Huynh / Nakamori / Ono Interval / Probabilistic Uncertainty and Non-classical Logics

2008
ISBN: 978-3-540-77664-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 46, 376 Seiten

Reihe: Advances in Intelligent and Soft Computing

This book contains the proceedings of the first International Workshop on Interval/Probabilistic Uncertainty and Non Classical Logics, Ishikawa, Japan, March 25-28, 2008. The workshop brought together researchers working on interval and probabilistic uncertainty and on non-classical logics. It is hoped this workshop will lead to a boost in the much-needed collaboration between the uncertainty analysis and non-classical logic communities, and thus, to better processing of uncertainty.

Autoren/Hrsg.

Huynh, Van-Nam

Nakamori, Yoshiteru

Ono, Hiroakira

Lawry, Jonathan

Kreinovich, Vkladik

Kreinovich, Vladik

Nguyen, Hung T.

Inhaltsverzeichnis

1;Preface;6
2;Contents;12
3;List of Contributors;16
4;Part I Keynote Addresses;21
4.1;An Algebraic Approach to Substructural Logics – An Overview;22
4.2;On Modeling of Uncertainty Measures and Observed Processes;24
4.2.1;Introduction;24
4.2.2;Some Uncertainty Measures Derived from Coarse Data;25
4.2.2.1;Belief Functions;25
4.2.2.2;Possibility Measures;27
4.2.3;Canonical Borel-$\sigma$ Fields and Continuous Lattices;28
4.2.4;Discrete Random Sets;29
4.2.5;The Space of Closed Sets as a Continuous Lattice;30
4.2.6;The Space of USC Functions as a Continuous Lattice;31
4.2.7;Concluding Remarks;33
4.2.8;References;33
5;Part II Statistics under Interval Uncertainty and Imprecise Probability;36
5.1;Fast Algorithms for Computing Statistics under Interval Uncertainty: An Overview;38
5.1.1;Computing Statistics Is Important;38
5.1.2;Interval Uncertainty;39
5.1.3;Estimating Statistics Under Interval Uncertainty: A Problem;40
5.1.4;Mean;40
5.1.5;Variance: Computing the Exact Range Is Difficult;40
5.1.6;Linearization;41
5.1.7;Linearization Is Not Always Acceptable;41
5.1.8;First Class: Narrow Intervals;42
5.1.9;Second Class: Slightly Wider Intervals;43
5.1.10;Third Class: Single Measuring Instrument;43
5.1.11;Fourth Class: Several MI;44
5.1.12;Fifth Class: Privacy Case;44
5.1.13;Sixth Class: Non-detects;45
5.1.14;Results;46
5.1.15;Conclusion;47
5.1.16;References;47
5.2;Trade-Off between Sample Size and Accuracy: Case of Static Measurements under Interval Uncertainty;51
5.2.1;General Formulation of the Problem;51
5.2.2;In Different Practical Situations, This General Problem Can Take Different Forms;52
5.2.3;A Realistic Formulation of the Trade-Off Problem;53
5.2.4;Solving the Trade-Off Problem in the General Case;55
5.2.5;How Does the Cost of a Measurement Depend on Its Accuracy?;56
5.2.6;Trade-Off between Accuracy and Sample Size in Different Cost Models;60
5.2.7;Conclusion;62
5.2.8;References;62
5.3;Trade-Off between Sample Size and Accuracy: Case of Dynamic Measurements under Interval Uncertainty;64
5.3.1;Formulation of the Problem;64
5.3.2;First Objective: Measuring the Average Value of a Varying Quantity;65
5.3.3;Second Objective: Measuring the Actual Dependence of the Measured Quantity on Space Location and/or on Time;68
5.3.4;Case Study: In Brief;73
5.3.5;Conclusions;74
5.3.6;References;74
5.4;Estimating Quality of Support Vector Machines Learning under Probabilistic and Interval Uncertainty: Algorithms and Computational Complexity;76
