Studies in Fuzziness and Soft Computing

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Studies in Fuzziness and Soft Computing

Volume 382

Series Editor

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The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and relatedfields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Contact the series editor by e-mail:[email protected]

Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

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Gleb Beliakov

•

Simon James

•

Jian-Zhang Wu

Discrete Fuzzy Measures

Computational Aspects

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Gleb Beliakov Deakin University Burwood, VIC, Australia

Simon James Deakin University Burwood, VIC, Australia Jian-Zhang Wu

Ningbo University Ningbo, Zhejiang, China

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing

ISBN 978-3-030-15304-5 ISBN 978-3-030-15305-2 (eBook)

https://doi.org/10.1007/978-3-030-15305-2

Library of Congress Control Number: 2019933708 © Springer Nature Switzerland AG 2020

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

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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, express 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.

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To

Gelui Patricia, Chaquen and So

ﬁa

Gleb Beliakov

Rachel

Simon James

Fang Chen and Melinda

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Preface

This book is about computational aspects of fuzzy measures—nonadditive mea-sures deﬁned on discrete sets X. Our main focus is on multicriteria decision making and the aggregation of inputs, two areas where a ﬁnite set of inputs need to be combined into an overall representative value. In contrast to many alternative ways of aggregating inputs, such as weighted means, aggregation based on fuzzy mea-sures allows one to incorporate mutual dependency of the inputs—their redundancy and complementarity. This makes fuzzy measures a valuable tool for modelling systems where the inputs such as decision criteria are correlated.

Fuzzy measures are known under different names, in particular, as capacities and (normed) cooperative games. They constitute a special class of set functions (that is, functions deﬁned on all subsets of a given set, called the power set, denoted by 2X ¼ PðXÞ), which are characterised by monotonicity (with respect to set inclu-sion) and normalisation conditions. We shall use the terms fuzzy measures and capacities interchangeably in this book. Many results presented here are also applicable to broader classes of set functions, games in particular, but we set them in the context of our main object of study—capacities.

What makes fuzzy measures so valuable is their ability to model the various ways inputs can interact, by assigning importance weights not just to individual inputs, but to all coalitionsC. Thus, an input may be unimportant individually but gain importance in the presence of other inputs, and vice versa. The central notion of monotonicity has important semantics: increasing the value of any criterion (e.g. utility, preference) cannot decrease the total aggregate value.

The flexibility of fuzzy measures when modelling interaction comes at a sig-niﬁcant cost: the exponential number of coalitions whose contributions need to be quantiﬁed. This gives rise to two problems: their interpretation and elicitation. If a fuzzy measure based model is to be understood by domain experts, the large number of capacity values need to be combined into some sort of characteristic indices, such as the overall importance of an input in all coalitions, or the overall interaction of a pair of inputs. On the other hand, if a fuzzy measure is to be

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speciﬁed, either by the experts or by machine learning techniques, it has to be done through a few desirability criteria and in a computationally efﬁcient way.

Both problems are addressed in this book. After presenting a broad introduction to the area of aggregation in Chap. 1, and deﬁning there the main mathematical concepts, we proceed to discussion of the main types of fuzzy measures in Chap.2. The problem of interpretation of fuzzy measures by means of several summative indices is addressed in Chap.3. Here, we present some classical concepts such as the Shapley value and probabilistic interaction indices, as well as more recent developments related to nonadditivity and nonmodularity. Chapter4is devoted to many alternative ways of representing fuzzy measures, in particular, through Möbius and interaction representations, matrix-vector representation and marginal contributions representation. This chapter also deals with linear transforms, con-veniently represented in matrix form.

Chapter5discusses different fuzzy integrals, also called nonlinear or nonadditive integrals, with respect to fuzzy measures. The Choquet integral plays a central role here; however, a broad collection of alternative integrals that have been developed in recent times are also presented in detail.

The following two chapters deal with various simplifications which reduce the large number of parameters that characterise fuzzy measures. In Chap. 6, we consider symmetric fuzzy measures. In this case, the Choquet integral becomes the popular Ordered Weighted Averaging function (OWA). This chapter discusses many types of OWA and special techniques for learning OWA weights from empirical data and other specifications. The Sugeno integral with respect to sym-metric fuzzy measures also coincides with a special class of functions called the ordered weighted maximum and minimum. In Chap.7, we present a range of other simplification strategies called, collectively, k-order fuzzy measures. Here, the interaction among the inputs (in one sense or another) is limited to coalitions of smaller cardinalities (up to k elements). This technique reduces the number of parameters to be specified or learned, and sometimes reduces the number of monotonicity constraints. The latter is crucial for the development of efficient computational algorithms.

