Possibility Theory For The Design Of Information Fusion Systems (information Fusion And Data Science)

This practical guidebook describes the basic concepts, the mathematical developments, and the engineering methodologies for exploiting possibility theory for the computer-based design of an information fusion system where the goal is decision support for industries in smart ICT (information and communications technologies). This exploitation of possibility theory improves upon probability theory, complements Dempster-Shafer theory, and fills an important gap in this era of Big Data and Internet of Things. The book discusses fundamental possibilistic concepts: distribution, necessity measure, possibility measure, joint distribution, conditioning, distances, similarity measures, possibilistic decisions, fuzzy sets, fuzzy measures and integrals, and finally, the interrelated theories of uncertainty..uncertainty. These topics form an essential tour of the mathematical tools needed for the latter chapters of the book. These chapters present applications related to decision-making and pattern recognition schemes, and finally, a concluding chapter on the use of possibility theory in the overall challenging design of an information fusion system. This book will appeal to researchers and professionals in the field of information fusion and analytics, information and knowledge processing, smart ICT, and decision support systems. Preface 6 Contents 8 Chapter 1: Introduction to Possibility Theory 12 1.1 Introduction 12 1.2 Information Concept 14 1.2.1 Information Element Definition 14 1.2.2 Intrinsic Information Imperfection Types 18 1.3 Possibilistic Information Concept 21 References 22 Chapter 2: Fundamental Possibilistic Concepts 23 2.1 Introduction 23 2.2 Possibility Distributions Concept 24 2.2.1 Defining a Possibility Distribution 24 2.2.2 Possibility Distribution Models 26 2.2.3 Possibility Distributions Discounting 31 2.2.4 Possibilistic Extension Principle 32 2.2.5 Specificity Concept and Minimal Specificity Principle (MSP) 33 2.3 Possibility and Necessity Measures 35 2.3.1 Possibility Measure 36 2.3.2 Necessity Measure 40 2.3.3 Duality Relevant Properties of Possibility and Necessity Measures 42 2.3.4 Relative Possibility and Necessity Measures of Ambiguous Events 43 2.3.5 Important Properties of Possibility/Necessity Degrees of Matching 46 2.4 Subnormal Possibility Distributions 48 2.4.1 Possibility Distributions Normalization Methods 50 2.4.1.1 Ordinal Normalization 50 2.4.1.2 Numerical (or Ratio) Normalization 51 2.4.1.3 Inconsistency Shifting Normalization 51 2.4.2 Dubois ́s Alternative Necessity Measure 52 2.4.3 Normal Versus Subnormal Distributions Properties 54 References 56 Chapter 3: Joint Possibility Distributions and Conditioning 57 3.1 Introduction 57 3.2 Joint and Marginal Possibility Distributions 59 3.3 Cylindrical Extension of Non-interactive Possibilistic Variables 62 3.3.1 Projections of a Cylindrical Extension 65 3.3.2 Joint Possibility and Necessity Measures 66 3.4 Conditioning Under the Knowledge of the Joint Possibility Distribution 69 3.4.1 Zadeh ́s Conditioning Rule 71 3.4.2 Hisdal ́s Conditioning Rule 73 3.4.3 Dempster ́s Conditioning Rule 76 3.4.4 Nguyen ́s Conditioning Rule 77 3.4.5 Causal Link Conditioning 80 3.5 Conditioning and Belief Revision 82 3.5.1 Crisp Event-Based Possibilistic Revision 82 3.5.2 Unreliable Crisp Event-Based Possibilistic Revision 86 3.5.2.1 Unreliability as a Constraint 86 3.5.2.2 Unreliability as a Certainty Degree 88 