Wiley Handbook Of Granular Computing Sep 2008 ISBN 0470035544 pdf
HANDBOOK
OF GRANULAR
COMPUTING
Edited by
Witold Pedrycz
University of Alberta, Canada
and Polish Academy of Sciences, Warsaw, Poland
Andrzej Skowron
Warsaw University, Poland
Vladik Kreinovich
University of Texas, USA
A
Publication
HANDBOOK
OF GRANULAR
COMPUTING
HANDBOOK
OF GRANULAR
COMPUTING
Edited by
Witold Pedrycz
University of Alberta, Canada
and Polish Academy of Sciences, Warsaw, Poland
Andrzej Skowron
Warsaw University, Poland
Vladik Kreinovich
University of Texas, USA
A
Publication
C 2008
Copyright
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Library of Congress Cataloging-in-Publication Data
Pedrycz, Witold, 1953–
Handbook of granular computing / Witold Pedrycz, Andrzej Skowron, Vladik Kreinovich.
p. cm.
Includes index.
ISBN 978-0-470-03554-2 (cloth)
1. Granular computing–Handbooks, manuals, etc. I. Skowron, Andrzej. II. Kreinovich,
Vladik. III. Title.
QA76.9.S63P445 2008
006.3–dc22
2008002695
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN: 978-0-470-03554-2
Typeset in 9/11pt Times by Aptara Inc., New Delhi, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
Contents
Preface
ix
Foreword
xiii
Biographies
Part One
1
Fundamentals and Methodology of Granular Computing Based on Interval
Analysis, Fuzzy Sets and Rough Sets
Interval Computation as an Important Part of Granular Computing:
An Introduction
Vladik Kreinovich
xv
1
3
2
Stochastic Arithmetic as a Model of Granular Computing
Ren´e Alt and Jean Vignes
33
3
Fundamentals of Interval Analysis and Linkages to Fuzzy Set Theory
Weldon A. Lodwick
55
4
Interval Methods for Non-Linear Equation Solving Applications
Courtney Ryan Gwaltney, Youdong Lin, Luke David Simoni,
and Mark Allen Stadtherr
81
5
Fuzzy Sets as a User-Centric Processing Framework of Granular Computing
Witold Pedrycz
97
6
Measurement and Elicitation of Membership Functions
¨ ¸en
Taner Bilgic¸ and I˙. Burhan Turks
7
Fuzzy Clustering as a Data-Driven Development Environment for Information
Granules
Paulo Fazendeiro and Jos´e Valente de Oliveira
141
153
8
Encoding and Decoding of Fuzzy Granules
Shounak Roychowdhury
171
9
Systems of Information Granules
Frank H¨oeppner and Frank Klawonn
187
10
Logical Connectives for Granular Computing
Erich Peter Klement, Radko Mesiar, Andrea Mesiarov´a-Zem´ankov´a,
and Susanne Saminger-Platz
205
Contents
vi
11
Calculi of Information Granules. Fuzzy Relational Equations
Siegfried Gottwald
225
12
Fuzzy Numbers and Fuzzy Arithmetic
Luciano Stefanini, Laerte Sorini, and Maria Letizia Guerra
249
13
Rough-Granular Computing
Andrzej Skowron and James F. Peters
285
14
Wisdom Granular Computing
Andrzej Jankowski and Andrzej Skowron
329
15
Granular Computing for Reasoning about Ordered Data: The Dominance-Based
Rough Set Approach
Salvatore Greco, Benedetto Matarazzo, and Roman Slowi´nski
347
A Unified Approach to Granulation of Knowledge and Granular Computing
Based on Rough Mereology: A Survey
Lech Polkowski
375
16
17
A Unified Framework of Granular Computing
Yiyu Yao
401
18
Quotient Spaces and Granular Computing
Ling Zhang and Bo Zhang
411
19
Rough Sets and Granular Computing: Toward Rough-Granular Computing
Andrzej Skowron and Jaroslaw Stepaniuk
425
20
Construction of Rough Information Granules
Anna Gomoli´nska
449
21
Spatiotemporal Reasoning in Rough Sets and Granular Computing
Piotr Synak
471
Part Two
Hybrid Methods and Models of Granular Computing
22
A Survey of Interval-Valued Fuzzy Sets
Humberto Bustince, Javier Montero, Miguel Pagola, Edurne Barrenechea,
and Daniel G´omez
23
Measurement Theory and Uncertainty in Measurements: Application of Interval
Analysis and Fuzzy Sets Methods
Leon Reznik
489
491
517
24
Fuzzy Rough Sets: From Theory into Practice
Chris Cornelis, Martine De Cock, and Anna Maria Radzikowska
533
25
On Type 2 Fuzzy Sets as Granular Models for Words
Jerry M. Mendel
553
26
Design of Intelligent Systems with Interval Type-2 Fuzzy Logic
Oscar Castillo and Patricia Melin
575
Contents
vii
27
Theoretical Aspects of Shadowed Sets
Gianpiero Cattaneo and Davide Ciucci
603
28
Fuzzy Representations of Spatial Relations for Spatial Reasoning
Isabelle Bloch
629
29
Rough–Neural Methodologies in Granular Computing
Sushmita Mitra and Mohua Banerjee
657
30
Approximation and Perception in Ethology-Based Reinforcement Learning
James F. Peters
671
31
Fuzzy Linear Programming
Jaroslav Ram´ık
689
32
A Fuzzy Regression Approach to Acquisition of Linguistic Rules
Junzo Watada and Witold Pedrycz
719
33
Fuzzy Associative Memories and Their Relationship to Mathematical
Morphology
Peter Sussner and Marcos Eduardo Valle
34
Fuzzy Cognitive Maps
E.I. Papageorgiou and C.D. Stylios
Part Three
35
Applications and Case Studies
Rough Sets and Granular Computing in Behavioral Pattern Identification and
Planning
Jan G. Bazan
733
755
775
777
36
Rough Sets and Granular Computing in Hierarchical Learning
Sinh Hoa Nguyen and Hung Son Nguyen
801
37
Outlier and Exception Analysis in Rough Sets and Granular Computing
Tuan Trung Nyuyen
823
38
Information Access and Retrieval
Gloria Bordogna, Donald H. Kraft, and Gabriella Pasi
835
39
Granular Computing in Medical Informatics
Giovanni Bortolan
847
40
Eigen Fuzzy Sets and Image Information Retrieval
Ferdinando Di Martino, Salvatore Sessa, and Hajime Nobuhara
863
41
Rough Sets and Granular Computing in Dealing with Missing Attribute Values
Jerzy W. Grzymala-Busse
873
42
Granular Computing in Machine Learning and Data Mining
Eyke H¨ullermeier
889
Contents
viii
43
On Group Decision Making, Consensus Reaching, Voting, and Voting Paradoxes
under Fuzzy Preferences and a Fuzzy Majority: A Survey and a Granulation
Perspective
Janusz Kacprzyk, Sl awomir Zadro˙zny, Mario Fedrizzi, and Hannu Nurmi
907
44
FuzzJADE: A Framework for Agent-Based FLCs
Vincenzo Loia and Mario Veniero
931
45
Granular Models for Time-Series Forecasting
Marina Hirota Magalh˜aes, Rosangela Ballini, and Fernando Antonio
Campos Gomide
949
46
Rough Clustering
Pawan Lingras, S. Asharaf, and Cory Butz
969
47
Rough Document Clustering and The Internet
Hung Son Nguyen and Tu Bao Ho
987
48
Rough and Granular Case-Based Reasoning
Simon C.K. Shiu, Sankar K. Pal, and Yan Li
1005
49
Granulation in Analogy-Based Classification
Arkadiusz Wojna
1037
50
Approximation Spaces in Conflict Analysis: A Rough Set Framework
Sheela Ramanna
1055
51
Intervals in Finance and Economics: Bridge between Words and Numbers,
Language of Strategy
Manuel Tarrazo
52
Granular Computing Methods in Bioinformatics
Julio J. Vald´es
Index
1069
1093
1113
Preface
In Dissertio de Arte Combinatoria by Gottfried Wilhelm Leibniz (1666), one can find the following
sentences: ‘If controversies were to arise, there would be no more need of disputation between two
philosophers than between two accountants. For it would suffice to take their pencils in their hands, and
say to each other: “Let us calculate” ’ and in New Essays on Human Understanding (1705) [1], ‘Languages
are the best mirror of the human mind, and that a precise analysis of the signification of words would
tell us more than anything else about the operations of the understanding.’ Much later, methods based
on fuzzy sets, rough sets, and other soft computing paradigms allowed us to understand that for calculi
of thoughts discussed by Leibniz, it is necessary to develop tools for approximate reasoning about
vague, non-crisp concepts. For example, human is expressing higher level perceptions using vague, nonBoolean concepts. Hence, for developing truly intelligent methods for approximate reasoning about such
concepts in two-valued accessible for intelligent systems languages should be developed. One can gain in
searching for solutions of tasks related to perceptions by using granular computing (GC). This searching
in GC becomes feasible because GC-based methods use the fact that the solutions satisfy non-Boolean
specifications to a satisfactory degree only. Solutions in GC can often be constructed more efficiently
than in the case of methods searching for detailed, purely numeric solutions. Relevant granulation leads
to efficient solutions that are represented by granules matching specifications to satisfactory degrees.
In an inductive approach to knowledge discovery, information granules provide a means of encapsulating perceptions about objects of interest [2–7].
No matter what problem is taken into consideration, we usually cast it into frameworks that facilitate
observations about clusters of objects with common features and lead to problem formulation and problem
solving with considerable acuity. Such frameworks lend themselves to problems of feature selection and
feature extraction, pattern recognition, and knowledge discovery. Identification of relevant features of
objects contained in information granules makes it possible to formulate hypotheses about the significance
of the objects, construct new granules containing sample objects during interactions with the environment,
use GC to measure the nearness of complex granules, and identify infomorphisms between systems of
information granules.
Consider, for instance, image processing. In spite of the continuous progress in the area, a human
being assumes a dominant and very much uncontested position when it comes to understanding and
interpreting images.
Surely, we do not focus our attention on individual pixels but rather transform them using techniques
such as non-linear diffusion and group them together in pixel windows (complex objects) relative to
selected features. The parts of an image are then drawn together in information granules containing
objects (clusters of pixels) with vectors of values of functions representing object features that constitute
information granule descriptions. This signals a remarkable trait of humans that have the ability to
construct information granules, compare them, recognize patterns, transform and learn from them, arrive
at explanations about perceived patterns, formulate assertions, and construct approximations of granules
of objects of interest.
