Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 1

By Smarandache, Florentin

Book Id: WPLBN0002828224
Format Type: PDF eBook:
File Size: 4.38 MB
Reproduction Date: 7/17/2013

Title:	Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 1
Author:	Smarandache, Florentin
Volume:	Volume 1
Language:	English
Subject:	Non Fiction, Education, Dezert-Smarandache Theory (DSmT)
Collections:	Authors Community, Mathematics
Historic Publication Date:	2013
Publisher:	World Public Library

Member Page:	Florentin Smarandache
Citation APA MLA Chicago Smarandache, B. F., & Dezert, J. (2013). Advances and Applications of DSmT for Information Fusion (Collected Works) : Volume 1. Retrieved from http://gutenberg.us/

Description
This book is devoted to an emerging branch of Information Fusion based on new approach for modeling the fusion problematic when the information provided by the sources is both uncertain and (highly) conflicting. This approach, known in literature as DSmT (standing for Dezert-Smarandache Theory), proposes new useful rules of combinations. We gathered in this volume a presentation of DSmT from the beginning to the latest development. Part 1 of this book presents the current state-of-the-art on theoretical investigations while Part 2 presents several applications of this new theory. We hope that this first book on DSmT will stir up some interests to researchers and engineers working in data fusion and in artificial intelligence. Many simple but didactic examples are proposed throughout the book. As a young emerging theory, DSmT is probably not exempt from improvements and its development will continue to evolve over the years. We just want through this book to propose a new look at the Information Fusion problematic and open a new track to attack the combination of information.

Summary
The Dezert-Smarandache Theory (DSmT) of plausible and paradoxical reasoning proposed by the authors in recent years can be considered as an extension of the classical Dempster-Shafer theory (DST) but includes fundamental differences with the DST. DSmT allows to formally combine any types of independent sources of information represented in term of belief functions, but is mainly focused on the fusion of uncertain, highly conflicting and imprecise sources of evidence. DSmT is able to solve complex static or dynamic fusion problems beyond the limits of the DST framework, specially when conflicts between sources become large and when the refinement of the frame of the problem under consideration.

Table of Contents
Preamble xi Prefaces xiii I Advances on DSmT 1 1 Presentation of DSmT 3 1.1 Introduction . . . . . 3 1.2 Short introduction to the DST . . . . 5 1.2.1 Shafer’s model and belief functions . . . 5 1.2.2 Dempster’s rule of combination . . . 5 1.2.3 Alternatives to Dempster’s rule of combination . . . 6 1.2.4 The discounting of sources of evidence . . . 10 1.3 Foundations of the DSmT . . . . 11 1.3.1 Notion of free and hybrid DSm models . . . 11 1.3.2 Notion of hyper-power set D_ . . . 13 1.3.3 Generalized belief functions . . . . 15 1.3.4 The classic DSm rule of combination . . . 16 1.3.5 The hybrid DSm rule of combination . . . 17 1.3.6 On the refinement of the frames . . . 18 1.3.7 On the combination of sources over different frames . . . 20 1.4 Comparison of different rules of combinations . . . 21 1.4.1 First example . . . . . 21 1.4.2 Second example . . . . 25 1.4.3 Third example . . . . 26 1.4.4 Fourth example . . . . 27 1.4.5 Fifth example . . . . . 27 1.5 Summary . . . . . 