3.6 Conditioning and Possibilistic Medical Diagnosis 88 References 90 Chapter 4: Possibilistic Similarity Measures 92 4.1 Introduction 92 4.2 Taxonomy of Similarity Measures 95 4.2.1 Metric-Based Similarity Measures 96 4.2.1.1 Metric Distance Measures 96 Minkowski Distance 97 Canberra Distance 99 Hausdorff Distance 100 4.2.1.2 Metric Similarity Measures 101 4.2.2 Set-Based Similarity Measures 103 4.3 Fuzzy Sets Theory and Similarity Measures 107 4.3.1 Metric-Based Similarity Measures of Fuzzy Sets 111 4.3.2 Set-Based Similarity Measures of Fuzzy Sets 113 4.3.3 Implication-Based Similarity Measures of Fuzzy Sets 120 4.4 Possibility Distributions Similarity Measures 125 4.4.1 Definition, Possibilistic Similarity Measures 127 4.4.2 Metric-Based Possibilistic Similarity Measures 130 4.4.3 Set-Based Possibilistic Similarity Measures 131 4.4.4 Informational-Based Possibilistic Similarity Measures 133 4.4.4.1 Possibilistic Degree of Matching 133 4.4.4.2 Information Closeness Index 134 4.4.4.3 Information Affinity Index 138 4.4.4.4 Possibilistic Similarity Index 139 4.4.4.5 Huete Similarity Measures 142 References 143 Chapter 5: The Interrelated Uncertainty Modeling Theories 145 5.1 Introduction 145 5.2 The Monotone Measures Theory 148 5.2.1 Sugeno Monotone Measure Definition 148 5.2.2 Distinguished Classes of Monotone Measures 150 5.3 Uncertainty Theories in the Framework of Monotone Measures Theory 151 5.4 Evidence-Possibility Transformations 156 5.4.1 Transforming a b.p.a into a Possibility Distribution 157 5.4.2 Transforming a Possibility Distribution into a b.p.a 158 5.5 Probability-Possibility Transformations 159 5.5.1 Probability-Possibility Consistency Concepts 160 5.5.1.1 Probability-Possibility Consistency Principle 161 5.5.1.2 Least Commitment Principle 162 5.5.1.3 Zadeh ́s Consistency Principle 162 5.5.1.4 Preference Preservation Principle 163 5.5.2 Probability-Possibility Transformation Methods 163 5.5.2.1 Ratio Scale Transformation 163 5.5.2.2 Maximal Specificity (Pr π) Transformation 164 5.5.2.3 Pignistic (π Pr) Transformation 167 5.5.2.4 Dubois-Prade ́s Symmetric Transformation 168 References 172 Chapter 6: Possibility Integral 173 6.1 Introduction 173 6.2 Aggregation Functions 175 6.3 Monotone Measures and Fuzzy Integrals 177 6.3.1 Monotone Measures Definition 178 6.3.2 Special Monotone Measures 180 6.3.2.1 The Sugeno λ-measure 180 6.3.2.2 Cardinality-Based Monotone Measures 181 6.3.2.3 Prioritization Monotone Measure 182 6.4 Discrete Choquet Integral 183 6.4.1 Important Properties of the Discrete Choquet Integral 186 6.4.2 Discrete Choquet Integral for Some Types of Monotone Measures 187 6.5 Discrete Sugeno Integral 192 6.5.1 Important Properties of the Discrete Sugeno Integral 194 6.5.2 Discrete Sugeno Integral for Some Monotone Measures 196 6.5.3 Twofold Integral 196 6.6 Possibility Integral 197 6.6.1 Possibilistic Choquet Integral 198 6.6.2 Possibilistic Sugeno Integral 200 6.6.3 Subnormal Possibilistic Sugeno Integral 204 6.7 Application of the Possibility Integral to Pattern Recognition 206 References 211 Chapter 7: Fusion Operators and Decision-Making Criteria in the Framework of Possibility Theory 212 7.1 Introduction 212 7.2 Possibility Distributions Fusion 212 7.2.1 Conjunctive Possibility Distributions Fusion 213 7.2.2 Disjunctive Possibility Distributions Fusion 215 7.2.3 Trade-Off Possibility Distributions Fusion 216 7.2.3.1 Weighted Possibilistic Fusion Operator 217 7.2.3.2 Consistency-Driven Possibilistic Fusion Operator 217 7.2.3.3 Adaptive Fusion