Interval / Probabilistic Uncertainty and Non-classical Logics (eBook)

Preface 6
Contents 12
List of Contributors 16
Part I Keynote Addresses 21
An Algebraic Approach to Substructural Logics – An Overview 22
On Modeling of Uncertainty Measures and Observed Processes 24
Introduction 24
Some Uncertainty Measures Derived from Coarse Data 25
Belief Functions 25
Possibility Measures 27
Canonical Borel-$/sigma$ Fields and Continuous Lattices 28
Discrete Random Sets 29
The Space of Closed Sets as a Continuous Lattice 30
The Space of USC Functions as a Continuous Lattice 31
Concluding Remarks 33
References 33
Part II Statistics under Interval Uncertainty and Imprecise Probability 36
Fast Algorithms for Computing Statistics under Interval Uncertainty: An Overview 38
Computing Statistics Is Important 38
Interval Uncertainty 39
Estimating Statistics Under Interval Uncertainty: A Problem 40
Mean 40
Variance: Computing the Exact Range Is Difficult 40
Linearization 41
Linearization Is Not Always Acceptable 41
First Class: Narrow Intervals 42
Second Class: Slightly Wider Intervals 43
Third Class: Single Measuring Instrument 43
Fourth Class: Several MI 44
Fifth Class: Privacy Case 44
Sixth Class: Non-detects 45
Results 46
Conclusion 47
References 47
Trade-Off between Sample Size and Accuracy: Case of Static Measurements under Interval Uncertainty 51
General Formulation of the Problem 51
In Different Practical Situations, This General Problem Can Take Different Forms 52
A Realistic Formulation of the Trade-Off Problem 53
Solving the Trade-Off Problem in the General Case 55
How Does the Cost of a Measurement Depend on Its Accuracy? 56
Trade-Off between Accuracy and Sample Size in Different Cost Models 60
Conclusion 62
References 62
Trade-Off between Sample Size and Accuracy: Case of Dynamic Measurements under Interval Uncertainty 64
Formulation of the Problem 64
First Objective: Measuring the Average Value of a Varying Quantity 65
Second Objective: Measuring the Actual Dependence of the Measured Quantity on Space Location and/or on Time 68
Case Study: In Brief 73
Conclusions 74
References 74
Estimating Quality of Support Vector Machines Learning under Probabilistic and Interval Uncertainty: Algorithms and Computational Complexity 76
Formulation of the Problem 76
How to Take into Account Probabilistic and Interval Uncertainty: Formulation of the Problem and Linearized Algorithms for Solving This Problem 81
In General, Estimating Quality of SVM Learning under Interval Uncertainty Is NP-Hard 85
Conclusion 87
References 88
Imprecise Probability as an Approach to Improved Dependability in High-Level Information Fusion 89
Introduction 89
Information Fusion 90
High-Level Information Fusion 91
Level 2 -- Situation Assessment 91
Level 3 -- Impact Assessment 93
Dependable High-Level Information Fusion 93
High-Level Information Fusion as a Service 94
Reliability 94
Fault 95
Safety 95
Imprecise Probability - Dependable High-Level Information Fusion 97
Application Domains 98
Defense 99
Manufacturing 99
Precision Agriculture 100
Discussion and Future Work 100
Conclusions 101
References 101
Part III Uncertainty Modelling and Reasoning in Knowledge-Based Systems 104
Label Semantics as a Framework for Granular Modelling 106
Introduction to Granular Modelling 106
Underlying Philosophy of Vagueness 107
Label Semantics 109
Ordering Labels 112
Granular Models in Label Semantics 114
Mass Relational Models 115
Linguistic Decision Trees 116
Linguistic Attribute Hierarchies 118
Conclusions 120
References 120
Approximating Reasoning for Fuzzy-Based Information Retrieval 122
Introduction 122
Fuzzy Retrieval Framework 123
Fuzzy Representations of Uncertainty Objects and Queries 124
Fuzzy Retrieval 125
Proposition Extraction 125
Approximating Reasoning 127
Fuzzy-Based Additional Relation Discovery 128
Performance Evaluation 130
Conclusion 132
References 132
Probabilistic Constraints for Inverse Problems 134
Introduction 134
Inverse Problems 135
Continuous Constraint Satisfaction Problems 136
Constraint Approach to Inverse Problems 137
Probabilistic Reasoning 139
Probabilistic Approach to Inverse Problems 139
Probabilistic Interval Computations 140
Probabilistic Constraint Reasoning 141
Probabilistic Constraint Approach to Inverse Problems 143
Conclusions and Future Work 146
References 146
The Evidential Reasoning Approach for Multi-attribute Decision Analysis under Both Fuzzy and Interval Uncertainty 148
Introduction 148
