Anne-Laure Jousselme

A new distance between two bodies of evidence

Abstract We present a measure of performance (MOP) for identification algorithms based on the evidential theory of Dempster–Shafer. As an MOP, we introduce a principled distance between two basic probability assignments (BPAs) (or two bodies of evidence) based on a quantification of the similarity between sets. We give a geometrical interpretation of BPA and show that the proposed distance satisfies all the requirements for a metric. We also show the link with the quantification of Dempsters weight of conflict proposed by George and Pal. We compare this MOP to that described by Fixsen and Mahler and illustrate the behaviors of the two MOPs with numerical examples.

Distances in evidence theory: Comprehensive survey and generalizations

The purpose of the present work is to survey the dissimilarity measures defined so far in the mathematical framework of evidence theory, and to propose a classification of these measures based on their formal properties. This research is motivated by the fact that while dissimilarity measures have been widely studied and surveyed in the fields of probability theory and fuzzy set theory, no comprehensive survey is yet available for evidence theory. The main results presented herein include a synthesis of the properties of the measures defined so far in the scientific literature; the generalizations proposed naturally lead to additions to the body of the previously known measures, leading to the definition of numerous new measures. Building on this analysis, we have highlighted the fact that Dempsters conflict cannot be considered as a genuine dissimilarity measure between two belief functions and have proposed an alternative based on a cosine function. Other original results include the justification of the use of two-dimensional indexes as (cosine; distance) couples and a general formulation for this class of new indexes. We base our exposition on a geometrical interpretation of evidence theory and show that most of the dissimilarity measures so far published are based on inner products, in some cases degenerated. Experimental results based on Monte Carlo simulations illustrate interesting relationships between existing measures.

Measuring ambiguity in the evidence theory

In the framework of evidence theory, ambiguity is a general term proposed by Klir and Yuan in 1995 to gather the two types of uncertainty coexisting in this theory: discord and nonspecificity. Respecting the five requirements of total measures of uncertainty in the evidence theory, different ways have been proposed to quantify the total uncertainty, i.e., the ambiguity of a belief function. Among them is a measure of aggregate uncertainty, called AU, that captures in an aggregate fashion both types of uncertainty. But some shortcomings of AU have been identified, which are that: 1) it is complicated to compute; 2) it is highly insensitive to changes in evidence; and 3) it hides the distinction between the two types of uncertainty that coexist in every theory of imprecise probabilities. To overcome the shortcomings, Klir and Smith defined the TU1 measure that is a linear combination of the AU measure and the nonspecificity measure N. But the TU1 measure cannot solve the problem of computing complexity, and brings a new problem with the choice of the linear parameter delta. In this paper, an alternative measure to AU for quantifying ambiguity of belief functions is proposed. This measure, called Ambiguity Measure (AM), besides satisfying all the requirements for general measures also overcomes some of the shortcomings of the AU measure. Indeed, AM overcomes the limitations of AU by: 1) minimizing complexity for minimum number of focal points; 2) allowing for sensitivity changes in evidence; and 3) better distinguishing discord and nonspecificity. Moreover, AM is a special case of TU1 that does not need the parameter delta

Robust combination rules for evidence theory

Dempsters rule of combination in evidence theory is a powerful tool for reasoning under uncertainty. Since Zadeh highlighted the counter-intuitive behaviour of Dempsters rule, a plethora of alternative combination rules have been proposed. In this paper, we propose a general formulation for combination rules in evidence theory as a weighted sum of the conjunctive and disjunctive rules. Moreover, with the aim of automatically accounting for the reliability of sources of information, we propose a class of robust combination rules (RCR) in which the weights are a function of the conflict between two pieces of information. The interpretation given to the weight of conflict between two BPAs is an indicator of the relative reliability of the sources: if the conflict is low, then both sources are reliable, and if the conflict is high, then at least one source is unreliable. We show some interesting properties satisfied by the RCRs, such as positive belief reinforcement or the neutral impact of vacuous belief, and establish links with other classes of rules. The behaviour of the RCRs over non-exhaustive frames of discernment is also studied, as the RCRs implicitly perform a kind of automatic deconditioning through the simple use of the disjunctive operator. We focus our study on two special cases: (1) RCR-S, a rule with symmetric coefficients that is proved to be unique and (2) RCR-L, a rule with asymmetric coefficients based on a logarithmic function. Their behaviours are then compared to some classical combination rules proposed thus far in the literature, on a few examples, and on Monte Carlo simulations.

Uncertainty in a situation analysis perspective

This paperproposes a discussion on the role of iincerfainty in situation analysis. An overview of the princi- pal typologies of uncertainty foundin the recent literuture is presented. This wide array of uncet-tainty conceptions is a consequence of the intrinsic richness and ambiguity of nat- ural language, but also a consequence of the complexplivs- ical nature of information. Definitions of a liniited number of concepts are proposed in order to better understand the diflerent facets of uncertainty. The benefits sought are: (I) the avoidance of untimely uses of dejniriorls and models of uncertainty. (2) clarifications allowing links with the al- ready well developed logics of knowledge and belief; and (3) guidelines for the selection of the appropriate mathe- matical model to process uncertainty-based information.