As another example, consider a collection of time series. From our perspective we can describe them
in a semiqualitative manner by pointing at specific regions of such signals. Specialists can effortlessly
interpret ECG signals. They distinguish some segments of such signals and interpret their combinations.
Preface
x
Experts can seamlessly interpret temporal readings of sensors and assess the status of the monitored
system. Again, in all these situations, the individual samples of the signals are not the focal point of the
analysis and the ensuing signal interpretation. We always granulate all phenomena (no matter if they
are originally discrete or analog in their nature). Time is another important variable that is subjected
to granulation. We use milliseconds, seconds, minutes, days, months, and years. Depending on specific
problem we have in mind and who the user is, the size of the information granules (time intervals) can
vary quite dramatically. To the high-level management, time intervals of quarters of year or a few years
can be meaningful temporal information granules on basis of which one develops any predictive model.
For those in charge of everyday operation of a dispatching plant, minutes and hours could form a viable
scale of time granulation. For the designer of high-speed integrated circuits and digital systems, the
temporal information granules concern nanoseconds, microseconds, and, perhaps, milliseconds. Even
such commonly encountered and simple examples are convincing enough to lead us to ascertain that (a)
information granules are the key components of knowledge representation and processing, (b) the level
of granularity of information granules (their size, to be more descriptive) becomes crucial to problem
description and an overall strategy of problem solving, (c) there is no universal level of granularity of
information; the size of granules is problem oriented and user dependent.
What has been said so far touched a qualitative aspect of the problem. The challenge is to develop a
computing framework within which all these representation and processing endeavors can be formally
realized. The common platform emerging within this context comes under the name of granular computing. In essence, it is an emerging paradigm of information processing that has its roots in Leibnitz’s
ideas [1] in Cantor’s set theory, Zadeh’s fuzzy information granulation [8], and Pawlak’s disovery of
elementary sets [9] (see also [10–14]).
While we have already noticed a number of important conceptual and computational constructs built
in the domain of system modeling, machine learning, image processing, pattern recognition, and data
compression in which various abstractions (and ensuing information granules) came into existence, GC
becomes innovative and intellectually proactive in several fundamental ways:
r The information granulation paradigm leads to formal frameworks that epitomize and synthesize what
has been done informally in science and engineering for centuries.
r With the emergence of unified frameworks for granular processing, we get a better grasp as to the
role of interaction between various, possibly distributed, GC machines and visualize infomorphisms
between them that facilitate classification and approximate reasoning.
r GC brings together the existing formalisms of set theory (interval analysis), fuzzy sets, and rough sets
under the same roof by clearly visualizing some fundamental commonalities and synergies.
r Interestingly, the inception of information granules is highly motivated. We do not form information
granules without reason. Information granules are an evident realization of the fundamental paradigm
of scientific discovery.
This volume is one of the first, if not the first, comprehensive compendium on GC. There are several
fundamental goals of this project. First, by capitalizing on several fundamental and well-established
frameworks of fuzzy sets, interval analysis, and rough sets, we build unified foundations of computing
with information granules. Second, we offer the reader a systematic and coherent exposure of the concepts,
design methodologies, and detailed algorithms. In general, we decided to adhere to the top-down strategy
of the exposure of the material by starting with the ideas along with some motivating notes and afterward
proceeding with the detailed design that materializes in specific algorithms, applications, and case studies.
We have made the handbook self-contained to a significant extent. While an overall knowledge of
GC and its subdisciplines would be helpful, the reader is provided with all necessary prerequisites. If
suitable, we have augmented some parts of the material with a step-by-step explanation of more advanced
concepts supported by a significant amount of illustrative numeric material.
We are strong proponents of the down-to-earth presentation of the material. While we maintain a
certain required level of formalism and mathematical rigor, the ultimate goal is to present the material so
Preface
xi
that it also emphasizes its applied side (meaning that the reader becomes fully aware of direct implications
of the presented algorithms, modeling, and the like).
This handbook is aimed at a broad audience of researchers and practitioners. Owing to the nature of
the material being covered and the way it is organized, we hope that it will appeal to the well-established
communities including those active in computational intelligence (CI), pattern recognition, machine
learning, fuzzy sets, neural networks, system modeling, and operations research. The research topic can
be treated in two different ways. First, as one the emerging and attractive areas of CI and GC, thus attracting
researchers engaged in some more specialized domains. Second, viewed as an enabling technology whose
contribution goes far beyond the communities and research areas listed above, we envision a genuine
interest from a vast array of research disciplines (engineering, economy, bioinformatics, etc).
We also hope that the handbook will also serve as a highly useful reference material for graduate
students and senior undergraduate students in a variety of courses on CI, artificial intelligence, pattern
recognition, data analysis, system modeling, signal processing, operations research, numerical methods,
and knowledge-based systems.
In the organization of the material we followed a top-down approach by splitting the content into
four main parts. The first one, fundamentals and methodology, covers the essential background of the
leading contributing technologies of GC, such as interval analysis, fuzzy sets, and rough sets. We also
offer a comprehensive coverage of the underlying concepts along with their interpretation. We also
elaborate on the representative techniques of GC. A special attention is paid to the development of
granular constructs, say, fuzzy sets, that serve as generic abstract constructs reflecting our perception of
the world and a way of an effective problem solving. A number of highly representative algorithms (say,
cognitive maps) are presented. Next, in Part II, we move on the hybrid constructs of GC where a variety
of symbiotic developments of information granules, such as interval-valued fuzzy sets, type-2 fuzzy sets
and shadowed sets, are considered. In the last part, we concentrate on a diversity of applications and case
studies.
W. Pedrycz gratefully acknowledges the support from Natural Sciences and Engineering Research
Council of Canada and Canada Research Chair program. Andrzej Skowron has been supported by the
grant from the Ministry of Scientific Research and Information Technology of the Republic of Poland.
Our thanks go to the authors who enthusiastically embraced the idea and energetically agreed to share
their expertise and research results in numerous domains of GC. The reviewers offered their constructive
thoughts on the submissions, which were of immense help and contributed to the quality of the content
of the handbook.
We are grateful for the truly professional support we have received from the staff of John Wiley,
especially Kate Griffiths and Debbie Cox, who always provided us with words of encouragement and
advice that helped us keep the project on schedule.
Editors-in-Chief
Edmonton – Warsaw – El Paso
May 2007
References
[1] G.W. Leibniz. New Essays on Human Understanding (1705). Cambridge University Press, Cambridge, UK,
1982.
[2] L.A. Zadeh. Fuzzy sets and information granularity. In: M.M. Gupta, R.K. Ragade, and R.R. Yager (eds),
Advances in Fuzzy Set Theory and Applications. North-Holland, Amsterdam, 1979, 3–18.
[3] L.A. Zadeh. Toward a generalized theory of uncertainty (GTU) – an outline. Inf. Sci., 172 (2005) 1–40.
[4] Z, Pawlak. Information systems-theoretical foundations. Inf. Syst. 6(3) (1981) 205–218.
[5] J.F. Peters and A. Skowron. Zdzisl aw Pawlak: Life and work, transaction on rough sets V. Springer Lect. Not.
Comput. Sci. 4100 (2006) 1–24..
[6] Z. Pawlak and A. Skowron. Rudiments of rough sets. Inf. Sci. 177(1) (2007) 3–27.
xii
Preface
[7] A. Bargiela and W. Pedrycz. Granular Computing: An Introduction. Kluwer Academic Publishers, Dordercht,
2003.
[8] L.A. Zadeh. Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy
logic. Fuzzy Sets Syst. 90 (1997) 111–127.
[9] Z. Pawlak. Rough sets. In: Theoretical Aspects of Reasoning About Data-Theory and Decision Library, Series D:
System Theory, Knowledge Engineering and Problem Solving, Vol. 9. Kluwer Academiic Publishers, Dordrecht,
(1991).
[10] J. Hobbs. Granulation. In: Proceedings of the 9th IJCAI 85, Los Angeles, California, August 18–23, 1985,
pp. 432–435.
[11] Z. Pawlak. Rough sets. Int. J. Comput. Inf. Sci. 11 (1982) 341–356.
[12] Z. Pawlak. Rough Sets. Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordercht,
1991.
[13] W. Pedrycz (ed). Granular Computing: An Emerging Paradigm. Physica-Verlag, Heidelberg, 2001.
[14] S.K. Pal, L. Polkowski, and A. Skowron (eds). Rough-Neural Computing: Techniques for Computing with Words.
Cognitive Technologies, Springer-Verlag, Heidelberg, 2004.
Foreword
Granular Computing – co-authored by professors A. Bargiela and W. Pedrycz, and published in 2003
– was the first book on granular computing [1]. It was a superlative work in all respects. Handbook of
Granular Computing is a worthy successor. Significantly, the co-editors of the handbook, Professors
Pedrycz, Skowron, and Kreinovich are, respectively, the leading contributors to the closely interrelated
fields of granular computing, rough set theory, and interval analysis – an interrelationship which is
accorded considerable attention in the handbook. The articles in the handbook are divided into three
groups: foundations of granular computing, interval analysis, fuzzy set theory, and rough set theory;
hybrid methods and models of granular computing; and applications and case studies. One cannot but
be greatly impressed by the vast panorama of applications extending from medical informatics and data
mining to time-series forecasting and the internet. Throughout the handbook, the exposition is aimed at
reader friendliness and deserves high marks in all respects.
What is granular computing? The preface and the chapters of this handbook provide a comprehensive
answer to this question. In the following, I take the liberty of sketching my perception of granular
computing – a perception in which the concept of a generalized constraint plays a pivotal role. An earlier
view may be found in my 1998 paper ‘Some reflections on soft computing, granular computing and their
roles in the conception, design and utilization of information/intelligent systems’ [2].
Basically, granular computing differs from conventional modes of computation in that the objects
of computation are not values of variables but information about values of variables. Furthermore,
information is allowed to be imperfect; i.e., it may be imprecise, uncertain, incomplete, conflicting, or
partially true. It is this facet of granular computing that endows granular computing with a capability
to deal with real-world problems which are beyond the reach of bivalent-logic-based methods which
are intolerant of imprecision and partial truth. In particular, through the use of generalized-constraintbased semantics, granular computing has the capability to compute with information described in natural
language.
Granular computing is based on fuzzy logic. There are many misconceptions about fuzzy logic. To
begin with, fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision. Fuzzy logic
is inspired by two remarkable human capabilities. First, the capability to reason and make decisions
in an environment of imprecision, uncertainty, incompleteness of information, and partiality of truth.
