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

This book presents the recent theoretical advances and applications of the Dezert-Smarandache Theory (DSmT) of plausible and paradoxical reasoning for information fusion. DSmT proposes a new mathematical framework to deal with the combination of uncertain, imprecise and highly conflicting sources of information expressed in terms of generalized basic belief functions. DSmT works beyond the limits of the Dempster-Shafer Theory and proposes a new general rule of combination which does not require a normalization step and works with any models (free DSm, hybrid DSm and Shafer's model) whatever the degree of conflict between sources is. The DSmT is well adapted for static and dynamic fusion problematics and allows to work on any finite frames (discrete, continuous and/or hybrid). Therefore it can combine belief functions even if the refinement of the frame of discernment is inaccessible because of the vague, relative, and imprecise intrinsic nature of its elements. Part 1 of this book presents in details the last investigations on DSmT but also some related theoretical works with other approaches for information fusion. Part 2 of the book presents interesting applications of DSmT for solving both academic fusion problems and real-world fusion problems. Collected works are by S. Corgne, F. Dambreville, M. Daniel, D. De Brucq, J. Dezert, M. Farooq, L. Hubert-Moy, A.-L. Jousselme, S. Kadambe, M. Khoshnevisan, P. D. Konstantinova, P. Maupin, G. Mercier, T. A. Semerdjiev, F. Smarandache, H. Sun, A. P. Tchamova. The book has been launched in June at the Fusion 2004 Conference in Stockholm, Sweden. A second volume about new applications and developments of DSmT (Dezert-Smarandache Theory of plausible, uncertain, and paradoxist information) will be published next year. Anybody is invited to contribute papers or chapters for this second collective book. Deadline: Fall 2005. Contributed papers should be sent to Dr. Jean Dezert (jean.dezert@onera.fr, jdezert@yahoo.com, ONERA - French National Establishment for Aerospace Research, BP 72 F-92322, Chatillon Cedex, France) and Prof. Florentin Smarandache (smarand@unm.edu, University of New Mexico, 200 College Road, Gallup, NM 87301, USA). Front Cover......Page 1 Title Page ......Page 2 Copyright ......Page 3 Contents ......Page 4 Preamble......Page 14 Prefaces......Page 16 Part I Advances on DSmT......Page 20 1.1 Introduction......Page 22 1.2.2 Dempster's rule of combination......Page 24 1.2.3 Alternatives to Dempster's rule of combination......Page 25 1.2.4 The discounting of sources of evidence......Page 29 1.3.1 Notion of free and hybrid DSm models......Page 30 1.3.2 Notion of hyper-power set......Page 32 1.3.3 Generalized belief functions......Page 34 1.3.4 The classic DSm rule of combination......Page 35 1.3.5 The hybrid DSm rule of combination......Page 36 1.3.6 On the re nement of the frames......Page 37 1.3.7 On the combination of sources over di erent frames......Page 39 1.4.1 First example......Page 40 1.4.2 Second example......Page 44 1.4.3 Third example......Page 45 1.4.5 Fifth example......Page 46 1.5 Summary......Page 48 1.6 References......Page 50 2.1 Introduction......Page 56 2.3 Example of the rst hyper-power sets......Page 57 2.4.1 Memory size requirements and complexity......Page 58 2.4.2 Monotone Boolean functions......Page 59 2.4.3 Generation of MBF......Page 61 2.5 Conclusion......Page 64 2.6 References......Page 65 3.1 Introduction to matrix calculus for belief functions......Page 68 3.2.1 Order based on the enumeration of isotone Boolean functions......Page 70 3.2.2 Ordering based on the DSm cardinality......Page 71 3.2.3 Ordering based on the intrinsic informational content......Page 75 3.3 Conclusion......Page 78 3.4 References......Page 79 4.1 Introduction......Page 80 4.3.1 De nition of the free-DSm model......Page 81 4.3.2 Example of a free-DSm model......Page 82 4.4.1 De nition......Page 83 4.4.2 Example 1 : hybrid DSm model with an exclusivity constraint......Page 84 4.4.3 Example 2 : hybrid DSm model with another exclusivity constraint......Page 85 4.4.5 Example 