Rough Set Theory and Granular Computing

302 Pages·2003·14.19 MB·English

by  Zdzislaw Pawlak (auth.)

M. Inuiguchi, S. Hirano, S. Tsumoto (Eds.)
Rough Set Theory and Granular Computing
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Masahiro Inuiguchi
Shoji Hirano
Shusaku Tsumoto (Eds.)
Rough Set Theory and
Granular Computing
Springer
Dr. Masahiro Inuiguchi Prof. Shoji Hirano
Osaka University Prof. Shusaku Tsumoto
Graduate School of Engineering Shimane Medical University
Yamada-Oka, Suita 2-1 Enya-Cho 89-1
565-0871 Osaka 693-8501 Izumo
Japan Shimane-ken
e-mail: [email protected] Japan
e-mail: [email protected]
ISBN 978-3-642-05614-7 ISBN 978-3-540-36473-3 (eBook)
DOI 10.1007/978-3-540-36473-3
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Foreword
After 20 years of pursuing rough set theory and its applications a look on its
present state and further prospects is badly needed. The monograph Rough
Set Theory and Granular Computing edited by Masahiro Inuiguchi, Shoji
Hirano and Shusaku Tsumoto meets this demand. It presents the newest
developments in this area and gives fair picture of the state of the art in this
domain.
Firstly, in the keynote papers by Zdzislaw Pawlak, Andrzej Skowron and
Sankar K. Pal the relationship of rough sets with other important methods
of data analysis - Bayes theorem, neuro computing and pattern recognition -
is thoroughly examined. Next, several interesting generalizations of the the
ory and new directions of research are presented. Furthermore application
of rough sets in data mining, in particular, rule induction methods based
on rough set theory is presented and discussed. Further important issue dis
cussed in the monograph is rough set based data analysis, including study of
decisions making in conflict situations. Last but not least, some recent engi
neering applications of rough set theory are given. They include a proposal of
rough set processor architecture organization for fast implementation of ba
sic rough set operations and discussion of results concerning advanced image
processing for unmanned aerial vehicle.
Thus the monograph beside presenting wide spectrum of ongoing research
in this area also points out new emerging areas of study and applications,
which makes it a valuable source of information to all interested in this do
main.
It is my great pleasure to congratulate the Authors and the Editors on
this excellent monograph.
Warsaw, July, 2002 Zdzislaw Pawlak
Preface
Rough set theory is a computational model of approximate reasoning pro
posed by Z. Pawlak in 1982, which captures a given concept as two approxi
mations represented by measurements.
Pawlak reached this idea from discussions with medical doctors that mea
surements given by human beings may not define correctly a concept given
by nature. Especially, he focused on the characteristics of medical diagnostic
reasoning: one is a set of manifestations, say L, which is sufficient to diag
nose flu. The other one is a set of symptoms observed in patients of flu, say
U. From the viewpoint of set of objects, a set of patients belonging to flu,
denoted by X, has a subset which satisfies L. Also, this set, X, is covered by
U. Pawlak called Land U a lower approximation and upper approximation
of X, both which give approximations of the concept flu. In other words, a
concept can be captured by lower and upper approximaiton of its supporting
set X such that L ~ X ~ U in which Land U satisfies a set of measurements
given by human beings.
This intuitive ideas inspire Pawlak to rough sets, whose paper was first
published in 1982, twenty years ago from now. Since rough set theory gives
the deep insights to uncertainty reasoning, it has been widely spreading over
various fields such as mathematics, logic, information science, decision sci
ence, economics and medical decision support.
Although rough sets have been studied independently of other soft com
puting, L.A. Zadeh gave a new insight into this field that rough set theory is
a crisp-set based granular computing after he proposed information granular
ity in 1996. From his insights, Lin and fuzzy people started their studies on
granular computing. Acutually, Rough sets apply fundamentals of set class
fication, including equivalence classes, as "information granules" to database
analysis, which can be viewed as a crisp-set based granular computing. In
the rough set community, a large amount of fundamental issues have been
intensively studied from this and now rough set theory has become one of
the most advanced area in granular computing.