The last, but very important, chapter in this book is Chap. 8. Here, we deal directly with many computational aspects of fuzzy measures, informed by the previous chapters. The problem of learning fuzzy measures from observed or desired data is discussed and translated into optimisation problems. In particular, due to very large numbers of monotonicity and other constraints, we prefer for-mulation of the learning problem as a linear programming problem. In this setting, we make use of efficient numerical methods, which handle large and sparse matrices of constraints. Still, larger numbers of decision criteria require simpli fi-cation strategies, and we present learning methods based on k-order simplifications. Many of the computational methods mentioned in this book have been imple-mented and available as software libraries. We mention here the RFMTool and Kappalab packages for R, and WOWA and AOTools packages in C++. It is hoped that efficient implementations of fuzzy measure algorithms, in particular, the

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associated learning problems, together with this text, will facilitate wider spread of sophisticated and powerful aggregation methods based on the theory of capacities. This book is oriented towards computational aspects, and therefore we present many results as statements of facts and refer to alternative sources for proofs. We should mention a few recent monographs which this book nicely complements, and where many more theoretical aspects or alternative aggregation methods are pre-sented. First is the book by M. Grabisch, Set Functions, Games and Capacities in Decision Making, 2016 (see the detailed references in the main text), where many theoretical aspects are addressed. Further theory and results pertaining to fuzzy integrals and fuzzy measures are also found in V. Torra, Y. Narukawa and M. Sugeno’s Non-Additive Measures: Theory and Applications, 2014, and Z. Wang and G. J. Klir’s Generalized Measure Theory, 2009. The book by G. Beliakov, H. Bustince and T. Calvo, A Practical Guide to Averaging Functions, 2016, pre-sents many alternative averaging functions from classical means to penalty-based functions and aggregation on lattices. The book by M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap, Aggregation Functions, 2009, also covers many distinct aspects of aggregation, including fuzzy integrals, and is highly recommended. Another two books are by S. James, An Introduction to Data Analysis using Aggregation Functions in R, 2016 and M. Gagolewski, Data Fusion: Theory, Methods, and Applications, 2015, which address applications and multidimensional aggregation in particular. Finally, the recent book by J. Dujmovic, Soft Computing Evaluation Logic: The LSP Decision Method and Its Applications, 2018, also offers a different perspective on aggregation in engineering problems.

G. Beliakov wishes to acknowledge support from his family he enjoyed during preparation of this book, as well as fruitful collaboration and friendship with his co-authors Simon and Wu.

S. James would like to acknowledge the support and friendship of colleagues during his 2018 academic study programme—in particular, Marek Gagolewski, Aoi Honda and Luigi Troiano, along with their institutions. Radko Mesiar and Michel Grabisch also have both played a big role in terms of inspiring his interest in fuzzy measures and fuzzy integrals and so their time and guidance (through works or otherwise) has always been appreciated. The support of friends and family, espe-cially Bronwen, Nathan and Rachel, should of course also be acknowledged when it comes to large projects like this, which may sometimes dominate his time and attention. Lastly, his co-authors, Gleb and Wu, with whom it has been a great privilege to collaborate.

J.-Z. Wu would like to thank Gleb Beliakov for his warm-hearted invitation to Deakin University and supervision throughout a 1-year academic visit. Over this period, they have enjoyed countless discussions with new ideas continuously arising, many of which have resulted in chapters of this book. Simon, his co-author, has provided many new ideas and representations on fuzzy measures and integrals over the course of his visit, and made great improvements to the content and quality

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of this book. The company and support of his family has given him the encour-agement to be able to write this book. Their many excursions in the wonderful land of Australia with Gleb’s family and other friends will deﬁnitely be treasured long into the future.

Melbourne, Australia and Ningbo, China Gleb Beliakov

December 2018 Simon James

Jian-Zhang Wu

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Studies in Fuzziness and Soft Computing