And second, the capability to perform a wide variety of physical and mental tasks based on perceptions,
without any measurements and any computations. The basic concepts of graduation and granulation form
the core of fuzzy logic, and are the principal distinguishing features of fuzzy logic. More specifically,
in fuzzy logic everything is or is allowed to be graduated, i.e., be a matter of degree or, equivalently,
fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a
clump of attribute values drawn together by indistinguishability, similarity, proximity, or functionality.
The concept of a generalized constraint serves to treat a granule as an object of computation. Graduated
granulation, or equivalently fuzzy granulation, is a unique feature of fuzzy logic. Graduated granulation
is inspired by the way in which humans deal with complexity and imprecision.
The concepts of graduation, granulation, and graduated granulation play key roles in granular computing. Graduated granulation underlies the concept of a linguistic variable, i.e., a variable whose values
are words rather than numbers. In retrospect, this concept, in combination with the associated concept of
a fuzzy if–then rule, may be viewed as a first step toward granular computing.
xiv
Foreword
Today, the concept of a linguistic variable is used in almost all applications of fuzzy logic. When I
introduced this concept in my 1973 paper ‘Outline of a new approach to the analysis of complex systems
and decision processes’ [3], I was greeted with scorn and derision rather than with accolades. The derisive
comments reflected a deep-seated tradition in science – the tradition of according much more respect
to numbers than to words. Thus, in science, progress is equated to progression from words to numbers.
In fuzzy logic, in moving from numerical to linguistic variables, we are moving in a countertraditional
direction. What the critics did not understand is that in moving in the countertraditional direction, we
are sacrificing precision to achieve important advantages down the line. This is what is called ‘the fuzzy
logic gambit.’ The fuzzy logic gambit is one of the principal rationales for the use of granular computing.
In sum, to say that the Handbook of Granular Computing is an important contribution to the literature
is an understatement. It is a work whose importance cannot be exaggerated. The coeditors, the authors,
and the publisher, John Wiley, deserve our thanks, congratulations, and loud applause.
Lotfi A. Zadeh
Berkeley, California
References
[1] A. Bargiela and W. Pedrycz. Granular Computing: An Introduction. Kluwer Academic Publishers, Dordercht,
2003.
[2] L.A. Zadeh. Some reflections on soft computing, granular computing and their roles in the conception, design
and utilization of information/intelligent systems. Soft Comput. 2 (1998) 23–25.
[3] L.A. Zadeh. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans.
Syst. Man Cybern. SMC-3 (1973) 28–44.
Biographies
Witold Pedrycz (M’88-SM’90-F’99) received the MSc, PhD, and DSci from the Silesian University of
Technology, Gliwice, Poland. He is a professor and Canada Research Chair in computational intelligence
in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada.
He is also with the Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland.
His research interests encompass computational intelligence, fuzzy modeling, knowledge discovery
and data mining, fuzzy control including fuzzy controllers, pattern recognition, knowledge-based neural
networks, granular and relational computing, and software engineering. He has published numerous
papers in these areas. He is also an author of 11 research monographs. Witold Pedrycz has been a member
of numerous program committees of IEEE conferences in the area of fuzzy sets and neurocomputing. He
serves as an editor-in-chief of IEEE Transactions on Systems Man and Cybernetics – Part A and associate
editor of IEEE Transactions on Fuzzy Systems. He is also an editor-in-chief of information sciences. Dr.
Pedrycz is a recipient of the prestigious Norbert Wiener Award from the IEEE Society of Systems, Man,
and Cybernetics as well as K.S. Fu Award from the North American Fuzzy Information Society.
Andrzej Skowron received the PhD and DSci from the University of Warsaw in Poland. In 1991 he
received the Scientific Title of Professor. He is a Full Professor in the Faculty of Mathematics, Computer
Science and Mechanics at Warsaw University. Andrzej Skowron is the author of numerous scientific
publications and editor of many books and special issues of scientific journals. His areas of expertise
include reasoning with incomplete information, approximate reasoning, soft computing methods and
applications, rough sets, rough mereology, granular computing, synthesis and analysis of complex objects,
intelligent agents, knowledge discovery systems, and advanced data mining techniques, decision support
systems, adaptive and autonomous systems. He was the supervisor of more than 20 PhD theses. He
was also involved in several national and international research and commercial projects relating to
data mining (fraud detection and web mining), control of unmanned vehicles, medical decision support
systems, and approximate reasoning in distributed environments among many others.
Since 1995 he is the editor-in-chief of Fundamenta Informaticae journal and a member of editorial
boards of several others journals including Knowledge Discovery and Data. He is the coeditor-in-chief
of the journal LNCS Transactions on Rough Sets published by Springer. Andrzej Skowron was the
president of the International Rough Set Society from 1996 to 2000. He served or is currently serving on
the program committees of almost 100 international conferences and workshops as program committee
member, program chair, or cochair. He has delivered numerous invited talks at international conferences,
including a plenary talk at the 16th IFIP World Computer Congress (Beijing, 2000). Throughout his
career, Andrzej Skowron has won many awards for his achievements, including awards from the Ministry
of Science, the Rector of Warsaw University, the Ministry of Education, Mazur’s Award of the Polish
Mathematical Society, and Janiszewski’s Award of the Polish Mathematical Society. In 2003 he received
the title of Honorary Professor from Chongqing University of Post and Telecommunication (China). In
2005 he received the ACM Recognition of Service Award for contributions to ACM and the award from
International Rough Sets Society for the outstanding research results.
Dr. Vladik Kreinovich received his MSc in mathematics and computer science from St Petersburg
University, Russia, in 1974 and PhD from the Institute of Mathematics, Soviet Academy of Sciences,
xvi
Biographies
Novosibirsk, in 1979. In 1975–1980, he worked with the Soviet Academy of Sciences, in particular, in
1978–1980, with the Special Astrophysical Observatory (representation and processing of uncertainty in
radioastronomy). In 1982–1989, he worked on error estimation and intelligent information processing
for the National Institute for Electrical Measuring Instruments, Russia. In 1989, he was a visiting scholar
at Stanford University. Since 1990, he is with the Department of Computer Science, University of Texas
at El Paso. Also, he served as an invited professor in Paris (University of Paris VI), Hong Kong, St
Petersburg, Russia, and Brazil.
His main interests include representation and processing of uncertainty, especially interval computations and intelligent control. He has published 3 books, 6 edited books, and more than 700 papers.
He is member of the editorial board of the international journal Reliable Computing (formerly, Interval
Computations) and several other journals. He is also the comaintainer of the international website on
interval computations, http://www.cs.utep.edu/interval-comp.
He is foreign member of the Russian Academy of Metrological Sciences, recipient of the 2003 El Paso
Energy Foundation Faculty Achievement Award for Research awarded by the University of Texas at El
Paso, and a corecipient of the 2005 Star Award from the University of Texas System.
Ren´e Alt is a professor of computer sciences at the Pierre et Marie Curie University in Paris (UPMC). He
received his master diploma in mathematics from UPMC in 1968, the Doctorate in Computer Sciences
(PhD) of UPMC in 1971, and was Docteur es Sciences from UPMC in 1981. He has been professor of
computer sciences at the University of Caen (France) from 1985 to 1991. He was head of the faculty
of computer sciences of UPMC from 1997 to 2001 and vice president of the administrative council of
UPMC from 2002 to 2006. Ren´e Alt’s fields of interest are the numerical solution of differential equations,
computer arithmetic, round-off error propagation, validation of numerical software, parallel computing,
and image processing.
S. Asharaf received the BTech from the Cochin University of Science and Technology, Kerala, and the
Master of Engineering from the Indian Institute of Science, where he is working toward a PhD. His
research interests include data clustering, soft computing, and support vector machines. He is one of the
recipients of IBM best PhD student award in 2006.
Rosangela Ballini received her BSc degree in applied mathematics from the Federal University of S˜ao
Carlos (UFSCar), SP, Brazil, in 1996. In 1998, she received the MSc degree in mathematics and computer
science from the University of S˜ao Paulo (USP), SP, Brazil, and the PhD degree in electrical engineering
from the State University of Campinas (Unicamp), SP, Brazil, in 2000. Currently, she is professor of the
Department of Economic Theory, Institute of Economics (IE), Unicamp. Her research interests include
time series forecasting, neural networks, fuzzy systems, and non-linear optimization.
Mohua Banerjee received her BSc (Hons) degree in mathematics, and the MSc, MPhil, and PhD degrees
in pure mathematics from the University of Calcutta in 1985, 1987, 1989, and 1995, respectively. During
1995–1997, she was a research associate at the Machine Intelligence Unit, Indian Statistical Institute,
Calcutta. In 1997, she joined the Department of Mathematics and Statistics, Indian Institute of Technology,
Kanpur, as lecturer, and is currently Assistant Professor in the same department. She was an associate of
The Institute of Mathematical Sciences, Chennai, India, during 2003–2005. Her main research interests
lie in modal logics and rough sets. She has made several research visits to institutes in India and abroad.
She is a member of the Working Group for the Center for Research in Discrete Mathematics and its
Applications (CARDMATH), Department of Science and Technology (DST), Government of India. She
serves in the reviewer panel of many international journals. Dr. Banerjee was awarded the Indian National
Science Academy Medal for Young Scientists in 1995.
Edurne Barrenechea is an assistant lecturer at the Department of Automatics and Computation, Public
University of Navarra, Spain. Having received an MSc in computer science at the Pais Vasco University
in 1990. She worked as analyst programmer in Bombas Itur from 1990 to 2001 and then she joined the
Public University of Navarra as associate lecturer. She obtained the PhD in computer science in 2005.
Biographies
xvii
Her research interests are fuzzy techniques for image processing, fuzzy sets theory, interval type-2 fuzzy
sets theory, neural networks, and industrial applications of soft computing techniques. She is member of
the European Society for Fuzzy Logic and Technology (EUSFLAT).
Jan G. Bazan is an Assistant Professor in the Institute of Mathematics at the University of Rzeszow in
Poland. He received his PhD degree in 1999 from the University of Warsaw in Poland. His recent research
interests focus on rough set theory, granular computing, knowledge discovery, data mining techniques,
reasoning with incomplete information, approximate reasoning, decision support systems, and adaptive
systems. He is the author or coauthor of more than 40 scientific publications and he was involved in
several national and international research projects relating to fraud detection, web mining, risk pattern
detection, and automated planning of the treatment among other topics.