5.4.1;Formulation of the Problem;76
5.4.2;How to Take into Account Probabilistic and Interval Uncertainty: Formulation of the Problem and Linearized Algorithms for Solving This Problem;81
5.4.3;In General, Estimating Quality of SVM Learning under Interval Uncertainty Is NP-Hard;85
5.4.4;Conclusion;87
5.4.5;References;88
5.5;Imprecise Probability as an Approach to Improved Dependability in High-Level Information Fusion;89
5.5.1;Introduction;89
5.5.2;Information Fusion;90
5.5.3;High-Level Information Fusion;91
5.5.3.1;Level 2 -- Situation Assessment;91
5.5.3.2;Level 3 -- Impact Assessment;93
5.5.4;Dependable High-Level Information Fusion;93
5.5.4.1;High-Level Information Fusion as a Service;94
5.5.4.2;Reliability;94
5.5.4.3;Fault;95
5.5.4.4;Safety;95
5.5.5;Imprecise Probability - Dependable High-Level Information Fusion;97
5.5.6;Application Domains;98
5.5.6.1;Defense;99
5.5.6.2;Manufacturing;99
5.5.6.3;Precision Agriculture;100
5.5.7;Discussion and Future Work;100
5.5.8;Conclusions;101
5.5.9;References;101
6;Part III Uncertainty Modelling and Reasoning in Knowledge-Based Systems;104
6.1;Label Semantics as a Framework for Granular Modelling;106
6.1.1;Introduction to Granular Modelling;106
6.1.2;Underlying Philosophy of Vagueness;107
6.1.3;Label Semantics;109
6.1.3.1;Ordering Labels;112
6.1.4;Granular Models in Label Semantics;114
6.1.4.1;Mass Relational Models;115
6.1.4.2;Linguistic Decision Trees;116
6.1.4.3;Linguistic Attribute Hierarchies;118
6.1.5;Conclusions;120
6.1.6;References;120
6.2;Approximating Reasoning for Fuzzy-Based Information Retrieval;122
6.2.1;Introduction;122
6.2.2;Fuzzy Retrieval Framework;123
6.2.3;Fuzzy Representations of Uncertainty Objects and Queries;124
6.2.4;Fuzzy Retrieval;125
6.2.4.1;Proposition Extraction;125
6.2.4.2;Approximating Reasoning;127
6.2.5;Fuzzy-Based Additional Relation Discovery;128
6.2.6;Performance Evaluation;130
6.2.7;Conclusion;132
6.2.8;References;132
6.3;Probabilistic Constraints for Inverse Problems;134
6.3.1;Introduction;134
6.3.2;Inverse Problems;135
6.3.3;Continuous Constraint Satisfaction Problems;136
6.3.3.1;Constraint Approach to Inverse Problems;137
6.3.4;Probabilistic Reasoning;139
6.3.4.1;Probabilistic Approach to Inverse Problems;139
6.3.5;Probabilistic Interval Computations;140
6.3.6;Probabilistic Constraint Reasoning;141
6.3.6.1;Probabilistic Constraint Approach to Inverse Problems;143
6.3.7;Conclusions and Future Work;146
6.3.8;References;146
6.4;The Evidential Reasoning Approach for Multi-attribute Decision Analysis under Both Fuzzy and Interval Uncertainty;148
6.4.1;Introduction;148
6.4.2;The FIER Approach for MADA under Fuzzy Uncertainty;149
6.4.2.1;The New FIER Distributed Modelling Framework using the Fuzzy Belief Structure;149
6.4.2.2;The New FIER Algorithm under Both Interval Probabilistic and Fuzzy Uncertainties;151
6.4.3;Fuzzy Expected Utilities for Characterising Alternatives;153
6.4.4;Application of the FIER Approach to a New Product Screening Problem;155
6.4.5;Concluding Remarks;157
6.4.6;References;158
6.5;Modelling and Computing with Imprecise and Uncertain Properties in Object Bases;160
6.5.1;Introduction;160
6.5.2;Combination of Probabilities and Fuzzy Sets;162
6.5.2.1;Probabilistic Interpretation of Relations on Fuzzy Sets;162
6.5.2.2;Algebra on Fuzzy Probabilistic Triples;163