On the other hand, in granular computing, new theory and methods are
proposed involving not only rough sets but also fuzzy sets, multisets, neural
networks, belief networks, modal logic, rule induction methods.
In twenty years from the first publication, the rough set theory has achieved
not only theoretical advances, but also has been applied to many real world
problems, such as data mining, medical decision support, image analysis and
sensor fusion.
This monograph is composed of 28 articles each of which presents novel
approaches and new results in the theory, methods and applications related to
rough sets and granular computing. The papers are classified into five chap-
VIII
ters. Chapter 1 collects three of key ideas on rough sets are given by three
honorary authors. First, Pawlak gives a new ideas on the correspondence be
tween bayesian reasoning rough sets. Secondly, Skowron introduces the ideas
on approximation space in rough neurocomputing. Finally, Pal presents the
combination of soft computing and pattern recognition. Chapter 2 collects
new theoretical advances in rough sets and granular computing, including
generalization of rough sets and fuzzy multisets, interval probability, intro
duction of fracal dimension into information systems, foundations on granular
computing and approximate bayesian networks. Chapter 3 gives the papers
on application of rough sets and granular computing to data mining, which
includes induction of high order decision rules, theoretical studies on associ
ation rules, semantic association rule induction, rough clustering, detection
of data dependencies, improvement of rule idncution, combination of rough
sets and inductive logic programming, and combination of rough sets and
genetic programming. Chapter 4 presents the papers on application of rough
sets and granular computing to conflict analysis and data analysis, including
rough set based conflict analysis, conflicts resolving, fuzzy clustering, rough
set based possibility distributions, interval pairwise comparisons. Chapter 5
collects the papers on application of rough sets and granular computing to
real world problems, which includes sensor fusion, rough set processor, adapt
able components in software engineering and analysis of image sequences.
Finally, we thank all the authors for their contributions to this project.
Especially, we are much obliged to Professors Zdzislaw Pawlak, Andrzej
Skowron, Lech Polkowski and Janusz Kacprzyk for their help and comments
on editing this book.
Suita and Izumo, Japan, Masahiro Inuiguchi
November 2002 Shoji Hirano
Shusaku Tsumoto
Contents
Bayes' Theorem - the Rough Set Perspective. . . . . . . . . . . . . . . . . 1
Zdzislaw Pawlak
1 Introduction................................................. 1
2 Bayes' Theorem ............................................. 2
3 Information Systems and Approximation of Sets . . . . . . . . . . . . . . . . . 2
4 Decision Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5 Decision Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6 Decision Rules in Information Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 6
7 Properties of Decision Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
8 Decision Tables and Flow Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
9 Illustrative Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
10 Conclusion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
Approximation Spaces in Rough Neurocomputing ... . . . . . . ... 13
Andrzej Skowron
1 Introduction................................................. 13
2 Approximation Spaces in Rough Set Theory. . . . . . . . . . . . . . . . . . . .. 14
3 Generalizations of Approximation Spaces ....................... 15
4 Information Granule Systems and Approximation Spaces. . . . . . . . .. 16
5 Classifiers as Information Granules. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18
6 Approximation Spaces for Information Granules ................. 19
7 Approximation Spaces in Rough-Neuro Computing. . . . . . . . . . . . . .. 20
8 Conclusion.................................................. 21
References ..................................................... 22
Soft Computing Pattern Recognition: Principles, Integrations
and Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
Sankar K. Pal
1 Introduction................................................. 23
2 Relevance of Fuzzy Set Theory in Pattern Recognition. . . . . . . . . . .. 25
3 Relevance of Neural Network Approaches. . . . . . . . . . . . . . . . . . . . . .. 27
4 Genetic Algorithms for Pattern Recognition . . . . . . . . . . . . . . . . . . . .. 28
5 Integration and Hybrid Systems ............................... 29
6 Evolutionary Rough Fuzzy MLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30
7 Data mining and knowledge discovery. . . . . . . . . . . . . . . . . . . . . . . . .. 31
References ..................................................... 33
x
Part I. Generalizations and New Theories
