinctly reviews the hidden variable FIFs (HFIFs) that are constructed by projecting vector-valued FIF corresponding to a generalized interpolation data which approximate non-self-affine patterns [10–12, 46] (Fig. 2.6).  = {(xi , yi , z i ) ∈ I × R2 : i ∈ N N }, where {z i : Consider a generalized data set D i ∈ N N } are real parameters. The idea is to construct a fractal interpolation function g1 : I → R such that g1 (xi ) = yi for all i ∈ N N , project its graph into I × R in such a way that the projection is the graph of a function that interpolates {(xi , yi )}. Let R2 be

endowed with the Manhattan metric d M (x1 , y1 ), (x2 , y2 ) = |x1 − x2 | + |y1 − y2 |, which is induced by the l 1 -norm. For i ∈ N N −1 , let the contraction homeomorphisms L i : I → Ii ⊂ I be as given in (2.1), and the functions Fi : I × R2 → R2 be expressed as

t

t Fi (x, y) = Fi (x, y, z) = Fi1 (x, y, z), Fi2 (x, z) := Ai (y, z)t + pi (x), qi (x) , (2.19)

30

2 Fractal Functions

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0

0.1

0.2

0.3

0.4

(a)

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

(b)

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -0.1

0.5

0

0

0.1

0.2

0.3

0.4

0.5

(c)

0.6

0.7

0.8

0.9

1

-0.1

0

0.1

0.2

0.3

0.4

0.5

(d)

Fig. 2.6 Fractal interpolation functions of the data set given in Example 2.4

  αi βi , and pi , qi where t denotes the transpose, An are upper-triangular matrices 0 γi are suitable real-valued continuous functions so that the following conditions are satisfied for all i ∈ N N −1 : ∗ ∗ constant c1 > 0, (i) d M (F i (x, y, z), Fi (x , y,∗ z))∗ ≤

c1 |x − x | for some

(ii) d M Fi (x, y, z), Fi (x, y , z ) ≤ s d M (y, z)(y ∗ , z ∗ ) for 0 ≤ s < 1, (iii) Fi (x1 , y1 , z 1 ) = (yi , z i ) and Fi (x N , y N , z N ) = (yi+1 , z i+1 ). The variables αi , βi , and γi are chosen such that Ai 1 < 1 for

all i ∈ N N −1 . Define wi : I × R2 → I × R2 by wi (x, y, z) = L i (x), Fi (x, y, z) . It follows from the ∗ conditions on L i and Fi that wi are contraction maps with respect to the metric d M

∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ defined on I × R by d M (x, y, z), (x , y , z ) = |x − x | + θ d M (y, z), (y , z ) , i where θ = 1−a , and a = max{ x Nh−x : i ∈ N N −1 }, where h i = xi+1 − xi . Conse2c1 1 2 quently, the generalized IFS {I × R ; wi : i ∈ N N −1 } admits an attractor A ⊆ I × R2 . It follows from the generalized IFS theory that the aforementioned attractor A is the graph of the continuous vector-valued function g : I → R2 such that g(xi ) = (yi , z i ) for all i ∈ N N −1 . Letting g = (g1 , g2 ), it follows that g1 : I → R is a continuous function interpolating D and is called the (coalescence) hidden variable fractal interpolation function (see, for instance, [10–12]). Similarly, the projection {(x, g2 (x)) : x ∈ I } of the attractor A is self-affine and provides a fractal function g2 that interpolates the data {(xi , z i ) : i ∈ N N }. For graphical representation see the Figs. 2.7 and 2.8. Let G∗ be the set of continuous functions h : I → R2 such that h(x1 ) = (y 1 , z 1 ), h(x N ) = (y N , z N ) equipped with the metric d(h, h∗ ) = max{d M h(x), h∗ (x) : x ∈

2.4 Hidden Variable Fractal Interpolation Function

31

I }. To obtain a functional equation for g, we recall that g is the fixed point of the operator T∗ : G∗ → G∗ defined by

(T∗ h)(x) = Fi L i−1 (x), h L i−1 (x) , for x ∈ Ii , i ∈ N N −1 . Whence, the vector-valued function g satisfies the functional equation



t g L i (x) = Ai g(x) + pi (x), qi (x) , x ∈ I. The image T∗ g of the vector-valued function g = (g1 , g2 ) can be written componentwise as (T1 g1 , T2 g2 ), where T1 and T2 are the componentwise Read–Bajraktarevi´c operator. Then satisfy

g1 L i (x) = T1 g1 (L i (x)) = Fi1 (x, g1 (x), g2 (x)) = αi g1 (x) + βi g2 (x) + pi (x),

g2 L i (x) = T2 g2 (L i (x)) = Fi2 (x, g2 (x)) = γi g2 (x) + qi (x), x ∈ I. Let f ∈ C(I ) be a continuous function and consider the case: qi (x) = f ◦ L i (x) − αi b(x).

(2.20)

Here, b : I → R is a continuous map that fulfills the conditions b(x1 ) = f (x1 ), and b(x N ) = f (x N ) and b = f . This case is proposed by Barnsley [2] and Navascués [42] as a generalization of any continuous function. Here, the interpolation data are {(xi , f (xi )) : i ∈ N N }. We define the α-fractal function corresponding to f in the following. Definition 2.2 The continuous function f α : I → R whose graph is the attractor of the IFS defined by (2.13), (2.20) is referred to as the α-fractal function associated with f , with respect to b and the partition D.

Fig. 2.7 Fractal interpolation function

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

32

2 Fractal Functions

1.2

0.8 0.7

1

0.6 0.8

0.5

0.6

0.4 0.3

0.4

0.2 0.2

0

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

0.1

0.2

(a)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Fig. 2.8 Hidden variable fractal interpolation function for the data set {(0, 0, 0), (1/2, 2/3, 2/3), (1, 0, 0)} with α = {0.4, 0.4}, β = {0.4, 0.4}, and γ = {0.4, 0.4}

According to (2.14), f α satisfies the functional equation: f α (x) = f (x) + αi [( f α − b) ◦ L i−1 (x)] ∀x ∈ Ii , i ∈ N N −1 .

(2.21)

Note that for α = 0, f α = f . Thus, the aforementioned equation may be treated as an entire family of functions f α with f as its germ. By this method, one can define fractal analogues of any continuous function.

2.5 Classical Calculus on Fractal Interpolation Functions Fractal interpolation functions are not necessarily differentiable even though they are continuous everywhere. Hence, FIFs are sophisticated in approximating the rough curve and which precisely reconstruct the naturally occurring functions when compared with classical interpolants. Nevertheless, a smooth curve is needed to approximate some kind of functions which include the self-similarity under magnification. Barnsley and Harrington presented that the fractal interpolation function generated from the IFS of the form given in Eq. (2.13) can be integrated indefinitely, and the integral of FIF is also FIF and interpolates a certain set of data. Further, they have explored the construction of p-times frequently differentiable FIF when the derivative values that reach up to pth order are available at the initial endpoint of the interval. The example of differentiable functions cannot actually be considered as fractals, but they retain the name fractal interpolation function because of the flavor of the scaling, and the Hausdorff–Besicovitch dimension of their graphs are non-integer. Due to the sophisticated usage of such fractal interpolants in nonlinear approximation, there are continuous efforts that have been extended on FIF. Later, there are authors who have enhanced the calculus of FIF and described the principle of construction of p-times continuously differentiable FIF which interpolated the

2.5 Classical Calculus on Fractal Interpolation Functions

33

given data. As mentioned earlier, by suitable choices of ri and qi , differentiable FIFs may be constructed. Now, let us briefly review the theorems presented by Barnsley and Harrington [5]. Theorem 2.2 If f is the fractal interpolation function associated with {K ; (L i (x), Ri (x, y)) : i =  x1, 2, . . . , n} where Ri is given in Eq. (2.13) and, for a given yˆ1 , define fˆ(x) = yˆ1 + x1 f (t)dt. Then fˆ is the fractal interpolation function associated with {K ; (L i (x), Rˆ i (x, y)) : i = 1, 2, . . . , n}, where, for i = 1, 2, . . . , n, Rˆ i = ai αi y + qˆi (x), xi − xi−1 ai = , xn − x1



x

qˆi (x) = yˆi − ai αi yˆ1 + ai yˆi+1 = yˆ1 + yˆn = yˆ1 +

i 

qi (t)dt,

x1





xn

αk ( yˆn − yˆ1 ) +

ak k=1 n−1 k=1

fˆ(x) = yˆ1 + i



x

qk (t)dt ,

x1

x ak x1n qk (t)dt .  1 − n−1 k=1 ak αk

Proof Define



f (t)dt.

(2.22)

x1



fˆ(L i (x)) = yˆ1 + i  = yˆ1 + = yˆi +

L i (x)

x1 xi

f (t)dt 

L i (x)

f (t)dt + i

x1  L i (x) L i (x1 )  x

= yˆi + ai

f (t)dt

xi

f (t)dt f (L i (x))d x.

x1

The functional equation f (L i (x)) = Ri (x, f (x)) = αi f (x) + qi (x) gives that fˆ(L i (x)) = yˆi + ai = yˆi + ai



x

x  1x x1

By Eq. (2.22)

(αi f (x) + qi (x)) d x  x αi f (x)d x + ai qi (x)d x. x1

34

2 Fractal Functions

fˆ(L i (x)) = yˆi + ai αi ( fˆ(x) − yˆ1 ) + ai



x

qi (x)d x

x1

= ai αi fˆ(x) + yˆi − ai αi yˆ1 + ai



x

qi (x)d x

x1

= ai αi fˆ(x) + qˆi (x) fˆ(L i (x)) = Rˆ i (x, fˆ(x)). Hence fˆ is the attractor of the stated iterated function system. The continuity at the joints yi must be satisfied since the antiderivate fˆ is continuous. Take x = xn , L i (xn ) = xi+1  xn yˆi+1 = yˆi + ai αi ( yˆn − yˆ1 ) + ai qi (x)d x x1  xn yˆi+1 − yˆi = ai {αi ( yˆn − yˆ1 ) + qi (x)d x}. x1

We know that yˆi+1 = yˆ1 +

i

yˆi+1 = yˆ1 +

ˆk+1 k=1 ( y

i 

− yˆk )

  ak αk ( yˆn − yˆ1 ) +

xn

 qk (x)d x .

x1

k=1

Take i = n − 1 yˆn = yˆ1 +

n−1 

  ak αk ( yˆn − yˆ1 ) +

yˆn − yˆ1 =  ( yˆn − yˆ1 ) 1 −

n−1  k=1

 ak αk

ak αk ( yˆn − yˆ1 ) +

k=1

=

n−1 



yˆn = yˆ1 +

n−1 



xn

xn

ak

k=1

ak

k=1

 qk (x)d x

x1

k=1 n−1 

xn

qk (x)d x

x1

qk (x)d x

x1

x ak x1n qk (x)d x .  1 − n−1 k=1 ak αk

n−1 k=1

Corollary 2.1 Use the notation of Theorem 2.2 associated with FIF f . Then fˆ = f if and only if fˆ is FIF generated by the IFS {K ; (L i (x), Rˆ i (x, y))}, where Rˆ i = αˆ i y + qˆi (x), for i = 1, 2, . . . , n and αˆ i = ai αi , qˆi (x) = ai qi (x). The following theorem gives the existence of differentiable FIF and a method for the construction of C p -FIF. For a prescribed data set, an FIF with C p -continuity is

2.5 Classical Calculus on Fractal Interpolation Functions

35

obtained as the fixed point of IFS (2.13) with (2.20), where the scaling factors αi and the functions qi are chosen according to the following theorem. Theorem 2.3 Let {(xi , yi ) ∈ I × R : i ∈ Nn } be a given data set such that x1 < x2 < · · · < xn . Let L i (x) = ai x + bi , i ∈ Nn−1 , be the affine transformations satisfying Eq. (2.1) and let Ri (x, y) = αi y + qi (x), i ∈ Nn−1 , obey Eq. (2.2). Suppose p for some integer p ≥ 0, |αi | < ai and Ri ∈ C p (I ), i ∈ Nn−1 . Let (k) (xn ) qn−1 αi y + qi(k) (x) q1(k) (x1 ) , y = , y = , k = 1, 2, . . . , p. 1,k n,k k k k ai a1 − α1 an−1 − αn−1 (2.23) If Ri−1,k (xi , yi,k ) = Ri,k (x1 , y1,k ), i = 2, 3, . . . , n and k = 1, 2, . . . , p, then the IFS {K ; wi (x, y) : i ∈ Nn−1 } determines an FIF f ∈ C p (I ), and f (k) is the FIF determined by {K 0 ; wi (x, y) : i ∈ Nn−1 } , where K 0 is a non-empty closed rectangle in I × R, where f (k) denotes the kth derivative of f .

Ri,k (x, y) =

The equality proposed in the above theorem demands the resolution of systems of equations. Sometimes the system has no solution, mainly whenever some boundary conditions are imposed on the function (see [5]).

2.6 Concluding Remarks The classical problem of the interpolation theory investigates problem of the existence and reconstruction of the continuous function which fits the prescribed data. The traditional interpolation techniques generate smooth or piecewise differentiable functions despite the given data being irregular. Thus, existing non-fractal techniques may not be suitable to describe naturally occurring functions. Motivated by this situation, the fractal interpolation function based on the iterated function system is invented which is a generalization of classical interpolants. Fractal interpolation functions speak to a significant development to the traditional interpolation techniques in light of the fact that the interpolants considered are not really differentiable, and, in some cases, they are nowhere differentiable. In words of Barnsley: “they appear ideally suited for the approximation of naturally occurring functions which display some kind of geometrical self-similarity under magnification”. In this chapter, both the classical interpolation and the fractal interpolation functions are presented. Besides, a seminal result on differentiability of fractal interpolation functions is provided which lead to the creation of a subject called fractal splines. Further, various types of fractal interpolation functions, including a hidden variable fractal interpolation, α-fractal functions and the fractal interpolation function with variable scaling factors are succinctly presented in this chapter which will be discussed with fractional calculus in upcoming chapters.

Chapter 3

Fractional Calculus on Fractal Functions

3.1 Introduction The words fractional calculus were born from a communication between L’Hospital n and Leibniz in 1695. By denoting the nth derivative of f with respect to x as dd x nf , Leibniz had written a letter to L’Hospital. In his letter, Leibniz assumed that n takes the value from the positive integers, i.e., n ∈ N. L’Hospital replied by raising the n question of what meaning could be recognized to dd x nf if n is 1/2, a fraction. This question became a historical statement to pronounce the new name in the theory of mathematic and was also universally accepted as the first existence of the socalled fractional derivative. The name fractional calculus has stayed being used ever since, despite the fact that it is notable at this point that there is no motivation to confine n to the rational numbers. Without a doubt, any real number will do similarly too, even complex numbers might be permitted, yet this is well beyond the aim of this book. Like fractals, the notion of fractional calculus has also been applied to numerous fields of science. Fractional calculus has never experienced the popularity currently enjoyed by fractals since it does not lend itself to produce complex graphical structures. In [13], significant effort has been devoted to relating fractional calculus to fractal geometry. In addition, the relationship between fractional calculus, such as Riemann–Liouville fractional calculus, and the von Koch curve has been revealed. Further, this study found that the value of the fractal dimension of the von Koch curve is a linear function of its order of the Riemann–Liouville fractional calculus. In the literature of the approximation theory, diverse interpolation schemes are developed and reported in various places. However, when all classical approaches are directed toward the construction of smooth functions, the fact that numerous experimental and natural signals are unpleasant having a dense set of non-differentiable points was easily ignored often. So, to overcome this issue, Barnsley invented the fractal interpolation functions as a fixed point of a certain operator on a function space by using an iterated function system which specifies that the density of nondifferentiable points in a continuous function. The fractal interpolation techniques © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_3

37

38

3 Fractional Calculus on Fractal Functions

supply a general frame for the understanding of complexities that prevails in nature and the fractal interpolation functions are a generalization of the classical interpolation functions. Motivated by the Barnsley framework, many researchers have generalized the fractal interpolation functions to a different aspect. While the approximation of a continuous function in terms of a polynomial is given by the Weierstrass theorem, the approximation of a non-smooth function is important as objects in the universe, in general, abound with the class of functions of continuous everywhere and nowhere differentiable. In this motivating direction, many researchers pay more attention in recent years to the problem of how to portray non-smooth functions by fractal functions from different aspects. Hence, in order to analyze the irregularity (non-smoothness) of fractal interpolation functions, the Riemann–Liouville fractional calculus is applied and compared with its fractal dimension [13–24]. This chapter presents the Riemann–Liouville fractional calculus of different types of the fractal interpolation function. Several interesting definitions of fractional integrals are given in the literature, but the Riemann–Liouville fractional integral has been a precious cornerstone in fractional calculus ever since [25, 26]. The Riemann–Liouville fractional calculus is defined as follows: Let C([a, b]) be the set of all continuous functions on [a, b]. If f ∈ C([a, b]) then the Riemann–Liouville fractional integral of order β > 0 is defined by  x 1 (x − t)β−1 f (t)dt. (3.1) Iaβ f (x) = (β) a Then, the Riemann–Liouville fractional derivative of order β > 0 is defined by Daβ

1 f (x) = (n − β)



d dx

n  a

x

f (t)dt , (x − t)β−n+1

(3.2)

where n = [β] + 1, [·] denotes the integral part of β, and  (x) =

∞

e−t t x−1 dt, for all x ∈ R.

0

The expression in Eq. (3.2) is the most widely known definition of the fractional derivative; it is usually called the Riemann–Liouville fractional derivative. The Grunwald–Letnikov fractional derivative can be obtained from (3.2) under the same assumption on f (x) by performing repeatedly integration by parts and differentiation. While n + 1 times continuously differentiable function is necessary to define the Grunwald–Letnikov fractional derivative, in the perception of mathematics, the class of n + 1 times continuously differentiable functions is limited nevertheless the class of continuous functions is massive. In addition, the class of continuous functions is significant for application as the character of the majority of dynamical processes is smooth enough and does not tolerate discontinuities. Understanding this fact is important for the proper use of the methods of the fractional calculus in applications, especially because of the fact that the Riemann–Liouville definition (3.2) offers an excellent opportunity to weaken the conditions on the function f .

3.2 Linear Fractal Interpolation Function

39

3.2 Linear Fractal Interpolation Function Let f be a C p -linear fractal interpolation function determined by the IFS of the form {K ; wi : i ∈ Nn } wi

       x a 0 x b = i + i di ci αi y y

(3.3) (3.4)

with L i = ai x + bi , Ri = αi y + q(x), and qi (x) = ci x + di . Throughout this section, FIF generated by the IFS given in Eq. (3.3) is considered. Let us define the Riemann–Liouville fractional integral on f by Ixβ1 f (k) (xn ) =

1 (β)



xn

(xn − t)β−1 f (k) (t)dt

(3.5)

x1

β

with Ix1 f (k) (x1 ) = 0, where f (k) denotes the kth derivative of f , k = 1, 2, . . . , p. The following theorem guarantees the continuity of the fractional integral of the kth order derivative for the fractal interpolation function. Theorem 3.1 Let f be the C p -linear FIF determined by the IFS defined in Eq. (3.3). β If the FIF f is continuous on [x1 , xn ], then Ix1 f (k) (x) is continuous on [x1 , xn ], for k = 1, 2, . . . , p, n ∈ N. Proof Fix k, since f ∈ C p , for given  > 0, | f (k) (x) − f (k) (y)| < , whenever |x − y| < δ for all x, y ∈ [x1 , xn ]. Assume x < y, i.e., y = x + h, h > x0 ,   x   1  y   1   β (k) (y − t)β−1 f (k) (t)dt − (x − t)β−1 f (k) (t)dt  I x1 f (y) − I xβ1 f (k) (x) =  (β) x1 (β) x1   h  y  1 1 =  (y − t)β−1 f (k) (t)dt + (y − t)β−1 f (k) (t)dt (β) x1 (β) h   x  1 − (x − t)β−1 f (k) (t)dt  (β) x1   h  x  1 1 (y − t)β−1 f (k) (t)dt + (x − z)β−1 f (k) (z + h)dz =  (β) x1 (β) x1   x  1 − (x − t)β−1 f (k) (t)dt  (β) x1  h  x       1 (y − t)β−1   f (k) (t) dt + 1 (x − t)β−1   f (k) (t + h) − f (k) (t) dt ≤ |(β)| x1 (β) x1  1  β−1 ≤ Kx (h − x1 ) + (x − x1 )β , |(β)|

40

3 Fractional Calculus on Fractal Functions

   β−1  1 K x (h − x1 ) + (x − x1 )β where K = supt∈[x1 ,xn ]  f (k) (t) ; choose   = |(β)|    β  β β > 0. Therefore, Ix1 f (k) (y) − Ix1 f (k) (x) <   . Hence Ix1 f (k) (x) is continuous on [x1 , xn ]. Since k is arbitrary in 1, 2, . . . , p, therefore, the statement is true for all k. Theorem 3.2 Let f be C p -linear FIF determined by the IFS defined in Eq. (3.3). β Then the Riemann–Liouville fractional integral of f of order β, Ix1 f (k) (x), is (k) C p -linear FIF associated with {K ; wi : wi (x, y) = (L i (x), Rˆ i,β (x, y)), i ∈ Nn−1 }, where yˆ1,β = 0 and for each i ∈ Nn−1 , β > 0, β−k (k) (k) (x, y) = ai ri y + qˆi,β (x), Rˆ i,β

ai =

(xi+1 − xi ) , (xn − x1 ) β−k β (k) Ix1 qi (x),

(k) (k) (k) qˆi,β (x) = yˆi−1,β + f i−1,β (x) + ai

(k) yˆi,β = Ixβ1 f (k) (xi ),  xi−1 1 (k) f i,β (x) = [(L i (x) − t)β−1 − (xi−1 − t)β−1 ] f (k) (t)dt, (β) x1

for all k = 1, 2, . . . , p. Proof Let f be the C p -linear FIF determined by the IFS defined in Eq. (3.3). Hence f obeys the functional equation (2.12) and by Theorem 2.3, f (k) (L i (x)) =   r f (k) (x)+qi(k) (x) R x, f (k) (x) = i for all x ∈ I, i ∈ Nn−1 . ak i

β I x1 f (k) (L i (x)) =

= + = + = =

 L (x) i

1 (L i (x) − t)β−1 f (k) (t)dt (β) x1  x  x i−1 i−1 1 1 (xi−1 − t)β−1 f (k) (t)dt − (xi−1 − t)β−1 f (k) (t)dt (β) x1 (β) x1  x  L (x) i−1 i 1 1 (L i (x) − t)β−1 f (k) (t)dt + (L i (x) − t)β−1 f (k) (t)dt (β) x1 (β) xi−1  x  x i−1 i−1 1 1 (xi−1 − t)β−1 f (k) (t)dt + [(L i (x) − t)β−1 − (xi−1 − t)β−1 ] f (k) (t)dt (β) x1 (β) x1  L (x) i 1 (L i (x) − t)β−1 f (k) (t)dt (β) xi−1  L (x) i 1 (k) (k) (L i (x) − t)β−1 f (k) (t)dt yˆi−1,β + f i,β (x) + (β) xi−1  x 1 (k) (k) yˆi−1,β + f i,β (x) + (L (x) − L i (u))β−1 f (k) (L i (u))ai du. (β) x1 i

The functional equation f (k) (L i (x)) = Ri (x, f (k) (x)) = ri f (k) (x) + qi (x) and Theorem 2.3 gives

3.2 Linear Fractal Interpolation Function

41

 x (k)  β−1 ri f (k) (u) + qi (u) a β (k) (k) a (x − u) I x1 f (k) (L i (x)) = yˆi−1,β + f i,β (x) + i du k (β) x1 i ai β−k  x

a (k) (k) = yˆi−1,β + f i,β (x) + i (β) (k)

(k)

β−k

= yˆi−1,β + f i,β (x) + ai 

β−k  x

a (x − u)β−1 ri f (k) (u)du + i (β) x1 β

x1

(k)

(x − u)β−1 qi

(u)du

β−k β (k) I x1 qi (x)

ri I x1 f (k) (x) + ai

β β (k) I x1 f (k) (L i (x)) = Rˆ i,β x, I x1 f (k) (x) .