6.5.3;Fuzzy and Probabilistic Object Properties;164
6.5.3.1;FPOB Class Hierarchy;164
6.5.3.2;FPOB Attributes and Methods;165
6.5.3.3;FPOB Schema;166
6.5.4;Fuzzy and Probabilistic Object Base Instances and Class Extents;168
6.5.4.1;FPOB Instances;168
6.5.4.2;Probabilistic Extents of Classes;169
6.5.5;Selection Operation on Fuzzy and Probabilistic Object Bases;169
6.5.5.1;Syntax of Selection Conditions;169
6.5.5.2;Semantics of Selection Conditions;170
6.5.6;FPDB4O: A Fuzzy and Probabilistic Object Base Management System;172
6.5.6.1;Overview of FPDB4O;172
6.5.6.2;Implementation of FPOB Types and Schemas;173
6.5.6.3;Implementation of FPOB Instances;174
6.5.6.4;Implementation of FPOB Selection Operation;175
6.5.7;Conclusion;176
6.5.8;References;177
7;Part IV Rough Sets and Belief Functions;180
7.1;Several Reducts in Dominance-Based Rough Set Approach;182
7.1.1;Introduction;182
7.1.2;Dominance-Based Rough Set Approach;183
7.1.2.1;Decision Table with Dominance Relations;183
7.1.2.2;DRSA;184
7.1.2.3;VP-DRSA;186
7.1.3;Union-Based Reducts in DRSA;187
7.1.3.1;Discernibility Matrices for Reducts;189
7.1.4;Union-Based Reducts in VP-DRSA;190
7.1.5;Concluding Remarks;193
7.1.6;References;193
7.2;Topologies of Approximation Spaces of Rough Set Theory;195
7.2.1;Introduction;195
7.2.2;Preliminaries;195
7.2.2.1;Relations;195
7.2.2.2;Topologies;196
7.2.2.3;Uniformities;197
7.2.3;Definability in Rough Set Theory;197
7.2.3.1;Definability Based on Equivalences;197
7.2.3.2;Definability Based on Tolerances;198
7.2.3.3;Definability Based on Preorders;200
7.2.3.4;General Case;201
7.2.4;Topologies of Approximation Spaces;202
7.2.5;Hammer's Extended Topology;203
7.2.6;Conclusions;205
7.2.7;References;205
7.3;Uncertainty Reasoning in Rough Knowledge Discovery;206
7.3.1;Introduction;206
7.3.2;A Rough Sets-Based Method for Rule Induction;207
7.3.2.1;Preliminary;207
7.3.2.2;Decision Rules;208
7.3.2.3;An Algorithm for Computing Multiple Reducts;210
7.3.2.4;Rough Sets for Rule Induction;212
7.3.3;Uncertainty Reasoning for Classification;213
7.3.3.1;Matching Process;213
7.3.3.2;Dempster-Shafer (DS) Theory of Evidence;215
7.3.3.3;Defining Rule Mass Function;215
7.3.3.4;An Example;217
7.3.4;Conclusion;218
7.3.5;References;218
7.4;Semantics of the Relative Belief of Singletons;220
7.4.1;Introduction: A New Bayesian Approximation;220
7.4.1.1;Previous Work on Bayesian Approximation;220
7.4.1.2;Relative Belief of Singletons;221
7.4.1.3;Outline of the Paper;222
7.4.2;A Conservative Estimate;222
7.4.3;Dual Interpretation as Relative Plausibility of a Plausibility;223
7.4.3.1;Pseudo Belief Functions;223
7.4.3.2;Duality between Relative Belief and Plausibility;224
7.4.4;On the Existence Constraint;225
7.4.4.1;Example: The Binary Case;225
7.4.4.2;Region Spanned by a Bayesian Approximation;226
7.4.4.3;Zero Mass to Singletons as a Pathological Situation;226
7.4.5;A Low-Cost Proxy for other Bayesian Approximations;228
7.4.5.1;Convergence under Quasi-bayesianity;228
7.4.5.2;Convergence in the Ternary Case;229
7.4.6;Conclusions;231
7.4.7;References;231
7.5;A Lattice-Theoretic Interpretation of Independence of Frames;233
7.5.1;Introduction;233
7.5.2;Independence of Sources in Dempster's Combination;234
7.5.2.1;Dempster's Combination of Belief Functions;234
7.5.2.2;Independence of Sources;234