A proof for the positive definiteness of the Jaccard index matrix

In this paper we provide a proof for the positive definiteness of the Jaccard index matrix used as a weighting matrix in the Euclidean distance between belief functions defined in Jousselme et al. [13]. The idea of this proof relies on the decomposition of the matrix into an infinite sum of positive semidefinite matrices. The proof is valid for any size of the frame of discernment but we provide an illustration for a frame of three elements. The Jaccard index matrix being positive definite guaranties that the associated Euclidean distance is a full metric and thus that a null distance between two belief functions implies that these belief functions are strictly identical.

Approximation techniques for the transformation of fuzzy sets into random sets

With the recent rising of numerous theories for dealing with uncertain pieces of information, the problem of connection between different frames has become an issue. In particular, questions such as how to combine fuzzy sets with belief functions or probability measures often emerge. The alternative is either to define transformations between theories, or to use a general or unified framework in which all these theories can be framed. Random set theory has been proposed as such a unified framework in which at least probability theory, evidence theory, possibility theory and fuzzy set theory can be represented. Whereas the transformations of belief functions or probability distributions into random sets are trivial, the transformations of fuzzy sets or possibility distributions into random sets lead to some issues. This paper is concerned with the transformation of fuzzy membership functions into random sets. In practice, this transformation involves the creation of a large number of focal elements (subsets with non-null probability) based on the @a-cuts of the fuzzy membership functions. In order to keep a computationally tractable fusion process, the large number of focal elements needs to be reduced by approximation techniques. In this paper, we propose three approximation techniques and compare them to classical approximations techniques used in evidence theory. The quality of the approximations is quantified using a distance between two random sets.

Reducing Algorithm Complexity for Computing an Aggregate Uncertainty Measure

In the theory of evidence, two kinds of uncertainty coexist, nonspecificity and discord. An aggregate uncertainty (AU) measure has been defined to include these two kinds of uncertainty, in an aggregate fashion. Meyerowitz et al. proposed an algorithm for calculating AU and validated its practical usage. Although this algorithm was proven to be absolutely correct by Klir and Wierman, in some cases, it remains too complex. In fact, when the cardinality of the frame of discernment is very large, it can be impossible to calculate AU. Therefore, based on Klirs and Harmanecs seminal work, we give some justifications for restricting the computation of AU(Bel) to the core of the corresponding belief function, and we also propose an algorithm to calculate AU(Bel), the F-algorithm, which reduces the computational complexity of the original algorithm of Meyerowitz et al. We prove that this algorithm gives the same results as Meyerowitzs algorithm, and we outline conditions under which it reduces the computational complexity significantly. Moreover, we illustrate the use of the F-algorithm in computing AU in a practical scenario of target identification.

Combining belief functions and fuzzy membership functions

In several practical applications of data fusion and more precisely in object identification problems, we need to combine imperfect information coming from different sources (sensors, humans, etc.), the resulting uncertainty being naturally of different kinds. In particular, one information could naturally been expressed by a membership function while the other could best be represented by a belief function. Usually, information modeled in the fuzzy sets formalism (by a membership function) concerns attributes like speed, length, or Radar Cross Section whose domains of definition are continuous. However, the object identification problem refers to a discrete and finite framework (the number of objects in the data base is finite and known). This implies thus a natural but unavoidable change of domain. To be able to respect the intrinsic characteristic of uncertainty arising from the different sources and fuse it in order to identify an object among a list of possible ones in the data base, we need (1) to use a unified framework where both fuzzy sets and belief functions can be expressed, (2) to respect the natural discretization of the membership function through the change of domain (from attribute domain to frame of discernment). In this paper, we propose to represent both fuzzy sets and belief function by random sets. While the link between belief functions and random sets is direct, transforming fuzzy sets into random sets involves the use of α-cuts for the construction of the focal elements. This transformation usually generates a large number of focal elements often unmanageable in a fusion process. We propose a way to reduce the number of focal elements based on some parameters like the desired number of focal elements, the acceptable distance from the approximated random set to the original discrete one, or the acceptable loss of information.

A situation analysis toolbox: Application to coastal and offshore surveillance

In this paper, we present a toolbox to evaluate motion strategies in a realistic surveillance context for the purpose of enhancing decision support capabilities. Five components of the toolbox (Discretization, State Generation, State Searching, Behaviour Simulation and Visualization) implement the theoretical concepts put forward in previous works which outlined formal definitions of situation, situation awareness and situation analysis defined with the interpreted systems semantics.