Moreover,  (k)  (k) x1 , yˆ1,β = Rˆ i,β Rˆ i,β (x1 , 0) β−k

(k) = qˆi,β (x1 ) + ai

ri Ixβ1 f (k) (x1 ) β−k β (k) Ix1 qi (x1 )

(k) (k) = yˆi−1,β + f i,β (x1 ) + ai (k) 



β−k

ri Ixβ1 f (k) (x1 )

β−k

ri Ixβ1 f (k) (xn )

+ ai

(k) = yˆi−1,β .

β−k (k) Rˆ i,β xn , yˆn,β = qˆi,β (xn ) + ai ri Ixβ1 f (k) (xn ) β−k β (k) Ix1 qi (xn )

(k) (k) = yˆi−1,β + f i,β (xn ) + ai

+ ai

= Ixβ1 f (k) (L i (xn )) (k) = yˆi,β . (k) (x, y)), i ∈ Nn−1 } generates C p Hence, the IFS {K ; wi : wi (x, y) = (L i (x), Rˆ i,β β (k) ) : i ∈ Nn }. linear FIF, Ix1 f (k) (x), which interpolates the data set {(xi , yˆi,β

Example 3.1 Consider the FIF f associated with the data set {(0, 0), (1/3, 1/2), (2/ 3, 1/2), (1, 0)} with vertical scaling factors r1 = 3/5, r2 = −3/5, and r3 = 4/5. The fractal interpolation function f is determined by the IFS consisting of 1 1 3 x, R1 (x, y) = y + x, 3 5 2 1 1 1 −3 L 2 (x) = x + , R2 (x, y) = y+ , 3 3 5 2 2 1 1 1 4 L 3 (x) = x + , R3 (x, y) = y − x + . 3 3 5 2 2 L 1 (x) =

The graph of f is given in Fig. 3.1a. If k = 0 and β = 0.2, then the fractional integral of order 0.2 of FIF, Ix0.2 f , interpolates the data {(0, 0), (1/3, 185/1223), (2/3, 721/ 1 f is generated by the IFS containing the same 1986), (1, 51/700)}. In this case, Ix0.2 1 L i (x) for i = 1, 2, 3 and

42

3 Fractional Calculus on Fractal Functions

1.4

0.8

1.2

0.7

1

0.6

0.8

0.5

0.6

0.4

0.4

0.3

0.2 0

0.2

−0.2

0.1

−0.4 0

0.2

0.6

0.4

(a) f

0.8

1

0

0

0.2

0.4

0.8

0.6

1

(b) I x0.02 f

Fig. 3.1 Fractal interpolation function f associated with {(0, 0), (1/3, 1/2), (2/3, 1/2), (1, 0)} and its fractional integral of order 0.2

the

data

set

51 1.2 328 0 y+ x , (x, y) = Rˆ 1,0.2 681 140 55 1.2 153 −328 153 0 y+ x + (x + 1)0.2 − , Rˆ 2,0.2 (x, y) = 681 1223 350 350 971 1.2 1819 0.2 1613 691 0 y+ x + x + (x, y) = Rˆ 3,0.2 1076 4318 1516 2065 with vertical scaling factors r1 = 328/681, r2 = −328/681, and r3 = 691/1076. f is shown in Fig. 3.1b. Now look at k = 0 and β = 1, then The graph of Ix0.2 1 Ix11 f passes through the interpolation points {(0, 0), (1/3, 23/132), (2/3, 7/12), (1, 5/11)} . Further, Ix11 f is generated by the same L i (x) for i = 1, 2, 3 and 1 1 0 (x, y) = y + x 2 , Rˆ 1,1 5 12 1 23 −1 0 ˆ y+ x+ , R2,1 (x, y) = 5 6 132 1 7 4 1 0 y − x2 + x + , (x, y) = Rˆ 3,1 15 12 6 12 with vertical scaling factors r1 = 1/5, r2 = −1/5, and r3 = 4/15. The graph of Ix11 f is shown in Fig. 3.3a. Figure 3.2a–c illustrates the graph of the fractional derivative of f of order 0.2, 0.4, and 0.6, respectively. Figure 3.3b–d is the corresponding fractional integral of order 0.2 of the functions f 0.2 , f 0.4 , and f 0.6 , respectively, and their IFS’s are obtained from Theorem 3.2. (k) The results of Theorem 3.2 exhibit fˆ is FIF which interpolates the data {(xi , yˆi,β ): i ∈ Nn } provided the integral value of the fractal interpolation function f is known at the initial point. As a consequence, the following theorem ensures a similar result when the integral value of FIF f is given at the endpoint with certain interpolation data. Let us define the Riemann–Liouville fractional integral on f by

3.2 Linear Fractal Interpolation Function

43

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

0

0.2

0.4

(a)

0.6

0.8

1

−0.4

0

0.4

(b)

0.5

0.5

0.4

0.45

0.6

0.8

1

0.6

0.8

1

4 D 0. x0 f

0.4

0.3

0.35

0.2

0.3

0.1

0.25

0

0.2

−0.1

0.15

−0.2

0.1

−0.3 −0.4

0.2

2 D 0. x0 f

.05 0

0.2

0.6

0.4

(c)

0.8

1

0 0

0.2

6 D 0. x0 f

0.4

(d)

I x10

f

Fig. 3.2 The Riemann–Liouville fractional integral and derivative of FIF associated with data {(0, 0), (1/3, 1/2), (2/3, 1/2), (1, 0)} 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2

0.4

0.6

0.8

1

−0.2

0

0.2

0.7

0.5

0.6

0.4

0.5

0.3

0.4

0.6

0.8

1

0.8

1

0.2

0.3

0.1

0.2

0

0.1 0

−0.1

−0.1

−0.2

−0.2

0.4

(b) I x0.02 f 0. 2

(a) I x0.06 f

0

0.2

0.4

0.6

(c) I x0.02 f 0. 4

0.8

1

−0.3

0

0.2

0.4

0.6

(d) I x0.02 f 0. 6

Fig. 3.3 The Riemann–Liouville fractional integral and derivative of FIF associated with data {(0, 0), (1/3, 1/2), (2/3, 1/2), (1, 0)}

44

3 Fractional Calculus on Fractal Functions

Ixβn f (k) (x1 ) =

1 (β)



xn

(t − x1 )β−1 f (k) (t)dt

(3.6)

x1

β

with Ixn f (k) (xn ) = 0. Theorem 3.3 Let f be the C p -linear FIF determined by the IFS defined in Eq. (3.3), β then the Riemann–Liouville fractional integral of order β, Ixn f (k) (x), is C p -linear (k) FIF associated with {K ; wi : wi (x, y) = (L i (x), Rˆ i,β (x, y)), i ∈ Nn−1 }, where yˆn,β = 0 and for each i ∈ Nn−1 , β > 0, β−k (k) (k) (x, y) = ai ri y + qˆi,β (x), Rˆ i,β

ai =

(xi+1 − xi ) (xn − x1 ) β−k β (k) Ixn qi (x)

(k) (k) (k) (x) = yˆi,β + f i,β (x) + ai qˆi,β

(k) yˆi,β = Ixβn f (k) (xi )  xn 1 (k) f i,β (x) = [(t − L i (x))β−1 − (t − xi )β−1 ] f (k) (t)dt, (β) xi

for all k = 1, 2, . . . , p. Proof Let f be the C p -linear FIF determined by the IFS defined in Eq. (3.3), therefore it obeys the functional equation (2.12) and by Theorem 2.3 gives f (k) (L i (x)) =   r f (k) (x)+qi(k) (x) R x, f (k) (x) = i , for all x ∈ I, i ∈ Nn−1 . ak i

1 β I xn f (k) (L i (x)) = (β) = + = = =

 xn L i (x)

(t − L i (x))β−1 f (k) (t)dt

 xn

 xn 1 1 (t − xi )β−1 f (k) (t)dt − (t − xi )β−1 f (k) (t)dt (β) xi (β) xi  x  xn i 1 1 (t − L i (x))β−1 f (k) (t)dt + (t − L i (x))β−1 f (k) (t)dt (β) L i (x) (β) xi  x i 1 (k) (k) yˆi,β + f i,β (x) + (t − L i (x))β−1 f (k) (t)dt (β) L i (x)  xn 1 (k) (k) yˆi,β + f i,β (x) + (L i (u) − L i (x))β−1 f (k) (L i (u))ai du (β) x  xn (k) β−1 ri f (k) (u) + qi (u)  a (k) (k) yˆi,β + f i,β (x) + i du ai (u − x) (β) x aik β−k  x n

a (k) (k) = yˆi,β + f i,β (x) + i (β)

x β−k β β−k β (k) (k) (k) = yˆi,β + f i,β (x) + ai ri I xn f (k) (x) + ai I xn qi (x)

 β β (k) I xn f (k) (L i (x)) = Rˆ i,β x, I xn f (k) (x) .

Moreover,

β−k  x n

a (u − x)β−1 ri f (k) (u)du + i (β)

x

(k)

(u − x)β−1 qi

(u)du

3.2 Linear Fractal Interpolation Function

45

 β−k (k)  (k) x1 , yˆ1,β = qˆi,β (x1 ) + ai ri Ixβn f (k) (x1 ) Rˆ i,β β−k β (k) Ixn qi (x1 )

(k) (k) = yˆi−1,β + f i,β (xn ) + ai

β−k

+ ai

ri Ixβn f (k) (x1 )

= Ixβn f (k) (L i (x1 )), (k) = yˆi,β

 β−k (k)  (k) (xn ) + ai ri Ixβn f (k) (xn ) xn , yˆn,β = qˆi,β Rˆ i,β β−k β (k) Ixn qi (xn )

(k) (k) = yˆi−1,β + f i,β (xn ) + ai (k) = yˆi−1,β .

(k) (x, y)), i ∈ Nn−1 } generates C p Hence, the IFS {K ; wi : wi (x, y) = (L i (x), Rˆ i,β β (k) linear FIF Ixn f (x).

Theorems 3.2 and 3.3 explored that the Riemann–Liouville fractional integral fˆ of C p -linear FIF f is also a C p -linear fractal interpolation function. Further, fˆ β (k) ) : i ∈ Nn } when the value of fˆ is Ix1 f (k) (x1 ) = 0 interpolates the data set {(xi , yˆi,β β (k) or Ixn f (xn ) = 0. This observation leads us to develop the fractional calculus of the C p - linear fractal interpolation function. Proposition 3.1 Let f be the C p -linear FIF determined by the IFS defined in β α−β β Eq. (3.3), Dxα1 (Ix1 f (k) (x)) = Dx1 f (k) (x), α ≥ β ≥ 0, if and only if Ix1 f (k) (x) is (k) the C p -linear FIF associated with {K ; wi (x, y) = (L i (x), Rˆ i,β (x, y)) : i ∈ Nn−1 }, β−k β−k (k) k−β k−β α−β (k) (k) where Rˆ i,β (x, y) = ai rˆi y + ai qˆi,β (x), rˆi = ri ai , Dxα (qˆi,β (x)) = ai Dx1 1

qi(k) (x).

Proof Consider the IFS {K ; wi (x, y) : n ∈ Nn−1 } , where L i (x) = ai x + bi , Ri (x, y) = αi y + qi (x). Here |αi | < 1 and qi (x) ∈ C p (I ). Let f be the C p -linear FIF β α−β associated with IFS defined in Eq. (3.3). Assume that Dxα1 (Ix1 f (k) (x))=Dx1 f (k) (x), β (k) α ≥ β ≥ 0. Ix1 f (x) is the C p -linear FIF associated with β−k (k) (k) Rˆ i,β (x, y) = ai ri y + qˆi,β (x)

ai = (k)

(xi+1 − xi ) , (xn − x1 ) (k)

(k)

β−k

(k)

qˆi,β (x) = yˆi−1,β + f i−1,β (x) + ai I xβ1 qi (x),

 xi−2 

1 β−k (k) [(L i (x) − t)β−1 − (xi−2 − t)β−1 ] f (k) (t)dt Dxα1 ( Rˆ i,β (x, y)) = Dxα1 ai ri y + I xβ1 f (k) (xi−1 ) + Dxαi (β) xi 

β−k (k) + Dxα1 ai I xβ1 qi (x) ,

α−β (k) but Dxα1 ( Rˆ i,β (x, y)) = Ri (x, y) = ri y + Dx1 qi(k) (x), therefore it must occur when β−k β−k β (k) (k) Rˆ i,β (x, y) = ai rˆi y + ai qˆi,β (x). Conversely, if Ix1 f (k) (t) is a C p -linear fractal

β−k β−k (k) (k) interpolation function associated with Rˆ i,β (x, y) = ai rˆi y + ai qˆi,β (x), α−β (k) β (k) α ˆ (k) α Dx1 ( Ri,β (x, y)) = ri y + Dx1 qi (x), then it gives that Dx1 (Ix1 f (x)) = α−β Dx1 f (k) (x).

46

3 Fractional Calculus on Fractal Functions

Example 3.2 In this example, we most famous Takagi nowhere have 1discussed the k inf |2 x − m|. Consider the data set differentiable function T (x) = ∞ m∈Z k=0 2k {(0, 0), (1/2, 1/2), (1, 0)} with vertical scaling factors r1 = 1/2 and r2 = 1/2. The fractal interpolation function f is determined from the IFS consisting of 1 1 1 x, R1 (x, y) = y + x, 2 2 2 1 1 1 1 1 L 2 (x) = x + , R2 (x, y) = y − x + . 2 2 2 2 2 L 1 (x) =

The graph of f is shown in Fig. 3.4a and f is similar to the Takagi function T . If β = 0.2 and k = 0, then Ix0.2 f interpolates the data {(0, 0)(1/2, 1135/5746), 0 f is generated by the same L i (x) for i = 1, 2, 3 and (1, 286/1415)}. The FIF Ix0.2 1 269 0 Rˆ 1,0.2 (x, y) = y+ 618 269 0 Rˆ 2,0.2 (x, y) = y+ 618

1135 1.2 x , 2873 381 1.2 309 162 633 0.2 x + (x − (x + 1)1.2 ) + (x + 1)0.2 − , 788 1382 2690 2941

with vertical scaling factors r1 = r2 = 269 . The graph of Ix11 f 0.2 is shown in Fig. 3.4b. 618 1 Now look at k = 0 and β = 1, then Ix1 f passes through the interpolation points {(0, 0), (1/2, 3/8), (1, 1/2)} . The FIF Ix11 f is generated by the same L i (x) for i = 1, 2, 3 and 1 0 (x, y) = y + Rˆ 1,1 4 1 0 Rˆ 2,1 (x, y) = y − 4

1 2 x , 8 1 2 1 3 x + x+ 8 4 8

with vertical scaling factors r1 = r2 = 41 . The graph of Ix11 f is shown in Fig. 3.5d. f , passes through the interpolation The fractional integral of order 0.4 of f , Ix0.4 1 f is generated by the points {(0, 0), (1/2, 169/1108), (1, 394/2145)} . The FIF Ix0.4 1 same L i (x) for i = 1, 2, 3 and 0.7

0.3

0.6

0.25

0.5

0.2

0.4

0.15

0.3

0.1

0.2

0.05

0.1 0

0

0.2

0.4

(a) f

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

(b) I x0.12 f

Fig. 3.4 Demonstration of a fractal interpolation function f of the Takagi function T and its integral

3.2 Linear Fractal Interpolation Function

47

0.3

0.08

0.25

0.06

0.2

0.04

0.15

0.02

0.1

0

0.05

−0.02

0

−0.04

−0.05

0

0.2

0.6

0.4

0.8

1

−0.06

0

0.2

0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 0

0.4

0.6

0.8

1

0.6

0.8

1

4 (b) D 0. x1 f

2 (a) D 0. x1 f 0.6 0.5 0.4 0.3 0.2 0.1 0.2

0.6

0.4

(c)

6 D 0. x1 f

0.8

1

0

0

0.2

0.4

(d)

I x11

f

Fig. 3.5 The Riemann–Liouville fractional integral and derivative of FIF associated with the Takagi function T

651 0 Rˆ 1,0.4 (x, y) = y+ 1718 651 0 y+ Rˆ 2,0.4 (x, y) = 1718

1153 1.4 x , 3534 2477 0.4 533 472 1.4 533 241 x + (x + 1)0.4 − x − (x + 1)1.4 − , 1802 4480 431 3200 7067

651 with vertical scaling factors r1 = r2 = 1718 . The graph of Ix11 f 0.4 is shown in Fig. 3.5e. f, passes through the interpolation points The fractional integral of order 0.6 of f , Ix0.6 1 {(0, 0), (1/2, 349/3025), (1, 929/5179)} . The fractal interpolation function Ix0.6 f is 1 generated by the same L i (x) for i = 1, 2, 3 and

509 0 Rˆ 1,0.6 y+ (x, y) = 1543 509 0 Rˆ 2,0.6 y+ (x, y) = 1543

2018 1.6 x , 7415 494 1.6 548 1179 0.6 685 604 x − (x + 1)1.6 + x + (x + 1)0.6 − , 2163 581 200 581 2151

509 with vertical scaling factors r1 = r2 = 1543 . The graph of Ix0.6 f is shown in Fig. 3.6b. 1 Figure 3.5a–c illustrates the graph of the fractional derivative of f of order 0.2, 0.4, and 0.6, respectively. Figure 3.6c–e is the corresponding fractional integral of order 0.2 of the functions f 0.2 , f 0.4 , and f 0.6 , respectively.

Theorem 3.4 Let {(xi , yi ) ∈ I × R : i ∈ Nn } be a given data set such that x0 < x1 < x2 < · · · < xn . Let L i (x) = ai x + bi , i ∈ Nn−1 , be the affine transformations p and let Ri (x, y) = ri y + qi (x), i ∈ Ni−1 , obey the join-up condition. If |ri | < ai for some real p ≥ 0 and qi ∈ D(I ), i ∈ Ni−1 , let

48

3 Fractional Calculus on Fractal Functions 0.2

1.8 0.16 1.5 0.12

1.2 0.9

0.08

0.6 0.04 0.3 0

(a)

0

1

0.5

0

1

0.5

0

I x0.14 f

(b)

I x0.16 f

0.35

1 0.9

0.3

0.8 0.25

0.7 0.6

0.2

0.5 0.15

0.4 0.3

0.1

0.2 0.05

0.1 0

0

0.2

0.4

(c)

0.8

0.6

0

1

0.6

0.4

0.2

0

0.8

1

(d) I x0.12 f 0. 4

I x0.12 f 0. 2 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01

0

0.2

0.4

(e)

0.6

0.8

1

I x0.12 f 0. 6

Fig. 3.6 The Riemann–Liouville fractional integral and derivative of FIF associated with the Takagi function T

Ri,k (x, y) =

ri y + qi(k) (x) q1(k) (x1 ) qn(k) (xn ) , y = , y = , k ∈ [0, p]. (3.7) 1,k n,k aik aik − r1 aik − rn

If Ri−1,k (xi , yi,k ) = Ri,k (x1 , y1,k ), i = 2, 3, . . . , n and k ∈ [0, p], then the IFS {K ; wi (x,y) : i ∈ Nn−1 } determines an FIF f ∈ D(I ),and f (k) is an FIF determined by K 0 ; wi (x, y) = (L i (x), Ri,k (x, y)) : i ∈ Nn−1 , where K 0 is a non-empty closed rectangle in I × R. Proof By hypothesis and Proposition 3.1, the proof is analogous to the proof of Theorem 3.1.

3.3 Riemann–Liouville Fractional Calculus Quadratic FIF

49

3.3 Riemann–Liouville Fractional Calculus Quadratic FIF In Sect. 3.2, fractional calculus of the linear fractal interpolation function with a constant scaling factor is investigated. In continuation with Sect. 3.2, this section deals the Riemann–Liouville fractional calculus of the quadratic fractal interpolation function with both constant scaling and variable scaling factors. Hereafter, throughout this section, the function qi in (2.13) is taken as a polynomial of degree 2, i.e., qi (x) := x 2 + ci x + di

(3.8)

for each i ∈ Nn−1 . Then the constants ci and di are estimated by using the condition (2.2) as yi+1 − yi − αi (yn − y1 ) − (xn2 − x12 ) , xn − x1 xn (yi − x12 − αi y1 ) + x1 (αi yn + xn2 − yi+1 ) . di = xn − x1 ci =

(3.9) (3.10)

Initially, we discuss the Riemann–Liouville fractional calculus and the fractal dimension of the quadratic fractal interpolation function (QFIF) with constant scaling factors. The quadratic FIF with the constant scaling factor g satisfies g(L i (x)) = Ri (x, g(x)) = αi g(x) + x 2 + ci x + di , x ∈ Ii , i ∈ Nn−1 .

(3.11)

Let us begin with the fractional calculus of a quadratic FIF with constant scaling factors. By using (3.1), the Riemann–Liouville integral of order β > 0 of the quadratic FIF g is defined on the interval [x1 , xn ] by Ixβ1 g(x)



1 = (β)

x

(x − t)β−1 g(t)dt

with

x1

Ixβ1 g(x1 ) = 0 .

(3.12)

Theorem 3.5 Let g be the quadratic FIF associated with the IFS (2.13). Then, for β each β > 0, the function gˆ := Ix1 g defined in (3.12) is an FIF generated by the IFS ˆ i ) = yˆi,β for i ∈ Nn , where yˆ1,β = 0 and {(L i (x), Ri (x, y)) : i ∈ Nn−1 } such that g(x β

Ri (x, y) = ai αi y + qˆi (x), β

a (x − x1 )β qˆi (x) = yˆi,β + gˆi,β (x) + i (β + 1) yˆi+1,β =

i  

(x12 + ci x1 + di ) +

(2x1 + ci )(x − x1 ) 2(x − x1 )2 + β +1 (β + 1)(β + 2)

 ,

β

gˆ k,β (xn ) + αk ak yˆn,β

k=1 β



a (xn − x1 )β + k (β + 1)

 (x12 + ck x1 + dk ) +

⎞ (2x1 + ck )(xn − x1 ) 2(xn − x1 )2 ⎠ , ∀i ∈ Nn−1 , + β +1 (β + 1)(β + 2)

50

3 Fractional Calculus on Fractal Functions n−1

β

ak (xn −x1 ) k=1 gˆ k,β (x n ) + (β+1)

yˆn,β = gˆi,β (x) =



β 

(x12 + ck x1 + dk ) +

(2x1 +ck )(xn −x1 ) 2(xn −x1 )2 + (β+1)(β+2) β+1



 β 1 − n−1 k=1 ak αk  x  i (L i (x) − t)β−1 − (xi − t)β−1 g(t)dt.