4 : Shafer's model......Page 86 4.4.7 Example 6 : hybrid DSm model with two non-existential constraints......Page 87 4.4.8 Example 7 : hybrid DSm model with a mixed constraint......Page 88 4.5.1 Notations......Page 89 4.5.2 Programming of the......Page 90 4.5.3 The hybrid DSm rule of combination for 2 sources......Page 92 4.5.5 On the associativity of the hybrid DSm rule......Page 93 4.5.6 Property of the hybrid DSm Rule......Page 95 4.5.7 On the programming of the hybrid DSm rule......Page 97 4.5.8 Application of the hybrid DSm rule on previous examples......Page 98 4.5.9 Example with more general basic belief assignments......Page 108 4.5.10 The hybrid DSm rule versus Dempster's rule of combination......Page 111 4.6.1 Example 1......Page 112 4.6.2 Example 2......Page 113 4.6.3 Example 3......Page 114 4......Page 120 4.8 Conclusion......Page 121 4.9 References......Page 122 5.1 Introduction......Page 124 5.2.1 Counter-examples for Bayesian sources......Page 125 5.2.2 Counter-examples for more general sources......Page 127 5.3.1 Zadeh's example......Page 129 5.3.3 Generalization with......Page 133 5.4 Third in nite class of counter examples......Page 134 5.4.2 Example with......Page 135 5.4.4 Even more general......Page 136 5.5.2 Another example with......Page 137 5.5.5 Generalization......Page 138 5.6 Conclusion......Page 139 5.7 References......Page 140 6.1 Introduction......Page 142 6.2.1 General DSm rule of combination......Page 143 6.2.2 Examples......Page 145 6.3 Operations on sets......Page 146 6.4.1 DSm rules of combination......Page 149 6.4.2 Example with the DSm classic rule......Page 151 6.4.3 Example with the hybrid DSm rule......Page 153 6.5 Generalization of DSm rules for sets......Page 154 6.5.2 Some lemmas and a theorem......Page 155 6.5.3 An example with multiple-interval masses......Page 157 6.6 Conclusion......Page 159 6.7 References......Page 160 A Generalized Pignistic Transformation......Page 162 7.1 A short introduction to the DSm cardinality......Page 163 7.2 The Classical Pignistic Transformation (CPT)......Page 164 7.3.2......Page 165 7.4.2 Example for the 3D case......Page 167 7.5 Conclusion......Page 170 7.6 References......Page 171 8.1 Introduction......Page 174 8.2.1 Preliminary: about probability......Page 176 8.2.2 Dempster Shafer Theory......Page 180 8.2.3 Transferable Belief Model......Page 181 8.3 Dezert Smarandache Theory (DSmT)......Page 182 8.3.1 Dezert Smarandache model......Page 183 8.3.2 Fusion rule......Page 185 8.4.1 De nition......Page 186 8.5.1 A possible modal interpretation......Page 187 8.5.2 Deriving a fusion rule......Page 188 8.6.1 Modal logic......Page 190 8.6.2 A multi-modal logic......Page 194 8.6.3 Some multi-modal theorems......Page 195 8.6.4 Sensor fusion......Page 196 8.7 Logical interpretation of the Bayes inference......Page 202 8.7.1 De nitions......Page 203 8.7.2 Properties......Page 205 8.8 Conclusion......Page 208 8.9 References......Page 209 On conjunctive and disjunctive combination rules of evidence......Page 212 9.1 Introduction......Page 213 9.2.1 Source of information and multi-valued mappings......Page 215 9.2.3 Degree of belief......Page 216 9.2.4 The DS combination rule......Page 217 9.3.1 De nition of probability measure over the mapping space......Page 218 9.3.2 Derivation of the DS combination rule......Page 219 9.3.3 New explanations for the problems in DS combination rule......Page 220 9.3.4 Remark about \multi-valued mapping" in Shafer's paper......Page 222 9.4.1 Derivation of combination rule of probabilities......Page 223 9.4.2 Combination rule of probability measures in space......Page 225 9.5 The disjunctive combination rule......Page 227 9.6 Properties of conjunctive and disjunctive combination rules......Page 229 9.6.1 The combination rules of evidence......Page 230 9.6.2 Properties of combination rules of evidence......Page 232 9.6.3 Example......Page 235 9.7 Conclusion......Page 237 9.8 References......Page 238 10.1 Introduction......Page 242 10.2 Con ict in belief combination......Page 243 10.3.1 A system of di erent types of con icts......Page 244 10.3.2 Combination on generalized frames of