The FIER Approach for MADA under Fuzzy Uncertainty 149
The New FIER Distributed Modelling Framework using the Fuzzy Belief Structure 149
The New FIER Algorithm under Both Interval Probabilistic and Fuzzy Uncertainties 151
Fuzzy Expected Utilities for Characterising Alternatives 153
Application of the FIER Approach to a New Product Screening Problem 155
Concluding Remarks 157
References 158
Modelling and Computing with Imprecise and Uncertain Properties in Object Bases 160
Introduction 160
Combination of Probabilities and Fuzzy Sets 162
Probabilistic Interpretation of Relations on Fuzzy Sets 162
Algebra on Fuzzy Probabilistic Triples 163
Fuzzy and Probabilistic Object Properties 164
FPOB Class Hierarchy 164
FPOB Attributes and Methods 165
FPOB Schema 166
Fuzzy and Probabilistic Object Base Instances and Class Extents 168
FPOB Instances 168
Probabilistic Extents of Classes 169
Selection Operation on Fuzzy and Probabilistic Object Bases 169
Syntax of Selection Conditions 169
Semantics of Selection Conditions 170
FPDB4O: A Fuzzy and Probabilistic Object Base Management System 172
Overview of FPDB4O 172
Implementation of FPOB Types and Schemas 173
Implementation of FPOB Instances 174
Implementation of FPOB Selection Operation 175
Conclusion 176
References 177
Part IV Rough Sets and Belief Functions 180
Several Reducts in Dominance-Based Rough Set Approach 182
Introduction 182
Dominance-Based Rough Set Approach 183
Decision Table with Dominance Relations 183
DRSA 184
VP-DRSA 186
Union-Based Reducts in DRSA 187
Discernibility Matrices for Reducts 189
Union-Based Reducts in VP-DRSA 190
Concluding Remarks 193
References 193
Topologies of Approximation Spaces of Rough Set Theory 195
Introduction 195
Preliminaries 195
Relations 195
Topologies 196
Uniformities 197
Definability in Rough Set Theory 197
Definability Based on Equivalences 197
Definability Based on Tolerances 198
Definability Based on Preorders 200
General Case 201
Topologies of Approximation Spaces 202
Hammer's Extended Topology 203
Conclusions 205
References 205
Uncertainty Reasoning in Rough Knowledge Discovery 206
Introduction 206
A Rough Sets-Based Method for Rule Induction 207
Preliminary 207
Decision Rules 208
An Algorithm for Computing Multiple Reducts 210
Rough Sets for Rule Induction 212
Uncertainty Reasoning for Classification 213
Matching Process 213
Dempster-Shafer (DS) Theory of Evidence 215
Defining Rule Mass Function 215
An Example 217
Conclusion 218
References 218
Semantics of the Relative Belief of Singletons 220
Introduction: A New Bayesian Approximation 220
Previous Work on Bayesian Approximation 220
Relative Belief of Singletons 221
Outline of the Paper 222
A Conservative Estimate 222
Dual Interpretation as Relative Plausibility of a Plausibility 223
Pseudo Belief Functions 223
Duality between Relative Belief and Plausibility 224
On the Existence Constraint 225
Example: The Binary Case 225
Region Spanned by a Bayesian Approximation 226
Zero Mass to Singletons as a Pathological Situation 226
A Low-Cost Proxy for other Bayesian Approximations 228
Convergence under Quasi-bayesianity 228
Convergence in the Ternary Case 229
Conclusions 231
References 231
A Lattice-Theoretic Interpretation of Independence of Frames 233
Introduction 233
Independence of Sources in Dempster's Combination 234
Dempster's Combination of Belief Functions 234
Independence of Sources 234
Independence of Sources and Independence of Frames 235
An Algebraic Study of Independence 237
The Semi-modular Lattice of Frames 237
Lattices 237
Semi-modularity of the Lattice of Frames 238
Finite Lattice of Frames 239
A Lattice-Theoretic Interpretation of Independence 239
Independence on Lattices and Independence of Frames 240
Independence on Lattices 240
Lattice-Theoretic Independence on the Lattice of Frames 241
Evidential Independence Is Stronger than $/mathcal{I}^*_1$, $/mathcal{I}^*_2$ 241
Evidential Independence Is Opposed to $/mathcal{I}^*_3$ 243
Comments and Conclusions 244
References 245
Part V Non-classical Logics 248
Completions of Ordered Algebraic Structures: A Survey 250
Introduction 250
Preliminaries 251
Completion Methods 252
A General Template for Completions 254
Extending Additional Operations 255
Preservation of Identities 256
Comparing Completions 258
Exploring the Boundaries 259
Conclusions and Discussion 260
References 261
The Algebra of Truth Values of Type-2 Fuzzy Sets: A Survey 264
Introduction 264
Type-1 Fuzzy Sets 264
Interval-Valued Fuzzy Sets 265
Type-2 Fuzzy Sets 265