Operator 218 7.3 Decision-Making 220 7.3.1 Possibilistic Decision Criteria 222 7.3.1.1 Decision Rule Based on the Maximum of Possibility 222 7.3.1.2 Decision Rule Based on Maximizing the Necessity Measure 223 7.3.1.3 Decision Rule Based on Maximizing the Confidence Index 225 7.4 Fuzzy Pattern Matching (FPM) 226 7.4.1 Confidence Index and Uncertainty Quantification 229 References 233 Chapter 8: Possibilistic Concepts Applied to Soft Pattern Classification 235 8.1 Introduction 235 8.2 Pixel-Based Image Classification 236 8.2.1 Pixel-Based Methods within the Context of Limited Prior Knowledge 238 8.2.2 The IRPDL Method 239 8.2.2.1 Possibilistic Seeds Extraction 240 8.2.2.2 Possibilistic Knowledge Projection and Decisions 241 8.2.2.3 Possibilistic Knowledge Diffusion (PKD) Process 242 8.2.3 The Performance Evaluation of the Proposed IRPDL Approach 242 8.3 Spatial Unmixing Based on Possibilistic Similarity 245 8.3.1 Possibilistic Knowledge Representation 245 8.3.2 Possibility Distributions Estimation Based on Pr π Transformation 246 8.3.3 Possibilistic Similarity 247 8.3.3.1 Possibilistic Similarity Functions 248 8.3.3.2 Evaluation of the Similarity between Two Classes 249 8.3.4 A Possibilistic Approach of Pixel Unmixing by Endmembers 250 8.3.5 Performance on Synthetic Data 252 8.3.5.1 Estimation of Classes ́ Abundances in the Mixed Pixels 252 8.3.5.2 Evaluation of the Improvement in Overall Classification Accuracy 254 8.4 Blind Image Segmentation Using a Possibilistic Approach 255 8.4.1 Region-Based Approaches in Image Segmentation 256 8.4.2 Possibilistic Region Growing Approach 257 8.4.2.1 The Knowledge Projection Subsystem 258 8.4.2.2 Decision-Making Subsystem 260 8.4.2.3 Possibilistic Diffusion Subsystem 261 8.4.2.4 Evaluation Subsystem 262 8.4.3 Empirical Results from Synthetic Images 262 8.4.3.1 Possibilistic Map Initial Estimation 263 8.4.3.2 Evaluation of Different Knowledge Diffusion Techniques 264 8.4.3.3 Efficiency Assessment of Possibilistic Knowledge Diffusion 264 References 265 Chapter 9: The Use of Possibility Theory in the Design of Information Fusion Systems 267 9.1 Introduction 267 9.2 The General Context for the Design of a FIAT-Based System 267 9.2.1 What Is FIAT? 268 9.2.2 Situation Awareness 269 9.2.3 Analytics and Information Fusion 271 9.2.3.1 Information Fusion 271 9.2.3.2 Analytics 272 9.2.4 Generic FIAT Core Functions 275 9.2.5 An Integrating Framework to Support the Design of FIAT-Based System 277 9.3 Awareness Quality and Decision Support 278 9.4 Where Does Possibility Theory Fit Within the Design of FIAT Systems? 281 9.5 Conclusion 281 References 283 Index 285 Front Matter ....Pages i-x Introduction to Possibility Theory (Basel Solaiman, Éloi Bossé)....Pages 1-11 Fundamental Possibilistic Concepts (Basel Solaiman, Éloi Bossé)....Pages 13-46 Joint Possibility Distributions and Conditioning (Basel Solaiman, Éloi Bossé)....Pages 47-81 Possibilistic Similarity Measures (Basel Solaiman, Éloi Bossé)....Pages 83-135 The Interrelated Uncertainty Modeling Theories (Basel Solaiman, Éloi Bossé)....Pages 137-164 Possibility Integral (Basel Solaiman, Éloi Bossé)....Pages 165-203 Fusion Operators and Decision-Making Criteria in the Framework of Possibility Theory (Basel Solaiman, Éloi Bossé)....Pages 205-227 Possibilistic Concepts Applied to Soft Pattern Classification (Basel Solaiman, Éloi Bossé)....Pages 229-260 The Use of Possibility Theory in the Design of Information Fusion Systems (Basel Solaiman, Éloi Bossé)....Pages 261-278 Back Matter ....Pages 279-288