29 1.6 References . . . . . 31 2 The generation of hyper-power sets 37 2.1 Introduction . . . . . 37 2.2 Definition of hyper-power set D_ . . . . 38 2.3 Example of the first hyper-power sets . . . 38 2.4 The generation of D_ . . . . 39 2.4.1 Memory size requirements and complexity . . . 39 2.4.2 Monotone Boolean functions . . . . 40 2.4.3 Generation of MBF . . . . 42 2.5 Conclusion . . . . . 45 2.6 References . . . . . 46 Appendix: MatLab code for generating hyper-power sets . . . 48 3 Partial ordering on hyper-power sets 49 3.1 Introduction to matrix calculus for belief functions . . . 49 3.2 Ordering elements of hyper-power set for matrix calculus . . . 51 3.2.1 Order based on the enumeration of isotone Boolean functions . . . 51 3.2.2 Ordering based on the DSm cardinality . . . 52 3.2.3 Ordering based on the intrinsic informational content . . . 56 3.3 Conclusion . . . . . 59 3.4 References . . . . . 60 4 Combination of beliefs on hybrid DSm models 61 4.1 Introduction . . . . . 61 4.2 On the independence of the sources of evidences . . . 62 4.3 DSm rule of combination for free-DSm models . . . 62 4.3.1 Definition of the free-DSm model Mf (_) . . . 62 4.3.2 Example of a free-DSm model . . . . 63 4.3.3 Classical DSm rule for 2 sources for free-DSm models . . . 64 4.3.4 Classical DSm rule for k _ 2 sources for free-DSm models . . . 64 4.4 Presentation of hybrid DSm models . . . . 64 4.4.1 Definition . . . . . 64 4.4.2 Example 1 : hybrid DSm model with an exclusivity constraint . . . 65 4.4.3 Example 2 : hybrid DSm model with another exclusivity constraint . . . 66 4.4.4 Example 3 : hybrid DSm model with another exclusivity constraint . . . 67 4.4.5 Example 4 : Shafer’s model . . . . 67 4.4.6 Example 5 : hybrid DSm model with a non-existential constraint . . . 68 4.4.7 Example 6 : hybrid DSm model with two non-existential constraints . . . 68 4.4.8 Example 7 : hybrid DSm model with a mixed constraint . . . 69 4.5 DSm rule of combination for hybrid DSm models . . . 70 4.5.1 Notations . . . . . 70 4.5.2 Programming of the u (X ) function . . . 71 4.5.3 The hybrid DSm rule of combination for 2 sources . . . 73 4.5.4 The hybrid DSm rule of combination for k _ 2 sources . . . 74 4.5.5 On the associativity of the hybrid DSm rule . . . 74 4.5.6 Property of the hybrid DSm Rule . . . 76 4.5.7 On the programming of the hybrid DSm rule . . . 78 4.5.8 Application of the hybrid DSm rule on previous examples . . . 79 4.5.9 Example with more general basic belief assignments m1 (. ) and m2 (. ) . . . 89 4.5.10 The hybrid DSm rule versus Dempster’s rule of combination . . . 92 4.6 Dynamic fusion . . . . . 93 4.6.1 Example 1 . . . . . 93 4.6.2 Example 2 . . . . . 94 4.6.3 Example 3 . . . . . 95 4.7 Bayesian mixture of hybrid DSm models . . . 101 4.8 Conclusion . . . . . 102 4.9 References . . . . . 103 5 Counter-examples to Dempster’s rule of combination 105 5.1 Introduction . . . . . 105 5.2 First infinite class of counter examples . . . 106 5.2.1 Counter-examples for Bayesian sources . . . 106 5.2.2 Counter-examples for more general sources . . . 108 5.3 Second infinite class of counter examples . . . 110 5.3.1 Zadeh’s example . . . . 110 5.3.2 Generalization with _ = {_1, _2, _3} . . . 114 5.3.3 Generalization with _ = {_1, _2, _3, _4} . . . 114 5.3.4 More general . . . . . 115 5.3.5 Even more general . . . . 115 5.4 Third infinite class of counter examples . . . 115 5.4.1 Example with _ = {_1, _2, _3, _4} . . . 116 5.4.2 Example with _ = {_1, _2, _3, _4, _5} . . . 116 5.4.3 More general . . . . . 