Generalization of Rough Sets Using Weak Fuzzy Similarity
Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
Rolly Intan, Y. Y. Yao, Masao Mukaidono
1 Introduction................................................. 37
2 Weak Fuzzy Similarity Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
3 Generalized Rough Set Approximations. . . . . . . . . . . . . . . . . . . . . . . .. 41
4 Generalized Rough Membership Functions ...................... 43
5 An Illustrative Example ...................................... 44
6 Conclusions................................................. 46
References ..................................................... 46
Two Directions toward Generalization of Rough Sets ........ 47
Masahiro Inuiguchi and Tetsuzo Tanino
1 Introduction................................................. 47
2 The Original Rough Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
3 Distinction among Positive, Negative and Boundary Elements ..... 50
4 Approximations by Means of Elementary Sets ................... 54
5 Concluding Remarks ......................................... 56
References ..................................................... 56
Two Generalizations of Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
Sadaaki Miyamoto
1 Introduction................................................. 59
2 Preliminaries................................................ 60
3 Infinite Memberships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62
4 Generalization of Membership Sequence. . . . . . . . . . . . . . . . . . . . . . . .. 64
5 Conclusion.................................................. 67
References ..................................................... 67
Interval Probability and Its Properties ....................... 69
Hideo Tanaka, Kazutomi Sugihara, Yutaka Maeda
1 Introduction................................................. 69
2 Interval Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
3 Combination and Conditional Rules for IPF . . . . . . . . . . . . . . . . . . . .. 74
4 Numerical Example of Bayes' Formula. . . . . . . . . . . . . . . . . . . . . . . . .. 75
5 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
References ..................................................... 77
On Fractal Dimension in Information Systems . . . . . . . . . . . . . . .. 79
Leeh Polkowski
1 Introduction................................................. 79
2 Fractal Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
XI
3 Rough Sets and Topologies on Rough Sets ...................... 81
4 Fractals in Information Systems ............................... 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86
A Remark on Granular Reasoning and Filtration. . . . . . . . . . . .. 89
Tetsuya Murai, Michinori Nakata, Yoshiharu Sato
1 Introduction................................................. 89
2 Kripke Semantics and Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
3 Relative Filtration with Approximation. . . . . . . . . . . . . . . . . . . . . . . .. 92
4 Relative Filtration and Granular Reasoning . . . . . . . . . . . . . . . . . . . .. 94
5 Concluding Remarks ......................................... 96
References ..................................................... 96
Towards Discovery of Relevant Patterns from Parameterized
Schemes of Information Granule Construction. . . . . . . . . . . . . . .. 97
Andrzej Skowron, Jaroslaw Stepaniuk, James F. Peters
1 Introduction................................................. 97
2 Approximation Granules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
3 Rough-Fuzzy Granules ....................................... 101
4 Granule Decomposition ....................................... 103
References ..................................................... 106
Approximate Markov Boundaries and Bayesian Networks: Rough
Set Approach ................................................. 109
Dominik Slt;zak
1 Introduction ................................................. 109
2 Data Based Probabilistic Models ............................... 110
3 Approximate Probabilistic Models ............................. 115
4 Conclusions ................................................. 120
References ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Part II. Data Mining and Rough Sets
Mining High Order Decision Rules ........................... 125
Y. Y. Yao
1 Introduction ................................................. 125
2 Motivations ................................................. 126
3 Mining High Order Decision Rules ............................. 128
4 Mining Ordering Rules: an Illustrative Example .................. 131
5 Conclusion .................................................. 134
References ..................................................... 134
Association Rules from a Point of View of Conditional Logic . 137
Tetsuya Murai, Michinori Nakata, Yoshiharu Sato
1 Introduction ................................................. 137
2 Preliminaries ................................................ 137