Taner Bilgi¸c received his BSc and MSc in industrial engineering from the Middle East Technical University, Ankara, Turkey, in 1987 and 1990, respectively. He received a PhD in industrial engineering from
the University of Toronto in 1995. The title of his dissertation is ‘Measurement-Theoretic Frameworks
for Fuzzy Set Theory with Applications to Preference Modelling.’ He spent 2 years at the Enterprise
Integration Laboratory in Toronto as a research associate. Since 1997, he has been a faculty member at
the Department of Industrial Engineering at Bogazici University in Istanbul, Turkey.
Isabelle Bloch is a professor at ENST (Signal and Image Processing Department), CNRS UMR 5141
LTCI. Her research interests include three-dimensional (3D) image and object processing, 3D and fuzzy
mathematical morphology, decision theory, information fusion, fuzzy set theory, belief function theory,
structural pattern recognition, spatial reasoning, and medical imaging.
Gloria Bordogna received her Laurea degree in Physics at the Universit`a degli Studi di Milano, Italy, in
1984. In 1986 she joined the Italian National Research Council, where she presently holds the position
of a senior researcher at the Institute for the Dynamics of Environmental Processes. She is also a contract
professor at the faculty of Engineering of Bergamo University, where she teaches information retrieval and
geographic information systems. Her research activity concerns soft computing techniques for managing
imprecision and uncertainty affecting both textual and spatial information. She is coeditor of a special
issue of JASIS and three volumes published by Springer-Verlag on uncertainty and impression management in databases. She has published over 100 papers in international journals, in the proceedings of
international conferences, and in books. She participated at the program committee of international conferences such as FUZZIEEE, ECIR, ACM SIGIR, FQAS, EUROFUSE, IJCAI2007, ICDE 2007, and ACM
SAC ‘Information Access and Retrieval’ track and served as a reviewer of journals such as JASIST, IEEE
Transactions on Fuzzy Systems, Fuzzy Sets and Systems, and Information Processing and Management.
Giovanni Bortolan received the doctoral degree from the University of Padova, Padova, Italy in 1978. He
is senior researcher at the Institute of Biomedical Engineering, Italian National Research Council (ISIBCNR), Padova, Italy. He has published numerous papers in the areas of medical informatics and applied
fuzzy sets. He is actively pursuing research in medical informatics in computerized electrocardiography,
neural networks, fuzzy sets, data mining, and pattern recognition.
Humberto Bustince is an Associate Professor at the Department of Automatics and Computation, Public
University of Navarra, Spain. He holds a PhD degree in mathematics from Public University of Navarra
from 1994. His research interests are fuzzy logic theory, extensions of fuzzy sets (type-2 fuzzy sets and
Atanassov’s intuitionistic fuzzy sets), fuzzy measures, aggregation operators, and fuzzy techniques for
image processing. He is the author of more than 30 peer-reviewed research papers and is member of
IEEE and European Society for Fuzzy Logic and Technology (EUSFLAT).
Cory J. Butz received the BSc, MSc, and PhD degrees in computer science from the University of
Regina, Saskatchewan, Canada, in 1994, 1996, and 2000, respectively. His research interests include
uncertainty reasoning, database systems, information retrieval, and data mining.
xviii
Biographies
Oscar Castillo was awarded Doctor of Science (DSc) from the Polish Academy of Sciences. He is a
professor of computer science in the Graduate Division, Tijuana Institute of Technology, Tijuana, Mexico.
In addition, he is serving as research director of computer science and head of the research group on fuzzy
logic and genetic algorithms. Currently, he is president of Hispanic American Fuzzy Systems Association
(HAFSA) and vice president of International Fuzzy Systems Association (IFSA) in charge of publicity.
Professor Castillo is also vice chair of the Mexican Chapter of the Computational Intelligence Society
(IEEE). Professor Castillo is also general chair of the IFSA 2007 World Congress to be held in Cancun,
Mexico. He also belongs to the Technical Committee on Fuzzy Systems of IEEE and to the Task Force
on ‘Extensions to Type-1 Fuzzy Systems.’ His research interests are in type-2 fuzzy logic, intuitionistic
fuzzy logic, fuzzy control, neuro–fuzzy, and genetic–fuzzy hybrid approaches. He has published over 60
journal papers, 5 authored books, 10 edited books, and 150 papers in conference proceedings.
Gianpiero Cattaneo is a Full Professor in ‘dynamical system theory’ at the Universit`a di Milano, Bicocca.
Previously, he was an Associate Professor in ‘mathematical methods of physics’ (from 1974 to 1984)
and researcher of ‘theoretical physics’ (from 1968 to 1974). From 1994 to 1997, he was a regular visiting
professor at the London School of Economics (Department of Logic and Scientific Methods), where,
since 1998, he had a position of research associate at ‘The Centre for the Philosophy of Natural and Social
Science.’ From 1997 to 1999, he was Maitre de Conferences at the Nancy-Metz Academy and Maitre de
Conferences at ‘la Ecole Normale Superieure’ in Lyon: Laboratoire de l’Informatique du Parall`elisme.
He is member of the editorial board of the Transactions on Rough Sets, LNCS (Springer-Verlag), the
Scientific Committee of the ‘International Quantum Structures Association (IQSA)’; the International
Advisory Board of the ‘European School of Advanced Studies in Methods for Management of Complex
Systems’ (Pavia); International Federation of Information Processing (IFIP): Working group on cellular
automata. Moreover, he is scientific coordinator of a biannual 2006–2007 ‘Program of International
Collaboration’ between France and Italy, involving the universities of Nice, Marseille, Ecole Normale
Superieure de Lyon, Marne-la-Valle, Milano-Bicocca, and Bologna. He was a member of numerous
program committees of international conferences. His research activities, with results published on
international journals in more than 140 papers, are centered on topological chaos, cellular automata and
related languages, algebraic approach to fuzzy logic and rough sets, axiomatic foundations of quantum
mechanics, and realization of reversible gates by quantum computing techniques.
Davide Ciucci received a PhD in 2004 in computer science from the University of Milan. Since 2005, he
has held a permanent position as a researcher at the University of Milano-Bicocca, where he delivered a
course on fuzzy logic and rough sets. His research interests are about a theoretical algebraic approach to
imprecision, with particular attention to many-valued logics, rough sets, and their relationship. Recently,
he got involved in the semantic web area, with a special interest in fuzzy ontology and fuzzy description
logics. He has been a member committee of several conferences about rough and fuzzy sets, co-organizer
of a special session at the Joint Rough Set Symposium JRS07. His webpages, with a list of publications,
can be found at www.fislab.disco.unimib.it.
Chris Cornelis is a postdoctoral researcher at the Department of Applied Mathematics and Computer Science at Ghent University (Belgium) funded by the Research Foundation – Flanders. His research interests include various models of imperfection (fuzzy rough sets, bilattices and interval-valued
fuzzy sets); he is currently focusing on their application to personalized information access and web
intelligence.
Martine De Cock is a professor at the Department of Applied Mathematics and Computer Science at
Ghent University (Belgium). Her current research efforts are directed toward the development and the
use of computational intelligent methods for next-generation web applications.
E.I. Papageorgiou was born in Larisa in 1975, Greece. She obtained the physics degree in 1997, MSc
in medical physics in 2000, and PhD in computer science in July 2004 from the University of Patras.
From 2004 to 2006, she was a postdoctoral researcher at the Department of Electrical and Computer
Biographies
xix
Engineering, University of Patras (Greece), on developing new models and methodologies based on
soft computing for medical decision support systems. From 2000 to 2006, she was involved in several
research projects related to the development of new algorithms and methods for complex diagnostic
and medical decision support systems. Her main activities were the development of innovative learning
algorithms for fuzzy cognitive maps and intelligent expert systems for medical diagnosis and decisionmaking tasks. From 2004 to 2005, she was appointed as lecturer at the Department of Electrical and
Computer Engineering at the University of Patras. Currently, she is Assistant Professor at the Department
of Informatics and Computer Technology, Technological Educational Institute of Lamia, and adjunct
Assistant Professor at the University of Central Greece. She has coauthored more than 40 journals and
conference papers, book chapters, and technical reports, and has more than 50 citations to her works.
Her interests include expert systems, intelligent algorithms and computational intelligence techniques,
intelligent decision support systems, and artificial intelligence techniques for medical applications. Dr.
E.I. Papageorgiou was a recipient of a scholarship of Greek State Scholarship Foundation ‘I.K.Y.’ during
her PhD studies (2000–2004), and from 2006 to May 2007, she was also a recipient of the postdoctoral
research fellowship from the Greek State Scholarship Foundation ‘I.K.Y.’
Paulo Fazendeiro received the BS degree in mathematics and informatics in 1995 (with honors) and
the equivalent of MS degree in computer science in 2001, all from the University of Beira Interior,
Portugal. He is preparing his dissertation on the relationships between accuracy and interpretability of
fuzzy systems as a partial fulfillment of the requirements for the informatics engineering PhD degree.
He joined the University of Beira Interior in 1995, where he is currently a lecturer in the Informatics
Department. His research interests include application of fuzzy set theory and fuzzy systems, data mining,
evolutionary algorithms, multiobjective optimization, and clustering techniques with applications to
image processing. Dr. Fazendeiro is a member of the Portuguese Telecommunications Institute and the
Informatics Laboratory of the University of Algarve.
Mario Fedrizzi received the MSc degree in mathematics in 1973 from the University of Padua, Italy.
Since 1976, he has been an Assistant Professor; since 1981, an Associate Professor; and since 1986, a Full
Professor with Trento University, Italy. He served as a chairman of the Institute of Informatics from 1985
to 1991 and as a dean of the Faculty of Economics and Business Administration from 1989 to 1995. His
research focused on utility and risk theory, stochastic dominance, group decision making, fuzzy decision
analysis, fuzzy regression analysis, and consensus modeling in uncertain environments, decision support
systems. He has authored or coauthored books and more than 150 papers, which appeared in international
proceedings and journals, e.g., European Journal of Operational Research, Fuzzy Sets and Systems, IEEE
Transactions on Systems, Man and Cybernetics, Mathematical Social Sciences, Quality and Quantity,
and International Journal of Intelligent Systems. He was also involved in consulting activities in the areas
of information systems and DSS design and implementation, office automation, quality control, project
management, expert systems, and neural nets in financial planning. From 1995 to 2006, he was appointed
as chairman of a bank and of a real-estate company, and as a member of the board of directors of Cedacri,
the largest Italian banking information systems outsourcing company, and of Unicredit Banca.
