Metric Spaces of Fuzzy Sets: Theory and Applications

180 Pages·1994·10.782 MB·English

by  Phil Diamond, Peter Kloeden

METRIC SPACES
OF
FUZZY SETS
Theory and Applications
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METRIC SPACES
OF
FUZZY SETS
Theory and Applications
Phil Diamond
Department of Mathematics
The University of Queensland, Australia
Peter Kloeden
School of Computing and Mathematics
Deakin University, Australia
\ \ k W o r ld S c i e n t i f ic
w ir Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
METRIC SPACES OF FUZZY SETS: THEORY AND APPLICATIONS
Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by any means, electronic or mechanical, including photocopying, recording or any
information storage and retrieval system now known or to be invented, without
written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through
the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.
ISBN 981-02-1731-5
Printed in Singapore.
Preface
Fuzzy subsets of a given base space are characterized by a membership function from the
base space to the unit interval. Spaces of such fuzzy sets are thus function spaces, albeit
with specialised properties sometimes a little atypical of the common function spaces of
modern mathematics. Topological, in particular metric topological properties of spaces of
fuzzy sets have been researched extensively over the past two decades. This book is based
on many papers that have appeared in journals or conference proceedings of a diversity
of disciplines. At its core is our own work, particularly that characterising compactness
in metric spaces of fuzzy sets. The primary aim of the book is to provide a systematic
development of the theory of metric spaces of normal, upper semicontinuous fuzzy convex
fuzzy sets with compact support sets, mainly on the base space An additional aim is to
sketch selected applications in which these metric space results and methods are essential
for a thorough mathematical analysis.
Our book is distinctly mathematical in its orientation and style, in contrast with many of
the other books now available on fuzzy sets, which, although all making use of mathematical
formalism to some extent, are essentially motivated by and oriented towards more immediate
applications and related practical issues. The reader is assumed to have some previous
undergraduate level acquaintance with metric spaces and elementary functional analysis.
Chapter 1 is introductory, while the remaining chapters are organised into three parts.
Chapters 2 to 5 outline background material about metric spaces of nonempty compact and
nonempty compact convex subsets of lftn and Banach spaces, as well as the calculus of set
valued mappings of a real variable. Chapters 6 to 10 on the space of fuzzy sets £n and its
properties under various metrics, together with the calculus of fuzzy set valued functions
of a real variable, form the core of the book. Detailed mathematical proofs are presented
in these chapters. The final part of the book consisting of Chapters 11 to 15 introduces a
representative variety of applications of fuzzy sets and illustrates how the theory developed
in the earlier chapters can be used for their formulation and analysis. Each of the chapters
ends with bibliographical notes referring to an extensive biliography at the end of the book.
Although we have tried to include all apposite references to metric properties of fuzzy sets,
it is inevitable that some have been left out and we apologise in advance for these omissions.
An appendix summarising some basic metric space results has also been included at the
end of the book.
In writing this book we have received much encouragement, support and constructive
criticism from a large number of sources. In particular we thank C. Cabrelli, U. Molter, A.
Pokrovskii, E. Vrscay and H.-J. Zimmermann, and many others too numerous to name, as
well as each others institutions for their hospitality during a number of visits. The second
author also thanks the Mathematics Department of the University of Queensland for the
award of an Ethol Raybould Visiting Fellowship in September 1990, during which time
several chapters were written. They both thank Peter Adams for producing the figures in
v