7.5.2.3;Independence of Sources and Independence of Frames;235
7.5.2.4;An Algebraic Study of Independence;237
7.5.3;The Semi-modular Lattice of Frames;237
7.5.3.1;Lattices;237
7.5.3.2;Semi-modularity of the Lattice of Frames;238
7.5.3.3;Finite Lattice of Frames;239
7.5.3.4;A Lattice-Theoretic Interpretation of Independence;239
7.5.4;Independence on Lattices and Independence of Frames;240
7.5.4.1;Independence on Lattices;240
7.5.4.2;Lattice-Theoretic Independence on the Lattice of Frames;241
7.5.4.3;Evidential Independence Is Stronger than $\mathcal{I}^*_1$, $\mathcal{I}^*_2$;241
7.5.4.4;Evidential Independence Is Opposed to $\mathcal{I}^*_3$;243
7.5.5;Comments and Conclusions;244
7.5.6;References;245
8;Part V Non-classical Logics;248
8.1;Completions of Ordered Algebraic Structures: A Survey;250
8.1.1;Introduction;250
8.1.2;Preliminaries;251
8.1.3;Completion Methods;252
8.1.3.1;A General Template for Completions;254
8.1.3.2;Extending Additional Operations;255
8.1.4;Preservation of Identities;256
8.1.5;Comparing Completions;258
8.1.6;Exploring the Boundaries;259
8.1.7;Conclusions and Discussion;260
8.1.8;References;261
8.2;The Algebra of Truth Values of Type-2 Fuzzy Sets: A Survey;264
8.2.1;Introduction;264
8.2.2;Type-1 Fuzzy Sets;264
8.2.3;Interval-Valued Fuzzy Sets;265
8.2.4;Type-2 Fuzzy Sets;265
8.2.5;Automorphisms;267
8.2.6;Some Subalgebras of $\mathbf{M}$;269
8.2.6.1;The Subalgebra of Convex Normal Functions;269
8.2.6.2;The Subalgebra of Subsets;269
8.2.6.3;The Subalgebra of Points;269
8.2.6.4;The Subalgebra of Intervals of Constant Height;270
8.2.7;T-Norms on $\mathbf{M}$;270
8.2.8;Finite Type-2 Fuzzy Sets;271
8.2.9;Miscellany;271
8.2.10;Conclusions;273
8.2.11;References;273
8.3;Some Properties of Logic Functions over Multi-interval Truth Values;275
8.3.1;Introduction;275
8.3.2;Multi-interval Truth Values and Basic Properties;276
8.3.3;3-Valued Multi-interval Logic Functions;281
8.3.4;Conclusion;285
8.3.5;References;285
8.4;Possible Semantics for a Common Framework of Probabilistic Logics;287
8.4.1;Introduction;287
8.4.2;Probabilistic Logics;288
8.4.2.1;The Progic Framework;288
8.4.2.2;The Standard Semantics;289
8.4.3;Probabilistic Argumentation;290
8.4.3.1;Degrees of Support and Possibility;291
8.4.3.2;Possible Semantics for the Progic Framework;292
8.4.4;Conclusion;296
8.4.5;References;297
8.5;A Unified Formulation of Deduction, Induction and Abduction Using Granularity Based on VPRS Models and Measure-Based Semantics for Modal Logics;299
8.5.1;Introduction;299
8.5.2;Backgrounds;300
8.5.2.1;Rough Sets;300
8.5.2.2;Kripke Models for Modal Logic;301
8.5.2.3;Scott-Montague Models for Modal Logic;303
8.5.2.4;Measure-Based Semantics;303
8.5.3;A Unified Formulation of Deduction, Induction and Abduction Using Granularity;304
8.5.3.1;Background Knowledge by Kripke Models Based on Approximation Spaces;304
8.5.3.2;$\alpha$-Level Fuzzy Measure Models Based on Background Knowledge;305
8.5.3.3;Deduction;306
8.5.3.4;Induction;306
8.5.3.5;Abduction;308
8.5.4;Conclusion;309
8.5.5;References;309
8.6;Information from Inconsistent Knowledge: A Probability Logic Approach;310
8.6.1;Introduction;310
8.6.2;Notation and Definitions;311
8.6.3;Properties of $^{\eta}\triangleright_{\zeta}$;313
8.6.4;An Equivalent of $^{\eta}\triangleright_{\zeta}$ within Propositional Logic;315
8.6.5;The Function F_{\Gamma,\theta};319