117 5.4.4 Even more general . . . . 117 5.5 Fourth infinite class of counter examples . . . 118 5.5.1 Example with _ = {_1, . . . , _6} . . . 118 5.5.2 Another example with _ = {_1, . . . , _6} . . . 118 5.5.3 Generalization . . . . 119 5.5.4 Example with _ = {_1, _2, _3, _4} and ignorance . . . 119 5.5.5 Generalization . . . . 119 5.5.6 Example with a paradoxical source . . . 120 5.6 Conclusion . . . . . 120 5.7 References . . . . . 121 6 Fusion of imprecise beliefs 123 6.1 Introduction . . . . . 123 6.2 Combination of precise beliefs . . . . 124 6.2.1 General DSm rule of combination . . . 124 6.2.2 Examples . . . . . 126 6.3 Operations on sets . . . . . 127 6.4 Fusion of beliefs defined on single sub-unitary intervals . . . 130 6.4.1 DSm rules of combination . . . . 130 6.4.2 Example with the DSm classic rule . . . 132 6.4.3 Example with the hybrid DSm rule . . . 134 6.5 Generalization of DSm rules for sets . . . . 135 6.5.1 General DSm rules for imprecise beliefs . . . 136 6.5.2 Some lemmas and a theorem . . . . 136 6.5.3 An example with multiple-interval masses . . . 138 6.6 Conclusion . . . . . 140 6.7 References . . . . . 141 7 A Generalized Pignistic Transformation 143 7.1 A short introduction to the DSm cardinality . . . 144 7.2 The Classical Pignistic Transformation (CPT) . . . 145 7.3 A Generalized Pignistic Transformation (GPT) . . . 146 7.3.1 Definition . . . . . 146 7.3.2 P{.} is a probability measure . . . . 146 7.4 Some examples for the GPT . . . . 148 7.4.1 Example for the 2D case . . . . 148 7.4.2 Example for the 3D case . . . . 148 7.5 Conclusion . . . . . 151 7.6 References . . . . . 152 Appendix: Derivation of the GPT for the 3D free DSm model . . . 153 8 Probabilized logics related to DSmT and Bayes inference 155 8.1 Introduction . . . . . 155 8.2 Belief Theory Models . . . . . 157 8.2.1 Preliminary: about probability . . . 157 8.2.2 Dempster Shafer Theory . . . . 161 8.2.3 Transferable Belief Model . . . . 162 8.3 Dezert Smarandache Theory (DSmT) . . . 163 8.3.1 Dezert Smarandache model . . . . 164 8.3.2 Fusion rule . . . . . 166 8.4 Probability over logical propositions . . . . 167 8.4.1 Definition . . . . . 167 8.4.2 Property . . . . . 168 8.4.3 Partially defined probability . . . . 168 8.5 Logical interpretation of DSmT: an example . . . 168 8.5.1 A possible modal interpretation . . . 168 8.5.2 Deriving a fusion rule . . . . 169 8.6 Multi-modal logic and information fusion . . . 171 8.6.1 Modal logic . . . . . 171 8.6.2 A multi-modal logic . . . . 175 8.6.3 Some multi-modal theorems . . . . 176 8.6.4 Sensor fusion . . . . . 177 8.7 Logical interpretation of the Bayes inference . . . 183 8.7.1 Definitions . . . . . 184 8.7.2 Properties . . . . . 186 8.8 Conclusion . . . . . 189 8.9 References . . . . . 190 9 On conjunctive and disjunctive combination rules of evidence 193 9.1 Introduction . . . . . 194 9.2 Preliminary . . . . . 196 9.2.1 Source of information and multi-valued mappings . . . 196 9.2.2 Dempster’s combination rule of independent information sources . . . 197 9.2.3 Degree of belief . . . . 197 9.2.4 The DS combination rule . . . . 198 9.3 The DS combination rule induced by multi-valued mapping . . . 199 9.3.1 Definition of probability measure over the mapping space . . . 199 9.3.2 Derivation of the DS combination rule . . . 200 9.3.3 New explanations for the problems in DS combination rule . . . 201 9.3.4 Remark about “multi-valued mapping” in Shafer’s paper . . . 203 9.4 A new combination rule of probability measures over mapping space . . . 