Soft Methodology and Random Information Systems

772 Pages·2004·41.192 MB·English

by  Professor Dr. Miguel López-Díaz, Professor Dr. María Á. Gil, Dr. Przemysław Grzegorzewski, Professor Dr. Olgierd Hryniewicz, Professor Dr. Jonathan Lawry (auth.)

Soft Methodology and Random Information Systems
Advances in Soft Computing
Editor-in-chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw,Poland
E-mail:[e-mail]
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María Á .Gil(Eds.) Grzegorzewski,OlgierdHryniewicz,Jonathan
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Miguel Lopez-Diaz
Maria A. Gil
Przemyslaw Grzegorzewski
Olgierd Hryniewicz
Jonathan Lawry
Soft Methodology and
Random Information Systems
With 64 Figures
and 43 Tables
cajAstur'=
~Springer
Professor Dr. Miguel L6pez-Diaz
Professor Dr. Marla A. Gil
Universidad de Oviedo
Fac. Ciencias
Dpto. Estadistica e I.O.y.D.M.
calle Calvo Sotelo s/n
33071 Oviedo, Spain
Dr. Przemyslaw Grzegorzewski
Professor Dr. Olgierd Hryniewicz
PAN Warszawa
Inst. Systems Research
ul. N ewelska 6
01-447 Warszawa, Poland
Professor Dr. Jonathan Lawry
University Bristol
Dept. Engineering Mathematics
University Walk
BSS 1T R Bristol, United Kingdom
Library of Congress Control Number: 2004109049
ISSN 16-15-3871
ISBN 978-3-540-22264-4 ISBN 978-3-540-44465-7 (eBook)
DOI 10.1007/978-3-540-44465-7
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© Springer-Verlag Berlin Heidelberg 2004
Originally published by Springer-Verlag Berlin Heidelberg in 2004
The use of general descriptive names, registered names, etc.in this publication does not imply, even
in the absence of a specific statement, that such names are exempt from the relevant protective laws
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Preface
The analysis of experimental data resulting from some underlying random
process, is a fundamental part of most scientific research. Probability Theory
and Statistics have been developed as flexible tools for this analysis, and have
been applied successfully in many fields including Biology, Economics, Engi
neering, Medicine, Psychology, and others. However, traditional techniques in
Probability and Statistics were devised to model only a single source of uncer
tainty, namely randomness. In many real-life problems randomness may arise
in conjunction with uncertainty from other sources, making the development
of additional approaches essential.
Soft Computing is a field integrating a number of different methodolo
gies, including Fuzzy Logic, Neuro-Computing and Evolutionary Computing.
The use of soft techniques in Probability and Statistics allows the integrated
modelling of randomness and other types of uncertainty.
This volume is a collection of papers presented at the 2nd International
Conference on Soft methods in Probability and Statistics (SMPS'2004) held
in Oviedo and organized by the Department of Statistics, OR and DM of the
University of Oviedo. As such it provides an overview of the innovative new
research taking place within this emerging field.
The volume is divided into five parts. Part I contains papers by the six
keynote speakers. These include a paper by Lotfi Zadeh discussing the suit
ability of a Probability Theory based on Fuzzy Logic; two contributions on
random sets by Ilya Molchanov and Hung Nguyen, the former referring to their
expectation and the application to Image Analysis of these random elements,
and the latter on the convergence of distributions in terms of the Choquet
Integral; a survey on traditional limit theorems in Probability Theory for in
dependent Random Fuzzy Sets by Yukio Ogura and Shoumei Li; a paper by
Dan Ralescu on the applications of Fuzzy Theory to Decision Making involv
ing fuzzy probabilities and fuzzy data; an analysis of the application of fuzzy
knowledge-based model in a financial context, by Bonissone; a contribution
by Aloscha Klose and Rudolf Kruse reviewing current trends and challenges
of Data Mining.
VI Preface
Part II is devoted to topics connecting Soft Methodology and Probability.
Specifically, these relate to Probability Theory of different types of generalized
Random Elements (e.g. random sets, random fuzzy sets, and so on), Imprecise
Probabilities, Copulas, Conditioning, Stochastic Processes, Fuzzy Measures
and Integration.