Fernando Antonio Campos Gomide received the BSc degree in electrical engineering from the Polytechnic Institute of the Pontifical Catholic University of Minas Gerais (IPUC/PUC-MG) Belo Horizonte,
B
OF GRANULAR
COMPUTING
Edited by
Witold Pedrycz
University of Alberta, Canada
and Polish Academy of Sciences, Warsaw, Poland
Andrzej Skowron
Warsaw University, Poland
Vladik Kreinovich
University of Texas, USA
A
Publication
HANDBOOK
OF GRANULAR
COMPUTING
HANDBOOK
OF GRANULAR
COMPUTING
Edited by
Witold Pedrycz
University of Alberta, Canada
and Polish Academy of Sciences, Warsaw, Poland
Andrzej Skowron
Warsaw University, Poland
Vladik Kreinovich
University of Texas, USA
A
Publication
C 2008
Copyright
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
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Library of Congress Cataloging-in-Publication Data
Pedrycz, Witold, 1953–
Handbook of granular computing / Witold Pedrycz, Andrzej Skowron, Vladik Kreinovich.
p. cm.
Includes index.
ISBN 978-0-470-03554-2 (cloth)
1. Granular computing–Handbooks, manuals, etc. I. Skowron, Andrzej. II. Kreinovich,
Vladik. III. Title.
QA76.9.S63P445 2008
006.3–dc22
2008002695
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN: 978-0-470-03554-2
Typeset in 9/11pt Times by Aptara Inc., New Delhi, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
Contents
Preface
ix
Foreword
xiii
Biographies
Part One
1
Fundamentals and Methodology of Granular Computing Based on Interval
Analysis, Fuzzy Sets and Rough Sets
Interval Computation as an Important Part of Granular Computing:
An Introduction
Vladik Kreinovich
xv
1
3
2
Stochastic Arithmetic as a Model of Granular Computing
Ren´e Alt and Jean Vignes
33
3
Fundamentals of Interval Analysis and Linkages to Fuzzy Set Theory
Weldon A. Lodwick
55
4
Interval Methods for Non-Linear Equation Solving Applications
Courtney Ryan Gwaltney, Youdong Lin, Luke David Simoni,
and Mark Allen Stadtherr
81
5
Fuzzy Sets as a User-Centric Processing Framework of Granular Computing
Witold Pedrycz
97
6
Measurement and Elicitation of Membership Functions
¨ ¸en
Taner Bilgic¸ and I˙. Burhan Turks
7
Fuzzy Clustering as a Data-Driven Development Environment for Information
Granules
Paulo Fazendeiro and Jos´e Valente de Oliveira
141
153
8
Encoding and Decoding of Fuzzy Granules
Shounak Roychowdhury
171
9
Systems of Information Granules
Frank H¨oeppner and Frank Klawonn
187
10
Logical Connectives for Granular Computing
Erich Peter Klement, Radko Mesiar, Andrea Mesiarov´a-Zem´ankov´a,
and Susanne Saminger-Platz
205
Contents
vi
11
Calculi of Information Granules. Fuzzy Relational Equations
Siegfried Gottwald
225
12
Fuzzy Numbers and Fuzzy Arithmetic
Luciano Stefanini, Laerte Sorini, and Maria Letizia Guerra
249
13
Rough-Granular Computing
Andrzej Skowron and James F. Peters
285
14
Wisdom Granular Computing
Andrzej Jankowski and Andrzej Skowron
329
15
Granular Computing for Reasoning about Ordered Data: The Dominance-Based
Rough Set Approach
Salvatore Greco, Benedetto Matarazzo, and Roman Slowi´nski
347
A Unified Approach to Granulation of Knowledge and Granular Computing
Based on Rough Mereology: A Survey
Lech Polkowski
375
16
17
A Unified Framework of Granular Computing
Yiyu Yao
401
18
Quotient Spaces and Granular Computing
Ling Zhang and Bo Zhang
411
19
Rough Sets and Granular Computing: Toward Rough-Granular Computing
Andrzej Skowron and Jaroslaw Stepaniuk
425
20
Construction of Rough Information Granules
Anna Gomoli´nska
449
21
Spatiotemporal Reasoning in Rough Sets and Granular Computing
Piotr Synak
471
Part Two
Hybrid Methods and Models of Granular Computing
22
A Survey of Interval-Valued Fuzzy Sets
Humberto Bustince, Javier Montero, Miguel Pagola, Edurne Barrenechea,
and Daniel G´omez
23
Measurement Theory and Uncertainty in Measurements: Application of Interval
Analysis and Fuzzy Sets Methods
Leon Reznik
489
491
517
24
Fuzzy Rough Sets: From Theory into Practice
Chris Cornelis, Martine De Cock, and Anna Maria Radzikowska
533
25
On Type 2 Fuzzy Sets as Granular Models for Words
Jerry M. Mendel
553
26
Design of Intelligent Systems with Interval Type-2 Fuzzy Logic
Oscar Castillo and Patricia Melin
575
Contents
vii
27
Theoretical Aspects of Shadowed Sets
Gianpiero Cattaneo and Davide Ciucci
603
28
Fuzzy Representations of Spatial Relations for Spatial Reasoning
Isabelle Bloch
629
29
Rough–Neural Methodologies in Granular Computing
Sushmita Mitra and Mohua Banerjee
657
30
Approximation and Perception in Ethology-Based Reinforcement Learning
James F. Peters
671
31
Fuzzy Linear Programming
Jaroslav Ram´ık
689
32
A Fuzzy Regression Approach to Acquisition of Linguistic Rules
Junzo Watada and Witold Pedrycz
719
33
Fuzzy Associative Memories and Their Relationship to Mathematical
Morphology
Peter Sussner and Marcos Eduardo Valle
34
Fuzzy Cognitive Maps
E.I. Papageorgiou and C.D. Stylios
Part Three
35
Applications and Case Studies
Rough Sets and Granular Computing in Behavioral Pattern Identification and
Planning
Jan G. Bazan
733
755
775
777
36
Rough Sets and Granular Computing in Hierarchical Learning
Sinh Hoa Nguyen and Hung Son Nguyen
801
37
Outlier and Exception Analysis in Rough Sets and Granular Computing
Tuan Trung Nyuyen
823
38
Information Access and Retrieval
Gloria Bordogna, Donald H. Kraft, and Gabriella Pasi
835
39
Granular Computing in Medical Informatics
Giovanni Bortolan
847
40
Eigen Fuzzy Sets and Image Information Retrieval
Ferdinando Di Martino, Salvatore Sessa, and Hajime Nobuhara
863
41
Rough Sets and Granular Computing in Dealing with Missing Attribute Values
Jerzy W. Grzymala-Busse
873
42
Granular Computing in Machine Learning and Data Mining
Eyke H¨ullermeier
889
Contents
viii
43
On Group Decision Making, Consensus Reaching, Voting, and Voting Paradoxes
under Fuzzy Preferences and a Fuzzy Majority: A Survey and a Granulation
Perspective
Janusz Kacprzyk, Sl awomir Zadro˙zny, Mario Fedrizzi, and Hannu Nurmi
907
44
FuzzJADE: A Framework for Agent-Based FLCs
Vincenzo Loia and Mario Veniero
931
45
Granular Models for Time-Series Forecasting
Marina Hirota Magalh˜aes, Rosangela Ballini, and Fernando Antonio
Campos Gomide
949
46
Rough Clustering
Pawan Lingras, S. Asharaf, and Cory Butz
969
47
Rough Document Clustering and The Internet
Hung Son Nguyen and Tu Bao Ho
987
48
Rough and Granular Case-Based Reasoning
Simon C.K. Shiu, Sankar K. Pal, and Yan Li
1005
49
Granulation in Analogy-Based Classification
Arkadiusz Wojna
1037
50
Approximation Spaces in Conflict Analysis: A Rough Set Framework
Sheela Ramanna
1055
51
Intervals in Finance and Economics: Bridge between Words and Numbers,
Language of Strategy
Manuel Tarrazo
52
Granular Computing Methods in Bioinformatics
Julio J. Vald´es
Index
1069
1093
1113
Preface
In Dissertio de Arte Combinatoria by Gottfried Wilhelm Leibniz (1666), one can find the following
sentences: ‘If controversies were to arise, there would be no more need of disputation between two
philosophers than between two accountants. For it would suffice to take their pencils in their hands, and
say to each other: “Let us calculate” ’ and in New Essays on Human Understanding (1705) [1], ‘Languages
are the best mirror of the human mind, and that a precise analysis of the signification of words would
tell us more than anything else about the operations of the understanding.’ Much later, methods based
on fuzzy sets, rough sets, and other soft computing paradigms allowed us to understand that for calculi
of thoughts discussed by Leibniz, it is necessary to develop tools for approximate reasoning about
vague, non-crisp concepts. For example, human is expressing higher level perceptions using vague, nonBoolean concepts. Hence, for developing truly intelligent methods for approximate reasoning about such
concepts in two-valued accessible for intelligent systems languages should be developed. One can gain in
searching for solutions of tasks related to perceptions by using granular computing (GC). This searching
in GC becomes feasible because GC-based methods use the fact that the solutions satisfy non-Boolean
specifications to a satisfactory degree only. Solutions in GC can often be constructed more efficiently
than in the case of methods searching for detailed, purely numeric solutions. Relevant granulation leads
to efficient solutions that are represented by granules matching specifications to satisfactory degrees.
In an inductive approach to knowledge discovery, information granules provide a means of encapsulating perceptions about objects of interest [2–7].
No matter what problem is taken into consideration, we usually cast it into frameworks that facilitate
observations about clusters of objects with common features and lead to problem formulation and problem
solving with considerable acuity. Such frameworks lend themselves to problems of feature selection and
feature extraction, pattern recognition, and knowledge discovery. Identification of relevant features of
objects contained in information granules makes it possible to formulate hypotheses about the significance
of the objects, construct new granules containing sample objects during interactions with the environment,
use GC to measure the nearness of complex granules, and identify infomorphisms between systems of
information granules.
Consider, for instance, image processing. In spite of the continuous progress in the area, a human
being assumes a dominant and very much uncontested position when it comes to understanding and
interpreting images.
Surely, we do not focus our attention on individual pixels but rather transform them using techniques
such as non-linear diffusion and group them together in pixel windows (complex objects) relative to
selected features. The parts of an image are then drawn together in information granules containing
objects (clusters of pixels) with vectors of values of functions representing object features that constitute
information granule descriptions. This signals a remarkable trait of humans that have the ability to
construct information granules, compare them, recognize patterns, transform and learn from them, arrive
at explanations about perceived patterns, formulate assertions, and construct approximations of granules
of objects of interest.