discernment......Page 246 10.3.3 Reallocation of belief masses of con icts......Page 248 10.3.4 Summary of the idea of the minC combination......Page 249 10.4.1 Comparison of generalized frames of discernment......Page 250 10.4.2 Comparison of principles of combination......Page 251 10.4.3 Two steps of combination......Page 252 10.4.5 The special cases......Page 253 10.5 Examples......Page 254 10.7 References......Page 259 General Fusion Operators from Cox's Postulates......Page 262 11.1 About uncertainty......Page 263 11.1.1 Probabilistic modelling......Page 264 11.1.2 The mathematical theory of evidence......Page 266 11.1.3 Fuzzy logic......Page 267 11.1.4 Con dence measures......Page 269 11.2.2 Machine on con dence......Page 270 11.2.3 Operator......Page 271 11.3 T-norm......Page 272 11.3.2 T-norm description......Page 275 11.4 Conclusions......Page 277 11.5 References......Page 279 Part II Applications of DSmT......Page 282 12.1 Introduction......Page 284 12.2 The Tweety Penguin Triangle Problem......Page 285 12.3.1 The Pearl's analysis......Page 286 12.3.2 The weakness of the Pearl's analysis......Page 288 12.4 The Dempster-Shafer reasoning......Page 293 12.5 The Dezert-Smarandache reasoning......Page 300 12.6 Conclusion......Page 305 12.7 References......Page 306 Estimation of Target Behavior Tendencies using DSmT......Page 308 13.2 Statement of the Problem......Page 309 13.3.1 The fuzzi cation interface......Page 310 13.3.2 The behavior model......Page 311 13.3.4 State updating using DSmT......Page 313 13.5 Simulation study......Page 314 13.6 Comparison between DSm and Fuzzy Logic Approaches......Page 318 13.7 Conclusions......Page 319 13.8 References......Page 320 Generalized Data Association for Multitarget Tracking in Clutter......Page 322 14.2.1 Data Association......Page 323 14.3 The Attribute Contribution to GDA......Page 325 14.3.1 The Input Fuzzi cation Interface......Page 326 14.3.2 Tracks' Updating Procedures......Page 329 14.4 The Generalized Data Association Algorithm......Page 331 14.4.1 Kinematics probability term for generalized data association......Page 333 14.4.2 Attribute probability terms for generalized data association......Page 334 14.5.1 Simulation scenario1: Crossing targets......Page 335 14.5.2 Simulation scenario 2: Closely spaced targets......Page 336 14.6.1 Simulation results: Two crossing targets......Page 337 14.6.2 Simulation results: Four closely spaced targets......Page 338 14.6.3 Simulation results of GDA based on Dempster-Shafer theory......Page 339 14.7 Comparative analysis of the results......Page 340 14.9 References......Page 342 15.1 Introduction......Page 344 15.2.2 Association Problem no. 2......Page 345 15.3.2 The minimum con ict approach......Page 346 15.3.4 Tchamova's approach......Page 347 15.3.5 The entropy approaches......Page 348 15.3.6 Schubert's approach......Page 350 15.4 DSmT approaches for BAP......Page 352 15.5 Monte-Carlo simulations......Page 353 15.6 Conclusion......Page 354 15.7 References......Page 355 Neutrosophic Frameworks for Situation Analysis......Page 356 16.1 Introduction......Page 357 16.2 Situation analysis......Page 358 16.2.1 Situation awareness as a mental state......Page 359 16.2.2 Situation Analysis as a process......Page 360 16.3 Sources of uncertainty in Situation Analysis......Page 361 16.4.1 Allowing statements and reasoning about uncertainty......Page 364 16.4.2 Contextualization......Page 367 16.4.3 Enrichment of the universe of discourse......Page 369 16.4.4 Autoreference......Page 371 16.5.1 Neutrosophy......Page 372 16.5.2 Neutrosophic logic......Page 373 16.5.3 Dezert-Smarandache theory (DSmT)......Page 374 16.6 Possible worlds semantics for neutrosophic frameworks......Page 375 16.6.1 Kripke model......Page 376 16.6.2 Kripke structure for neutrosophic propositions......Page 378 16.6.3 Probability assignments and structures......Page 379 16.6.4 Connection between DSmT and neutrosophic logic in Kripke structures......Page 384 16.8 References......Page 385 Application of DSmT for Land Cover Change Prediction......Page 390 17.1 Introduction......Page 391 17.2.2 Hierarchization of the factors of