Soft Methodology and Random Information Systems PDF
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Soft Methodology and Random Information Systems Advances in Soft Computing Editor-in-chief Prof.JanuszKacprzyk SystemsResearchInstitute PolishAcademyofSciences ul.Newelska6 01-447Warsaw,Poland E-mail:[email protected] Furtherbooksofthisseriescanbefoundonourhomepage:springeronline.com BerndReuschandKarl-HeinzTemme(Eds.) MieczysławKłopotek,MaciejMichalewicz ComputationalIntelligenceinTheoryandPractice andSławomirT.Wierzchoń(Eds.) 2000.ISBN3-7908-1357-5 IntelligentInformationSystems2002 2002.ISBN3-7908-1509-8 RainerHampel,MichaelWagenknecht, NasredinChaker(Eds.) AndreaBonarini,FrancescoMasulli FuzzyControl andGabriellaPasi(Eds.) 2000.ISBN3-7908-1327-3 SoftComputingApplications 2002.ISBN3-7908-1544-6 HenrikLarsen,JanuszKacprzyk, SławomirZadrozny,TroelsAndreasen, LeszekRutkowski,JanuszKacprzyk(Eds.) HenningChristiansen(Eds.) NeuralNetworksandSoftComputing FlexibleQueryAnsweringSystems 2003.ISBN3-7908-0005-8 2000.ISBN3-7908-1347-8 JürgenFranke,GholamrezaNakhaeizadeh, RobertJohnandRalphBirkenhead(Eds.) IngridRenz(Eds.) DevelopmentsinSoftComputing TextMining 2001.ISBN3-7908-1361-3 2003.ISBN3-7908-0041-4 MieczysławKłopotek,MaciejMichalewicz TetsuzoTanino,TamakiTanaka, andSławomirT.Wierzchoń(Eds.) MasahiroInuiguchi IntelligentInformationSystems2001 Multi-ObjectiveProgrammingandGoal 2001.ISBN3-7908-1407-5 Programming 2003.ISBN3-540-00653-2 AntonioDiNolaandGiangiacomoGerla(Eds.) LecturesonSoftComputingandFuzzyLogic MieczysławKłopotek,SławomirT.Wierzchoń, 2001.ISBN3-7908-1396-6 KrzysztofTrojanowski(Eds.) IntelligentInformationProcessingandWebMining TadeuszTrzaskalikandJerzyMichnik(Eds.) 2003.ISBN3-540-00843-8 MultipleObjectiveandGoalProgramming 2002.ISBN3-7908-1409-1 AhmadLotfi,JonathanM.Garibaldi(Eds.) ApplicationsandScienceinSoft-Computing JamesJ.BuckleyandEsfandiarEslami 2004.ISBN3-540-40856-8 AnIntroductiontoFuzzyLogicandFuzzySets 2002.ISBN3-7908-1447-4 MieczysławKłopotek,SławomirT.Wierzchoń, KrzysztofTrojanowski(Eds.) AjithAbrahamandMarioKöppen(Eds.) IntellligentInformationProcessingand HybridInformationSystems WebMining 2002.ISBN3-7908-1480-6 2004.ISBN3-540-21331-7 PrzemysławGrzegorzewski,OlgierdHryniewicz, MiguelLópez-Díaz,María Á .Gil,Przemysław María Á .Gil(Eds.) Grzegorzewski,OlgierdHryniewicz,Jonathan SoftMethodsinProbability,Statistics Lawry andDataAnalysis SoftMethodologyandRandomInformation 2002.ISBN3-7908-1526-8 Systems 2004.ISBN3-540-22264-2 LechPolkowski RoughSets 2002.ISBN3-7908-1510-1 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 This work is subject to copyright. All rights are reserved, whether the whole or part oft he material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions ofthe German Copyright Law ofS eptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to Prosecution under the German Copyright Law. © 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 and regulations and free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting: Digital data supplied by the authors Printed on acid-free paper 62/3020Rw-5 4 3 2 10 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
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