As another example, consider a collection of time series. From our perspective we can describe them
in a semiqualitative manner by pointing at specific regions of such signals. Specialists can effortlessly
interpret ECG signals. They distinguish some segments of such signals and interpret their combinations.
Preface
x
Experts can seamlessly interpret temporal readings of sensors and assess the status of the monitored
system. Again, in all these situations, the individual samples of the signals are not the focal point of the
analysis and the ensuing signal interpretation. We always granulate all phenomena (no matter if they
are originally discrete or analog in their nature). Time is another important variable that is subjected
to granulation. We use milliseconds, seconds, minutes, days, months, and years. Depending on specific
problem we have in mind and who the user is, the size of the information granules (time intervals) can
vary quite dramatically. To the high-level management, time intervals of quarters of year or a few years
can be meaningful temporal information granules on basis of which one develops any predictive model.
For those in charge of everyday operation of a dispatching plant, minutes and hours could form a viable
scale of time granulation. For the designer of high-speed integrated circuits and digital systems, the
temporal information granules concern nanoseconds, microseconds, and, perhaps, milliseconds. Even
such commonly encountered and simple examples are convincing enough to lead us to ascertain that (a)
information granules are the key components of knowledge representation and processing, (b) the level
of granularity of information granules (their size, to be more descriptive) becomes crucial to problem
description and an overall strategy of problem solving, (c) there is no universal level of granularity of
information; the size of granules is problem oriented and user dependent.
What has been said so far touched a qualitative aspect of the problem. The challenge is to develop a
computing framework within which all these representation and processing endeavors can be formally
realized. The common platform emerging within this context comes under the name of granular computing. In essence, it is an emerging paradigm of information processing that has its roots in Leibnitz’s
ideas [1] in Cantor’s set theory, Zadeh’s fuzzy information granulation [8], and Pawlak’s disovery of
elementary sets [9] (see also [10–14]).
While we have already noticed a number of important conceptual and computational constructs built
in the domain of system modeling, machine learning, image processing, pattern recognition, and data
compression in which various abstractions (and ensuing information granules) came into existence, GC
becomes innovative and intellectually proactive in several fundamental ways:
r The information granulation paradigm leads to formal frameworks that epitomize and synthesize what
has been done informally in science and engineering for centuries.
r With the emergence of unified frameworks for granular processing, we get a better grasp as to the
role of interaction between various, possibly distributed, GC machines and visualize infomorphisms
between them that facilitate classification and approximate reasoning.
r GC brings together the existing formalisms of set theory (interval analysis), fuzzy sets, and rough sets
under the same roof by clearly visualizing some fundamental commonalities and synergies.
r Interestingly, the inception of information granules is highly motivated. We do not form information
granules without reason. Information granules are an evident realization of the fundamental paradigm
of scientific discovery.
This volume is one of the first, if not the first, comprehensive compendium on GC. There are several
fundamental goals of this project. First, by capitalizing on several fundamental and well-established
frameworks of fuzzy sets, interval analysis, and rough sets, we build unified foundations of computing
with information granules. Second, we offer the reader a systematic and coherent exposure of the concepts,
design methodologies, and detailed algorithms. In general, we decided to adhere to the top-down strategy
of the exposure of the material by starting with the ideas along with some motivating notes and afterward
proceeding with the detailed design that materializes in specific algorithms, applications, and case studies.
We have made the handbook self-contained to a significant extent. While an overall knowledge of
GC and its subdisciplines would be helpful, the reader is provided with all necessary prerequisites. If
suitable, we have augmented some parts of the material with a step-by-step explanation of more advanced
concepts supported by a significant amount of illustrative numeric material.
We are strong proponents of the down-to-earth presentation of the material. While we maintain a
certain required level of formalism and mathematical rigor, the ultimate goal is to present the material so
Preface
xi
that it also emphasizes its applied side (meaning that the reader becomes fully aware of direct implications
of the presented algorithms, modeling, and the like).
This handbook is aimed at a broad audience of researchers and practitioners. Owing to the nature of
the material being covered and the way it is organized, we hope that it will appeal to the well-established
communities including those active in computational intelligence (CI), pattern recognition, machine
learning, fuzzy sets, neural networks, system modeling, and operations research. The research topic can
be treated in two different ways. First, as one the emerging and attractive areas of CI and GC, thus attracting
researchers engaged in some more specialized domains. Second, viewed as an enabling technology whose
contribution goes far beyond the communities and research areas listed above, we envision a genuine
interest from a vast array of research disciplines (engineering, economy, bioinformatics, etc).
We also hope that the handbook will also serve as a highly useful reference material for graduate
students and senior undergraduate students in a variety of courses on CI, artificial intelligence, pattern
recognition, data analysis, system modeling, signal processing, operations research, numerical methods,
and knowledge-based systems.
In the organization of the material we followed a top-down approach by splitting the content into
four main parts. The first one, fundamentals and methodology, covers the essential background of the
leading contributing technologies of GC, such as interval analysis, fuzzy sets, and rough sets. We also
offer a comprehensive coverage of the underlying concepts along with their interpretation. We also
elaborate on the representative techniques of GC. A special attention is paid to the development of
granular constructs, say, fuzzy sets, that serve as generic abstract constructs reflecting our perception of
the world and a way of an effective problem solving. A number of highly representative algorithms (say,
cognitive maps) are presented. Next, in Part II, we move on the hybrid constructs of GC where a variety
of symbiotic developments of information granules, such as interval-valued fuzzy sets, type-2 fuzzy sets
and shadowed sets, are considered. In the last part, we concentrate on a diversity of applications and case
studies.
W. Pedrycz gratefully acknowledges the support from Natural Sciences and Engineering Research
Council of Canada and Canada Research Chair program. Andrzej Skowron has been supported by the
grant from the Ministry of Scientific Research and Information Technology of the Republic of Poland.
Our thanks go to the authors who enthusiastically embraced the idea and energetically agreed to share
their expertise and research results in numerous domains of GC. The reviewers offered their constructive
thoughts on the submissions, which were of immense help and contributed to the quality of the content
of the handbook.
We are grateful for the truly professional support we have received from the staff of John Wiley,
especially Kate Griffiths and Debbie Cox, who always provided us with words of encouragement and
advice that helped us keep the project on schedule.
Editors-in-Chief
Edmonton – Warsaw – El Paso
May 2007
References
[1] G.W. Leibniz. New Essays on Human Understanding (1705). Cambridge University Press, Cambridge, UK,
1982.
[2] L.A. Zadeh. Fuzzy sets and information granularity. In: M.M. Gupta, R.K. Ragade, and R.R. Yager (eds),
Advances in Fuzzy Set Theory and Applications. North-Holland, Amsterdam, 1979, 3–18.
[3] L.A. Zadeh. Toward a generalized theory of uncertainty (GTU) – an outline. Inf. Sci., 172 (2005) 1–40.
[4] Z, Pawlak. Information systems-theoretical foundations. Inf. Syst. 6(3) (1981) 205–218.
[5] J.F. Peters and A. Skowron. Zdzisl aw Pawlak: Life and work, transaction on rough sets V. Springer Lect. Not.
Comput. Sci. 4100 (2006) 1–24..
[6] Z. Pawlak and A. Skowron. Rudiments of rough sets. Inf. Sci. 177(1) (2007) 3–27.
xii
Preface
[7] A. Bargiela and W. Pedrycz. Granular Computing: An Introduction. Kluwer Academic Publishers, Dordercht,
2003.
[8] L.A. Zadeh. Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy
logic. Fuzzy Sets Syst. 90 (1997) 111–127.
[9] Z. Pawlak. Rough sets. In: Theoretical Aspects of Reasoning About Data-Theory and Decision Library, Series D:
System Theory, Knowledge Engineering and Problem Solving, Vol. 9. Kluwer Academiic Publishers, Dordrecht,
(1991).
[10] J. Hobbs. Granulation. In: Proceedings of the 9th IJCAI 85, Los Angeles, California, August 18–23, 1985,
pp. 432–435.
[11] Z. Pawlak. Rough sets. Int. J. Comput. Inf. Sci. 11 (1982) 341–356.
[12] Z. Pawlak. Rough Sets. Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordercht,
1991.
[13] W. Pedrycz (ed). Granular Computing: An Emerging Paradigm. Physica-Verlag, Heidelberg, 2001.
[14] S.K. Pal, L. Polkowski, and A. Skowron (eds). Rough-Neural Computing: Techniques for Computing with Words.
Cognitive Technologies, Springer-Verlag, Heidelberg, 2004.
Foreword
Granular Computing – co-authored by professors A. Bargiela and W. Pedrycz, and published in 2003
– was the first book on granular computing [1]. It was a superlative work in all respects. Handbook of
Granular Computing is a worthy successor. Significantly, the co-editors of the handbook, Professors
Pedrycz, Skowron, and Kreinovich are, respectively, the leading contributors to the closely interrelated
fields of granular computing, rough set theory, and interval analysis – an interrelationship which is
accorded considerable attention in the handbook. The articles in the handbook are divided into three
groups: foundations of granular computing, interval analysis, fuzzy set theory, and rough set theory;
hybrid methods and models of granular computing; and applications and case studies. One cannot but
be greatly impressed by the vast panorama of applications extending from medical informatics and data
mining to time-series forecasting and the internet. Throughout the handbook, the exposition is aimed at
reader friendliness and deserves high marks in all respects.
What is granular computing? The preface and the chapters of this handbook provide a comprehensive
answer to this question. In the following, I take the liberty of sketching my perception of granular
computing – a perception in which the concept of a generalized constraint plays a pivotal role. An earlier
view may be found in my 1998 paper ‘Some reflections on soft computing, granular computing and their
roles in the conception, design and utilization of information/intelligent systems’ [2].
Basically, granular computing differs from conventional modes of computation in that the objects
of computation are not values of variables but information about values of variables. Furthermore,
information is allowed to be imperfect; i.e., it may be imprecise, uncertain, incomplete, conflicting, or
partially true. It is this facet of granular computing that endows granular computing with a capability
to deal with real-world problems which are beyond the reach of bivalent-logic-based methods which
are intolerant of imprecision and partial truth. In particular, through the use of generalized-constraintbased semantics, granular computing has the capability to compute with information described in natural
language.
Granular computing is based on fuzzy logic. There are many misconceptions about fuzzy logic. To
begin with, fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision. Fuzzy logic
is inspired by two remarkable human capabilities. First, the capability to reason and make decisions
in an environment of imprecision, uncertainty, incompleteness of information, and partiality of truth.