204 9.4.1 Derivation of combination rule of probabilities p(i)_1_2..._n. . . 204 9.4.2 Combination rule of probability measures in space S . . . 206 9.5 The disjunctive combination rule . . . . 208 9.6 Properties of conjunctive and disjunctive combination rules . . . 210 9.6.1 The combination rules of evidence . . . 211 9.6.2 Properties of combination rules of evidence . . . 213 9.6.3 Example . . . . . 216 9.7 Conclusion . . . . . 218 9.8 References . . . . . 219 10 Comparison between DSm and MinC combination rules 223 10.1 Introduction . . . . . 223 10.2 Conflict in belief combination . . . . 224 10.3 The minC combination . . . . 225 10.3.1 A system of different types of conflicts . . . 225 10.3.2 Combination on generalized frames of discernment . . . 227 10.3.3 Reallocation of belief masses of conflicts . . . 229 10.3.4 Summary of the idea of the minC combination . . . 230 10.4 Comparison . . . . . 231 10.4.1 Comparison of generalized frames of discernment . . . 231 10.4.2 Comparison of principles of combination . . . 232 10.4.3 Two steps of combination . . . . 233 10.4.4 On the associativity of the combination rules . . . 234 10.4.5 The special cases . . . . 234 10.4.6 Comparison of expressivity of DSm and minC approaches . . . 235 10.5 Examples . . . . . 235 10.6 Conclusion . . . . . 240 10.7 References . . . . . 240 11 General Fusion Operators from Cox’s Postulates 243 11.1 About uncertainty . . . . . 244 11.1.1 Probabilistic modelling . . . . 245 11.1.2 The mathematical theory of evidence . . . 247 11.1.3 Fuzzy logic . . . . . 248 11.1.4 Confidence measures . . . . 250 11.2 Fusions . . . . . 251 11.2.1 Postulates . . . . . 251 11.2.2 Machine on confidence . . . . 251 11.2.3 Operator . . . . . 252 11.3 T-norm . . . . . 253 11.3.1 Independence-interdependence . . . 256 11.3.2 T-norm description . . . . 256 11.4 Conclusions . . . . . 258 11.5 References . . . . . 260 II Applications of DSmT 263 12 On the Tweety Penguin Triangle Problem 265 12.1 Introduction . . . . . 265 12.2 The Tweety Penguin Triangle Problem . . . 266 12.3 The fallacious Bayesian reasoning . . . . 267 12.3.1 The Pearl’s analysis . . . . 267 12.3.2 The weakness of the Pearl’s analysis . . . 269 12.4 The Dempster-Shafer reasoning . . . . 274 12.5 The Dezert-Smarandache reasoning . . . . 281 12.6 Conclusion . . . . . 286 12.7 References . . . . . 287 13 Estimation of Target Behavior Tendencies using DSmT 289 13.1 Introduction . . . . . 290 13.2 Statement of the Problem . . . . 290 13.3 Approach for Behavior Tendency Estimation . . . 291 13.3.1 The fuzzification interface . . . . 291 13.3.2 The behavior model . . . . 292 13.3.3 The amplitude state prediction . . . 294 13.3.4 State updating using DSmT . . . . 294 13.4 The decision criterion . . . . 295 13.5 Simulation study . . . . . 295 13.6 Comparison between DSm and Fuzzy Logic Approaches . . . 299 13.7 Conclusions . . . . . 300 13.8 References . . . . . 301 14 Generalized Data Association for Multitarget Tracking in Clutter 303 14.1 Introduction . . . . . 304 14.2 Basic Elements of Tracking Process . . . . 304 14.2.1 Data Association . . . . 304 14.2.2 Filtering . . . . . 306 14.3 The Attribute Contribution to GDA . . . . 306 14.3.1 The Input Fuzzification Interface . . . 307 14.3.2 Tracks’ Updating Procedures . . . . 310 14.4 The Generalized Data Association Algorithm . . . 312 14.4.1 Kinematics probability term for generalized data association . . . 314 14.4.2 Attribute probability terms for generalized data association . . . 315 14.5 Simulation scenarios . . . . . 