8.6.6;Conclusion;325
8.6.7;References;325
9;Part VI Fuzziness and Uncertainty Analysis in Applications;328
9.1;Personalized Recommendation for Traditional Crafts Using Fuzzy Correspondence Analysis with Kansei Data and OWA Operator;330
9.1.1;Introduction;330
9.1.2;Preliminaries;332
9.1.3;Fuzzy Correspondence Analysis Using Kansei Data;333
9.1.3.1;Analysis Based on the Average Data;333
9.1.3.2;Modeling Fluctuation of Subjective Evaluations;334
9.1.4;A Ranking Procedure for Personalized Recommendation;336
9.1.4.1;A Fitness Measure;337
9.1.4.2;OWA Operators;337
9.1.4.3;A Ranking Procedure;338
9.1.5;A Case Study for Yamanaka Lacquer;339
9.1.5.1;Identification of Kansei Features;340
9.1.5.2;Evaluated Objects and Results;340
9.1.6;Concluding Remarks;343
9.1.7;References;343
9.2;A Probability-Based Approach to Consumer Oriented Evaluation of Traditional Craft Items Using Kansai Data;345
9.2.1;Introduction;345
9.2.2;Preliminaries;346
9.2.2.1;OWA Operators and Linguistic Quantifiers;346
9.2.2.2;Formulation of the Problem;348
9.2.3;A Consumer-Oriented Evaluation Model;349
9.2.3.1;Generating Kansei Profiles;350
9.2.3.2;Evaluation Function;350
9.2.3.3;Rating Craft Patterns;353
9.2.4;Application to Kutani Porcelain;353
9.2.4.1;Gathering Data and Kansei Profiles;354
9.2.4.2;Consumer-Oriented Evaluation;354
9.2.4.3;Discussion;356
9.2.5;Conclusion;358
9.2.6;References;358
9.3;Using Interval Function Approximation to Estimate Uncertainty;360
9.3.1;Introduction;360
9.3.1.1;Interval Function;360
9.3.1.2;The Objective of This Paper;361
9.3.2;Least Squares Approximation;362
9.3.2.1;Basis of a Function Space;362
9.3.2.2;The Least Squares Principle;362
9.3.2.3;Discrete Algorithm;362
9.3.2.4;Time Series and Slicing-Window;363
9.3.3;Interval Function Approximation;363
9.3.3.1;Computational Challenges;364
9.3.3.2;An Inner Approximation Approach;364
9.3.3.3;Width Adjustment;364
9.3.3.4;Interval Least-Squares Approximation;364
9.3.3.5;Other Approaches to Obtain an Interval Approximation;365
9.3.4;Assessing Interval Function Approximation;365
9.3.5;Case Study: Forecasting the S & P 500 Index;366
9.3.5.1;The Model;366
9.3.5.2;The Data;367
9.3.5.3;Interval Rolling Least Squares Forecasts;367
9.3.5.4;Quality Comparisons;368
9.3.6;Conclusion;369
9.3.7;References;370
9.4;Interval Forecasting of Crude Oil Price;372
9.4.1;Introduction;372
9.4.1.1;Forecasting Crude Oil Price;372
9.4.1.2;Interval Computing;373
9.4.1.3;Motivation of This Work;374
9.4.2;The Model and Computational Methods;374
9.4.2.1;The Model;375
9.4.2.2;Interval Least Square Method;375
9.4.3;Data and Software for the Empirical Study;376
9.4.3.1;Data Source;376
9.4.3.2;Data Preprocessing;377
9.4.3.3;Software;377
9.4.4;Computational Results and Comparisons;377
9.4.4.1;The Error Measurements;377
9.4.4.2;Comparison with Actual Monthly Price Interval;378
9.4.4.3;Comparison with Actual Monthly Average Price;379
9.4.5;Conclusions and Future Work;381
9.4.6;References;381
9.5;Automatic Classification for Decision Making of the Severeness of the Acute Radiation Syndrome;384
9.5.1;Introduction;384
9.5.2;Feature Extraction;385
9.5.3;Automatic Classification;385
9.5.3.1;Bayesian Classification;385
9.5.3.2;Classification with Parzen Windows;388
9.5.4;Reliability of the Diagnosis Result;389
9.5.5;Conclusion and Outlook;391
9.5.6;References;391
10;Index;392