And second, the capability to perform a wide variety of physical and mental tasks based on perceptions,
without any measurements and any computations. The basic concepts of graduation and granulation form
the core of fuzzy logic, and are the principal distinguishing features of fuzzy logic. More specifically,
in fuzzy logic everything is or is allowed to be graduated, i.e., be a matter of degree or, equivalently,
fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a
clump of attribute values drawn together by indistinguishability, similarity, proximity, or functionality.
The concept of a generalized constraint serves to treat a granule as an object of computation. Graduated
granulation, or equivalently fuzzy granulation, is a unique feature of fuzzy logic. Graduated granulation
is inspired by the way in which humans deal with complexity and imprecision.
The concepts of graduation, granulation, and graduated granulation play key roles in granular computing. Graduated granulation underlies the concept of a linguistic variable, i.e., a variable whose values
are words rather than numbers. In retrospect, this concept, in combination with the associated concept of
a fuzzy if–then rule, may be viewed as a first step toward granular computing.
xiv
Foreword
Today, the concept of a linguistic variable is used in almost all applications of fuzzy logic. When I
introduced this concept in my 1973 paper ‘Outline of a new approach to the analysis of complex systems
and decision processes’ [3], I was greeted with scorn and derision rather than with accolades. The derisive
comments reflected a deep-seated tradition in science – the tradition of according much more respect
to numbers than to words. Thus, in science, progress is equated to progression from words to numbers.
In fuzzy logic, in moving from numerical to linguistic variables, we are moving in a countertraditional
direction. What the critics did not understand is that in moving in the countertraditional direction, we
are sacrificing precision to achieve important advantages down the line. This is what is called ‘the fuzzy
logic gambit.’ The fuzzy logic gambit is one of the principal rationales for the use of granular computing.
In sum, to say that the Handbook of Granular Computing is an important contribution to the literature
is an understatement. It is a work whose importance cannot be exaggerated. The coeditors, the authors,
and the publisher, John Wiley, deserve our thanks, congratulations, and loud applause.
Lotfi A. Zadeh
Berkeley, California
References
[1] A. Bargiela and W. Pedrycz. Granular Computing: An Introduction. Kluwer Academic Publishers, Dordercht,
2003.
[2] L.A. Zadeh. Some reflections on soft computing, granular computing and their roles in the conception, design
and utilization of information/intelligent systems. Soft Comput. 2 (1998) 23–25.
[3] L.A. Zadeh. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans.
Syst. Man Cybern. SMC-3 (1973) 28–44.
Biographies
Witold Pedrycz (M’88-SM’90-F’99) received the MSc, PhD, and DSci from the Silesian University of
Technology, Gliwice, Poland. He is a professor and Canada Research Chair in computational intelligence
in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada.
He is also with the Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland.
His research interests encompass computational intelligence, fuzzy modeling, knowledge discovery
and data mining, fuzzy control including fuzzy controllers, pattern recognition, knowledge-based neural
networks, granular and relational computing, and software engineering. He has published numerous
papers in these areas. He is also an author of 11 research monographs. Witold Pedrycz has been a member
of numerous program committees of IEEE conferences in the area of fuzzy sets and neurocomputing. He
serves as an editor-in-chief of IEEE Transactions on Systems Man and Cybernetics – Part A and associate
editor of IEEE Transactions on Fuzzy Systems. He is also an editor-in-chief of information sciences. Dr.
Pedrycz is a recipient of the prestigious Norbert Wiener Award from the IEEE Society of Systems, Man,
and Cybernetics as well as K.S. Fu Award from the North American Fuzzy Information Society.
Andrzej Skowron received the PhD and DSci from the University of Warsaw in Poland. In 1991 he
received the Scientific Title of Professor. He is a Full Professor in the Faculty of Mathematics, Computer
Science and Mechanics at Warsaw University. Andrzej Skowron is the author of numerous scientific
publications and editor of many books and special issues of scientific journals. His areas of expertise
include reasoning with incomplete information, approximate reasoning, soft computing methods and
applications, rough sets, rough mereology, granular computing, synthesis and analysis of complex objects,
intelligent agents, knowledge discovery systems, and advanced data mining techniques, decision support
systems, adaptive and autonomous systems. He was the supervisor of more than 20 PhD theses. He
was also involved in several national and international research and commercial projects relating to
data mining (fraud detection and web mining), control of unmanned vehicles, medical decision support
systems, and approximate reasoning in distributed environments among many others.
Since 1995 he is the editor-in-chief of Fundamenta Informaticae journal and a member of editorial
boards of several others journals including Knowledge Discovery and Data. He is the coeditor-in-chief
of the journal LNCS Transactions on Rough Sets published by Springer. Andrzej Skowron was the
president of the International Rough Set Society from 1996 to 2000. He served or is currently serving on
the program committees of almost 100 international conferences and workshops as program committee
member, program chair, or cochair. He has delivered numerous invited talks at international conferences,
including a plenary talk at the 16th IFIP World Computer Congress (Beijing, 2000). Throughout his
career, Andrzej Skowron has won many awards for his achievements, including awards from the Ministry
of Science, the Rector of Warsaw University, the Ministry of Education, Mazur’s Award of the Polish
Mathematical Society, and Janiszewski’s Award of the Polish Mathematical Society. In 2003 he received
the title of Honorary Professor from Chongqing University of Post and Telecommunication (China). In
2005 he received the ACM Recognition of Service Award for contributions to ACM and the award from
International Rough Sets Society for the outstanding research results.
Dr. Vladik Kreinovich received his MSc in mathematics and computer science from St Petersburg
University, Russia, in 1974 and PhD from the Institute of Mathematics, Soviet Academy of Sciences,
xvi
Biographies
Novosibirsk, in 1979. In 1975–1980, he worked with the Soviet Academy of Sciences, in particular, in
1978–1980, with the Special Astrophysical Observatory (representation and processing of uncertainty in
radioastronomy). In 1982–1989, he worked on error estimation and intelligent information processing
for the National Institute for Electrical Measuring Instruments, Russia. In 1989, he was a visiting scholar
at Stanford University. Since 1990, he is with the Department of Computer Science, University of Texas
at El Paso. Also, he served as an invited professor in Paris (University of Paris VI), Hong Kong, St
Petersburg, Russia, and Brazil.
His main interests include representation and processing of uncertainty, especially interval computations and intelligent control. He has published 3 books, 6 edited books, and more than 700 papers.
He is member of the editorial board of the international journal Reliable Computing (formerly, Interval
Computations) and several other journals. He is also the comaintainer of the international website on
interval computations, http://www.cs.utep.edu/interval-comp.
He is foreign member of the Russian Academy of Metrological Sciences, recipient of the 2003 El Paso
Energy Foundation Faculty Achievement Award for Research awarded by the University of Texas at El
Paso, and a corecipient of the 2005 Star Award from the University of Texas System.
Ren´e Alt is a professor of computer sciences at the Pierre et Marie Curie University in Paris (UPMC). He
received his master diploma in mathematics from UPMC in 1968, the Doctorate in Computer Sciences
(PhD) of UPMC in 1971, and was Docteur es Sciences from UPMC in 1981. He has been professor of
computer sciences at the University of Caen (France) from 1985 to 1991. He was head of the faculty
of computer sciences of UPMC from 1997 to 2001 and vice president of the administrative council of
UPMC from 2002 to 2006. Ren´e Alt’s fields of interest are the numerical solution of differential equations,
computer arithmetic, round-off error propagation, validation of numerical software, parallel computing,
and image processing.
S. Asharaf received the BTech from the Cochin University of Science and Technology, Kerala, and the
Master of Engineering from the Indian Institute of Science, where he is working toward a PhD. His
research interests include data clustering, soft computing, and support vector machines. He is one of the
recipients of IBM best PhD student award in 2006.
Rosangela Ballini received her BSc degree in applied mathematics from the Federal University of S˜ao
Carlos (UFSCar), SP, Brazil, in 1996. In 1998, she received the MSc degree in mathematics and computer
science from the University of S˜ao Paulo (USP), SP, Brazil, and the PhD degree in electrical engineering
from the State University of Campinas (Unicamp), SP, Brazil, in 2000. Currently, she is professor of the
Department of Economic Theory, Institute of Economics (IE), Unicamp. Her research interests include
time series forecasting, neural networks, fuzzy systems, and non-linear optimization.
Mohua Banerjee received her BSc (Hons) degree in mathematics, and the MSc, MPhil, and PhD degrees
in pure mathematics from the University of Calcutta in 1985, 1987, 1989, and 1995, respectively. During
1995–1997, she was a research associate at the Machine Intelligence Unit, Indian Statistical Institute,
Calcutta. In 1997, she joined the Department of Mathematics and Statistics, Indian Institute of Technology,
Kanpur, as lecturer, and is currently Assistant Professor in the same department. She was an associate of
The Institute of Mathematical Sciences, Chennai, India, during 2003–2005. Her main research interests
lie in modal logics and rough sets. She has made several research visits to institutes in India and abroad.
She is a member of the Working Group for the Center for Research in Discrete Mathematics and its
Applications (CARDMATH), Department of Science and Technology (DST), Government of India. She
serves in the reviewer panel of many international journals. Dr. Banerjee was awarded the Indian National
Science Academy Medal for Young Scientists in 1995.
Edurne Barrenechea is an assistant lecturer at the Department of Automatics and Computation, Public
University of Navarra, Spain. Having received an MSc in computer science at the Pais Vasco University
in 1990. She worked as analyst programmer in Bombas Itur from 1990 to 2001 and then she joined the
Public University of Navarra as associate lecturer. She obtained the PhD in computer science in 2005.
Biographies
xvii
Her research interests are fuzzy techniques for image processing, fuzzy sets theory, interval type-2 fuzzy
sets theory, neural networks, and industrial applications of soft computing techniques. She is member of
the European Society for Fuzzy Logic and Technology (EUSFLAT).
Jan G. Bazan is an Assistant Professor in the Institute of Mathematics at the University of Rzeszow in
Poland. He received his PhD degree in 1999 from the University of Warsaw in Poland. His recent research
interests focus on rough set theory, granular computing, knowledge discovery, data mining techniques,
reasoning with incomplete information, approximate reasoning, decision support systems, and adaptive
systems. He is the author or coauthor of more than 40 scientific publications and he was involved in
several national and international research projects relating to fraud detection, web mining, risk pattern
detection, and automated planning of the treatment among other topics.