,

1 (β) x1

β

Proof Given that g is a quadratic FIF with the IFS (2.13) and Ix1 g(x1 ) = 0, then Ixβ1 g(L i (x))

1 = (β)



L i (x)

(L i (x) − t)β−1 g(t)dt

x1  xi

 L i (x) 1 1 (xi − t)β−1 g(t)dt + (L i (x) − t)β−1 g(t)dt (β) x1 (β) xi  xi   1 + (L i (x) − t)β−1 − (xi − t)β−1 g(t)dt (β) x1  L i (x) 1 = yˆi,β + gˆ i,β (x) + (L i (x) − t)β−1 g(t)dt, (β) L i (x1 ) =

 xi  xi  1 1 β−1 β−1 where yˆi,β = (β) g(t)dt and gˆ i,β (x) = (β) − x1 (x i − t) x1 (L i (x) − t)  β−1 g(t)dt. By changing the variable t = L i (s) and using (2.1), it gives that (xi − t) Ixβ1 g(L i (x))

β



β



= yˆi,β

a + gˆ i,β (x) + i (β)

= yˆi,β

a + gˆ i,β (x) + i (β)

x

(x − s)β−1 g(L i (s))ds

x1 x

(x − s)β−1 (αi g(s) + qi (s))ds.

x1

As g is a quadratic FIF, thence (3.11) yields Ixβ1 g(L ˆ i (x))

β

αi ai = (β)



x

(x − s)

β−1

x1

β

a (x − x1 )β 2 (x1 + ci x1 + di ) g(s)ds + i (β + 1)

β

yˆi,β + gˆ i,β (x) +

β

ai (x − x1 )β+1 2a (x − x1 )β+2 (2x1 + ci ) + i . (β + 2) (β + 3) (3.13)

Using the notation qˆi (x) = yˆi,β + gˆi,β (x) +

β

ai (x−x1 )β (β+1)



(x12 + ci x1 + di ) +

2(x−x1 ) + (β+1)(β+2) , we get 2

β

Ixβ1 g(L ˆ i (x)) = ai αi gˆ + qˆi (x) = Ri (x, Ixβ1 g(x)).

(2x1 +ci )(x−x1 ) β+1

3.3 Riemann–Liouville Fractional Calculus Quadratic FIF

51

Thus, the Riemann–Liouville fractional integral of order β of the quadratic FIF is also an FIF which passes through the data {(xi , yˆi,β ) : i ∈ Nn }, and the ordinates yˆi,β ’s are obtained as follows. For each i ∈ Nn−1 , yˆi+1,β = yˆ1,β +

i  ( yˆk+1,β − yˆk,β ).

(3.14)

k=1

By taking x = xn in (3.13), i 



β

ak (xn − x1 )β  2 (x1 + ck x1 + dk ) (β + 1) k=1  2(xn − x1 )2 (2x1 + ck )(xn − x1 ) + . + β +1 (β + 1)(β + 2)

yˆi+1,β =

β

gˆ k,β (xn ) + αk ak yˆn,β +

By taking i = n − 1 in the above equation, yˆn,β is obtained as n−1 yˆn,β =

k=1

 gˆ k,β (xn ) +

β

ak (xn −x1 )β (β+1)



(x12 + ck x1 + dk ) + 1−

n−1

(2x1 +ck )(xn −x1 ) β+1

+

2(xn −x1 )2 (β+1)(β+2)

 .

β k=1 ak αk

This completes the proof. The above theorem gives the fractional integral of QFIF with the fractional integral predefined at the initial point of the interpolation data while the following corollary gives the fractional integral of QFIF with the fractional integral predefined at the endpoint of the interpolation data, namely, Ixβn g(x) =

1 (β)



xn

(t − x)β−1 g(t)dt

with

x

Ixβn g(xn ) = 0.

(3.15)

Corollary 3.1 Let g be the quadratic FIF associated with the IFS (2.13). Then, for β each β > 0, the function gˆ := Ixn g defined in (3.15) is an FIF generated by the IFS ˆ i ) = yˆi,β for i ∈ Nn , where yˆn,β = 0 and {(L i (x), Ri (x, y)) : i ∈ Nn−1 } such that g(x β

Ri (x, y) = ai αi y + qˆi (x), β

a (xn − x)β qˆi (x) = yˆi+1,β + gˆi+1,β (x) + i (β + 1) yˆi,β = yˆn,β +

n−1 

(xn2 + ci xn + di ) −

(2xn + ci )(xn − x) 2(xn − x)2 + β +1 (β + 1)(β + 2)

β

gˆ k+1,β (x1 ) + αk ak yˆ1,β

k=i β



a (xn − x1 )β + k (β + 1)



⎞

(2xn + ck )(xn − x1 ) 2(xn − x1 )2 ⎠ , ∀i ∈ Nn−1 , (xn2 + ck xn + dk ) − + β +1 (β + 1)(β + 2)

 ,

52

3 Fractional Calculus on Fractal Functions n−1

β

ak (xn −x1 ) k=1 gˆ k,β (x 1 ) + (β+1)

yˆ1,β = gˆi+1,β (x) =



 xn 

1 (β) xi+1

β 

(2xn +ck )(xn −x1 ) 2(xn −x1 )2 + (β+1)(β+2) β+1

(xn2 + ck xn + dk ) −



n−1 β k=1 ak αk (t − L i (x))β−1 − (t − xi+1 )β−1 g(t)dt. 1−

,

By using (3.2), the Riemann–Liouville derivative of order β > 0 of the quadratic FIF g is defined on the interval [x1 , xn ] by Dxβ1 g(x) =

1 (m − β)



d dx

m 

x

g(t)(x − t)m−β−1 dt

(3.16)

x1

β

with Dx1 g(x1 ) = 0, where m = min{x ∈ N : x ≥ β}. Theorem 3.6 Let g be the quadratic FIF associated with the IFS (2.13). Then the β fractional derivative g˜ = Dx1 g of order 0 < β < 2 of FIF g is an FIF whenever β αi < ai αi Ri (x, y) = β y + q˜i (x), ai   (1 − β)(x12 + ci x1 + di ) (2x1 + ci )(x − x1 ) 2(x − x1 )2 1 + + q˜i (x) = β (2 − β) (2 − β) (3 − β) ai (x − x1 )β m  xi j=1 (m − j − β) (L i (x) − t)−β−1 g(t)dt. + (m − β) x1 β

Proof By (3.16), Dx1 g(L i (x)) =  d m m−β Ix1 g(L i (x)). Now dx

1 (m−β)



 d m dx

m−β

m−β

I x1

g(L i (x)) = yˆi,m−β + gˆi,m−β (x) +

Dxβ1 g(L i (x)) =



d dx

= =

(m − β) β ai −β

αi β

ai

(m − β) x1

xi

(L i (x) − t)−β−1 g(t)dt +

x1

m

j=1 (m

Dxβ1 g(x) + 

+ ai (x − x1 )−β =

 x

(L i (x) − t)m−β−1 g(t)dt =

(x − s)m−β−1 (αi g(s) + qi (s))ds.

 xi m−β  x αi ai 1 (L i (x) − t)m−β−1 f (t)dt + (x − s)m−β−1 g(s)ds (m − β) x1 (m − β) x1   x (x − s)m−β−1 qi (s)ds

ai (m − β) x1 m  j=1 (m − j − β) αi

x1

m 

m−β

+

ai

 L i (x)

− j − β)

(m − β)



xi

αi β

ai

−β

Dxβ1 g(x) + ai

Dxβ1 qi (x)

(L i (x) − t)−β−1 g(t)dt

x1

(1 − β)(x12 + ci x1 + di ) (2x1 + ci )(x − x1 ) 2(x − x1 )2 + + (2 − β) (2 − β) (3 − β)

Dxβ1 g(x) + q˜i (x).



3.3 Riemann–Liouville Fractional Calculus Quadratic FIF β

β

Since αi < ai , Dx1 g(L i (x)) =

αi β ai

53

β

Dx1 g(x) + q˜i (x) satisfies the functional equation β

given in (3.11). Hence, the operator Dx1 admits an attractor which is also an FIF. This completes the proof.

3.4 Fractal Dimension The FIF g is called a linear fractal interpolation function, if g is associated with the following IFS L i (x) = ai x + bi , Ri (x, y) = αi y + ci x + di , i ∈ N N −1 .

(3.17)

In [15], Ruan et al. presented the relation between the order of fractional integral and the box dimension of linear fractal interpolation functions as follows. Ri (x, y)) : i ∈ Theorem 3.7 ([15]) Let f be the linear FIF determined by {(L i (x),  Nn−1 } where L i (x) = ai x + bi and Ri (x, y) = αi y + qi (x). Suppose i∈Nn−1 |αi | > 1 and dim B (Gqi ) = 1 for i ∈ Nn−1 . Then dim B (G f ) = D({ai , αi }) or 1, where D({ai , αi }) is the unique solution s of the equation

 i∈Nn−1

ais−1 |αi | = 1.

Theorem 3.8 ([19]) Use the notations and hypothesis of Theorem 3.7 and dim B (G f ) = D({ai , αi }). Then for any β ∈ (0, D({ai , αi }) − 1), dim B G fˆ = dim B G f − β holds, where qi are bounded variation on I and for any β ∈ (0, 2 − D({ai , αi })), dim B G f˜ = dim B G f + β holds, whenever dim B Gq˜i = 1 for all i ∈ Nn−1 . Here, fˆ denotes the fractional integral of f while f˜ refers to the fractional derivative of f . As a consequence of Theorems 3.7 and 3.8, the relation between the order of the Riemann–Liouville fractional integral and the fractal dimension of QFIF with constant scaling factors is proved as follows. Theorem 3.9 Let f be the quadratic FIF determined by {(L i (x),  Ri (x, y)) : i ∈ Nn−1 } where L i (x) = ai x + bi and Ri (x, y) = αi y + qi (x). Suppose i∈Nn−1 |αi | > 1 and dim B (G f ) = D({ai , αi }). Then for any β ∈ (0, D({ai , αi }) − 1), dim B G fˆ = dim B G f − β holds,

54

3 Fractional Calculus on Fractal Functions

where qi is quadratic on I and for any β ∈ (0, 2 − D({ai , αi })), dim B G f˜ = dim B G f + β holds, whenever dim B Gq˜i = 1 for all i ∈ Nn−1 . Here, fˆ denotes the fractional integral of f while f˜ refers to the fractional derivative of f . Proof Given that f is a quadratic FIF. Hence, f is associated with the IFS of the form L i (x) = ai x + bi , Ri (x, y) = αi y + qi (x), where qi (x) = x 2 + ci x + di with dim B (Gqi ) = 1 for all i ∈ N N −1 . Then the rest of the proof is analogous to the proofs of Theorems 3.7 and 3.8.

3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors In this section, we investigate the fractional calculus of a quadratic fractal interpolation function with variable scaling factors, i.e., αi ∈ C∞ (I ), satisfying αi ∞ < 1. The corresponding QFIF satisfies g(L i (x)) = Ri (x, g(x)) = αi (x)g(x) + x 2 + ci x + di , ∀x ∈ Ii , i ∈ Nn−1 . (3.18) Theorem 3.10 If g is the quadratic FIF with variable scaling factor αi (x) associated β ˆ i ) = yˆi,β with the IFS (2.13). Then, for each β > 0, gˆ = Ix1 g is a FIF such that g(x for i ∈ Nn , where yˆ1,β = 0 and β

Ri (x, y) = ai αi (x)y + qˆi (x),   β a (x − x1 )β 2(x − x1 )2 (2x1 + ci )(x − x1 ) (x12 + ci x1 + di ) + + + yˆi,β + gˆ i,β (x) qˆi (x) = i (β + 1) β +1 (β + 1)(β + 2) ∞    β β D (r) αi (x)I xβ+r g(x), + ai 1 r r=1  ∞   i   β β yˆn,β+r D (r) αk (xn ) yˆi+1,β = gˆ k,β (xn ) + ak r r=0 k=1   β a (xn − x1 )β (2x1 + ck )(xn − x1 ) 2(xn − x1 )2 + k (x12 + ck x1 + dk ) + + ∀i ∈ Nn−1 , (β + 1) β +1 (β + 1)(β + 2)  β   n−1  ak (xn − x1 )β 2(xn − x1 )2 (2x1 + ck )(xn − x1 ) (x12 + ck x1 + dk ) + + yˆn,β = (β + 1) β +1 (β + 1)(β + 2) k=1  ∞   n−1   β β β +gˆ k,β (xn ) + ak 1− ak αk (xn ), yˆn,β+r D (r) αk (xn ) r r=1 k=1  xi   1 gˆ i,β (x) = (L i (x) − t)β−1 − (xi − t)β−1 g(t)dt. (β) x1

3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors

55

Proof Given that g is a quadratic FIF with variable scaling factors, αi ∈ C(I ), satβ isfying αi ∞ = sup{|αi | : x ∈ I } < 1 and Ix1 g(x1 ) = 0. By the same argument of Theorem 3.5, it follows that Ixβ1 g(L i (x))

= yˆi,β

1 + gˆ i,β (x) + (β)



L i (x) L i (x1 )

(L i (x) − t)β−1 g(t)dt.

By changing the variable t = L i (s) and using (2.1), we get Ixβ1 g(L i (x)) = yˆi,β + gˆ i,β (x) +

β

ai (β) β

= yˆi,β

a + gˆ i,β (x) + i (β)

 

x

(x − s)β−1 g(L i (s))ds

x1 x

(x − s)β−1 (αi (s)g(s) + qi (s))ds.

x1

As g is a quadratic FIF with function scaling factors, by (2.16) β

Ixβ1 g(L ˆ i (x)) =ai

∞    β r =0

r

D (r ) αi (x)Ixβ+r g(x)+ 1

β

ai (x − x1 )β 2 (x1 + ci x1 + di ) (β + 1)

β

β

ai (x − x1 )β+1 2a (x − x1 )β+2 (2x1 + ci ) + i . (β + 2) (β + 3) (3.19)  2(x−x1 )2 i )(x−x 1 ) (x12 + ci x1 + di ) + (2x1 +cβ+1 + + (β+1)(β+2)

+ yˆi,β + gˆ i,β (x) + β

a (x−x )β

Using the notation qˆi (x)= i(β+1)1 β ∞ β  (r ) β+r yˆi,β + gˆ i,β (x) + ai r =1 r D αi (x)I x1 g(x), we get β

Ixβ1 g(L ˆ i (x)) = ai αi (x)gˆ + qˆi (x) = Ri (x, Ixβ1 g(x)). β

Clearly ||ai αi (x)|| < 1. Thus, the IFS {(L i (x), Ri (x, y)) : i ∈ Nn−1 } is hyperbolic and has an attractor gˆ which is a graph of the continuous function. Further, gˆ passes through the data {(xi , yˆi,β ) : i ∈ Nn } and the new set of yˆi,β ’s are obtained as follows. For each i ∈ Nn−1 , i  yˆi+1,β = yˆ1,β + ( yˆk+1,β − yˆk,β ). (3.20) k=1

Take x = xn in (3.19), we have yˆi+1,β =

i 



k=1 β

+

∞   β β

gˆ k,β (xn ) + ak

ak (xn − x1 )β (β + 1)



r =0

r

yˆn,β+r D (r ) αk (xn )

2(xn − x1 )2 (2x1 + ck )(xn − x1 ) (x12 + ck x1 + dk ) + + β +1 (β + 1)(β + 2)

 .

56

3 Fractional Calculus on Fractal Functions

Take i = n − 1, it gives yˆn,β as    β ak (xn − x1 )β 2(xn − x1 )2 (2x1 + ck )(xn − x1 ) (x12 + ck x1 + dk ) + + (β + 1) β +1 (β + 1)(β + 2) k=1   n−1 ∞    β β β 1− yˆn,β+r D (r ) αk (xn ) +gˆ k,β (xn ) + ak ak αk (xn ). r

yˆn,β =

n−1 

r =1

k=1

This completes the proof. Corollary 3.2 If g is the quadratic FIF with function scaling factor αi (x) associated β ˆ i ) = yˆi,β with the IFS (2.13). Then, for each β > 0, gˆ = Ixn g is an FIF such that g(x for i ∈ Nn , where yˆn,β = 0 and β

Ri (x, y) = ai αi (x)y + qˆi (x),   β a (xn − x)β 2(xn − x)2 (2xn + ci )(xn − x) (xn2 + ci xn + di ) − + qˆi (x) = i (β + 1) β +1 (β + 1)(β + 2) ∞    β β D (r) αi (x)I xβ+r + yˆi+1,β + gˆi+1,β (x) + ai g(x), 1 r r=0  ∞   n−1   β β yˆ1,β+r D (r) αk (x1 ) yˆi,β = yˆn,β + gˆ k+1,β (x1 ) + ak r k=i

β

a (xn − x1 )β + k (β + 1)



r=0



(xn2 + ck xn + dk ) −

2(xn − x1 )2 (2xn + ck )(xn − x1 ) + β +1 (β + 1)(β + 2)

 , ∀i ∈ Nn−1 ,

  2(xn − x1 )2 (2xn + ck )(xn − x1 ) (xn2 + ck xn + dk ) − + β +1 (β + 1)(β + 2) k=1     ∞ n−1   β β β 1− yˆ1,β+r D (r) αk (x1 ) +gˆ k+1,β (x1 ) + ak ak αk (x1 ), r r=1 k=1  xn   1 (t − L i (x))β−1 − (t − xi+1 )β−1 g(t)dt. gˆi+1,β (x) = (β) xi+1 yˆ1,β =

n−1 

β

ak (xn − x1 )β (β + 1)

Now, it is proved below that the Riemann–Liouville fractional derivative of a quadratic FIF is a fractal interpolation function. Theorem 3.11 Let g be the quadratic FIF with variable scaling factor αi (x) assoβ ciated with the IFS (2.13). Then, the fractional derivative g˜ = Dx1 g, of order β 0 < β < 2 of g, is also an FIF whenever αi (x) < ai and m = min{x ∈ N : x ≥ β}.

3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors

57

α (x) Ri (x, y) = i β y + q˜i (x), ai   (1 − β)(x12 + ci x1 + di ) (2x1 + ci )(x − x1 ) 2(x − x1 )2 1 q˜i (x) = β + + (2 − β) (2 − β) (3 − β) a (x − x1 )β i

m

+ +

j=1 (m − j − β)

(m − β) ∞  r = β +1

 x i x1

(L i (x) − t)−β−1 g(t)dt +

r =1

  β r −β Drx1 αi (x)I x1 g(x), r

where β = max{x ∈ Z : x ≤ β}. β

Proof By (3.16), Dx1 g(L i (x)) =  d m m−β Ix1 g(L i (x)). Now dx

1 (m−β)



m−β

m−β

I x1

g(L i (x)) = yˆi,m−β + gˆi,m−β (x) +

β    β

ai

 d m dx  x

(m − β) x1

 L i (x) x1

r

β−r

Drx1 αi (x)Dx1 g(x)

(L i (x) − t)m−β−1 g(t)dt =

(x − s)m−β−1 (αi (s)g(s) + qi (s))ds.

β Dx g(L i (x)) 1 ⎛ m−β     x x ai 1 d m⎝ i (L i (x) − t)m−β−1 g(t)dt + (x − s)m−β−1 αi (s)g(s)ds = dx (m − β) x1 (m − β) x1 ⎞ m−β  x a (x − s)m−β−1 qi (s)ds ⎠ + i (m − β) x1 m  1 β j=1 (m − j − β) xi −β β D α (x)g(x) + ai Dx qi (x) = (L i (x) − t)−β−1 g(t)dt + 1 β x1 i (m − β) x1 ai m   β

 ∞   β  β j=1 (m − j − β) xi r −β β−r = (L i (x) − t)−β−1 g(t)dt + Drx αi (x)I x g(x) + Drx αi (x)Dx g(x) 1 1 1 1 (m − β) r r x1 r =0 r = β +1 ⎞ ⎛ (1 − β)(x12 + ci x1 + di ) (2x1 + ci )(x − x1 ) 2(x − x1 )2 ⎠ −β + + + ai (x − x1 )−β ⎝ (2 − β) (2 − β) (3 − β) =

αi (x) β Dx g(x) + q˜i (x). 1 β ai

β

β

Here, αi (x) < ai and Dx1 g(L i (x)) =

αi (x) β β D x 1 g(x) ai

isfies the functional equation given in (3.18) with variable scaling factors β

β

+ q˜i (x) shows that Dx1 g satαi (x) . β ai

According to the IFS theory, the operator Dx1 has an attractor and the attractor is an FIF. Here, the maximum degree of q˜i (x) is 2 − β. Note 3.1 In q˜i , r runs over the non-negative integers, and hence Dxr 1 αi (x) may be replaced by Dr αi (x) (derivative in the classical sense). Example 3.3 Consider the interpolation points {(0, 1/5), (1/3, 1/2), (1/2, 1/3), (3/4, 3/4), (1, 1/2)}. In this example, the FIF refers to the linear FIF if qi is taken as the linear function of the form ci x + di and the FIF refers to the quadratic FIF if qi is taken as the quadratic function of the form x 2 + ci x + di .

58

3 Fractional Calculus on Fractal Functions

Table 3.1 Constants involved in the linear FIF with constant scaling factors L i (x) Ri (x, y) ai bi ci di 1/3 1/6 1/4 1/4

0 1/3 1/2 3/4

3/2 −4/15 4/15 13/40

1/10 13/30 7/30 7/10

Table 3.2 Constants involved in the QFIF with constant scaling factors Ri (x, y) ci di −17/20 −19/15 −11/15 −53/40

1/10 13/30 7/30 7/10

Table 3.3 Constants involved in linear FIF with variable scaling factors Ri (x, y) ci di 0.2400 −0.2667 0.2974 −0.2806

0.1600 0.4333 0.2538 0.7296

Suppose the scaling factors are taken as α1 = 1/2, α2 = 1/3, α3 = 1/2, and α4 = 1/4. Then the computed coefficients/constants ai , bi , ci , and di involved in the linear FIF are tabulated in Table 3.1. Similarly, for the same constants ai , bi , the constants ci , di given in (3.9) associated with the QFIF are tabulated in Table 3.2. The linear FIF and QFIF with these constant scaling factors associated with the interpolation data T that are generated through the constants listed in Tables 3.1 and 3.2 are depicted in Fig. 3.7a and c, respectively. If the scaling factors are taken as variable scaling factors α1 (x) = (1/2) sin(5x) + 0.2, α2 (x) = (1/3) cos(x), α3 (x) = (1/5) exp(−2x) + 0.2, and α4 (x) = (1/5) sin(x) exp(x) + 0.1, then the linear FIF and QFIF with these variable scaling factors associated with the interpolation data T that are generated through the constants listed in Tables 3.3 and 3.4 are depicted in Fig. 3.7b and d, respectively. In comparison with the graph of the linear fractal interpolation function with constant scaling factors given in Fig. 3.7a, we observe that there are visually pleasing changes in the graph of the linear FIF with variable scaling factors in Fig. 3.7b. By analyzing Fig. 3.7a with Fig. 3.7c, b with Fig. 3.7d, it is conspicuous that the

3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors

59

Table 3.4 Constant involved in QFIF with variable scaling factors Ri (x, y) ci di −0.7600 −1.2667 −0.7026 −1.2806

0.1600 0.4333 0.2538 0.7296

Fractal Interpolation Function with Variable Scaling

Fractal Interpolation Function 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2

0.6

0.4

0.8

1

−0.10

0.2

0.8

1

(b) Quadratic Fractal Interpolation Function with Variable Scaling

Quadratic Fractal Interpolation Function 0.8

0.8 0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0

−0.1 0

0.6

0.4

(a)

0.2

0.6

0.4

0.8

(c)

1

−0.1 0

0.2

0.6

0.4

0.8

1

(d)

Fig. 3.7 Fractal interpolation functions of the data set {(0, 1/5), (1/3, 1/2), (1/2, 1/3), (3/4, 3/4), (1, 1/2)}

irregularity of fractal functions associated with sub-figures (b) and (d) is moderately lower than (a) and (b), respectively. Further, in this case, one can notice that the smoothness of the quadratic FIF increases in each subinterval when the absolute value of the scaling factor αi approaches zero for all i.

