Khaled Mellouli

Propagating belief functions in qualitative Markov trees

Abstract This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempsters rule. The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables. We show that efficient implementation of Dempsters rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempsters rule can be implemented by propagating belief functions through the tree.

Assessing sensor reliability for multisensor data fusion within the transferable belief model

This paper presents a method for assessing the reliability of a sensor in a classification problem based on the transferable belief model. First, we develop a method for the evaluation of the reliability of a sensor when considered alone. The method is based on finding the discounting factor minimizing the distance between the pignistic probabilities computed from the discounted beliefs and the actual values of data. Next, we develop a method for assessing the reliability of several sensors that are supposed to work jointly and their readings are aggregated. The discounting factors are computed on the basis of minimizing the distance between the pignistic probabilities computed from the combined discounted belief functions and the actual values of data.

Belief function independence

In this paper, we study the notion of marginal independence between two sets of variables when uncertainty is expressed by belief functions as understood in the context of the transferable belief model (TBM). We define the concepts of non-interactivity and irrelevance, that are not equivalent. Doxastic independence for belief functions is defined as irrelevance and irrelevance preservation under Dempsters rule of combination. We prove that doxastic independence and non-interactivity are equivalent.

Belief function independence: II. The conditional case

Abstract In the companion paper [Int. J. Approx. Reasoning 29 (1) (2002) 47], we have emphasized the distinction between non-interactivity and doxastic independence in the context of the transferable belief model. The first corresponds to decomposition of the belief function, whereas the second is defined as irrelevance preserved under Dempster’s rule of combination. We had shown that the two concepts are equivalent in the marginal case. We proceed here with the conditional case. We show how the definitions generalize themselves, and that we still have the equivalence between conditional non-interactivity and conditional doxastic independence.

Inference in directed evidential networks based on the transferable belief model

Inference algorithms in directed evidential networks (DEVN) obtain their efficiency by making use of the represented independencies between variables in the model. This can be done using the disjunctive rule of combination (DRC) and the generalized Bayesian theorem (GBT), both proposed by Smets [Ph. Smets, Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, International Journal of Approximate Reasoning 9 (1993) 1-35]. These rules make possible the use of conditional belief functions for reasoning in directed evidential networks, avoiding the computations of joint belief function on the product space. In this paper, new algorithms based on these two rules are proposed for the propagation of belief functions in singly and multiply directed evidential networks.

Anytime propagation algorithm for min-based possibilistic graphs

This paper proposes a new anytime possibilistic inference algorithm for min-based directed networks. Our algorithm departs from a direct adaptation of probabilistic propagation algorithms since it avoids the transformation of the initial network into a junction tree which is known to be a hard problem. The proposed algorithm is composed of several, local stabilization, procedures. Stabilization procedures aim to guarantee that local distributions defined on each node are coherent with respect to those of its parents. We provide experimental results which, for instance, compare our algorithm with the ones based on a direct adaptation of probabilistic propagation algorithms.

An NSGA-II algorithm for the green vehicle routing problem

In this paper, we present and define the bi-objective Green Vehicle Routing Problem GVRP in the context of green logistics. The bi-objective GVRP states for the problem of finding routes for vehicles to serve a set of customers while minimizing the total traveled distance and the co2 emissions. We review emission factors and techniques employed to estimate co2 emissions and integrate them into the GVRP definition and model. We apply the NSGA-II evolutionary algorithm to solve GVRP benchmarks and perform statistical analysis to evaluate and validate the obtained results. The results show that the algorithm obtain good results and prove the explicit interest grant to emission minimization objective.

Naïve possibilistic network classifiers

Naive Bayesian network classifiers have proved their effectiveness to accomplish the classification task, even if they work under the strong assumption of independence of attributes in the context of the class node. However, as all of them are based on probability theory, they run into problems when they are faced with imperfection. This paper proposes a new approach of classification under the possibilistic framework with naive classifiers. To output the naive possibilistic network classifier, two procedures are studied namely the building phase, which deals with imperfect (imprecise/uncertain) dataset attributes and classes, and the classification phase, which is used to classify new instances that may be characterized by imperfect attributes. To improve the performance of our classifier, we propose two extensions namely selective naive possibilistic classifier and semi-naive possibilistic classifier. Experimental study has shown naive Bayes style possibilistic classifier, and is efficient in the imperfect case.

A theoretical framework for possibilistic independence in a weakly ordered setting

The notion of independence is central in many information processing areas, such as multiple criteria decision making, databases, or uncertain reasoning. This is especially true in the later case, where the success of Bayesian networks is basically due to the graphical representation of independence they provide. This paper first studies qualitative independence relations when uncertainty is encoded by a complete pre-order between states of the world. While a lot of work has focused on the formulation of suitable definitions of independence in uncertainty theories our interest in this paper is rather to formulate a general definition of independence based on purely ordinal considerations, and that applies to all weakly ordered settings. The second part of the paper investigates the impact of the embedding of qualitative independence relations into the scale-based possibility theory. The absolute scale used in this setting enforces the commensurateness between local pre-orders (since they share the same scale). This leads to an easy decomposability property of the joint distributions into more elementary relations on the basis of the independence relations. Lastly we provide a comparative study between already known definitions of possibilistic independence and the ones proposed here.

Information Affinity: A New Similarity Measure for Possibilistic Uncertain Information

This paper addresses the issue of measuring similarity between pieces of uncertain information in the framework of possibility theory. In a first part, natural properties of such functions are proposed and a survey of the few existing measures is presented. Then, a new measure so-called Information Affinity is proposed to overcome the limits of the existing ones. The proposed function is based on two measures, namely, a classical informative distance, e.g. Manhattan distance which evaluates the difference, degree by degree, between two normalized possibility distributions and the well known inconsistency measure which assesses the conflict between the two possibility distributions. Some potential applications of the proposed measure are also mentioned in this paper.