Part III focuses on the area of Soft Methodology and Statistics/Operations
Research, and includes topics such as Statistical Information Theory, Statistics
with Imprecise Data, Data Analysis, Decision Analysis, Optimization and
Networks.
Part IV deals with the development of mathematical tools for this soft
methodology with either immediate or potential stochastic implications.
Finally, Part V presents a number of diverse applications combining Soft
and Stochastic Methodologies in fields ranging from Biomedicine, Business,
Economics, Electricity and Geology to Hydrology.
The editors are very grateful to the contributing authors, as well as to
all the Program Committee members and referees of the SMPS'2004 confer
ence who took part in the reviewing process. They would also like to thank the
editorial staff of Springer-Verlag and the members of the Organizing Commit
tee for their hard work which made it possible to complete this book within
the limited time available. They are also especially grateful to Professor M.
Asuncion Lubiano for her tireless assistance in supervising this volume and
for her meticulous proofreading and patient care in handling and unifying the
presentation of all the papers in it.
The Editors also thank those who made the SMPS'2004 Conference possi
ble, especially Professors Lotfi Zadeh and Janusz Kacprzyk (for their patron
age of the SMPS series of conferences), the University of Oviedo (hosting the
Conference), and the Consejerfa de Educaci6n y Cultura of the Principality
of Asturias, the Spanish Ministry of Education and Science, the Consulting
Neo-Metrics (founded by alumni of Oviedo's Department of Statistics, OR
and DM), and the main Saving Bank in Asturias, cajAstur (through the Obra
Social and Cultural) for their financial support.
The publication of this volume has been made possible by the generous
financial support of cajAstur, who covered all the production and distribution
costs. This initiative is a valuable support to scientific research which is deeply
acknowledged.
Oviedo, May 2004 Miguel L6pez-Dfaz
Marfa Angeles Gil
Przemyslaw Grzegorzewski
Olgierd Hryniewicz
Jonathan Lawry
Contents
Part I Keynote Papers
Probability Theory Should be Based on Fuzzy Logic -
A Contentious View
Lotfi A. Zadeh................................................... 3
Applications of Random Sets in Image Analysis.
How to Average a Cat and a Dog?
flya S. Molchanov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Choquet Weak Convergence of Capacity Functionals
of Random Sets
Hung T. Nguyen ................................................. 19
On Limit Theorems for Random Fuzzy Sets Including
Large Deviation Principles
Yukio Ogura and Shoumei Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Fuzzy Probability and Some Applications
Dan A. Ralescu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Development and Maintenance of Fuzzy Models
in Financial Applications
Piero P. Bonissone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Information Mining with Semi-Supervised Learning
Aljoscha Klose and Rudolf Kruse .................................. 67
Part II Soft Methodology and Probability
On Statistical Inference with Random Sets
Ding Feng and Hung T. Nguyen ................................... 77
VIII Contents
Interval-Valued and Fuzzy-Valued Random Variables: From
Computing Sample Variances
to Computing Sample Covariances
Jan B. Beck, Vladik Kreinovich and Berlin Wu ...................... 85
Estimability and Admissibility in Sample Survey Models
Elbert A. Walker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Riesz Space and Fuzzy Upcrossing Theorems
Wen-Chi Kuo, Coenraad C.A. Labuschagne and Bruce A. Watson ..... 101
On Laws of Large Numbers for Exchangeable Random Sets
Hiroshi Inoue .................................................... 109
Strong and Weak Laws of Large Numbers for Weighted
Sums of Fuzzy Set-Valued Random Variables
Li Guan and Shoumei Li .......................................... 117
Assessing Debris Flow Hazard by Credal Nets
Alessandro Antonucci, Andrea Salvetti and Marco Zaffalon ............ 125
A Possibilistic Interpretation of the Expectation of a Fuzzy
Random Variable
Ines Couso, Enrique Miranda and Gert de Cooman .................. 133
Learning and Optimal Control of Imprecise Markov Decision
Processes by Dynamic Programming Using the Imprecise