3.6 Concluding Remarks The Riemann–Liouville fractional integral of the p-times continuously differentiable linear fractal interpolation function is explored in this chapter. While interpolating the data which is irregular in nature, various types of fractal interpolations other than the linear fractal interpolation function are explored and a suitable one is applied.

60

3 Fractional Calculus on Fractal Functions

Similarly, there is a need to explore fractional calculus of various types of fractal interpolations other than the linear interpolation function. Hence, a study of fractional calculus of the quadratic fractal interpolation function with both constant scaling and variable scaling factors is endeavored. The variations between the linear fractal interpolation function and the quadratic fractal interpolation functions have been elucidated by giving suitable examples. In addition, this chapter provides the fruitful result that the Riemann–Liouville fractional integral of linear and quadratic fractal interpolation functions are again fractal interpolation functions for a different set of data when the integral is predefined at the endpoints of the original data set.

Chapter 4

Fractal Interpolation Function for Countable Data

4.1 Introduction The problem of extending the theory of an iterated function system carried over to an infinite and countable iterated function system has been widely studied in the last decades. Such an extension is applied in the theory of sampling and reconstruction where a countable iterated function system contributes to a better approximation. It is notable that interpolation and approximation regularly show up as two sides of the same coin; results about one as often as possible infer results about the other. To substantiate this in the context of the fractal interpolation function, Barnsley perceived that a given continuous function f on a closed interval, say I , in R to R can be expressed as a class of fractal functions. This is accomplished by choosing a finite number of points {(xn , yn ) : n ∈ N N } on I × R and reconstructing f through the fractal interpolation function with an appropriately defined iterated function system. In the conventional setting of fractal interpolation and in practically the entirety of its expansions referenced previously, a fractal interpolation function is developed for a finite data set. As mentioned earlier, there are numerous natural phenomena where an infinite data set might be necessary, for example, in the theory of signal reconstruction and the sampling scheme. This idea inspired to investigate the existence of the fractal interpolation function for an infinite data set instead of a finite number of data points. Recently, the standard construction of the fractal interpolation function reached out from the finite data set to the case of a countable number of data points with the notion of the countable iterated function system [22, 49]. This development of the fractal interpolation function for a prescribed countable set of data frames the reason for the discussion of fractional calculus of the fractal interpolation function for countable data. Hence, this chapter introduces and describes the sequence of data and the corresponding interpolation function which is a generalization of the Secelean framework. The existence of the continuous function f which interpolates the sequence of prescribed data {(xn , yn ) : n ∈ N} is discussed, where (xn )∞ n=1 is a monois a bounded sequence of real numbers. In addition to tonic real sequence and (yn )∞ n=1 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_4

61

62

4 Fractal Interpolation Function for Countable Data

that, the prominent influence of free parameters in the shape of a fractal interpolation function is illustrated with suitable examples. Besides, the existence of a countable iterated function system is investigated when the fractal interpolation function for a sequence of data is given. Further, the existence of the Riemann–Liouville fractional integral and derivative of the fractal interpolation function of countable data set (or sequence of points) is established.

4.2 Existence of FIF for Countable Data Set This section provides the existence of the fractal interpolation function for a sequence of data, and also the existence of a countable iterated function system when the fractal interpolation function is given. ∞ Let (xn )∞ n=1 be a real, increasing, and bounded sequence and (yn )n=1 be a bounded ∗ sequence of real numbers. Let supn xn = x , then the sequence x1 < x2 < x3 < · · · partitions the closed interval I = [x1 , x ∗ ]. Let supn yn = y ∗ , inf n yn = y1 . Consider the sequence of data {(xn , yn ) : n ≥ 1}, where yn ∈ [y1 , y ∗ ] and intend to construct a continuous function which passes through the given data. That is, f : I −→ R such that f (xn ) = yn , for all n ∈ N and the graph of f is the attractor of CIFS. Let K = [x1 , x ∗ ] × [y1 , y ∗ ]. Define In = [xn−1 , xn ], L n : I −→ In , for n ≥ 2 be contractive homeomorphisms such that L n (x1 ) = xn−1 , L n (x ∗ ) = xn , |L n (s) − L n (t)| ≤ ln |s − t| , for all s, t ∈ I, for some ln ∈ [0, 1). Let Rn : K −→ [y1 , y ∗ ] be continuous with, for some rn ∈ [1, 0), Rn (x1 , y1 ) = yn−1 , Rn (x ∗ , y ∗ ) = yn , |Rn (s, y1 ) − Rn (s, y2 )| ≤ rn |y1 − y2 | , for all s ∈ I, y1 , y2 ∈ [y1 , y ∗ ] and n ≥ 2. For all n ≥ 2, define wn : K −→ K by wn (x, y) = (L n (x), Rn (x, y)), then the system {K ; wn : n ≥ 2} is CIFS. Define F : K (K ) −→ K (K ) by F (B) = ∞ n=1 wn (B), for all B ∈ K (K ), where the bar means the closure of the corresponding set. Theorem 4.1 The function F , as defined above, has a unique fixed point A which is the graph of a continuous interpolation function for the sequence of data {(xn , yn ) : n ≥ 1} .

4.2 Existence of FIF for Countable Data Set

63

Proof Let A be the attractor of the CIFS {K ; wn : n ≥ 2}, that is A = F (A ) =

∞ 

wn (A ).

2

For each N ∈ N, TN = {K ; wn : n = 2, 3, . . . , N } is a finite IFS, hence by Theorem 2.1, F has a unique attractor, denoted as A N . A =

∞ 

AN =

N =2

∞ 

F(A N ) =

N =2

∞  N 

wn (A N )

N =2 n=2

=

∞  N 

(L n (x), Rn (x, y)), for all (x, y) ∈ A N .

N =2 n=2

Assume that I˜ = {x ∈ I : (x, y) ∈ A , for some y ∈ [y1 , y ∗ ]}, then by Theorem  N L ( I˜). Meanwhile L n is a contraction on I , for each n ≥ 1.5, I˜ = ∞  N N =2 n=2 n 2 and n=2 L n (I ) = I . Hence, I˜ = I. Let us take Si = {(xi , y) ∈ A : i ∈ N}. Now, consider S1 = {(x1 , y) ∈ A : y ∈ [y1 , y ∗ ]}, clearly S1 ⊂ A , and hence S ⊂ ∞  N N =2 n=2 wn (A ), ⇒ S1 ⊆

∞ 

(L 2 (x1 ), R2 (x1 , y)) ∪ (L 3 (x1 ), R3 (x1 , y)) ∪ · · · ∪ (L N (x1 ), R N (x1 , y))

N =2

=

∞  N =2

(x1 , R2 (x1 , y)) ∪ (x2 , R3 (x1 , y)) ∪ · · · ∪ (x N −1 , R N (x1 , y)),

⇒ S1 ⊆ (x1 , R2 (x1 , y)) = w2 (x1 , R2 (x1 , y)),

and w2 (x1 , R2 (x1 , y)) ⊆ S1 ⇒ S1 = w2 (L 2 (x1 ), R2 (x1 , y)). Since, the function w2 is a strict contraction, S1 = (x1 , y1 ). Now, consider   S∗ = (x ∗ , y) ∈ A : y ∈ [y1 , y ∗ ] , w∗ = lim

N →∞

N 

wn (x ∗ , Rn (x ∗ , y)),

n=2

the same argument yields S∗ = {w∗ (x ∗ , Rn (x ∗ , y))} = {(x ∗ , y ∗ )}. So, for each i ≥ 1, (xi+1 , yi+1 ) = wi (S1 ) ∪ wi+1 (S∗ ). Consider δ = sup {|t1 − t2 | : (x, t1 ) ∈ A , (x, t2 ) ∈ A , for some x ∈ I }, attractor A is compact, and hence A has supremum. Suppose that the supremum is ˜ t2 ), then there exist two points (L −1 ˜ u 1 ), (L −1 ˜ u2) achieved at (x, ˜ t1 ) and (x, n ( x), n ( x), −1 −1 ˜ u 1 )) and t2 = Fn (L n (x, ˜ u 2 )). It gives that in attractor A such that t1 = Fn (L n (x,

64

4 Fractal Interpolation Function for Countable Data

δ = |t1 − t2 | = |Fn (L −1 ˜ u 1 )) − Fn (L −1 ˜ u 2 ))| ≤ q.|u 1 − u 2 | ≤ q.δ, n ( x, n ( x, with q ∈ [0, 1), therefore δ = 0. Hence, the only one y is associated to each x ∈ I . Therefore, A is the graph of a function f : I −→ [y1 , y ∗ ] and f (xn ) = yn , for each n ∈ N. Define C0 (I ) = { f | f : I −→ [y1 , y ∗ ] , f is continuous, f (x1 ) = y1 , f (x ∗ ) = y ∗ }; clearly C0 (I ) is a closed subspace of the Banach space of continuous real-valued functions g : I −→R, with supremum norm f ∞ = sup {| f (x)| : x ∈ I }, denoted as C(I ). Hence, C0 (I ) is a complete metric space; define T : C0 (I ) −→ C0 (I ) by −1 T g(x) = Fn (L −1 n (x), g(L n (x))) when x ∈ In . Meanwhile, T is a contraction mapping with a contraction ratio q and T is induced by the CIFS.      −1 −1 −1 T h − T g ∞ = sup Fn (L −1 n (x), h(L n (x))) − Fn (L n (x), g(L n (x))) : x ∈ In , n ≥ 2       −1 ≤ sup q. h(L −1 n (x)) − g(L n (x)) : x ∈ In , n ≥ 2 ≤ q. h − g ∞ .

Hence, T has a unique fixed point h˜ ∈ C0 (I ). The graph of h˜ is an attractor of the CIFS; it gives that h˜ = f , and hence f is a continuous interpolation function of the  sequence of data {(xn , yn ) : n ≥ 1}. Remark 4.1 1. If (xn )∞ n=1 is a decreasing and bounded sequence, then the sequence x1 > x2 > x3 > · · · partitions the closed interval I = [x ∗ , x1 ], where x ∗ = inf n xn . Hence, Theorem 4.1 is also valid for data {(xn , yn ) : n ∈ N}, where yn is a bounded sequence. ∞ 2. Let (xn )∞ n=1 be an increasing and bounded sequence. If (yn )n=1 is a convergent sequence then the above theorem is also stated as “there exists an interpolation function f corresponding to the given sequence of data such that the graph of f is the attractor of the associated CIFS”. Hence, Theorem 4.1 is the generalization in R of Theorem 2 presented by Secelean in [49]. Theorem 4.1 presents the existence of the fractal interpolation function for a given sequence of data, while the following corollary assures the existence of the countable iterated function system, if the fractal interpolation function is given. Corollary 4.1 For any g ∈ K (K ), let  > 0 be given. Choose CIFS {K ; wn : n ≥ 2}  , with contractive ratio r ∈ [0, 1) such that Hd (g, F (g)) ≤ . Then Hd (g, A ) ≤ 1−r where A is the attractor of CIFS.

4.2 Existence of FIF for Countable Data Set

65

Proof Hd (g, A ) = Hd (g, lim F ◦m (g)) = lim Hd (g, F ◦m (g)) m→∞

= lim Hd (g, m→∞

m→∞

∞ 

wn◦m (g)) = lim Hd (g, lim m→∞

n=2

k→∞

k 

wn◦m (g))

n=2

= lim lim Hd (g, F ◦m (g)) m→∞ k→∞

m

≤ lim lim

m→∞ k→∞

≤ lim lim ≤ lim

m→∞

≤

∞

m

t=1 m

m→∞ k→∞

Hd (F ◦(t−1) (g), F ◦t (g)) rtm−1 Hd (g, F(g))

t=1

max(rtm−1 )Hd (g, F (g))

t=1

r t−1 Hd (g, F (g)) ≤

t=1

 . 1−r 

Remark 4.2 1. For any g∈C ˜ 0 (I ), let  > 0 be given. Choose CIFS {K ; wn : n ≥ 2} T :C0 (I ) −→ C0 (I ) associated with FIF f˜ such that g˜ − T (g) ˜ ∞ ≤ , where     . is defined in Theorem 4.1. Then, by above Corollary, g˜ − f˜ ≤ 1−r ∞

∞ Let (xn )∞ n=1 be a real, increasing, and bounded sequence. Let (yn )n be a bounded sequence of real numbers. In = [xn−1 , xn ], n ≥ 2; L n : I −→ In defined by L n (x) = an x + en , where

xn − xn−1 , x ∗ − x1 x ∗ xn−1 − x1 xn en = . x ∗ − x1

an =

Clearly, L n becomes a homeomorphic between I and In , for any n ≥ 2, such that L n (x1 ) = xn−1 , L n (x ∗ ) = xn . Now consider Rn (x, y) = cn x + rn y + f n , if Rn obeys the endpoint condition Rn (x1 , y1 ) = yn−1 , Rn (x ∗ , y ∗ ) = yn and rn ∈ [0, 1), then the equations Rn (x1 , y1 ) = yn−1 = cn x1 + rn y1 + f n , Rn (x ∗ , y ∗ ) = yn = cn x ∗ + rn y ∗ + f n

66

4 Fractal Interpolation Function for Countable Data 0

−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9

0

(a) r n =

2

1.5

1

0.5

1 2

Fig. 4.1 Approximation of FIF associated with CIFS as in Eq. (4.1)

give (yn − yn−1 ) − rn (y ∗ − y1 ) , x ∗ − x1 f n = yn−1 − rn y1 − cn x1 ,

cn =

for n ≥ 2. If rn is predefined, then the above discussion generates CIFS of the form wn



   x x a 0 e = n + n , fn cn rn y y ∗

n−1 n (y −y1 ) where an = xxn −x , cn = (yn −yn−1x ∗)−r , en = ∗ −x −x1 1 − cn x1 , and all wn ’s obey the endpoint conditions.

x ∗ xn−1 −x1 xn , x ∗ −x1

f n = yn−1 − rn y1

−1 ∞ Example 4.1 If xn = ( 2n−1 )∞ n=1 , yn = ( n )n=1 with r n = 1/2, for all n ∈ N, then n CIFS interpolating the data set {(xn , yn ) : n ∈ N} is



1 x n 2 −n wn = 1 − y (n 2 −n)

  2n 2 −3n−1 x 2 −n . + n 2n−2n−1 1 y 2 2

0 1 2

(4.1)

n −n

Hence, L n (x) = n 21−n x + 2n n−3n−1 , Rn (x, y) = ( (n 21−n) − 21 )x + 21 y + n n−2n−1 . The 2 −n 2 −n pictorial representation of the approximation process of the attractor of CIFS is given in Fig. 4.1. 2

2

4.2 Existence of FIF for Countable Data Set

67

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1 −1.5

−1.5

−2

−2

−2.5

−2.5 −3

0

0.5

1.5

1

(a) n = 10, r n =

2

−3

(b) n = 100, r n =

1

1 0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2

−2

−2.5

−2.5

0

0.5

1.5

1

(c) n = 1000, r n =

1

0.5

1 2

0.5

−3

0

1 2

2

−3

0

0.5

1.5

2

1.5

2

1 2

1

(d) n = 10000, r n =

1 2

Fig. 4.2 Approximation of FIF associated with CIFS as in Eq. (4.2)

Example 4.2 If yn = ((−1)n )∞ n=1 , x n remains the same as in the above Example 4.1 with rn = 1/2, for all n ∈ N, then CIFS of the form



  2n 2 −3n−1  1 0 x x 2 −n n n 2 −n = + . wn y y −2(−1)n−1 − 1 21 3(−1)n−1 + 23

(4.2)

Hence, 2n 2 − 3n − 1 1 x + , n2 − n n2 − n 1 3 Rn (x, y) = (−2(−1)n−1 − 1)x + y + 3(−1)n−1 + . 2 2 L n (x) =

The graphical representation of approximation of FIF associated with CIFS as in Eq. (4.2) is given in Fig. (4.2).

68

4 Fractal Interpolation Function for Countable Data

4.3 Fractional Calculus on Interpolation Function of Sequence of Data This section reveals the integer order integral of FIF for the sequence of data, if the value of the integral is known at the initial endpoint or final endpoint. After that, these results are implemented into a fractional order integral. In order to validate the results, numerical examples are illustrated. For the rest of this section, CIFS of the form L n (x) = an x + en , Rn (x, y) = rn y + qn (x), n ≥ 2

(4.3)

is considered, where −1 < rn < 1, and qn : I −→ R, n ≥ 2 are continuous functions that satisfy qn (x1 ) = yn − rn y1 , qn (x ∗ ) = yn+1 − rn y ∗ . The interpolation function f of the above CIFS, with respect to the sequence of data {(xn , yn ) : n ≥ 1}, is known as linear FIF. x Theorem 4.2 If fˆ(x) = yˆ1 + x1 f (t)dt, then fˆ is FIF associated with {(L n (x), Rˆ n (x, y)) : n ≥ 2}, where Rˆ n (x, y) = an rn y + qˆn (x), xn − xn−1 an = ∗ , x − x1



x

qˆn (x) = yˆn−1 − an rn yˆ1 + an

qn (t)dt,

x1

yˆn = fˆ(xn ) = yˆ1 + yˆ ∗ = yˆ1 +

n

n

 ai ri ( yˆ ∗ − yˆ1 ) +

i=2

x∗

 qi (t)dt ,

x1

 x∗

ai x1 qi (t)dt  . 1 − supn nn=2 ai ri

supn

i=2

Proof fˆ(L n (x)) = yˆ1 + = yˆ1 +



L n (x)

x1  xn−1 x1

f (t)dt  f (t)dt +

L n (x)

f (t)dt.

xn−1

x Use the change of variable t = L n (t) then fˆ(L n (x)) = yˆn−1 + an x1 f (L n (t))dt and the functional equation f (L n (x)) = Rn (x, f (x)) = rn f (x) + qn (x) gives

4.3 Fractional Calculus on Interpolation Function of Sequence of Data



fˆ(L n (x)) = yˆn−1 + an

69

x

(rn f (t) + qn (t))dt  x  x = yˆn−1 + an rn f (t)dt + an qn (t))dt x1 x1  x = yˆn−1 + an rn ( fˆ(x) − yˆ1 ) + an qn (t))dt. x1

x1

Here, the existence of yˆn−1 , for all n ≥ 2, follows of f on [x1 , x ∗ ].  from the continuity  ∗ x If x = x ∗ , fˆ(L n (x ∗ )) = yˆn − yˆn−1 = an rn ( yˆ ∗ − yˆ1 ) + x1 qn (t)dt . But yˆn =    x∗ n n yˆ1 + i=1 ( yˆi − yˆi−1 ) = yˆ1 + i=1 ai ri ( yˆ ∗ − yˆ1 ) + x1 qi (t)dt .  ˆ  Theorem 4.2 reveals that f is FIF that interpolates the sequence of data (xn , yˆn ) : n ≥ 2 , if the integral value of FIF f is known at the initial point. Simultaneously, the following corollary ensures the similar result when the integral value of FIF f is known at the endpoint.  x∗ Corollary 4.2 If fˆ(x) = yˆ ∗ − x f (t)dt, then fˆ is FIF associated with {(L n (x), Rˆ n (x, y)) : n ≥ 2}, where Rˆ n (x, y) = an rn y + qˆn (x), xn − xn−1 an = ∗ , x − x1



qˆn (x) = yˆn − an rn yˆ ∗ − an

x∗

qn (t)dt,

x

yˆn = fˆ(xn ) = yˆ ∗ − ∗

yˆ1 = yˆ −

n

 ∗ ai ri ( yˆ − yˆ1 ) +

n

 qi (t)dt ,

x1

i=2

 x∗ ai x1 qi (t)dt n . 1 − supn i=2 ai ri

supn

x∗

n=2

Proof Similar arguments of the above theorem yield fˆ(L n (x)) = yˆ ∗ −



x∗

f (t)dt

L n (x)  x∗

= yˆn − an

(rn f (t) + qn (t))dt

x



x∗

= yˆn − an rn



x∗

f (t)dt − an

x

= yˆn − an rn ( yˆ ∗ − fˆ(x)) − an

qn (t))dt

x



x x

∗

qn (t))dt,

70

4 Fractal Interpolation Function for Countable Data

   x∗ if x = x1 , fˆ(L n (x1 )) = yˆn−1 − yˆn = −an rn ( yˆ ∗ − yˆ1 ) − x qn (t))dt . But yˆn =    x∗ n n yˆ ∗ − i=2 ( yˆi − yˆi−1 ) = yˆ ∗ − i=1 ai ri ( yˆ ∗ − yˆ1 ) + x1 qi (t)dt .  Now, let us define the fractional integral of FIF f (k) by Ixβ1 f (k) (x ∗ ) =

1 (β)



x∗

(x ∗ − t)β−1 f (k) (t)dt,

x1

β

with Ix1 f (k) (x1 ) = 0. Theorem 4.3 Let f be the C p -linear FIF determined by the CIFS defined in β Eq. (4.3), then Ix1 f (k) (x) is the C p -linear FIF associated with {K ; wn = (L n (x), (k) Rˆ n,β (x, y)) : n ≥ 2}, where yˆ1,β = 0 and for each n ≥ 2 (k) (k) (x, y) = anβ rn y + qˆn,β (x), Rˆ n,β

(xn − xn−1 ) , (x ∗ − x1 )

an =

(k) (k) (k) (x) = yˆn−1,β + f n−1,β (x) + anβ Ixβ1 qn(k) (x), qˆn,β (k) = Ixβ1 f (k) (xn ), yˆn,β  xn−1 1 (k) f n,β (x) = [(L n (x) − t)β−1 − (xn−1 − t)β−1 ] f (k) (t)dt, (β) x1

for all k = 1, 2, . . . , p. Proof As f is a C p -linear FIF determined by the CIFS of  the form (4.3), f satisfies the functional equation f (k) (L n (x)) = R x, f (k) (x) . Hence, f (k) (L n (x)) = rn f (k) (x) + qn(k) (x), for all x ∈ I, n ≥ 2. I xβ1 f (k) (L n (x)) = = + =

1 (β) 1 (β) 1 (β) 1 (β)

1 + (β) (k)

  

L n (x) x1 xn−1

(xn−1 − t)β−1 f (k) (t)dt −

x1 xn−1

x1  xn−1



(L n (x) − t)β−1 f (k) (t)dt

x1

L n (x)

(k)

(k)

(k)

= yˆn−1,β + f n,β (x) + (k)

xn−1

(xn−1 − t)β−1 f (k) (t)dt

x1



(L n (x) − t)β−1 f (k) (t)dt

xn−1

= yˆn−1,β + f n,β (x) + (k)



L n (x) 1 (L n (x) − t)β−1 f (k) (t)dt (β) xn−1  xn−1 1 (xn−1 − t)β−1 f (k) (t)dt + [(L n (x) − t)β−1 − (xn−1 − t)β−1 ] f (k) (t)dt (β) x1

(L n (x) − t)β−1 f (k) (t)dt +

= yˆn−1,β + f n,β (x) + (k)

1 (β)

(k)

= yˆn−1,β + f n,β (x) +

1 (β) 1 (β) an (β) β

an (β)



L n (x)

(L n (x) − t)β−1 f (k) (t)dt

xn−1  x

(L n (x) − L n (u))β−1 f (k) (L n (u))an du

 

x1 x x1 x x1

  (an (x − u))β−1 rn f (k) (u) + qn(k) (u) du (x − u)β−1 rn f (k) (u)du +

β

an (β)



x x1

(x − u)β−1 qn(k) (u)du

4.3 Fractional Calculus on Interpolation Function of Sequence of Data (k)

71

(k)

= yˆn−1,β + f n,β (x) + anβ rn I xβ1 f (k) (x) + anβ I xβ1 qn(k) (x)   (k) = Rˆ n,β x, I xβ1 f (k) (x) .

