General combination rules for qualitative and quantitative beliefs

Martin and Osswald \cite{Martin07} have recently proposed many generalizations of combination rules on quantitative beliefs in order to manage the conflict and to consider the specificity of the responses of the experts. Since the experts express themselves usually in natural language with linguistic labels, Smarandache and Dezert \cite{Li07} have introduced a mathematical framework for dealing directly also with qualitative beliefs. In this paper we recall some element of our previous works and propose the new combination rules, developed for the fusion of both qualitative or quantitative beliefs

A Simple Proportional Conflict Redistribution Rule

One proposes a first alternative rule of combination to WAO (Weighted Average Operator) proposed recently by Josang, Daniel and Vannoorenberghe, called Proportional Conflict Redistribution rule (denoted PCR1). PCR1 and WAO are particular cases of WO (the Weighted Operator) because the conflicting mass is redistributed with respect to some weighting factors. In this first PCR rule, the proportionalization is done for each non-empty set with respect to the non-zero sum of its corresponding mass matrix - instead of its mass column average as in WAO, but the results are the same as Ph. Smets has pointed out. Also, we extend WAO (which herein gives no solution) for the degenerate case when all column sums of all non-empty sets are zero, and then the conflicting mass is transferred to the non-empty disjunctive form of all non-empty sets together; but if this disjunctive form happens to be empty, then one considers an open world (i.e. the frame of discernment might contain new hypotheses) and thus all conflicting mass is transferred to the empty set. In addition to WAO, we propose a general formula for PCR1 (WAO for non-degenerate cases).Comment: 21 page

Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion (Collected works), Vol. 2

This second volume dedicated to Dezert-Smarandache Theory (DSmT) in Information Fusion brings in new fusion quantitative rules (such as the PCR1-6, where PCR5 for two sources does the most mathematically exact redistribution of conflicting masses to the non-empty sets in the fusion literature), qualitative fusion rules, and the Belief Conditioning Rule (BCR) which is different from the classical conditioning rule used by the fusion community working with the Mathematical Theory of Evidence. Other fusion rules are constructed based on T-norm and T-conorm (hence using fuzzy logic and fuzzy set in information fusion), or more general fusion rules based on N-norm and N-conorm (hence using neutrosophic logic and neutrosophic set in information fusion), and an attempt to unify the fusion rules and fusion theories. The known fusion rules are extended from the power set to the hyper-power set and comparison between rules are made on many examples. One defines the degree of intersection of two sets, degree of union of two sets, and degree of inclusion of two sets which all help in improving the all existing fusion rules as well as the credibility, plausibility, and communality functions. The book chapters are written by Frederic Dambreville, Milan Daniel, Jean Dezert, Pascal Djiknavorian, Dominic Grenier, Xinhan Huang, Pavlina Dimitrova Konstantinova, Xinde Li, Arnaud Martin, Christophe Osswald, Andrew Schumann, Tzvetan Atanasov Semerdjiev, Florentin Smarandache, Albena Tchamova, and Min Wang

Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion (Collected Works), Vol. 4

The fourth volume on Advances and Applications of Dezert-Smarandache Theory (DSmT) for information fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics. The contributions (see List of Articles published in this book, at the end of the volume) have been published or presented after disseminating the third volume (2009, http://fs.unm.edu/DSmT-book3.pdf) in international conferences, seminars, workshops and journals. First Part of this book presents the theoretical advancement of DSmT, dealing with Belief functions, conditioning and deconditioning, Analytic Hierarchy Process, Decision Making, Multi-Criteria, evidence theory, combination rule, evidence distance, conflicting belief, sources of evidences with different importance and reliabilities, importance of sources, pignistic probability transformation, Qualitative reasoning under uncertainty, Imprecise belief structures, 2-Tuple linguistic label, Electre Tri Method, hierarchical proportional redistribution, basic belief assignment, subjective probability measure, Smarandache codification, neutrosophic logic, Evidence theory, outranking methods, Dempster-Shafer Theory, Bayes fusion rule, frequentist probability, mean square error, controlling factor, optimal assignment solution, data association, Transferable Belief Model, and others. More applications of DSmT have emerged in the past years since the apparition of the third book of DSmT 2009. Subsequently, the second part of this volume is about applications of DSmT in correlation with Electronic Support Measures, belief function, sensor networks, Ground Moving Target and Multiple target tracking, Vehicle-Born Improvised Explosive Device, Belief Interacting Multiple Model filter, seismic and acoustic sensor, Support Vector Machines, Alarm classification, ability of human visual system, Uncertainty Representation and Reasoning Evaluation Framework, Threat Assessment, Handwritten Signature Verification, Automatic Aircraft Recognition, Dynamic Data-Driven Application System, adjustment of secure communication trust analysis, and so on. Finally, the third part presents a List of References related with DSmT published or presented along the years since its inception in 2004, chronologically ordered

Evidential Bagging: Combining Heterogeneous Classifiers in the Belief Functions Framework

International audienceIn machine learning, Ensemble Learning methodologies are known to improve predictive accuracy and robustness. They consist in the learning of many classifiers that produce outputs which are finally combined according to different techniques. Bagging, or Bootstrap Aggre-gating, is one of the most famous Ensemble methodologies and is usually applied to the same classification base algorithm, i.e. the same type of classifier is learnt multiple times on bootstrapped versions of the initial learning dataset. In this paper, we propose a bagging methodology that involves different types of classifier. Classifiers' probabilist outputs are used to build mass functions which are further combined within the belief functions framework. Three different ways of building mass functions are proposed; preliminary experiments on benchmark datasets showing the relevancy of the approach are presented