Dirichlet Model
Matthias C.M. Troffaes ........................................... 141
Representation of Incomplete Probabilistic Information
Cedric Baudrit, Didier Dubois and Helene Fargier .................... 149
Range of Entropy for Credal Sets
Joaquin Abellan and Serafin Moral ................................. 157
From the Comparison of Ordered Lists to the Construction
of Copulas
Bernard De Baets and Hans De Meyer ............................. 165
Compatibility of Three Bivariate Quasi-Copulas:
Applications to Copulas
Jose Antonio Rodriguez-Lallena and Manuel Ubeda-Flores ............. 173
Transformations of Copulas and Quasi-Copulas
Erich Peter Klement, Radko Mesiar and Endre Pap .................. 181
Compositions of Copulas and Quasi-Copulas
Fabrizio Durante and Carlo Sempi ................................. 189
Contents IX
Countably Generated Idempotent Copulas
Angela Albanese and Carlo Sempi .... ·. ............................. 197
1-Lipschitz Aggregation Operators, Quasi-Copulas
and Copulas with Given Diagonals
Erich Peter Klement and Anna Kolesarova .......................... 205
Few Remarks Concerning a Concept to Define
Normally Distributed Random Fuzzy Sets
Volker Kriitschmer and Joachim Oberlinger ......................... 212
Fuzzy Integrals on Partially-Known Measures
M. Jorge Bolanos and Fernando Reche ............................. 219
Characterization of the Scalar (s,t)-Additive Measures:
Probabilities and Cardinalities
Jaume Casasnovas ............................................... 227
On Pareto Process
K. Jayakumar ................................................... 235
Representation of Probabilities on IFS Events
Beloslav Riecan .................................................. 243
Possibilistic Time Processes and Soft Decoding
Andrea Sgarro ................................................... 249
Some Aspects of Conditioning in a Coherent Setting
Giulianella Coletti and Romano Scozzafava .......................... 257
The Role of Zero Probabilities in Dealing
with Zero Frequency Problems
Romano Scozzafava and Barbara Vantaggi ........................... 265
Generalized Stochastic Transitivity for Probabilistic Relations
Hans De Meyer and Bernard De Baets ............................. 273
The Concept of Non-Symmetric Independence on
MY-Algebras
Martin Kalina and Olga Nanasiova ................................. 281
On Limit Theorems for t-Normed Sums
of Fuzzy Random Variables
Pedro Teran ..................................................... 289
X Contents
Part III Soft Methodology and Statistics/OR
Phi-Residual Based on Minimum Phi-Divergence Estimator
in the Loglinear Model of Symmetry
Arjun K. Gupta, True T. Nguyen and Leandro Pardo ................. 299
Bahadur Efficiency of the Phi-Divergence Test Statistic
Leandro Pardo ................................................... 307
Digitalization of Observations Permits Efficient Estimation
in Continuous Models
Domingo Morales, Leandro Pardo and Igor Vajda .................... 315
Kef> Divergence Statistics in Multinomial Populations
Teresa Perez and Julio Angel Pardo ................................ 323
On the Selection of the Best Rukhin's Statistic
for the Uniform Exact Distribution Function
Yolanda Marhuenda, Domingo Morales, Julio Angel Pardo
and Maria del Carmen Pardo ...................................... 331
On Testing Hypotheses with Divergence Statistics
Isabel Molina and Domingo Morales ................................ 339
Testing a Hierarchical Sequence of Linear Logistic Models
Julio Angel Pardo and Maria del Carmen Pardo .................... 347
Inference for Three-Dimensional Contingency Tables
Based on (PI-Divergences
Julio Angel Pardo ................................................ 355
Information Theory and Complexity in Probability 'and
Statistics
Flemming Tops{iJe ................................................ 363
Measures of Multivariate Association and Dependence
Apostolos Batsidis and Konstantinos Zografos ....................... 371
Phi-Divergence Test Statistics for Trends in Binary Responses
Maria Luisa Menendez ........................................... 379
A Random Set View on Fuzzy Random Variables
Wolfgang Niither ................................................. 387
Fuzzy Information and Bayesian Statistics
Dietmar Hareter and Reinhard Viertl ............................... 392