β

Moreover, for each k, f (k) is continuous on [x1 , x ∗ ], hence Ix1 f (k) (x) is continβ uous on [x1 , x ∗ ]. Further, limn→∞ xn = x ∗ , so it must give limn→∞ Ix1 f (k) (xn ) = β (k) ∗ (k) Ix1 f (x ). Therefore, yˆn,β exist for all n.  (k)  (k) x1 , yˆ1,β = Rˆ n,β Rˆ n,β (x1 , 0) (k) (x1 ) + anβ rn Ixβ1 f (k) (x1 ) = qˆn,β (k) (k) + f n,β (x1 ) + anβ Ixβ1 qn(k) (x1 ) + anβ rn Ixβ1 f (k) (x1 ) = yˆn−1,β (k) , = yˆn−1,β

 (k)  ∗ (k) (x ∗ ) + anβ rn Ixβ1 f (k) (x ∗ ) x , yˆβ∗ = qˆn,β Rˆ n,β (k) (k) ∗ + f n,β (x ) + anβ Ixβ1 qn(k) (x ∗ ) + anβ rn Ixβ1 f (k) (x ∗ ) = yˆn−1,β

= Ixβ1 f (k) (L n (x ∗ )) (k) = yˆn,β . β

Hence, Ix1 f (k) (x) is the C p -linear fractal interpolation function associated with (k) (x, y)) : n ≥ 2}.  {K ; wn = (L n (x), Rˆ n,β As a consequence of Theorem 4.3, let us define the fractional integral of FIF f (k) β with the terminal endpoint value Ix ∗ f (k) (x ∗ ) = 0 as β

Ix ∗ f (k) (x1 ) =

1 (β)



x∗

(t − x1 )β−1 f (k) (t)dt,

x1

then the following result is proved by Theorem 4.3. Corollary 4.3 Let f be the C p -linear FIF determined by the CIFS defined in β Eq. (4.3), then Ix ∗ f (k) (x) is the C p -linear FIF associated with {K ; wn = (L n (x), (k) Rˆ n,β (x, y)) : n ≥ 2}, where yˆβ∗ = 0 and for each n ≥ 2, (k) (k) (x, y) = anβ rn y + qˆn,β (x), Rˆ n,β

an =

(xn − xn−1 ) , (x ∗ − x1 ) β

(k) (k) (k) qˆn,β (x) = yˆn,β + f n,β (x) + anβ Ix ∗ qn(k) (x), β

(k) = Ix ∗ f (k) (xn ), yˆn,β  x∗ 1 (k) [(t − L n (x))β−1 − (t − xn )β−1 ] f (k) (t)dt, f n,β (x) = (β) xn−1

72

4 Fractal Interpolation Function for Countable Data

for all k = 1, 2, . . . , p. Proof Similar arguments of Theorem 4.3 yield I x ∗ f (k) (L n (x)) =

1 (β)

=

1 (β)

β

+

1 (β) (k)

β

Ix ∗

  

x∗ L n (x) x∗

(L n (x) − t)β−1 f (k) (t)dt

(t − xn )β−1 f (k) (t)dt −

xn xn

1 (β)

(t − L n (x))β−1 f (k) (t)dt +

L n (x) β (k) f n,β (x) + anβ rn I x ∗

= yˆn,β +   β (k) (k) f (L n (x)) = Rˆ n,β x, I x ∗ f (k) (x) .



x∗ xn

1 (β)

(t − xn )β−1 f (k) (t)dt 

x∗

(t − L n (x))β−1 f (k) (t)dt

xn

β

f (k) (x) + anβ I x ∗ qn(k) (x)

β

Moreover, for each k, f (k) is continuous on [x1 , x ∗ ], hence Ix ∗ f (k) (x) is continβ uous on [x1 , x ∗ ]. Further, limn→∞ xn = x ∗ , so it must give limn→∞ Ix ∗ f (k) (xn ) = β (k) exist for all n. Ix ∗ f (k) (x ∗ ) = 0. Therefore, yˆn,β  β (k)  (k) x1 , yˆ1,β = qˆn,β (x1 ) + anβ rn Ix ∗ f (k) (x1 ) Rˆ n,β β

(k) (k) + f n,β (x1 ) + anβ Ix ∗ qn(k) (x1 ) + anβ rn Ixβ1 f (k) (x1 ) = yˆn−1,β β

= Ix ∗ f (k) (L n (x1 )) 

 ∗

(k) , = yˆn,β

β (k) (k) x ∗ , yˆβ = qˆn,β (x ∗ ) + anβ rn Ix ∗ f (k) (x ∗ ) Rˆ n,β β

(k) (k) ∗ + f n,β (x ) + anβ Ix ∗ qn(k) (x ∗ ) = yˆn−1,β (k) . = yˆn−1,β β

Hence, Ix1 f (k) (x) is the C p -linear fractal interpolation function generated by the IFS (k) (x, y)) : n ≥ 2}.  {K ; wn = (L n (x), Rˆ n,β Corollary 4.4 Let f be the C p -linear FIF determined by the CIFS in Eq. (4.3). β α−β β Then Dxα1 (Ix1 f (k) (x)) = Dx1 f (k) (x), α ≥ β ≥ 0, if and only if Ix1 f (k) (x) is the (k) C p -linear FIF generated by the CIFS {K ; wn = (L n (x), Rˆ n,β (x, y)) : n ≥ 2}, where β β β β α−β (k) (k) (k) α ˆ Rn,β (x, y) = an rˆn y + an qˆn,β (x), rˆn = rn an , and Dx1 (ˆrn,β (x)) = an Dx1 qn(k) (x). Proof Consider CIFS {K ; wn = (L n (x), Fn (x, y)) : n ≥ 2} , where L n (x) = an x + bn , Rn (x, y) = αn y + qn (x). Here, |αn | < 1 and qn (x) ∈ C p (I ). Let f be the C p β α−β β linear FIF. Assume that Dxα1 (Ix1 f (k) (x)) = Dx1 f (k) (x), α ≥ β ≥ 0. Ix1 f (k) (x) is p the C -linear FIF associated with

4.3 Fractional Calculus on Interpolation Function of Sequence of Data

73

(k) (k) (x, y) = anβ rn y + qˆn,β (x), Rˆ n,β

an =

(xn − xn−1 ) , (x ∗ − x1 )

(k) (k) (k) qˆn,β (x) = yˆn−1,β + f n−1,β (x) + anβ I xβ1 qn(k) (x),   (k) (x, y)) = Dxα1 anβ rn y + I xβ1 f (k) (xn−1 ) Dxα1 ( Rˆ n,β     xn−2  1 [(L n (x) − t)β−1 − (x − t)β−1 ] f (k) (t)dt + Dxα1 anβ I xβ1 qn(k) (x) . + Dxα1 (β) x1

α−β (k) But Dxα1 ( Rˆ n,β (x, y)) = Rn (x, y) = rn y + Dx1 qn(k) (x), therefore it must occur when β β (k) (k) (x, y) = an rˆn y + an qˆn,β (x). Rˆ n,β β

Conversely, if Ix1 f (k) (t) is the C p -linear fractal interpolation function associβ β (k) α−β (k) (k) ated with Rˆ n,β (x, y) = an rˆn y + an qˆn,β (x), Dxα1 ( Rˆ n,β (x, y)) = rn y + Dx1 qn(k) (x), β α−β  it gives that Dxα1 (Ix1 f (k) (x)) = Dx1 f (k) (x). Example 4.3 If FIF f generated by the CIFS of the form 2n 2 − 3n − 1 1 x + , n2 − n n2 − n 1 1 n 2 − 2n − 1 1 − )x + y + , Rn (x, y) = ( 2 (n − n) 2 2 n2 − n L n (x) =

and f passes through the data set {(xn , yn ) : n ∈ N}, where xn = ( 2n−1 )∞ n=1 and yn = n −1 ∞ ˆ ( n )n=1 . If we choose yˆ1 = y1 , then FIF f generated by the CIFS of the form y 1 + yˆn−1 + 2(n 2 − n) 2(n 2 − n)     2 1 1 1 2 n − 2n − 1 + 2 ( − )(x − 1) + (x − 1) , n − n 2(n 2 − n) 4 n2 − n

Rˆ n (x, y) =

n−1 24i 2 −1(13+2π 2 )−2 where yˆn−1 = −1 + i=2 . Then fˆ interpolates the data 8(i 2 −i)   (xn , yˆn−1 ), n ∈ N , when fˆ value is chosen at the initial endpoint yˆ1 = −1; the graph of fˆ is presented in Fig. 4.3a. Figure 4.3c reveals the fractional integral of order 0.5 of fˆ with yˆ1,β = 0 and a contraction factor rn = 1/2. If we choose yˆ ∗ = y ∗ , then FIF fˆ generated by the CIFS of the form Rˆ n (x, y) =

y 1 + yˆn − 2 − n) n −n

2(n 2



    2 1 n − 2n − 1 1 (4 − x 2 ) + (2 − x) , − 2 − n) 4 n −n

2(n 2

n i 2 (10+2π 2 )−i(28+2π 2 )−2 where yˆn = − i=2 . Here, fˆ interpolates the countable system 8(i 2 −i)   (xn , yˆn ) : n ∈ N , when fˆ value is chosen at the terminal endpoint as yˆ1 = 0; the graph of fˆ is presented in Fig. 4.3b. Further, Fig. 4.3d reveals the fractional integral of order 0.5 of fˆ with yˆβ∗ = 0 and a contraction ratio rn = 1/2.

74

4 Fractal Interpolation Function for Countable Data

−0.1

1

−0.2

0.9

−0.3

0.8

−0.4

0.7

−0.5

0.6 −0.6

0.5

−0.7

0.4

−0.8

0.3

−0.9

0.2

−1

0

0.5

1

1.5

2

0.1

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0.4

−0.05 −0.1

0.35

−0.15

0.3

−0.2 −0.25

0.25

−0.3

0.2

−0.35 −0.4

0.15

−0.45

0.1

−0.5 −0.55

0

0.5

1

1.5

2

0.05

Fig. 4.3 Fractional and ordinary integral of FIF associated with CIFS as in Example 4.3

Example 4.4 If FIF f generated by the CIFS of the form

Fig. 4.4 Approximation of FIF associated with CIFS as in Eq. (4.4)

4.3 Fractional Calculus on Interpolation Function of Sequence of Data

75

Fig. 4.5 Fractional, ordinary integral and fractional derivative of FIF for CIFS as in Example 4.4

2n 2 − 3n − 1 1 x+ , −n n2 − n   1 (−1)n−1 1 (−1)n−1 (3n + 1) 5 x + Rn (x, y) = − y + + , (n 2 − n) 2 3 n2 − n 6 L n (x) =

n2

(4.4)

)∞ and f passes through the data set {(xn , yn ) : n ∈ N}, where xn = ( 2n−1 n=1 and yn = n (−1)n ∞ ( n )n=1 . The pictorial representation of the respective attractor’s approximation process is given in Fig. 4.4. If we choose yˆ1 = −1, then FIF fˆ generated by the CIFS of the form

76

4 Fractal Interpolation Function for Countable Data

Fig. 4.6 Fractional integral and fractional derivative of FIF for CIFS as in Example 4.4

Rˆ n (x, y) = yˆn−1 + +

1 n2 − n



(−1)n−1 1 − n2 − n 2



x2 − 1 2



 +

  (−1)n−1 (3n + 1) 5 + (x − 1) n2 − n 6

y 1 + , 3(n 2 − n) 3(n 2 − n)

n−1 1  (−1)n−1 (1−6i) 1788  where yˆn−1 = −1 + i=2 + 1327 . Then fˆ interpolates the data i 2 −i 2(i 2 −i)   (xn , yˆn−1 ) : n ∈ N , when fˆ value is chosen at the initial endpoint yˆ1 = −1, and the graph of fˆ presented in Fig. 4.5a, c reveals the fractional integral of order 0.5 with rn = 1/3 and yˆ1,β = 0. If we choose yˆ ∗ = 1/2, then FIF fˆ generated by the CIFS of the form L n (x) remains the same as found above and 

(−1)n−1 1 − n2 − n 2 y 1 + − , 3(n 2 − n) 6(n 2 − n)

Rˆ n (x, y) = yˆn −

1 n2 − n



4 − x2 2



 +

  (−1)n−1 (3n + 1) 5 + (2 − x) n2 − n 6

  n (−1)n−1 (1−6i) 1 1788 . Here, fˆ interpolates the data where yˆn = 21 − i=2 + 2 2 i −i 2(i −i) 1327   (xn , yˆn ) : n ∈ N , when fˆ value is chosen at the terminal endpoint as yˆ ∗ = 0; the graph of fˆ is presented in Fig. 4.5b. Further, Fig. 4.5d reveals the fractional integral of order 0.5 of fˆ, when yˆβ∗ = 0, and Fig. 4.5e provides the fractional derivative of order 0.5 with rn = 1/3 of FIF f . Finally, Fig. 4.6a, b presents, respectively, the fractional β integral of order 0.5, rn = 1/3, with initial and final endpoint values of Ia f (k) (x)

4.3 Fractional Calculus on Interpolation Function of Sequence of Data

77

defined as yˆ1,β = yˆβ∗ = 0, where FIF is considered as in Example 4.2. Figure 4.6c presents the fractional derivative of FIF f and Fig. 4.6d reveals the fractional derivaβ tive of order 0.8 of Ix1 f (k) (x) with β = 0.5 and rn = 1/2.

4.4 Concluding Remarks The sampling theory and interpolation concepts are intertwined; undoubtedly, at a fundamental level both are generally very much the same. A traditional problem of the sampling theory concerns the reproduction of signals from a finite discrete set of samples. Despite the fact that the traditional outcomes on this issue usually focus on the perfect reconstruction of signals under appropriate presumptions, a more broad methodology is to look for a rough as opposed to the ideal reconstruction on the some constrained signals. Besides, one has to work with an infinite number of sample points instead of working only with a finite number of data points. Hence, this chapter proposed a newer strategy in the interpolation theory for reconstructing the function from its availability in a countable number of data points with the notion of a countable iterated function system. The existence of the countable iterated function system has been investigated when the fractal interpolation function is predefined. Further, the classical integral and the Riemann–Liouville fractional integral of FIF for a countable system of data have been explored in this chapter. Besides, the effects of free variables in the shape of FIF for the reconstruction of intricate functions have been demonstrated.

Chapter 5

Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

5.1 Introduction Multifractal measures have shown significant differences among normal, interictal, and ictal EEGs. The proposed scheme is demonstrated with high accuracy through suitable graphical methods and statistical tools called the one-way Analysis of Variance (ANOVA) test with Box Plot. It is shown that the designed methods perform significantly in the detection of epileptic seizures in EEG signals. The EEG data are further tested for linearity by using the Normal Probability Plot and have proved that Epileptic EEG has significant nonlinearity whereas Healthy EEG is distributed normally and similar to the Gaussian Linear Process.

5.2 Experimental Signals 5.2.1 Synthetic Weierstrass Signals Theorem 5.1 (Weierstrass Sine and Cosine Functions [69]) Suppose 1 < s < 2 and λ > 1. Define f s : [0, 1] −→ R and f c : [0, 1] −→ R by f s (t) =

∞ 

  λ(s−2)k sin λk t

k=1

and

f c (t) =

∞ 

  λ(s−2)k cos λk t .

k=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_5

79

80

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.1 Weierstrass sine curves

Then, provided λ is large enough, dim B (graph f s ) = dim B (graph f c ) = s. Note that this function is everywhere continuous but nowhere differentiable. Also observe that the Hausdorff dimension or fractal dimension of the graph of the above Weierstrass functions is at most s [69]. Synthetic waveforms are generated using the Weierstrass sine and cosine functions f s : [0, 1] −→ R and f c : [0, 1] −→ R, respectively, given as follows: f s (t) =

M 

  λ(s−2)k sin λk t

k=1

and

f c (t) =

M 

  λ(s−2)k cos λk t ,

k=1

where 1 < s < 2, λ > 1 and M ∈ N. Now we fix the parameters as λ = 5 and M = 400 and get the Weierstrass sine and cosine waveforms for s = 1.1, 1.3, 1.5, 1.7, and 1.9 as in Figs. 5.1 and 5.2, respectively.

5.2 Experimental Signals

81

Fig. 5.2 Weierstrass cosine curves

5.2.2 Clinical EEG Signals The data measured by the Scalp and Intracranial EEGs are used for clinical and research purposes. In cognitive neuroscience, EEG is used to investigate the neural correlates of mental activity from low-level perceptual and motor processes to higherorder cognition (attention, memory, reading, etc.). In neurology, the main diagnostic application of EEG is for epilepsy but this technique is also used to investigate much other pathology such as sleep-related disorders, Alzheimer’s disease, and brain tumors. The EEG clinical data consists of five different sets (A, B, C, D & E), each single-channel EEG segment containing 4096 samples of 23.6 s duration, which were obtained from the EEG Database available with Clinic of Epileptology of the University Hospital of Bonn, Bonn, Germany [78, 79]. The EEG data was recorded with the International 10–20 system and digitized at a sampling rate (sampling frequency) of 173.61 Hz. Note that the time series has the spectral bandwidth of the acquisition system, which is 0.5–85 Hz. The first two Data sets consist of EEG segments taken from surface EEG recordings that were carried out on five Healthy (Normal) persons with eyes open ( A) and eyes closed (B). The Data for the last three sets (C, D & E) was taken from five Epileptic Patients undergoing presurgical evaluations. The third and the fourth data sets consist of intracranial EEG recordings during seizure-free intervals (Interictal periods) from opposite (Contralateral) to the Epileptogenic zone (C) and within (Ipsi Lateral) the Epileptogenic zone (D) of the Epileptic Patients’ Brain, respectively. The data in the

82

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Table 5.1 Summary of the clinical EEG signals Data set Subject’s state Electrode type A B C

D E

Healthy, normal (eyes open) Healthy, normal (eyes closed) Epileptic, interictal (seizure free)

Scalp

Epileptic, interictal (seizure free) Epileptic, ictal (seizure activity)

Intracranial

Scalp Intracranial

Intracranial

Electrode placement International 10–20 system International 10–20 system Opposite (contralateral) of epileptogenic zone Within (ipsi lateral) epileptogenic zone Within (ipsi lateral) epileptogenic zone

Fig. 5.3 Sample EEG signals from both healthy subjects and epileptic patients

last set (E) was recorded during seizure activity (Ictal periods) using depth electrodes placed within (Ipsi Lateral) the Epileptogenic zone of the Epileptic Patients’ Brains. A summary of the Five Data sets is given in Table 5.1. A sample of EEG epochs from each of the five data sets is plotted in Fig. 5.3.

5.3 Statistical Methods

83

5.3 Statistical Methods 5.3.1 ANOVA Test One-way ANOVA (Analysis of Variance) test is one of the statistical tools to analyze the mean and variance of the given data set. ANOVA uses variances to decide whether the means are different. If the observed differences are high, then it is considered to be statistically significant. The p-value can be obtained using analysis of variance between groups. In this test, if the p-value is near zero, this casts doubt on the null hypothesis and suggests that at least one sample mean is significantly different from the other sample means. That is, if the p-value decreases to zero, then we decide that the significant differences between the data sets increase.

5.3.2 Box Plot The Box Plot technique significantly shows the difference between the data sets. The Box Plot produces a box and a whisker plot for each set of data. The box has lines at the lower quartile, median, and upper quartile values. The whiskers are lines extending from each end of the box to show the extent of the rest of the data. Outliers are data with values beyond the ends of the whiskers. If there is no data outside the whisker, a dot is placed at the bottom whisker.

5.3.3 Normal Probability Plot The purpose of a Normal Probability Plot is to graphically assess whether the data could come from a normal distribution. The data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. If the data are normal, the plot will be linear. Departures from this straight line indicate departures from normality. Other types of distribution will introduce curvature in the plot.

5.4 Development of Multifractal Analysis in EEG Signal Classification This section explores the three different methods namely, Modified, Improved, and Advanced forms of Generalized Fractal Dimensions in order to discriminate the Healthy and Epileptic EEGs. We compare our designed GFD Methods with the GFD to distinguish the Healthy and Epileptic EEGs through graphically using Fractal Spectra, Range of Fractal Spectra, and the graph plotted with the absolute values of

84

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

the entropy against the log values of the corresponding scaling factor, and statistically using the one-way ANOVA test, Box Plot, and Normal Probability Plot. Finally, we conclude that there are significant differences between the Healthy and Epileptic Signals in the designed methods than the GFD. Also, the Normal Probability Plot shows that the Epileptic EEG Data has significant nonlinearity but the normal and interictal EEGs are dominated by the Gaussian Linear Process.

5.4.1 Modified Generalized Fractal Dimensions In order to modify our GFD form, we define a probability distribution of a given Time Series as follows. The total range of the Signal Time Series is divided into N V bins such that NV =

Vmax − Vmin r

where Vmax and Vmin are the maximum and the minimum values of the signal received in the experiments, respectively; and r is the uncertainty factor that may be dependent on the measuring device used to record the signal. The time domain of the Signal is also divided into Nt intervals such that Nt =

tmax − tmin r

where tmax and tmin are the maximum and the minimum time in the time domain of the signal received in the experiments, respectively; and r is the uncertainty factor. For the fixed ith bin of size r in the range, the probability that the signal passes through the jth interval of length r in the time domain is given by pi j = lim

Nt →∞

Ni j , Nt

j = 1, 2, . . . , Nt

where Ni j is the number of times the signal passes through the jth time interval in the ith bin of size r . Now the probability that the signal passes through the ith bin of size r in the range of the signal is given by  Nt p Mi = lim

N V →∞

j=1

NV

pi j

,

i = 1, 2, . . . , N V .