land cover change......Page 392 17.3.1 Basic belief assignment......Page 394 17.4 Land cover prediction with DSmT......Page 396 17.4.1 Mass belief assignment......Page 397 17.4.2 Results......Page 398 17.6 References......Page 400 Power and Resource Aware Distributed Smart Fusion......Page 402 18.1 Introduction......Page 403 18.2.1 Discovery of missing information......Page 404 18.2.3 Feature discrimination......Page 406 18.2.4 Measures of value of information......Page 408 18.2.5 Fusion using DSmT......Page 409 18.3.1 Simulated network of radar sensors......Page 410 18.3.2 A real network of spatially DSN with disparate sensors......Page 418 18.5 References......Page 427 Biographies of contributors......Page 430 Papers Collected From Researchers In Fusion Information, Such As: Florentin Smarandache, Jean Dezert, Hongshe Dang, Chongzhao Han, Frederic Dambreville, Milan Daniel, Mohammad Khoshnevisan, Sukanto Bhattacharya, Albena Tchamova, Tzvetan Semerdjiev, Pavlina Konstantinova, Hongyan Sun, Mohammad Farooq, John J. Sudano, Samuel Corgne, Gregoire Mercier, Laurence Hubert-moy, Anne-laure Jousselme, Patrick Maupin And Others On Dezert-smarandache Theory Of Plausible And Paradoxical Reasoning (dsmt).. The Principal Theories Available Until Now For Data Fusion Are The Probability Theory, The Fuzzy Set Theory, The Possibility Theory, The Hint Theory And The Theory Of Evidence. Since Last Two Years J. Dezert And F. Smarandache Are Actively Developing A New Theory Of Plausible And Paradoxical Reasoning, Called Dsmt (acronym For Dezert-smarandache Theory), For Information Fusion Of Uncertain And Highly Conflicting Sources Of Information. The Dsmt Can Be Interpreted As A Generalization Of The Dempster-shafer Theory (dst) But Goes Far Beyond The Dst. The Free-dsmt Model, Which Assumes That The Ultimate Refinement Of The Frame Of Discernment Of The Fusion Problem Is Not Accessible Due To The Intrinsic Nature Of Its Elements, Is Opposite To The Shafer's Model (on Which Is Based The Dst) Assuming The Exhaustivity And Exclusivity Of All Elements Of The Frame Of Discernment. The Dsmt Proposes A New Theoretical Framework For Data Fusion Based On Definition Of Hyper-power Sets And A New Simple Commutative And Associative Rule Of Combination. Recently, It Has Been Discovered, Through A New Dsm Hybrid Rule Of Combination, That Dsmt Can Be Also Extended To Problems Involving Hybrid-models (models Including Some Exclusivity And/or Non-existentially Constraints). This New Important Theoretical Result Offers Now To The Dsmt A Wider Class Of Fusion Applications And Allows Potentially To Attack The Next Generation Of Complex Dynamical/temporal Fusion Problems. Dsmt Can Also Provide A Theoretical Issue For The Fusion Of Neutrosophic Information (extension Of Fuzzy Information Proposed By F. Smarandache In Nineties - See Http://www.gallup.unm.edu/~smarandache/firstneutconf.htm For Details About The Neutrosophy Logic And Neutrosophy Set Theory). This book presents the foundations, advances and some applications of a new theory of paradoxical and plausible reasoning developed by Jean Dezert and Florentin Smarandache, known as DSmT. This theory proposes a general method for combining uncertain, highly conflicting and imprecise data, provided by independent sources of information. It can be considered as a generalization of classical Dempster-Shafer mathematical theory of evidence, overcoming its inherent constraints, closely related with the acceptance of the law of the excluded middle. Refuting that principle, DSmT proposes a formalism to describe, analyze and combine all the available information, allowing the possibility for paradoxes between the elements of the frame of discernment. It is adapted to deal with each model of fusion occurring, taking into account all possible integrity constraints of the problem under consideration, due to the true nature and granularity of the concepts involved. This theory shows through the considered applications that conclusions drawn from it provides coherent results, which agree with the human reasoning and improves performances with respect to Dempster-Shafer theory.