Taner Bilgi¸c received his BSc and MSc in industrial engineering from the Middle East Technical University, Ankara, Turkey, in 1987 and 1990, respectively. He received a PhD in industrial engineering from
the University of Toronto in 1995. The title of his dissertation is ‘Measurement-Theoretic Frameworks
for Fuzzy Set Theory with Applications to Preference Modelling.’ He spent 2 years at the Enterprise
Integration Laboratory in Toronto as a research associate. Since 1997, he has been a faculty member at
the Department of Industrial Engineering at Bogazici University in Istanbul, Turkey.
Isabelle Bloch is a professor at ENST (Signal and Image Processing Department), CNRS UMR 5141
LTCI. Her research interests include three-dimensional (3D) image and object processing, 3D and fuzzy
mathematical morphology, decision theory, information fusion, fuzzy set theory, belief function theory,
structural pattern recognition, spatial reasoning, and medical imaging.
Gloria Bordogna received her Laurea degree in Physics at the Universit`a degli Studi di Milano, Italy, in
1984. In 1986 she joined the Italian National Research Council, where she presently holds the position
of a senior researcher at the Institute for the Dynamics of Environmental Processes. She is also a contract
professor at the faculty of Engineering of Bergamo University, where she teaches information retrieval and
geographic information systems. Her research activity concerns soft computing techniques for managing
imprecision and uncertainty affecting both textual and spatial information. She is coeditor of a special
issue of JASIS and three volumes published by Springer-Verlag on uncertainty and impression management in databases. She has published over 100 papers in international journals, in the proceedings of
international conferences, and in books. She participated at the program committee of international conferences such as FUZZIEEE, ECIR, ACM SIGIR, FQAS, EUROFUSE, IJCAI2007, ICDE 2007, and ACM
SAC ‘Information Access and Retrieval’ track and served as a reviewer of journals such as JASIST, IEEE
Transactions on Fuzzy Systems, Fuzzy Sets and Systems, and Information Processing and Management.
Giovanni Bortolan received the doctoral degree from the University of Padova, Padova, Italy in 1978. He
is senior researcher at the Institute of Biomedical Engineering, Italian National Research Council (ISIBCNR), Padova, Italy. He has published numerous papers in the areas of medical informatics and applied
fuzzy sets. He is actively pursuing research in medical informatics in computerized electrocardiography,
neural networks, fuzzy sets, data mining, and pattern recognition.
Humberto Bustince is an Associate Professor at the Department of Automatics and Computation, Public
University of Navarra, Spain. He holds a PhD degree in mathematics from Public University of Navarra
from 1994. His research interests are fuzzy logic theory, extensions of fuzzy sets (type-2 fuzzy sets and
Atanassov’s intuitionistic fuzzy sets), fuzzy measures, aggregation operators, and fuzzy techniques for
image processing. He is the author of more than 30 peer-reviewed research papers and is member of
IEEE and European Society for Fuzzy Logic and Technology (EUSFLAT).
Cory J. Butz received the BSc, MSc, and PhD degrees in computer science from the University of
Regina, Saskatchewan, Canada, in 1994, 1996, and 2000, respectively. His research interests include
uncertainty reasoning, database systems, information retrieval, and data mining.
xviii
Biographies
Oscar Castillo was awarded Doctor of Science (DSc) from the Polish Academy of Sciences. He is a
professor of computer science in the Graduate Division, Tijuana Institute of Technology, Tijuana, Mexico.
In addition, he is serving as research director of computer science and head of the research group on fuzzy
logic and genetic algorithms. Currently, he is president of Hispanic American Fuzzy Systems Association
(HAFSA) and vice president of International Fuzzy Systems Association (IFSA) in charge of publicity.
Professor Castillo is also vice chair of the Mexican Chapter of the Computational Intelligence Society
(IEEE). Professor Castillo is also general chair of the IFSA 2007 World Congress to be held in Cancun,
Mexico. He also belongs to the Technical Committee on Fuzzy Systems of IEEE and to the Task Force
on ‘Extensions to Type-1 Fuzzy Systems.’ His research interests are in type-2 fuzzy logic, intuitionistic
fuzzy logic, fuzzy control, neuro–fuzzy, and genetic–fuzzy hybrid approaches. He has published over 60
journal papers, 5 authored books, 10 edited books, and 150 papers in conference proceedings.
Gianpiero Cattaneo is a Full Professor in ‘dynamical system theory’ at the Universit`a di Milano, Bicocca.
Previously, he was an Associate Professor in ‘mathematical methods of physics’ (from 1974 to 1984)
and researcher of ‘theoretical physics’ (from 1968 to 1974). From 1994 to 1997, he was a regular visiting
professor at the London School of Economics (Department of Logic and Scientific Methods), where,
since 1998, he had a position of research associate at ‘The Centre for the Philosophy of Natural and Social
Science.’ From 1997 to 1999, he was Maitre de Conferences at the Nancy-Metz Academy and Maitre de
Conferences at ‘la Ecole Normale Superieure’ in Lyon: Laboratoire de l’Informatique du Parall`elisme.
He is member of the editorial board of the Transactions on Rough Sets, LNCS (Springer-Verlag), the
Scientific Committee of the ‘International Quantum Structures Association (IQSA)’; the International
Advisory Board of the ‘European School of Advanced Studies in Methods for Management of Complex
Systems’ (Pavia); International Federation of Information Processing (IFIP): Working group on cellular
automata. Moreover, he is scientific coordinator of a biannual 2006–2007 ‘Program of International
Collaboration’ between France and Italy, involving the universities of Nice, Marseille, Ecole Normale
Superieure de Lyon, Marne-la-Valle, Milano-Bicocca, and Bologna. He was a member of numerous
program committees of international conferences. His research activities, with results published on
international journals in more than 140 papers, are centered on topological chaos, cellular automata and
related languages, algebraic approach to fuzzy logic and rough sets, axiomatic foundations of quantum
mechanics, and realization of reversible gates by quantum computing techniques.
Davide Ciucci received a PhD in 2004 in computer science from the University of Milan. Since 2005, he
has held a permanent position as a researcher at the University of Milano-Bicocca, where he delivered a
course on fuzzy logic and rough sets. His research interests are about a theoretical algebraic approach to
imprecision, with particular attention to many-valued logics, rough sets, and their relationship. Recently,
he got involved in the semantic web area, with a special interest in fuzzy ontology and fuzzy description
logics. He has been a member committee of several conferences about rough and fuzzy sets, co-organizer
of a special session at the Joint Rough Set Symposium JRS07. His webpages, with a list of publications,
can be found at www.fislab.disco.unimib.it.
Chris Cornelis is a postdoctoral researcher at the Department of Applied Mathematics and Computer Science at Ghent University (Belgium) funded by the Research Foundation – Flanders. His research interests include various models of imperfection (fuzzy rough sets, bilattices and interval-valued
fuzzy sets); he is currently focusing on their application to personalized information access and web
intelligence.
Martine De Cock is a professor at the Department of Applied Mathematics and Computer Science at
Ghent University (Belgium). Her current research efforts are directed toward the development and the
use of computational intelligent methods for next-generation web applications.
E.I. Papageorgiou was born in Larisa in 1975, Greece. She obtained the physics degree in 1997, MSc
in medical physics in 2000, and PhD in computer science in July 2004 from the University of Patras.
From 2004 to 2006, she was a postdoctoral researcher at the Department of Electrical and Computer
Biographies
xix
Engineering, University of Patras (Greece), on developing new models and methodologies based on
soft computing for medical decision support systems. From 2000 to 2006, she was involved in several
research projects related to the development of new algorithms and methods for complex diagnostic
and medical decision support systems. Her main activities were the development of innovative learning
algorithms for fuzzy cognitive maps and intelligent expert systems for medical diagnosis and decisionmaking tasks. From 2004 to 2005, she was appointed as lecturer at the Department of Electrical and
Computer Engineering at the University of Patras. Currently, she is Assistant Professor at the Department
of Informatics and Computer Technology, Technological Educational Institute of Lamia, and adjunct
Assistant Professor at the University of Central Greece. She has coauthored more than 40 journals and
conference papers, book chapters, and technical reports, and has more than 50 citations to her works.
Her interests include expert systems, intelligent algorithms and computational intelligence techniques,
intelligent decision support systems, and artificial intelligence techniques for medical applications. Dr.
E.I. Papageorgiou was a recipient of a scholarship of Greek State Scholarship Foundation ‘I.K.Y.’ during
her PhD studies (2000–2004), and from 2006 to May 2007, she was also a recipient of the postdoctoral
research fellowship from the Greek State Scholarship Foundation ‘I.K.Y.’
Paulo Fazendeiro received the BS degree in mathematics and informatics in 1995 (with honors) and
the equivalent of MS degree in computer science in 2001, all from the University of Beira Interior,
Portugal. He is preparing his dissertation on the relationships between accuracy and interpretability of
fuzzy systems as a partial fulfillment of the requirements for the informatics engineering PhD degree.
He joined the University of Beira Interior in 1995, where he is currently a lecturer in the Informatics
Department. His research interests include application of fuzzy set theory and fuzzy systems, data mining,
evolutionary algorithms, multiobjective optimization, and clustering techniques with applications to
image processing. Dr. Fazendeiro is a member of the Portuguese Telecommunications Institute and the
Informatics Laboratory of the University of Algarve.
Mario Fedrizzi received the MSc degree in mathematics in 1973 from the University of Padua, Italy.
Since 1976, he has been an Assistant Professor; since 1981, an Associate Professor; and since 1986, a Full
Professor with Trento University, Italy. He served as a chairman of the Institute of Informatics from 1985
to 1991 and as a dean of the Faculty of Economics and Business Administration from 1989 to 1995. His
research focused on utility and risk theory, stochastic dominance, group decision making, fuzzy decision
analysis, fuzzy regression analysis, and consensus modeling in uncertain environments, decision support
systems. He has authored or coauthored books and more than 150 papers, which appeared in international
proceedings and journals, e.g., European Journal of Operational Research, Fuzzy Sets and Systems, IEEE
Transactions on Systems, Man and Cybernetics, Mathematical Social Sciences, Quality and Quantity,
and International Journal of Intelligent Systems. He was also involved in consulting activities in the areas
of information systems and DSS design and implementation, office automation, quality control, project
management, expert systems, and neural nets in financial planning. From 1995 to 2006, he was appointed
as chairman of a bank and of a real-estate company, and as a member of the board of directors of Cedacri,
the largest Italian banking information systems outsourcing company, and of Unicredit Banca.
Fernando Antonio Campos Gomide received the BSc degree in electrical engineering from the Polytechnic Institute of the Pontifical Catholic University of Minas Gerais (IPUC/PUC-MG) Belo Horizonte,
B