(5.1)

Then, Modified Renyi Fractal Dimensions or Modified Generalized Fractal Dimensions (MGFD) of order q ∈ (−∞, ∞) such that q = 1 for the known probability distribution, denoted by M Dq , can be defined as

5.4 Development of Multifractal Analysis in EEG Signal Classification

 NV q p Mi 1 log2 i=1 M Dq = lim . r →0 q − 1 log2 r

85

(5.2)

Here, M Dq is defined in terms of generalized Renyi Entropy with the probability given in Eq. (5.1). Equation (5.2) is called the Modified form of the GFD as in Eq. (1.19).

5.4.1.1

Limiting Cases of Modified Generalized Fractal Dimensions

The two limiting cases in the Modified GFD Method, when q = −∞ and q = ∞, are   log2 p Mmin M D−∞ = lim r →0 log2 r   log2 p Mmax M D∞ = lim r →0 log2 r where p Mmin = min{ p M1 , p M2 , . . . , p M NV }, p Mmax = max{ p M1 , p M2 , . . . , p M NV }. 5.4.1.2

Range of Modified Generalized Fractal Dimensions

The two limit cases, M D−∞ and M D∞ , define the Range of Modified Generalized Fractal Dimensions of a given Fractal Time Series as R M G F D = M D−∞ − M D∞ .

(5.3)

5.4.2 Improved Generalized Fractal Dimensions In order to improve our GFD Method, we define a probability distribution of a given Fractal Time Series as follows. The total range of the Signal Time Series is divided into N V × Nt bins (boxes) such that Vmax − Vmin tmax − tmin and Nt = NV = r r where Vmax and Vmin are the maximum and the minimum values of the signal received in the experiments, and tmax and tmin are the maximum and minimum time of the

86

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

experiments, respectively; and r is the uncertainty factor that may be dependent on the measuring device used to record the signal. Now the probability that the signal passes through the i jth bin (box) of size r is given by p Ii j =

lim

N V ,Nt →∞

Ni j , N V × Nt

i = 1, 2, . . . , N V and j = 1, 2, . . . , Nt

(5.4)

where Ni j is the number of times the signal passes through the i jth bin of size r . Then, Improved Renyi Fractal Dimensions or Improved Generalized Fractal Dimensions (IGFD) of order q ∈ (−∞, ∞) such that q = 1 for the known probability distribution, denoted by I Dq , can be defined as 1 log2 I Dq = lim r →0 q − 1

 N V  Nt i=1

j=1

log2 r

q

p Ii j

.

(5.5)

Here, I Dq is also defined in terms of generalized Renyi Entropy with the probability given in Eq. (5.4). Equation (5.5) is called the Improved form of Generalized Fractal Dimensions as in Eq. (1.19).

5.4.2.1

Limiting Cases of Improved Generalized Fractal Dimensions

The two limit cases in the Improved GFD Method, when q = −∞ and q = ∞, are   log2 p Imin I D−∞ = lim r →0 log2 r   log2 p Imax I D∞ = lim r →0 log2 r where p Imin = p Imax =

5.4.2.2

min

{ p Ii j },

max

{ p Ii j }.

1≤i≤N V & 1≤ j≤Nt

1≤i≤N V & 1≤ j≤Nt

Range of Improved Generalized Fractal Dimensions

The two limit cases, I D−∞ and I D∞ , define the Range of Improved Generalized Fractal Dimensions of a given Fractal Time Series as R I G F D = I D−∞ − I D∞ .

(5.6)

5.4 Development of Multifractal Analysis in EEG Signal Classification

87

5.4.3 Advanced Generalized Fractal Dimensions In order to design the advanced form of GFD, we define a probability distribution of a given Fractal Time Series by the following construction. The total range of the Signal Time Series is divided into N intervals (bins) such that Vmax − Vmin NA = r where Vmax and Vmin are the maximum and the minimum values of the signal received in the experiments, respectively; and r is the uncertainty factor that may be dependent on the measuring device used to record the signal. Now the probability that the signal passes through the ith interval of length r in the range of Fractal Time Series is given by p Ai = lim

T →∞

Ni , T

i = 1, 2, . . . , N A

(5.7)

where Ni is the number of times the signal passes through the ith range interval of length r , and T is the total number of values in the given data set. Then, Advanced Renyi Fractal Dimensions or Advanced Generalized Fractal Dimensions (AGFD) of order q ∈ (−∞, ∞) such that q = 1 for the known probability distribution, denoted by ADq , can be defined as NA q p Ai 1 log2 i=1 . ADq = lim r →0 q − 1 log2 r

(5.8)

Here, ADq is also defined in terms of generalized Renyi Entropy with the probability given in Eq. (5.7). Equation (5.8) is called the Advanced form of Generalized Fractal Dimensions as in Eq. (1.19).

5.4.3.1

Limiting Cases of Advanced Generalized Fractal Dimensions

The two limit cases in the Advanced GFD Method, when q = −∞ and q = ∞, are   log2 p Amin AD−∞ = lim r →0 log2 r   log2 p Amax AD∞ = lim r →0 log2 r where

88

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

p Amin = min{ p A1 , p A2 , . . . , p A N A }, p Amax = max{ p A1 , p A2 , . . . , p A N A }. 5.4.3.2

Range of Advanced Generalized Fractal Dimensions

The two limit cases, AD−∞ and AD∞ , define the Range of Advanced Generalized Fractal Dimensions of a given Fractal Time Series as R AG F D = AD−∞ − AD∞ .

(5.9)

5.4.4 Methods to Analyze the Fractal Time Signals For a given probability distribution of given Fractal Time Signals, the GFD function, Dq , is called Fractal Spectrum. This Fractal Spectrum describes information about both the amplitudes and frequencies of the given Signal. So we consider that it is a significant tool to characterize the Signals. We can calculate GFD (Dq ), MGFD (M Dq ), IGFD (I Dq ), and AGFD (ADq ) for the given Time Signals by using Eqs. (1.19), (5.2), (5.5), and (5.8), respectively, through the graph plotted for the absolute values of the Renyi Entropy against the log values of the corresponding scaling factor. Also, we can determine Range of GFD (RG F D ), Range of MGFD (R M G F D ), Range of IGFD (R I G F D ), and Range of AGFD (R AG F D ) for the given Fractal Time Signals by using Eqs. (1.22), (5.3), (5.6), and (5.9), respectively. From these measures, we compare the Fractal Time Signals. In order to discriminate the fractal signals, set  q 1 log2 pi q −1 i=1

from Eq. (1.19)

(5.10)

V  1 q log2 p Mi q −1 i=1

from Eq. (5.2)

(5.11)

N

RE N =

N

M RE N =

V  t  1 q log2 p Ii j q −1 i=1 j=1

N

I RE N =

N

and NA  1 q log2 ARE N = p Ai q −1 i=1

from Eq. (5.5)

(5.12)

from Eq. (5.8)

(5.13)

Then plot the graph of R E N , M R E N , I R E N , and A R E N versus corresponding log(r ) for the given signals and compare these graphs to analyze and classify the fractal signals.

5.4 Development of Multifractal Analysis in EEG Signal Classification

89

Fig. 5.4 Three samples of generalized fractal spectra for normal, interictal, and ictal EEGs

5.4.5 Results and Discussions In this section, we have analyzed the Epileptic seizure through various forms of GFD by using the five representative EEG signals, which are described in Sect. 5.2.2, such as Healthy EEG Data Sets A & B, Epileptic Interictal EEG Data Sets C & D, and Epileptic Ictal EEG Data Set E. The computations in this section are performed through MATLAB software. The probability distribution of 20 representative Clinical EEG segments from each of the five data sets taken from the Healthy subjects and the Epileptic patients during Interictal and Ictal periods were obtained, and corresponding Generalized Fractal Dimensions, Modified Generalized Fractal Dimensions, Improved Generalized Fractal Dimensions, and Advanced Generalized Fractal Dimensions for q varying from 2 to 50 were computed. Generalized Fractal Spectra, Modified Generalized Fractal Spectra, Improved Generalized Fractal Spectra, and Advanced Generalized Fractal Spectra of three sample EEG segments from each of the five data sets are depicted in Figs. 5.4, 5.5, 5.6, and 5.7, respectively.

90

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.5 Three samples of modified generalized fractal spectra for normal, interictal, and ictal EEGs

Then the values of Range of GFD (RG F D ), Range of MGFD (R M G F D ), Range of IGFD (R I G F D ), and Range of AGFD (R AG F D ) for all the given Clinical EEG Signals are obtained, which are shown in Tables 5.2, 5.3, 5.4, and 5.5, respectively. The graphs which are plotted the values of Range of GFD, Range of MGFD, Range of IGFD, and Range of AGFD against the Data Set Numbers are shown in Fig. 5.8. For q = 0.5, all the values of R E N , M R E N , I R E N , and A R E N as in Eqs. (5.10), (5.11), (5.12), and (5.13) with the corresponding scaling factors r for all the representative EEG signals from both the Healthy subjects and the Epileptic patients during Interictal and Ictal periods were calculated. Then the values of R E N , M R E N , I R E N , and A R E N against the corresponding values of log(r ) for all Normal, Interictal, and Ictal EEGs were plotted. The plotted graphs for GFD, MGFD, IGFD, and AGFD Methods for a sample EEG segment are depicted in Fig. 5.9. In order to test the mean differences among Normal, Interictal, and Ictal EEG segments statistically, a repeated measure one-way Analysis of Variance (ANOVA) is performed on Fractal Spectra for GFD, MGFD, IGFD, and AGFD Methods, using a standard tool of statistical analysis (MATLAB Statistical Tool Box), as shown in

5.4 Development of Multifractal Analysis in EEG Signal Classification

91

Fig. 5.6 Three samples of improved generalized fractal spectra for normal, interictal, and ictal EEGs

Table 5.6. Similarly, an ANOVA test is performed on the Range of Fractal Spectra for GFD, MGFD, IGFD, and AGFD Methods as shown in Table 5.7. The Box Plots of Fractal Spectra for GFD, MGFD, IGFD, and AGFD Methods among Normal, Interictal, and Ictal EEG signals are drawn as shown in Fig. 5.10. Likewise, the Box Plots of the Range values of Fractal Spectra for GFD, MGFD, IGFD, and AGFD Methods among Normal, Interictal, and Ictal EEG signals are obtained as shown in Fig. 5.11. The representative EEG Data are tested by the Normal Probability Plot to examine that the data is distributed normally or not. The Normal Probability Plots for five representative EEG Data are depicted in Fig. 5.12. Figures 5.4, 5.5, 5.6, and 5.7 shows that, as q increases, there is a significant difference among the values of curves plotted for Normal, Interictal, and Ictal EEGs in the Modified GFD, Improved GFD, and Advanced GFD Methods than in the GFD Method for the sample EEG segments. For all the representative EEG segments we observed that, as q increases, the values of Dq for Normal, Interictal, and Ictal EEGs would coincide in the GFD Method. But in all of our designed GFD Methods,

92

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.7 Three samples of advanced generalized fractal spectra for normal, interictal, and ictal EEGs

there is a significant difference among the values of M Dq , I Dq , and ADq for each of the Normal, Interictal, and Ictal EEGs even for high values of q. Note that the Interictal EEG, which was taken opposite to the Epileptogenic zone, has fewer values of Fractal Spectra than the Interictal EEG taken within the Epileptogenic zone. It reveals that there are many disturbances of seizure onset in the opposite region than in the Epileptogenic region of the Epileptic Patient’s Brain. In Tables 5.2, 5.3, 5.4, and 5.5, the Range value of Advanced Fractal Spectra significantly differs among the Normal, Interictal, and Ictal EEGs than that of the other three methods. Figure 5.8 also shows that the Range values of Advanced GFD have a specific amount of differences between the Normal and Epileptic EEGs, when compared with the other methods. In Fig. 5.9, R E N , M R E N , I R E N , and A R E N against the corresponding values of log(r ) for Normal, Interictal, and Ictal EEGs were plotted. We observed that, by using the GFD Method, the values of R E N for Normal, Interictal, and Ictal EEGs would be mingled in Fig. 5.9a and the linearity of values in the GFD Method is very low.

5.4 Development of Multifractal Analysis in EEG Signal Classification

93

Table 5.2 Range values of generalized fractal spectra for Normal, Interictal, and Ictal EEGs Data set no. Data set type A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5.2610 4.7053 9.1098 13.9069 4.9505 4.6417 4.0683 6.3030 4.7899 5.2240 5.0953 6.3295 4.7292 4.3784 4.9232 22.7800 4.9163 4.3153 4.1489 4.7241

2.8690 2.4051 2.1110 2.8270 3.1634 2.8017 2.2648 2.6418 2.4303 4.1558 5.3701 2.9589 2.4347 3.4656 3.1217 2.2922 3.0646 3.2375 2.2081 2.2087

1.9389 5.4074 2.4267 6.4996 3.9884 3.3231 3.8477 3.3989 5.1106 2.0017 4.9128 2.8026 3.9401 3.3420 3.5863 2.4982 2.9656 5.2213 2.6963 3.6049

2.2521 1.7608 6.3487 3.2954 2.2089 2.0752 4.6456 5.9069 3.2363 2.5532 3.0152 1.3554 3.4925 1.9538 12.2176 4.2750 12.6599 2.9174 2.8078 9.1438

1.4137 1.2215 1.5707 1.7179 1.3259 1.1445 1.2177 1.7383 1.8354 1.5883 1.7712 1.2985 1.3353 1.5864 1.5858 1.2069 1.3083 1.3441 1.8172 1.3308

Also, we noticed that, in our designed Methods of GFD, the values of M R E N , I R E N , and A R E N versus log(r ) for Ictal EEG is significantly different from the values of M R E N , I R E N , and A R E N versus log(r ) for Normal and Interictal EEGs as in Fig. 5.9b-d, and the designed methods have a high degree of linearity than the GFD Method. Since only the linearity of such graphs in Fig. 5.9 decides the dimensional values of the EEG signals, our designed methods give the accurate Dimensional values than the GFD Method in the discrimination of Healthy and Epileptic EEGs. The ANOVA test also supports our designed methods statistically than the GFD Method. The p-value in Table 5.6—(a) is greater than the p-values in Table 5.6—(b), (c), and (d), which is zero. Hence, the values corresponding to Normal, Interictal, and Ictal EEGs significantly differed in all our three designed methods than the GFD Method. The p-value in Table 5.7—(d) is much lesser than the p-values in Table 5.7—(a), (b), and (c). This shows that the Range values of Fractal Spectra for Normal, Interictal, and Ictal EEGs essentially differ in the Advanced GFD Method than that of all other methods. It is also observed from the Box-Whiskers Plots of Fig. 5.10 that there is significant variability in the Fractal Spectra of all our designed

94

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Table 5.3 Range values of modified generalized fractal spectra for normal, interictal, and ictal EEGs Data set no. Data set type A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

9.0400 8.3583 16.2924 24.7345 8.8537 8.3836 7.0987 11.3958 8.5765 9.3429 8.8238 11.3577 8.6227 7.9831 8.9766 41.1862 8.5139 7.7940 7.4201 8.3332

6.7401 5.6062 5.2593 6.6559 6.9392 6.3552 5.4796 6.3552 5.5487 9.2385 11.6815 7.0398 5.5228 8.4350 7.1748 5.0391 6.6439 7.2536 5.3626 5.3134

4.4979 12.8975 5.3742 15.7514 9.3027 7.2379 9.2045 8.6798 11.9806 4.3908 11.8184 6.6536 8.1998 7.2540 8.4873 5.9968 6.9128 11.3089 6.4502 8.4328

4.7472 3.7104 15.1047 6.9265 5.1743 4.1876 10.7032 13.2877 7.4564 5.9515 6.8991 3.1750 8.0271 4.0861 27.3733 9.9223 28.5961 7.1440 6.4690 20.9705

3.3455 2.6423 3.8706 4.0873 3.3573 2.7085 2.8249 4.1585 4.0647 3.5308 4.1827 2.7668 2.7697 3.9091 3.9245 3.0277 2.8176 3.1331 3.9948 3.1263

methods among the Normal, Interictal, and Ictal EEGs when compared with the GFD Method. Then the Box-Whiskers Plots of Fig. 5.11 also show that there is an essential variability in the Range values of the Advanced GFD Method than that of the other methods. In addition to that, in Fig. 5.12, the plots for Normal and Interictal EEGs are linear whereas the plot for Epileptic Ictal EEG is deviated from the approximate linear line. Hence, it is proved that during seizure activity, Ictal EEG has significant nonlinearity presence while Normal and Interictal EEGs are powered by the Gaussian Linear Process. Note that Ictal EEG has sufficient nonlinearity but the Interictal EEG Data during seizure-free intervals resembles a Linear Process. Thus, Figs. 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10 and 5.11, and Tables 5.2, 5.3, 5.4, 5.5, 5.6, and 5.7 are the evidence that the Modified Method, Improved Method, and Advanced Method of GFD play an efficient role than the GFD Method in the detection or recognition of Healthy and Epileptic EEGs. Especially, the Advanced form of GFD is a more consistent method in the analysis of Epileptic EEG signals.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

95

Table 5.4 Range values of improved generalized fractal spectra for normal, interictal, and ictal EEGs Data set no. Data set type A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

15.2991 14.1455 27.5731 41.9510 14.9840 13.7672 12.0138 19.3279 14.5462 15.8117 14.9333 18.8490 14.5929 13.5105 15.1918 68.1257 14.4088 13.1904 12.5577 14.1030

11.4316 9.4879 8.9201 11.2643 11.7693 10.7554 9.2737 10.7554 9.4109 15.6352 19.7696 11.8703 9.3466 14.2752 12.1425 8.5281 10.9375 12.2759 9.0953 8.9923

7.5579 22.0151 9.0953 26.7727 15.8117 12.2759 15.5776 14.7845 20.3633 7.4471 20.0012 11.2848 13.9371 12.3032 14.3638 10.1709 11.6991 18.8490 10.9162 14.2613

8.0340 6.3199 25.5630 11.7223 8.7569 7.0870 18.1532 21.6212 12.6464 10.0723 11.6760 5.3734 13.5850 6.9303 46.1563 16.8648 48.3957 12.0905 10.9717 35.5671

5.6619 4.4815 6.5266 6.9173 5.6818 4.5838 4.7808 7.0377 6.8791 5.9535 6.9372 4.7027 4.6976 6.6157 6.6562 5.1352 4.7685 5.3024 6.7608 5.3024

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification Denoising of EEG signals is an important task in signal processing. The noises must be corrected or reduced before the subsequent decision analysis. This section presents a wavelet-based denoising method for the recovery of EEG signals contaminated by non-stationary noises and investigates the recognition of healthy and epileptic EEG signals by using multifractal measures such as Generalized Fractal Dimensions. The multifractal measures show the significant differences among normal, interictal, and epileptic ictal EEGs with denoising by the wavelet transform as the preprocessing step. The denoised artifact-free EEG presents a very good improvement in the identification rate of an epileptic seizure. The proposed scheme illustrates with high accuracy through suitable graphical and statistical tools and performs an important role in epileptic seizure detection.

96

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Table 5.5 Range values of advanced generalized fractal spectra for normal, interictal, and ictal EEGs Data set no. Data set type A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10.5769 9.6204 18.6905 28.4834 10.1569 9.5717 8.2427 13.0187 9.8442 10.7180 10.2841 13.0084 9.7997 9.0728 10.2019 47.0518 9.9229 8.8985 8.5123 9.6244

2.8690 2.4051 2.1110 2.8270 3.1634 2.8017 2.2648 2.6418 2.4303 4.1558 5.3701 2.9589 2.4347 3.4656 3.1217 2.2922 3.0646 3.2375 2.2081 2.2087

1.9389 5.4074 2.4267 6.4996 3.9884 3.3231 3.8477 3.3989 5.1106 2.0017 4.9128 2.8026 3.9401 3.3420 3.5863 2.4982 2.9656 5.2213 2.6963 3.6049

2.2521 1.9735 6.3487 3.2954 2.2089 2.0752 4.6456 5.9069 3.2363 2.5532 3.0152 1.3554 3.4925 1.9538 12.2176 4.2750 2.9656 2.9174 2.8078 9.1438

1.4137 1.2215 1.5707 1.7179 1.3259 1.1445 1.2177 1.7383 1.8354 1.5883 1.7712 1.2985 1.3353 1.5864 1.5858 1.2069 1.3083 1.3441 1.8172 1.3308

5.5.1 Discrete Wavelet Transform Wavelet analysis is a method which relies on the introduction of an appropriate basis and characterization of the signal by the distribution of amplitude in the basis. The wavelet transform which gives us a powerful tool to confront very diverse problems in applied sciences offers a well-suited technique to detect and analyze the complex events occurring in different scales. Wavelet transforms are widely applied in many biomedical engineering fields for solving various real-life problems. Especially, wavelet transform forms a general mathematical tool for biomedical signal processing with many applications in EEG data analysis as well. Discrete Wavelet Transform (DWT) analyzes the signal at different frequency bands, with different resolutions by decomposing the signal into a coarse approximation and detailed information. DWT employs two sets of functions called scaling functions and wavelet functions, which are associated with low-pass and high-pass filters, respectively. The decomposition of the signal into different

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

(a) GFD Method

(b) Modified GFD Method

(c) Improved GFD Method

(d) Advanced GFD Method

97

Fig. 5.8 Range of generalized fractal spectra for normal, interictal, and ictal EEGs

frequency bands is simply obtained by successive high-pass and low-pass filtering of the time domain signal. The continuous wavelet transform (CWT) of a signal, x(t), is the integral of the signal multiplied by scaled and shifted versions of a wavelet function  [80, 81] and is defined by,    ∞ t −b 1 dt (5.14) x(t) √  C W T (a, b) = a |a| −∞ where a and b are the so-called scaling (reciprocal of frequency) and time localization of shifting parameters, respectively. Calculating wavelet coefficients at every possible scale is computationally a very expensive task. Instead, if the scales and shifts are selected based on powers of 2, so-called dyadic scales and positions, then the wavelet analysis will be much more efficient. Such analysis is obtained from the DWT which is defined as

98

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

(a) GFD Method

(b) Modified GFD Method

(c) Improved GFD Method

(d) Advanced GFD Method

Fig. 5.9 Linearity of generalized fractal spectra for normal, interictal, and ictal EEGs

1

DW T ( j, k) = |2 j |





∞

x(t) −∞

t − 2jk 2j

 dt

(5.15)

where a and b are replaced by 2 j and k2 j , respectively. In 1989, Mallat [82] developed an efficient way for implementing this scheme by passing the signal through a series of low-pass (LP) and high-pass (HP) filter pairs named as quadrature mirror filters. In the first step of the DWT, the signal is simultaneously passed through a LP and a HP filter with the cut-off frequency being one-fourth of the sampling frequency. The outputs from the low- and high-pass filters are referred to as approximation (A1 ) and detail (D1 ) coefficients of the first level, respectively. The output signals having half the frequency bandwidth of the original signal can be downsampled by 2 according to Nyquist rule. The same procedure can be repeated for the first-level approximation and the detail coefficients to get the second-level coefficients. At each step of this decomposition process, the frequency resolution is doubled through filtering, and the time resolution is halved through downsampling.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

99

Table 5.6 One-way ANOVA tables for generalized fractal spectra among normal, interictal, and ictal EEGs. SS—sum of squares, df—degrees of freedom, MS—mean square, and F—F-ratio (a) GFD method ANOVA table Source SS df MS F Prob>F Columns 1.0728 Error 11.4662 Total 12.539 (b) Modified GFD method ANOVA table Source

SS

Columns 138.869 Error 7.577 Total 146.445 (c) Improved GFD method ANOVA table Source

SS

Columns 381.424 Error 15.966 Total 397.39 (d) Advanced GFD method ANOVA table

4 240 244

0.2682 0.04778

5.61

0.0002

df

MS

F

Prob>F

4 240 244

34.7171 0.0316

1099.72

0

df

MS

F

Prob>F

4 240 244

95.3559 0.0665

1433.34

0

Source

SS

df

MS

F

Prob>F

Columns Error Total

71.0927 1.3554 72.4481

4 240 244

17.7732 0.0056

3147.05

0

In the third-level wavelet decomposition of a signal, the coefficients A1 , D1 , A2 , D2 , A3 , and D3 represent the frequency content of the original signal within the bands 0 − f s /4, f s /4 − f s /2, 0 − f s /8, f s /8 − f s /4, 0 − f s /16, and f s /16 − f s /8, respectively, where f s is the sampling frequency of the original signal.