316 14.5.1 Simulation scenario1: Crossing targets . . . 316 14.5.2 Simulation scenario 2: Closely spaced targets . . . 317 14.6 Simulation results . . . . . 318 14.6.1 Simulation results: Two crossing targets . . . 318 14.6.2 Simulation results: Four closely spaced targets . . . 319 14.6.3 Simulation results of GDA based on Dempster-Shafer theory . . . 320 14.7 Comparative analysis of the results . . . . 321 14.8 Conclusions . . . . . 323 14.9 References . . . . . 323 15 On Blackman’s Data Association Problem 325 15.1 Introduction . . . . . 325 15.2 Blackman’s Data Association Problem . . . 326 15.2.1 Association Problem no. 1 . . . . 326 15.2.2 Association Problem no. 2 . . . . 326 15.3 Attempts for solutions . . . . 327 15.3.1 The simplest approach . . . . 327 15.3.2 The minimum conflict approach . . . 327 15.3.3 Blackman’s approach . . . . 328 15.3.4 Tchamova’s approach . . . . 328 15.3.5 The entropy approaches . . . . 329 15.3.6 Schubert’s approach . . . . 331 15.4 DSmT approaches for BAP . . . . 333 15.5 Monte-Carlo simulations . . . . 334 15.6 Conclusion . . . . . 335 15.7 References . . . . . 336 16 Neutrosophic Frameworks for Situation Analysis 337 16.1 Introduction . . . . . 338 16.2 Situation analysis . . . . . 339 16.2.1 Situation awareness as a mental state . . . 340 16.2.2 Situation Analysis as a process . . . 341 16.2.3 A general model of a distributed system . . . 342 16.3 Sources of uncertainty in Situation Analysis . . . 342 16.4 Ontological principles in Situation Analysis . . . 345 16.4.1 Allowing statements and reasoning about uncertainty . . . 345 16.4.2 Contextualization . . . . 348 16.4.3 Enrichment of the universe of discourse . . . 350 16.4.4 Autoreference . . . . . 352 16.5 Neutrosophic frameworks for Situation Analysis . . . 353 16.5.1 Neutrosophy . . . . . 353 16.5.2 Neutrosophic logic . . . . 354 16.5.3 Dezert-Smarandache theory (DSmT) . . . 355 16.6 Possible worlds semantics for neutrosophic frameworks . . . 356 16.6.1 Kripke model . . . . . 357 16.6.2 Kripke structure for neutrosophic propositions . . . 359 16.6.3 Probability assignments and structures . . . 360 16.6.4 Connection between DSmT and neutrosophic logic in Kripke structures . . . 365 16.7 Conclusion . . . . . 366 16.8 References . . . . . 366 17 Application of DSmT for Land Cover Change Prediction 371 17.1 Introduction . . . . . 372 17.2 Determination of information sources . . . . 373 17.2.1 Identification of the driving factors of land cover change . . . 373 17.2.2 Hierarchization of the factors of land cover change . . . 373 17.3 Land cover prediction with the Dempster-Shafer Theory . . . 375 17.3.1 Basic belief assignment . . . . 375 17.3.2 Conflict managing with the source weakness . . . 377 17.4 Land cover prediction with DSmT . . . . 377 17.4.1 Mass belief assignment . . . . 378 17.4.2 Results . . . . . 379 17.5 Conclusion . . . . . 381 17.6 References . . . . . 381 18 Power and Resource Aware Distributed Smart Fusion 383 18.1 Introduction . . . . . 384 18.2 Description of proposed research . . . . 385 18.2.1 Discovery of missing information . . . 385 18.2.2 Measure of consistency . . . . 387 18.2.3 Feature discrimination . . . . 387 18.2.4 Measures of value of information . . . 389 18.2.5 Fusion using DSmT . . . . 390 18.3 Experimental details and results . . . . 391 18.3.1 Simulated network of radar sensors . . . 391 18.3.2 A real network of spatially DSN with disparate sensors . . . 399 18.4 Conclusions . . . . . 408 18.5 References . . . . . 408 Biographies of contributors 411