vi
electronic format suitable for the text, and Martin Sharry for general advice when compiling
this book in lATfjjX. The first author is indebted to Marion Diamond for her encouragement
and support over many years, while the second expresses his gratitude to Karin Wahl for
continuing refuge in Tubingen.
Phil Diamond and Peter Kloederr
Brisbane and Geelong, August If
Contents
Preface v
1 Fuzzy Sets 1
1.1 Definition of a Fuzzy Set 1
1.2 Basic Operations on Fuzzy Sets 3
1.3 Bibliographical Notes 5
2 Spaces of Subsets of 9fcn 7
2.1 Introduction 7
2.2 Algebraic Operations on Subsets 7
2.3 The Hausdorff Metric 8
2.4 Compact Subsets of 3fcn 10
2.5 Bibliographical Notes 11
3 Compact Convex Subsets of 9fcn 13
3.1 Support Functions 13
3.2 Steiner Centroid and Parametrization 15
3.3 L Metrics 17
R
3.4 A Banach Space of Asymmetry Classes 18
3.5 Bibliographical Notes 20
4 Set Valued Mappings 21
4.1 Continuity and Measurability 21
4.2 Differentiation 25
4.3 Integration 29
4.4 Bibliographical Notes 32
5 Crisp Generalizations 33
5.1 Star Shaped Sets 33
5.2 Subsets of a Banach Space 34
5.3 Bibliographical Notes 35
6 The Space Sn 37
6.1 Definitions and Basic Properties . . . • 37
6.2 Useful Subsets of 2>n and £* 43
6.3 Parametrization by a Single Valued Mapping 44
vii
viii CONTENTS
6.4 The Bobylev Characterization of Fuzzy Sets 45
6.5 Bibliographical Notes 49
7 Metrics on Sn 51
7.1 Definitions and Basic Properties 51
7.2 Completeness 54
7.3 Separability 62
7.4 Convergence Relationships 64
7.5 Bibliographical Notes 70
8 Compactness Criteria 71
8.1 Introduction 71
8.2 Compact Subsets in (£* , <£«>) 73
8.3 Compact Subsets in (£n , dp) . 76
8.4 Bibliographical Notes . 79
9 Generalizations 81
9.1 Fuzzy Star Shaped Fuzzy Sets 81
9.2 Banach Base Space 85
9.3 Higher Order Fuzzy Sets 86
9.4 Bibliographical Notes 90
10 Fuzzy Set Valued Mappings of Real Variables 91
10.1 Continuity and Measurability 91
10.2 Differentiation 94
10.3 Integration 102
10.4 Bibliographical Notes 108
11 Fuzzy Random Variables 109
11.1 Definitions 109
11.2 Statistical Limit Theorems 110
11.3 Bibliographical Notes 112
12 Computational Methods 115
12.1 Estimation and Least Squares 115
12.2 Fuzzy Kriging 118
12.3 Interpolation and Splines 120
12.4 Bernstein Approximation , . . . . 124
12.5 Bibliographical Notes 126
13 Fuzzy Differential Equations 129
13.1 Introduction 129
13.2 Existence and Uniqueness of Solutions 131
13.3 Solutions as Fuzzy Dynamical Systems 133
13.4 Bibliographical Notes 136
CONTENTS ix
14 Optimization Under Uncertainty 137
14.1 Pozzy Constraints 137
14.2 Robust Kuhn-Tucker Conditions 139
14.3 Fuzzy Optimal Control 140
14.4 Bibliographical Notes 142
15 Fuzzy Iterations and Image Processing 143
15.1 Iterated Fuzzy Systems, Fractal Compression 143
15.2 Chaotic Fuzzy Mappings 148
15.3 Bibliographical Notes 153
Appendix on Metric Spaces 155
Bibliography 161
Symbols and Abbreviations 171
Index 173
Chapter 1
Fuzzy Sets
1.1 Definition of a Fuzzy Set
The idea of a fuzzy set is really quite simple. It was first proposed by Lotfi Zadeh,
an electrical engineer, in the 1960s as a means of handling uncertainty that is due to
imprecision or vagueness rather than to randomness. A typical example of what is
meant by this is a "set of real numbers much greater than l w .
Fuzzy sets are considered with respect to a nonempty base set X of elements of
interest. The essential idea is that each element x € X is assigned a membership
grade u(x) taking values in [0,1], with u(x) = 0 corresponding to non-membership,
0 < u(x) < 1 to partial membership, and u(x) = 1 to full membership. According to
Zadeh a fuzzy subset of X is a nonempty subset {(z, u(x)) : x £ X} of X x [0,1] for
some function u : X —• [0,1]. The function u itself is often used synonomously for
the fuzzy set.
For instance, the function u : R 1 —> [0,1] with
( 0 if x <l
^ ( x - l) if K x < 1 00 (1.1)
1 if 100 < x
provides an example of a fuzzy set of real numbers x ^> 1 (see Fig. 1). There are of
course many other reasonable choices of membership grade function.
1 100
Fig. 1.1. A fuzzy set of real numbers x > 1.
The only membership possibilities for an ordinary or crisp subset A of X are
non-membership and full membership. Such a set can thus be identified with the
1