5.5.2 Wavelet Denoising of Signals In the wavelet decomposition of signals using DWT, the more outliers and noises in the original signals are placed in approximation coefficients but not in the detail coefficients at each level of decomposition. Also, we have concentrated to decompose the approximation coefficients for a further level of decomposition so that the remaining noises are disposed of for subsequent decompositions. Then, we have

100

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Table 5.7 One-way ANOVA tables for the range values of generalized fractal spectra among normal, interictal, and ictal EEGs. SS—sum of squares, df—degrees of freedom, MS—mean square, and F—F-ratio (a) GFD method ANOVA table Source SS df MS F Prob>F Columns 274.59 Error 625.181 Total 899.771 (b) Modified GFD method ANOVA table Source

SS

Columns 793.75 Error 2527.23 Total 3320.98 (c) Improved GFD method ANOVA table Source

SS

Columns 2245.92 Error 7077.13 Total 9323.05 (d) Advanced GFD method ANOVA table Source SS Columns 1743.55 Error 1852.07 Total 3595

4 95 99

68.6475 6.5809

10.43

4.77778e-007

df

MS

F

Prob>F

4 95 99

198.438 26.602

7.46

2.86581e-005

df

MS

F

Prob>F

4 95 99

561.479 7 4.496

7.54

2.5653e-005

df 4 95 99

MS 435.889 19.495

F 22.36

Prob>F 4.95715e-013

retrieved the signal by unifying all detail coefficients with the last-level approximation coefficients of corresponding frequency bands. In this way, we have denoised the signals by using the discrete wavelet transform in the preprocessing step.

5.5.3 Results and Discussions In this section, simulations are carried out in MATLAB software to analyze the Epileptic seizure through DWT and GFD by using the four representative EEG signals, which are described in Sect. 5.2.2, such as Healthy EEG Data Set A, Epileptic Interictal EEG Data Sets C & D, and Epileptic Ictal EEG Data Set E.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

(a) GFD Method

(b) Modified GFD Method

(c) Improved GFD Method

(d) Advanced GFD Method

101

Fig. 5.10 Box plots of generalized fractal spectra among normal, interictal, and ictal EEGs

In the first step of the proposed denoising scheme, the representative EEG epochs were analyzed using DWT. As in the description of DWT in Eq. 5.15, third-level wavelet decomposition using Daubechies wavelets is applied to all representative normal, interictal, and ictal EEG segments for both healthy subjects and epileptic patients as the preprocessing step. The representative EEG signals of all categories are simultaneously passed through a LP and a HP filter with the cut-off frequency being one-fourth of the sampling frequency, and the experimental signals of all categories up to the third level are decomposed as shown in Fig. 5.13. In the third-level wavelet decomposition of a signal, the coefficients A1 , D1 , A2 , D2 , A3 , and D3 represent the frequency content of the original EEG signals of all four categories. The wavelet decomposition along with the corresponding frequency bands of the approximation and detail coefficients at level 3 of all representative EEG segments were obtained as in Fig. 5.13. Also, approximation (A3 ) and detail (D1 , D2 & D3 ) coefficients of sample normal, interictal, and ictal EEG signals are presented in Figs. 5.14, 5.15, 5.16, and 5.17, respectively.

102

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

(a) GFD Method

(b) Modified GFD Method

(c) Improved GFD Method

(d) Advanced GFD Method

Fig. 5.11 Box plots of the range values of generalized fractal spectra among normal, interictal, and ictal EEGs

All the representative original normal, interictal, and ictal EEG signals from healthy and epileptic subjects were denoised by using wavelet decomposition at level 3. The original and denoised normal, interictal, and ictal EEG signals for both healthy and epileptic subjects were depicted in Figs. 5.18, 5.19, 5.20, and 5.21. The performances of the denoising process using wavelet decomposition are analyzed in Figs. 5.22, 5.23, 5.24, and 5.25 for EEG signals of all categories. In the second step, GFD values of the original EEGs and denoised EEGs were computed. The probability distribution of representative clinical EEG segments from each of the four data sets taken from the healthy subjects and the epileptic patients during interictal and ictal periods were obtained; and the corresponding GFD for q varying from 2 to 100 were computed. Likewise, the probability distribution of the corresponding denoised signals for all representative normal, interictal, and ictal EEG epochs from both healthy and epileptic subjects were calculated; and the corresponding GFD for q varying from 2 to 100 were computed. For instance, Generalized Fractal Spectra of sample EEG segments and their corresponding denoised EEG segments were depicted in Figs. 5.26 and 5.27.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

103

(a) Normal EEG (Eyes Open)

(b) Normal EEG (Eyes Closed))

(c) Interictal EEG (Opposite)

(d) Interictal EEG (Within)

(e) Ictal EEG

Fig. 5.12 Normal probability plots for normal, interictal, and ictal EEGs

104

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.13 Wavelet decomposition of EEG signals at level 3

Fig. 5.14 Wavelet decomposition of a sample healthy EEG

Further, the differences in the GFD values of each q for the Normal and Interictal EEGs with the Epileptic Ictal EEG in both Original and Denoised cases are tabulated in Tables 5.8 and 5.9, respectively, as well as illustrated graphically in Fig. 5.28 to analyze the Denoising Process. In order to test the mean differences among normal, interictal, and ictal EEG segments statistically, a repeated measure one-way Analysis of Variance (ANOVA) is performed on Generalized Fractal Spectra of the original EEGs and their corresponding denoised EEG segments from each of the four data sets, using a standard tool of statistical analysis (MATLAB Statistical Tool Box), as shown in Table 5.10.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

Fig. 5.15 Wavelet decomposition of a sample interictal (opposite to epileptogenic zone) EEG

Fig. 5.16 Wavelet decomposition of a sample interictal (within epileptogenic zone) EEG

105

106

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.17 Wavelet decomposition of a sample epileptic ictal EEG

Fig. 5.18 Original and denoised signals of a sample healthy EEG

Figures 5.18, 5.19, 5.20, and 5.21 show that the original signals are denoised in a better manner by comparing the original EEG signals with the denoised EEG signals. Also, Figs. 5.22, 5.23, 5.24, and 5.25 exhibit the performance of the denoising process in normal, interictal, and ictal EEG signals. In these figures, the denoised signal is superimposed on the original signal in order to reveal that the denoising performance rate is good in all categories of EEGs. The noises are removed increasingly at each level of wavelet decomposition of EEG signals. This is proved in Figs. 5.22, 5.23, 5.24, and 5.25 by showing the differences between the original coefficients and the automatic thresholded coefficients of decomposition. In these figures, the denoised

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

107

Fig. 5.19 Original and denoised signals of a sample interictal (opposite to epileptogenic zone) EEG

Fig. 5.20 Original and denoised signals of a sample interictal (within epileptogenic zone) EEG

Fig. 5.21 Original and denoised signals of a sample epileptic ictal EEG

108

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.22 Performance of the denoising process for a sample healthy EEG. In the upper figure, denoised signal (yellow color) is superimposed on the original signal (red color). In the middle figure, original coefficients are represented at three levels. In the lower figure, thresholded coefficients after denoising are represented at three levels

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification Fig. 5.23 Performance of the denoising process for a sample interictal (opposite to epileptogenic zone) EEG. In the upper figure, denoised signal (yellow color) is superimposed on the original signal (red color). In the middle figure, original coefficients are represented at three levels. In the lower figure, thresholded coefficients after denoising are represented at three levels

109

110

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.24 Performance of the denoising process for a sample interictal (within epileptogenic zone) EEG. In the upper figure, denoised signal (yellow color) is superimposed on the original signal (red color). In the middle figure, original coefficients are represented at three levels. In the lower figure, thresholded coefficients after denoising are represented at three levels

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification Fig. 5.25 Performance of the denoising process for a sample epileptic ictal EEG. In the upper figure, denoised signal (yellow color) is superimposed on the original signal (red color). In the middle figure, original coefficients are represented at three levels. In the lower figure, thresholded coefficients after denoising are represented at three levels

111

112

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.26 Generalized fractal spectra of normal, interictal, and ictal original EEGs

Fig. 5.27 Generalized fractal spectra of normal, interictal, and ictal denoised EEGs

signal in yellow color is superimposed on the original signal in red color. These figures clearly show that the sharpened spikes in original signals are removed and they become smoothened in the denoised signals while compared with the original signals for all four categories of EEGs. Also, the denoised thresholded coefficients show that the denoising performance is gradually increased at each level of decomposition when compared to the original coefficients for all four categories of EEGs. To prove that the above wavelet-based denoising technique plays an important role in the classification of healthy and epileptic EEG signals, now, we analyze GFD Spectra for original and denoised EEGs. Figure 5.27 shows that, as q increases, there is a significant difference in the healthy, interictal, and epileptic EEGs among the values of curves plotted for the denoised EEGs than the original EEG for normal, interictal, and ictal cases as in Fig. 5.26. For all the representative EEG segments we observed that, as q increases, the values of Dq for normal, interictal, and ictal EEGs would coincide in the original EEGs, as shown in Fig. 5.26. Specifically, the GFD curve for healthy and ictal EEGs are mingled in Fig. 5.26. But in denoised EEGs, there are significant differences among the values of Dq for normal, interictal, and ictal cases even for high values of q, as in Fig. 5.27. Particularly, there are significant differences in interictal and ictal EEGs taken from epileptic patients.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

113

Table 5.8 Analysis of the denoising process by the differences in GFD values for original EEG signals q Dq (Set A) − Dq (Set E) Dq (Set C) − Dq (Set E) Dq (Set D) − Dq (Set E) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80

0.9981300 0.1841500 0.0415180 −0.0022661 −0.0156180 −0.0181960 −0.0170990 −0.0149900 −0.0128330 −0.0109400 −0.0093798 −0.0081305 −0.0071457 −0.0063756 −0.0057761 −0.0053102 −0.0049484 −0.0046671 −0.0044482 −0.0042774 −0.0041440 −0.0040395 −0.0039574 −0.0038926 −0.0038414 −0.0038008 −0.0037684 −0.0037425 −0.0037217 −0.0037049 −0.0036912 −0.0036801 −0.0036710 −0.0036635 −0.0036573 −0.0036520 −0.0036476 −0.0036439 −0.0036407 −0.0036380

0.31616 0.15502 0.12296 0.11424 0.11271 0.11352 0.11496 0.11642 0.11769 0.11872 0.11954 0.12016 0.12061 0.12095 0.12118 0.12133 0.12143 0.12148 0.12149 0.12148 0.12146 0.12142 0.12137 0.12132 0.12126 0.12120 0.12114 0.12108 0.12103 0.12097 0.12091 0.12086 0.12081 0.12076 0.12072 0.12067 0.12063 0.12059 0.12055 0.12051

0.515990 0.151410 0.087214 0.063650 0.052208 0.045643 0.041415 0.038443 0.036208 0.034435 0.032967 0.031714 0.030619 0.029646 0.028771 0.027979 0.027258 0.026599 0.025994 0.025439 0.024929 0.024458 0.024024 0.023623 0.023252 0.022909 0.022591 0.022295 0.022020 0.021763 0.021524 0.021301 0.021092 0.020896 0.020712 0.020539 0.020377 0.020223 0.020079 0.019942 (continued)

114

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Table 5.8 (continued) q 82 84 86 88 90 92 94 96 98 100

Dq (Set A) − Dq (Set E)

Dq (Set C) − Dq (Set E)

Dq (Set D) − Dq (Set E)

−0.0036356 −0.0036335 −0.0036317 −0.0036301 −0.0036286 −0.0036273 −0.0036261 −0.0036250 −0.0036240 −0.0036231

0.12047 0.12044 0.12041 0.12037 0.12034 0.12032 0.12029 0.12026 0.12024 0.12021

0.019812 0.019690 0.019573 0.019463 0.019358 0.019257 0.019162 0.019071 0.018984 0.018900

Table 5.9 Analysis of the denoising process by the differences in GFD values for denoised EEG signals q Dq (Set A) − Dq (Set Dq (Set C) − Dq (Set Dq (Set D) − Dq (Set E) E) E) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

1.63550 0.41581 0.25321 0.22003 0.21521 0.21668 0.21912 0.22132 0.22308 0.22444 0.22549 0.22629 0.22691 0.22740 0.22778 0.22807 0.22830

0.33945 0.19579 0.16339 0.15181 0.14744 0.14605 0.14596 0.14643 0.14710 0.14781 0.14849 0.14911 0.14966 0.15013 0.15054 0.15089 0.15119

0.53054 0.20084 0.15166 0.13951 0.13733 0.13826 0.14002 0.14185 0.14350 0.14490 0.14606 0.14702 0.14781 0.14845 0.14899 0.14943 0.14979 (continued)

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification Table 5.9 (continued) q 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

115

Dq (Set A) − Dq (Set E)

Dq (Set C) − Dq (Set E)

Dq (Set D) − Dq (Set E)

0.22849 0.22863 0.22874 0.22882 0.22888 0.22893 0.22896 0.22898 0.22899 0.22900 0.22899 0.22898 0.22897 0.22895 0.22893 0.22891 0.22889 0.22886 0.22883 0.22881 0.22878 0.22875 0.22872 0.22869 0.22866 0.22863 0.22860 0.22857 0.22854 0.22851 0.22848 0.22845 0.22842

0.15145 0.15166 0.15185 0.15201 0.15214 0.15225 0.15235 0.15242 0.15249 0.15255 0.15259 0.15263 0.15266 0.15268 0.15270 0.15272 0.15273 0.15273 0.15274 0.15274 0.15274 0.15273 0.15273 0.15272 0.15271 0.15270 0.15269 0.15268 0.15267 0.15266 0.15265 0.15263 0.15262

0.15010 0.15035 0.15056 0.15074 0.15088 0.15101 0.15111 0.15120 0.15127 0.15133 0.15138 0.15142 0.15145 0.15147 0.15149 0.15151 0.15152 0.15153 0.15153 0.15153 0.15153 0.15153 0.15152 0.15152 0.15151 0.15150 0.15149 0.15148 0.15147 0.15146 0.15145 0.15143 0.15142

116

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

Fig. 5.28 Graphical analysis of the denoising process by the differences in GFD values for original and denoised EEG signals

Tables 5.8 and 5.9, and Fig. 5.28 exhibit clearly about the denoising performance rate in the discrimination among the normal, interictal, and epileptic EEGs by the analysis in GFD values for original and denoised EEG signals. In Table 5.9, the differences in GFD values of the normal and interictal EEGs with the epileptic ictal EEG for the denoised case are very high when compared to the original cases as in Table 5.8, as illustrated graphically in Fig. 5.28. The differences are demonstrated apparently in Fig. 5.28. The ANOVA test supports the same observations statistically. The p-value in Table 5.10(a) is greater than the p-values in Table 5.10(b). Hence, the values corresponding to normal, interictal, and ictal EEGs differed significantly in the denoised EEGs than in the original EEGs. From these observations, significant differences were found between the GFD measures for denoised signals of the four EEG data sets, and these differences allowed us to detect seizure activity of Epileptic patients with high accuracy. Without denoising by DWT as a preprocessing step, it is shown that the detection rate was significantly reduced.

5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification

117

Table 5.10 One-way ANOVA tables for generalized fractal spectra of the original EEGs and their corresponding denoised EEGs. SS—sum of squares, df—degrees of freedom, MS—mean square, and F—F-ratio (a) Original EEG signals ANOVA table Source SS df MS F Prob>F Columns 0.92121 Error 8.13243 Total 9.05364 (b) Denoised EEG signals ANOVA table

3 392 395

0.30707 0.02075

14.8

3.78799e-009

Source

SS

df

MS

F

Prob>F

Columns Error Total

3.1641 10.0231 13.1872

3 392 395

1.05469 0.02557

41.25

0

Hence, Figs. 5.22, 5.23, 5.24, 5.25, 5.26, 5.27 and 5.28, and Tables 5.8, 5.9, and 5.10 are the evidence that the wavelet denoising and GFD play an efficient role in the classification of healthy and epileptic EEGs.

5.6 Concluding Remarks In this chapter, we have designed the Modified form, Improved form, and Advanced form of GFD, which were developed from the concept of GFD for the recognition of Seizure-free (Normal) and Epileptic (Interictal and Ictal) EEGs. We have calculated Fractal Spectra (Dq , M Dq , I Dq , and ADq ), the Range of Fractal Spectra (RG F D , R M G F D , R I G F D , and R AG F D ), and the values of R E N , M R E N , I R E N , and A R E N for the Normal, Interictal, and Ictal EEGs using GFD, MGFD, IGFD, and AGFD Methods. Finally, we have compared Fractal Spectra, the Range of Fractal Spectra, and the values of R E N , M R E N , I R E N , and A R E N against log(r ) for Normal, Interictal, and Ictal EEGs to distinguish the Normal, Interictal, and Ictal EEGs through the graphical methods. From these observations, we have to make a decision that our designed methods are well organized when compared with the GFD Method. Mainly Advanced form of GFD is a more reliable method among all of our methods in the analysis of Epileptic EEG signals. The statistical tool namely, ANOVA test illustrated that there are significant differences in the values corresponding to the Normal, Interictal, and Ictal EEGs in our proposed methods than the GFD Method. In addition to that, we have designed a technique based on wavelet denoising to improve the preprocessing of EEG signals and studied the classification of healthy

118

5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification

and epileptic EEG signals by using multifractal measures such as Generalized Fractal Dimensions. Wavelet decomposition of EEG into sub-bands through DWT proved that approximation and detail coefficients provided the best classification rates by graphically and statistically using GFD as a classification measure. It is shown that the denoising process enhances the differences among the GFD values for normal, interictal, and epileptic ictal EEGs. EEG signal recordings taken from both healthy and epileptic patients during interictal and ictal periods showed that wavelet denoising and the GFD classification method are very accurate by graphical and statistical methods. Also, the Normal Probability Plot shows that the Epileptic EEG Data has significant nonlinearity but the normal and interictal EEGs are dominated by the Gaussian Linear Process.

Chapter 6

Fuzzy Multifractal Analysis in ECG Signal Classification

6.1 Introduction The fuzzy-based multifractal theory for signals is developed in order to define Fuzzy Generalized Fractal Dimensions by introducing the fuzzy membership function in the classical GFD method, and it is used for the classification of chaotic behaviors in the fractal waveforms. In this chapter, we have also proposed a fuzzy multifractal measure for biomedical signals to identify the age-group of subjects. The proposed method discriminates the young and elderly subjects by applying Fuzzy Generalized Fractal Dimensions with the Gaussian fuzzy membership function in ECG Signals through the cardiac inter-heartbeat time interval dynamics.

6.2 Fuzzy Multifractal Analysis for Fractal Signals In this section, we develop a fuzzy multifractal theory for defining the Fuzzy GFD (FGFD) by using the fuzzy membership function in the classical GFD method in order to analyze and classify the Weierstrass curves.

6.2.1 Fuzzy Renyi Entropy Similar to the definition of the Renyi entropy, we define the Fuzzy Renyi Entropy of order q, where q ≥ 0 and q = 1, on the given set S as ⎛ ⎛ ⎞q ⎞ N   1 ⎝ F R Eq = μ(x)⎠ ⎠ log2 ⎝ 1−q i=1 x∈S

(6.1)

i

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3_6

119

120

6 Fuzzy Multifractal Analysis in ECG Signal Classification

where μ : S −→ [0, 1] is the fuzzy membership function on the set S with N Partitions S1 , S2 , . . . , S N .

6.2.1.1

Some Particular Cases of Fuzzy Renyi Entropy

• If q = 0, then

F R E 0 = log2 N

which is called the Fuzzy Hartley entropy of the given fuzzy membership function. • Note that if q approaches 1, it can be shown that F R E q converges to F R E 1 , which is defined as ⎞⎞ ⎛ ⎛ N    ⎝ F R E1 = − μ(x) log2 ⎝ μ(x)⎠⎠ i=1

x∈Si

x∈Si

which is called the Fuzzy Shannon entropy of the given fuzzy membership function.

6.2.2 Fuzzy Generalized Fractal Dimensions for Signals Here, we introduce Fuzzy Generalized Fractal Dimensions (FGFD) for quantifying the fuzziness of multifractal signals. Now we define Fuzzy Generalized Fractal Dimensions for fractal waveforms by the following construction. The total range of the signal S is divided into N intervals (bins) S1 , S2 , . . . , S N such that Vmax − Vmin N= r where Vmax and Vmin are the maximum and the minimum values of the signal received in the experiments, respectively; and r is the uncertainty factor that may be depend on the measuring device used to record the signals. Define a fuzzy membership function, μ : S −→ [0, 1], on the set of signal values S, partitioned by N intervals S1 , S2 , . . . , S N . Then, the Fuzzy Renyi Fractal Dimensions or Fuzzy Generalized Fractal Dimensions (FGFD) of order q ∈ (−∞, ∞) such that q = 1 for the given fuzzy membership function on the set of signal values S, denoted by F Dq , can be defined as

F Dq = lim

r →0

1 q −1

log2



N i=1

 x∈Si

log2 r

μ(x)

q  .

Here, F Dq is defined in terms of the generalized Fuzzy Renyi Entropy.

(6.2)

6.2 Fuzzy Multifractal Analysis for Fractal Signals

6.2.2.1

121

Some Special Cases of Fuzzy Generalized Fractal Dimensions

1. If q = 0, then F D0 = −

log2 N log2 r

which is nothing but the Fractal Dimension. 2. As q −→ 1, F Dq converges to F D1 , which is given by N   F D1 = lim

i=1

x∈Si

μ(x) log2

r →0

 x∈Si

log2 r

μ(x)

 .