Automorphisms 267
Some Subalgebras of $/mathbf{M}$ 269
The Subalgebra of Convex Normal Functions 269
The Subalgebra of Subsets 269
The Subalgebra of Points 269
The Subalgebra of Intervals of Constant Height 270
T-Norms on $/mathbf{M}$ 270
Finite Type-2 Fuzzy Sets 271
Miscellany 271
Conclusions 273
References 273
Some Properties of Logic Functions over Multi-interval Truth Values 275
Introduction 275
Multi-interval Truth Values and Basic Properties 276
3-Valued Multi-interval Logic Functions 281
Conclusion 285
References 285
Possible Semantics for a Common Framework of Probabilistic Logics 287
Introduction 287
Probabilistic Logics 288
The Progic Framework 288
The Standard Semantics 289
Probabilistic Argumentation 290
Degrees of Support and Possibility 291
Possible Semantics for the Progic Framework 292
Conclusion 296
References 297
A Unified Formulation of Deduction, Induction and Abduction Using Granularity Based on VPRS Models and Measure-Based Semantics for Modal Logics 299
Introduction 299
Backgrounds 300
Rough Sets 300
Kripke Models for Modal Logic 301
Scott-Montague Models for Modal Logic 303
Measure-Based Semantics 303
A Unified Formulation of Deduction, Induction and Abduction Using Granularity 304
Background Knowledge by Kripke Models Based on Approximation Spaces 304
$/alpha$-Level Fuzzy Measure Models Based on Background Knowledge 305
Deduction 306
Induction 306
Abduction 308
Conclusion 309
References 309
Information from Inconsistent Knowledge: A Probability Logic Approach 310
Introduction 310
Notation and Definitions 311
Properties of $^{/eta}/triangleright_{/zeta}$ 313
An Equivalent of $^{/eta}/triangleright_{/zeta}$ within Propositional Logic 315
The Function F_{/Gamma,/theta} 319
Conclusion 325
References 325
Part VI Fuzziness and Uncertainty Analysis in Applications 328
Personalized Recommendation for Traditional Crafts Using Fuzzy Correspondence Analysis with Kansei Data and OWA Operator 330
Introduction 330
Preliminaries 332
Fuzzy Correspondence Analysis Using Kansei Data 333
Analysis Based on the Average Data 333
Modeling Fluctuation of Subjective Evaluations 334
A Ranking Procedure for Personalized Recommendation 336
A Fitness Measure 337
OWA Operators 337
A Ranking Procedure 338
A Case Study for Yamanaka Lacquer 339
Identification of Kansei Features 340
Evaluated Objects and Results 340
Concluding Remarks 343
References 343
A Probability-Based Approach to Consumer Oriented Evaluation of Traditional Craft Items Using Kansai Data 345
Introduction 345
Preliminaries 346
OWA Operators and Linguistic Quantifiers 346
Formulation of the Problem 348
A Consumer-Oriented Evaluation Model 349
Generating Kansei Profiles 350
Evaluation Function 350
Rating Craft Patterns 353
Application to Kutani Porcelain 353
Gathering Data and Kansei Profiles 354
Consumer-Oriented Evaluation 354
Discussion 356
Conclusion 358
References 358
Using Interval Function Approximation to Estimate Uncertainty 360
Introduction 360
Interval Function 360
The Objective of This Paper 361
Least Squares Approximation 362
Basis of a Function Space 362
The Least Squares Principle 362
Discrete Algorithm 362
Time Series and Slicing-Window 363
Interval Function Approximation 363
Computational Challenges 364
An Inner Approximation Approach 364
Width Adjustment 364
Interval Least-Squares Approximation 364
Other Approaches to Obtain an Interval Approximation 365
Assessing Interval Function Approximation 365
Case Study: Forecasting the S & P 500 Index
The Model 366
The Data 367
Interval Rolling Least Squares Forecasts 367
Quality Comparisons 368
Conclusion 369
References 370
Interval Forecasting of Crude Oil Price 372
Introduction 372
Forecasting Crude Oil Price 372
Interval Computing 373
Motivation of This Work 374
The Model and Computational Methods 374
The Model 375
Interval Least Square Method 375
Data and Software for the Empirical Study 376
Data Source 376
Data Preprocessing 377
Software 377
Computational Results and Comparisons 377
The Error Measurements 377
Comparison with Actual Monthly Price Interval 378
Comparison with Actual Monthly Average Price 379
Conclusions and Future Work 381
References 381
Automatic Classification for Decision Making of the Severeness of the Acute Radiation Syndrome 384
Introduction 384
Feature Extraction 385
Automatic Classification 385
Bayesian Classification 385
Classification with Parzen Windows 388
Reliability of the Diagnosis Result 389
Conclusion and Outlook 391
References 391
Index 392