This is called as the Fuzzy Information Dimension. 3. If q = 2, then F Dq is called the Fuzzy Correlation Dimension. 4. If a fuzzy membership function, μ : S −→ [0, 1], on the set of signal values S partitioned by N intervals S1 , S2 , . . . , S N , is defined as μ(x) = 1/N then Fuzzy Generalized Fractal Dimensions are the same as classical Fuzzy Generalized Fractal Dimensions, i.e., F Dq = Dq . In this case, classical GFD is a particular case of the Fuzzy GFD.

6.2.2.2

Gaussian Fuzzy Membership Function

The symmetric Gaussian fuzzy membership function, g : S −→ [0, 1], on S depends on two parameters namely, mean (x) and standard deviation (σ ) as given by g(x; x, σ ) = e

−(x−x)2 2σ 2

.

(6.3)

6.3 Fuzzy Generalized Fractal Dimensions for Deterministic Fractal Waveforms In this section, simulations are carried out in MATLAB software to analyze the irregularity of fractal waveforms through fuzzy GFD by using the synthetic Weierstrass cosine signals with different chaotic nature, which are described in Sect. 5.2.1. The Weierstrass cosine curves with s = 1.1, 1.3, 1.5, 1.7, and 1.9, as described in Sect. 5.2.1 of this chapter, were taken in this study to analyze the efficiency of the developed fuzzy multifractal theory and Fuzzy GFD. The probability distributions of each of the five representative synthetic Weierstrass cosine waveforms were obtained

122

6 Fuzzy Multifractal Analysis in ECG Signal Classification

Fig. 6.1 Graph of Gaussian fuzzy membership function for Weierstrass cosine waveforms

Fig. 6.2 Generalized fractal spectra for Weierstrass cosine waveforms

and the corresponding Generalized Fractal Dimensions were also computed for the obtained probability distribution. The fuzzy membership values of each of the five representative signals were obtained by using the Gaussian fuzzy membership function (g). Then, Fuzzy Generalized Fractal Dimensions for all five representative waveforms were determined by using obtained Gaussian fuzzy membership values. The graph of the Gaussian fuzzy membership function for all five representative Weierstrass cosine waveforms with s = 1.1, 1.3, 1.5, 1.7, and 1.9 are depicted in Fig. 6.1. The computed classical GFD values are plotted against their order value q which varies from 2 to 100 for all five representative waveforms as in Fig. 6.2. Similarly, the determined Fuzzy GFD values are plotted against their order value q which varies from 2 to 100 for all waveforms as shown in Fig. 6.3. In addition to that, box plots of GFD values and Fuzzy GFD values are plotted for all five representative Weierstrass cosine waveforms with s = 1.1, 1.3, 1.5, 1.7, and 1.9 and are demonstrated, respectively, in Figs. 6.4 and 6.5.

6.3 Fuzzy Generalized Fractal Dimensions for Deterministic Fractal Waveforms Fig. 6.3 Fuzzy generalized fractal spectra for Weierstrass cosine waveforms

Fig. 6.4 Box plot of GFD for Weierstrass cosine waveforms

Fig. 6.5 Box plot of FGFD for Weierstrass cosine waveforms

123

124

6 Fuzzy Multifractal Analysis in ECG Signal Classification

Figure 6.1 shows that if the complexity of the Weierstrass cosine function increases, i.e., if s increases, then there is an increase in the variations of the values of their corresponding fuzzy membership function. The Weierstrass cosine function with s = 1.9 has a higher variation of membership values than the Weierstrass waveforms with s = 1.1, 1.3, 1.5, and 1.7, as demonstrated in Fig. 6.1. In Fig. 6.2, the GFD spectra of Weierstrass cosine functions with s = 1.1, 1.3, 1.5, 1.7, and 1.9 are not classified correctly in the classical multifractal dimension theory. In Fig. 6.3, the Fuzzy GFD spectra of Weierstrass cosine functions with s = 1.1, 1.3, 1.5, 1.7, and 1.9 are accurately classified by using the fuzzy multifractal dimension theory. The Fuzzy GFD values of Weierstrass cosine functions increase accordingly as s increases from 1.1 to 1.9, i.e., the Fuzzy GFD values increase gradually as the complexity of the waveforms increases. But the same could not happen in the classical GFD method as clearly shown in Fig. 6.2. Especially, the values of Fuzzy GFD for the Weierstrass cosine function with s = 1.9 are significantly greater than the values of Fuzzy GFD for the waveform with s = 1.1, 1.3, 1.5, and 1.7. But in the GFD case, the GFD values for Weierstrass cosine function with s = 1.9 are less than the values of GFD for s = 1.1 and 1.3. In addition to that, the box plots for both GFD and Fuzzy GFD methods support that the Fuzzy GFD accurately classifies the Weierstrass waveforms with different range of complexities. Hence, Figs. 6.1, 6.2, 6.3, 6.4, and 6.5 are the evidence that the fuzzy multifractal analysis plays an efficient role in the quantification of the complexity of chaotic waveforms by using Fuzzy Generalized Fractal Dimensions than the classical GFD method. Especially, the special case 4 in Sect. 6.2.2 shows that Fuzzy GFD is a generalized version of the classical GFD by choosing the suitable fuzzy membership function.

6.4 Experimental ECG Data Two groups of five young (21–34 years old) and five elderly (68–81 years old) rigorously screened healthy subjects underwent 120 min of continuous supine resting while continuous electrocardiogram (ECG) signals were collected. All the subjects remained in a resting state in sinus rhythm while watching a particular movie to help maintain wakefulness. Each subgroup of subjects includes equal numbers of men and women. The continuous ECG was digitized at 250 Hz. Each heartbeat was annotated using an automated arrhythmia detection algorithm, and each beat annotation was verified by visual inspection. The R-R interval (inter-heartbeat interval) time series for each subject was then computed [72]. A sample of ECG epochs from each of the two age-groups are plotted in Fig. 6.6.

6.5 Fuzzy Generalized Fractal Dimensions for Clinical ECG Signals

125

Fig. 6.6 Sample ECG epochs from elderly and young healthy subjects

6.5 Fuzzy Generalized Fractal Dimensions for Clinical ECG Signals Simulations are carried out in MATLAB software using the ECG Signals described in Sect. 6.4. The representative biomedical ECG Data were taken in this study to analyze the efficiency of the developed fuzzy multifractal measure. The probability distributions of each of the R-R interval time series of clinical ECG Data were obtained and the corresponding Generalized Fractal Dimensions were also computed for the obtained probability distribution. Likewise, the membership values of each of the representative time series were obtained by using the Gaussian fuzzy membership function (g). Then, Fuzzy Generalized Fractal Dimensions for all representative waveforms were determined by using obtained Gaussian membership values. The computed classical GFD values are plotted against their order values q which varies from 2 to 100 for all representative interval time series as in Fig. 6.7. Similarly, the determined Fuzzy GFD values are plotted against their order values q which varies from 2 to 100 for all time series as shown in Fig. 6.8. In order to test the mean differences among Elderly and Young ECG time series statistically, ANOVA Tables and Box Plots of GFD values and Fuzzy GFD values are constructed for all R-R interval time series of clinical ECG Data, and are demonstrated in Table 6.1 and Figs. 6.9 and 6.10. In Fig. 6.8, the Fuzzy GFD spectra of the R-R interval time series significantly classified the Elderly and Young Subjects than the GFD spectra of the R-R interval time series as in Fig. 6.7. In Table 6.1, ANOVA Test also supports our designed methods statistically than the GFD method. The box plots for the Fuzzy GFD method in Fig. 6.10 show the differences in mean variance than the box plots for the GFD method as in Fig. 6.9. Thus, Figs. 6.7, 6.8, 6.9, and 6.10 and Table 6.1 are the evidences that the fuzzy multifractal analysis plays an efficient role in the age-related discrimination by using Fuzzy Generalized Fractal Dimensions than the classical GFD method.

126

6 Fuzzy Multifractal Analysis in ECG Signal Classification

Fig. 6.7 Generalized fractal dimension spectra for R-R interval time series of elderly and young subjects

Fig. 6.8 Fuzzy generalized fractal dimensions spectra for R-R interval time series of elderly and young subjects

6.6 Concluding Remarks

127

Table 6.1 One-way ANOVA tables for GFD and fuzzy GFD spectrum of elderly and young subjects (a) GFD method Source SS df MS F Prob>F Columns 0.09191 Error 1.3199 Total 1.41182 (b) Fuzzy GFD method

1 36 37

0.09191 0.3666

2.51

0.1221

Source

SS

df

MS

F

Prob>F

Columns Error Total

0.93736 4.95819 5.89555

1 36 37

0.93736 0.13773

6.81

0.0131

Fig. 6.9 Notched box plots for GFD spectra of elderly and young subjects

6.6 Concluding Remarks In this chapter, we have developed a fuzzy multifractal theory in order to define Fuzzy Generalized Fractal Dimensions (FGFD) by introducing the fuzzy membership function in the classical the Generalized Fractal Dimensions method. It is shown that the defined Fuzzy GFD method accurately classifies the complexity of the chaotic waveforms such as Weierstrass functions by comparing graphically with the classical GFD method. Hence, the fuzzy multifractal analysis performs specifically than the classical multifractal analysis. Also, the defined Fuzzy GFD is a generalized method of the classical GFD. In this study, we have developed a fuzzy multifractal theory in order to discriminate the two age-groups (young and elderly subjects) by using Fuzzy Generalized Fractal Dimensions with the Gaussian fuzzy membership function in ECG Signals through

128

6 Fuzzy Multifractal Analysis in ECG Signal Classification

Fig. 6.10 Notched box plots for fuzzy GFD spectra of elderly and young subjects

the cardiac inter-heartbeat time series. Further, the ANOVA test has been performed which statistically supported the classification rate between the two age-groups. Thus, the simulation results show that the performance of fuzzy multifractal analysis is significant when compared to the classical multifractal analysis pertaining to agerelated problems.

References

1. Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1983) 2. Barnsley, M.F.: Fractals Everywhere. Academic, Dublin (1988). (2nd edn, Morgan Kaufmann 1993; 3rd edn, Dover Publications, 2012) 3. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 313–747 (1981) 4. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(1), 303–329 (1986) 5. Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57(1), 14–34 (1989) 6. Barnsley, M.F., Hegland, M., Massopust, P.: Numerics and fractals. Bull. Inst. Math. 9(3), 389–430 (2014) 7. Barnsley, M.F., Hurd, L.P.: Fractal Image Compression. AK Peters Ltd., Wellesly (1993) 8. Massopust, P.R.: Fractal functions. Fractal Surfaces and Wavelets. Academic, Cambridge (1994) 9. Banerjee, S., Hassan, M.K., Mukherjee, S., Gowrisankar, A.: Fractal Patterns in Nonlinear Dynamics and Applications. CRC Press, Boca Raton (2019) 10. Barnsley, M.F., Elton, J., Hardin, D., Massopust, P.: Hidden variable fractal interpolation functions. SIAM J. Math. Anal. 20(5), 1218–1242 (1989) 11. Bouboulis, P., Dalla, L.: Hidden variable vector valued fractal interpolation functions. Fractals 13(3), 227–232 (2005) 12. Chand, A.K.B., Katiyar, S.K., Viswanathan, P.: Approximation using hidden variable fractal interpolation functions. J. Fractal Geom. 2(1), 81–114 (2015) 13. Tatom, F.B.: The relationship between fractional calculus and fractal. Fractals 3(1), 217–229 (1995) 14. Yao, K., Su, W.Y., Zhou, S.P.: On the connection between the order of fractional calculus and the dimensions of a fractal function. Chaos, Solitons Fractals 23, 621–629 (2005) 15. Ruan, H.-J., Su, W.-Y., Yao, K.: Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory 161, 187–197 (2009) 16. Liang, Y.S., Su, W.Y.: Fractal dimensions of fractional integral of continuous functions. Acta Math. Sin. 32(12), 1494–1508 (2016) 17. Xiao, E.W., Jun, H.D.: Box dimension of Hadamard fractional integral of continuous functions of bounded and unbounded variation. Fractals 25(3), 1750035 (2017) 18. Li, Y., Liang, Y.: Upper bound estimation of fractal dimension of fractional calculus of continuous functions. Adv. Anal. 2(2), 121–128 (2017) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Banerjee et al., Fractal Functions, Dimensions and Signal Analysis, Understanding Complex Systems, https://doi.org/10.1007/978-3-030-62672-3

129

130

References

19. Liang, Y.S., Zhang, Q.: A type of fractal interpolation functions and their fractional calculus. Fractals 24(2), 1650026 (2016) 20. Liang, Y.S.: Progress on estimation of fractal dimensions of fractional calculus of continuous functions. Fractals 27(05), 1950084 (2019) 21. Liang, Y.S.: Estimation of fractal dimension of Weyl fractional integral of certain continuous functions. Fractals 28(02), 2050030 (2020) 22. Gowrisankar, A., Uthayakumar, R.: Fractional calculus on fractal interpolation function for a sequence of data with countable iterated function system. Mediterr. J. Math. 13(6), 3887–3906 (2016) 23. Gowrisankar, A., Prasad, M.G.P.: Riemann-Liouville calculus on quadratic fractal interpolation function with variable scaling factors. J. Anal. 27(2), 347–363 (2019) 24. Liang, Y.S.: Estimation of fractal dimension of fractional calculus of the holder continuous functions. Fractals (2020). https://doi.org/10.1142/S0218348X20501236 25. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) 26. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Sydney (2006) 27. Ross, B.: Fractional Calculus and Its Applications. Springer, Berlin (1975) 28. Secelean, N.A.: Countable iterated function systems. Far East J. Dyn. Syst. 3(2), 149–167 (2001) 29. Secelean, N.A.: Approximation of the attractor of a countable iterated function system. Gen. Math. 17(3), 221–231 (2009) 30. Secelean, N.A.: Generalized F-iterated function systems on product of metric spaces. J. Fixed Point Theory Appl. 17, 575–595 (2015) 31. Secelean, N.A., Dariusz, W.: ψF-contractions: not necessarily nonexpansive picard operators. RM 70, 415–431 (2016) 32. Uthayakumar, R., Gowrisankar, A.: Fractals in product fuzzy metric space. Fractals, Wavelets, and their Applications. Springer Proceedings in Mathematics and Statistics, vol. 92, pp. 157– 164 (2014) 33. Uthayakumar, R., Gowrisankar, A., Attractor and self-similar group of generalized fuzzy contraction mapping in fuzzy metric space. Cogent Math. (Taylor & Francis) 2(1), 1024579, 1–12 (2015) 34. Secelean, N.A.: The existence of the attractor of countable iterated function systems. Mediterr. J. Math. 9, 61–79 (2012) 35. Easwaramoorthy, D., Uthayakumar, R.: Analysis on fractals in fuzzy metric spaces. Fractals 19(3), 379–386 (2011) 36. Uthayakumar, R., Easwaramoorthy, D.: Hutchinson-Barnsley operator in fuzzy metric spaces. Int. J. Math. Comput. Sci., World Acad. Sci., Eng. Techol 5(8), 1418–1422 (2011) 37. Gowrisankar, A., Easwaramoorthy, D.: Local countable iterated function systems. In: ICAMS2017 - Trends in Mathematics - Springer Book Series, vol. 1, pp. 169–175 (2018) 38. Uthayakumar, R., Gowrisankar, A.: Mid-sagittal plane detection in magnetic resonance image based on multifractal techniques. IET Image Proc. 10(10), 751–762 (2016) 39. Raja Mohan, C., Gowrisankar, A., Uthayakumar, R., Jayakumar, K.: Morphology dependent electrical property of chitosan film and modeling by fractal theory. Eur. Phys. J. Spec. Top. 228, 233–243 (2019) 40. Punam Kumari, P., Gowrisankar, A., Saha, A., Santo, B.: Dynamical properties and fractal patterns of nonlinear waves in solar wind plasma. Phys. Scr. 95(6), 065603 (2020) 41. Fataf, N.A.A., Gowrisankar, A., Santo, B.: In search of self-similar chaotic attractors based on fractal function with variable scaling. Phys. Scr. 95(7), 075206 (2020) 42. Navascues, M.A.: Fractal polynomial interpolation. Zeitschrift fíur Analysis und ihre Anwendungen 25(2), 401–418 (2005) 43. Navascues, M.A., Sebastian, M.V.: Smooth fractal interpolation. J. Inequalities Appl. 20, 78734 (2004) 44. Navascues, M.A.: Non-Smooth polynomial. Int. J. Math. Anal. 1(4), 159–174 (2007)

References

131

45. Chand, A.K.B., Vijender, N.: Monotonicity preserving rational quadratic fractal interpolation functions. Adv. Numer. Anal. 17, 504825 (2014) 46. Chand, A.K.B., Kapoor, G.P.: Hidden variable bivariate fractal interpolation surfaces. Fractals 11(3), 277–288 (2003) 47. Wang, H.Y., Yu, J.S.: Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–8 (2013) 48. Navascues, M.A.: A fractal approximation to periodicity. Fractals 14(4), 315–325 (2006) 49. Secelean, N.A.: The fractal interpolation for countable systems of data. Publikacije Elektrotehnickog Fakulteta - Serija: Matematika 14, 11–19 (2003) 50. Balasubramani, N., Guru Prem Prasad, M., Natesan, S.: Fractal quintic spline solutions for fourth-order boundary-value problems. Int. J. Appl. Comput. Math. 5, 130 (2019) 51. Balasubramani, N., Guru Prem Prasad, M., Natesan, S., Constrained and convex interpolation through rational cubic fractal interpolation surface. Comput. Appl. Math. 37, 6308–6331 (2018) 52. Balasubramani, N., Guru Prem Prasad, M., Natesan, S.: Fractal cubic spline methods for singular boundary-value problems. Int. J. Appl. Comput. Math. 6, 47 (2020) 53. Balasubramani, N.: Shape preserving rational cubic fractal interpolation function. J. Comput. Appl. Math. 319, 277–295 (2017) 54. Chand, A.K.B., Reddy, K.M.: Constrained fractal interpolation functions with variable scaling. Sib. Electron. Math. Rep. 15, 60–73 (2018) 55. Tsvetkov, V.P., Mikheyev, S.A., Tsvetkov, I.V.: Fractal phase space and fractal entropy of instantaneous cardiac rhythm. Chaos, Solitons Fractals 108, 71–76 (2018) 56. Luor, D.C.: Fractal interpolation functions with partial self similarity. J. Math. Anal. Appl. 464, 911–923 (2018) 57. Chand, A.K.B., Navascues, M.A., Viswanathan, P., Katiyar, S.K.: Fractal trigonometric polynomials for restricted range approximation. Fractals 24(2), 1650022 (2016) 58. Katiyar, S.K., Chand, A.K.B., Saravana Kumar, G.: A new class of rational cubic spline fractal interpolation function and its constrained aspects. Appl. Math. Comput. 346, 319–335 (2019) 59. Navascues, M.A., Katiyar, S.K., Chand, A.K.B.: Multivariate affine fractal interpolation. Fractals (2020). https://doi.org/10.1142/S0218348X20501364 60. Easwaramoorthy, D., Uthayakumar, R.: Estimating the complexity of biomedical signals by multifractal analysis. In: Proceedings of the IEEE Students’ Technology Symposium, pp. 6–11. IEEE Xplore Digital Library, IEEE, USA (2010) 61. Easwaramoorthy, D., Uthayakumar, R.: Analysis of EEG signals using advanced generalized fractal dimensions, Proceedings of the International Conference on Computing, Communication and Networking Technologies, pp. 1–6. IEEE Xplore Digital Library, IEEE, USA (2010) 62. Easwaramoorthy, D., Uthayakumar, R.: Analysis of biomedical EEG signals using wavelet transforms and multifractal analysis, Proceedings of the IEEE International Conference on Communication Control and Computing Technologies, pp. 544–549. IEEE Xplore Digital Library, IEEE, USA (2010) 63. Easwaramoorthy, D., Uthayakumar, R.: Improved generalized fractal dimensions in the discrimination between healthy and epileptic EEG signals. J. Comput. Sci. 2(1), 31–38 (2011) 64. Uthayakumar, R., Easwaramoorthy, D.: Multifractal-wavelet based denoising in the classification of healthy and epileptic EEG signals. Fluct. Noise Lett. 11(4), 1250034 (2012) 65. Uthayakumar, R., Easwaramoorthy, D.: Generalized fractal dimensions in the recognition of noise free images, Proceedings of the International Conference on Computing, Communication and Networking Technologies, pp. 1–5. IEEE Xplore Digital Library, IEEE, USA (2012) 66. Uthayakumar, R., Easwaramoorthy, D., Multifractal analysis indenoising of color images. In: Proceedings of the International Conference on Emerging Trends in Science, Engineering and Technology, pp. 228–234. IEEE Xplore Digital Library, IEEE, USA (2012) 67. Uthayakumar, R., Easwaramoorthy, D.: Epileptic seizure detection in EEG signals using multifractal analysis and wavelet transform. Fractals 21(2), 1350011 (2013) 68. Uthayakumar, R., Easwaramoorthy, D.: Fuzzy generalized fractal dimensions for chaotic waveforms. In: Chaos, Complexity and Leadership 2012, Springer Proceedings in Complexity, pp. 411–422 (2014)

132

References

69. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, England (2003) 70. Grassberger, P.: Generalized dimensions of strange attractors. Phys. Lett. A 97, 227–320 (1983) 71. Hentschel, H.G.E., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Physica 8D, 435–444 (1983) 72. Iyengar, N., Peng, C.K., Morin, R., Goldberger, A.L., Lipsitz, L.A.: Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am. J. Physiol.-Regul., Integr. Comp. Physiol. 271(4), 1078–1084 (1996) 73. Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics: Integrability. Chaos and Patterns. Springer, Heidelberg (2003) 74. Mario, S., Roberta, F., Alessandro, P., Carlo, S.: Electrocardiogram pattern recognition and analysis based on artificial neural networks and support vector machines: a review. J. Healthc. Eng. 4(4), 465–504 (2013) 75. Renyi, A.: On a new axiomatic theory of probability. Acta Math. Hung. 6, 285–335 (1955) 76. Rishi, R.S., Pachori, R.B.: Baseline wander and power line interference removal from ECG signals using eigenvalue decomposition. Biomed. Signal Process. Control 45, 33–49 (2018) 77. Shannon, C.E.: The Mathematical Theory of Communication. University of Illinois Press, Champaign (1998) 78. Andrzejak, R.G., Lehnertz, K., Mormann, F., Rieke, C., David, P., Elger, C.E.: Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state. Phys. Rev. E 64, 061907-1–061907-8 (2001) 79. EEG Signals, Department of Epileptology at the University Hospital of Bonn, Bonn, Germany. Website: http://www.epileptologie-bonn.de/ 80. Ocak, H.: Optimal classification of epileptic seizures in EEG using wavelet analysis and genetic algorithm. Sig. Process. 88, 1858–1867 (2008) 81. Ocak, H.: Automatic detection of epileptic seizures in EEG using discrete wavelet transform and approximate entropy. Expert Syst. Appl. 36, 2027–2036 (2009) 82. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989) 83. Hassan, M.K., Rahman, M.M.: Percolation on a multifractal scale-free planar stochastic lattice and its universality class. Phys. Rev. 92, 040101(R) (2015) 84. Dayeen, F.R., Hassan, M.K.: Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice. Chaos, Solitons Fractals 91, 228–234 (2016)