New Distance Measures of Evidence Based on Belief Intervals

Abstract

Abstract.A distance or dissimilarity of evidence represents the degree of dissimilarity between bodies of evidence, which has been widely used in the applications based on belief functions theory. In this paper, new distance measures are proposed based on belief intervals [Bel, P l]. For a basic belief assignment (BBA), the belief intervals of different focal elements are first calculated, respectively, which can be considered as interval numbers. Then, according to the distance of interval numbers, we can calculate the distance values between the corresponding belief intervals of the same focal elements in two given BBAs. Based on these distance values of belief intervals, new distance measures of evidence can be obtained using Euclidean and Chebyshev approaches, respectively. Some experiments and related analyses are provided to show the rationality and efficiency of the proposed measures.

Decision-Making with Belief Interval Distance

“…In [23], we have proved that d BI (x, y) is a true distance metric because it satisfies the properties of non-negativity x)), and the triangle inequality (d(x, y) + d(y, z) ≥ d(x, z), for any BBAs x, y and z defined on 2 Θ . The choice of Wasserstein's distance in d BI definition is justified by the fact that Wasserstein's distance is a true distance metric and it fits well with our needs because we have to compute a distance between [Bel 1 (X), P l 1 (X)] and [Bel 2 (X), P l 2 (X)].…”

“…measure is theoretically not satisfactory at all because the transformation is lossy since we cannot retrieve m(·) from P (·) when some focal elements of m(·) are not singletons. In the next section, we propose a better justified decision scheme based on the belief interval distance [23,24].…”

“…In our previous works [23,24], we have defined a Euclidean belief interval distance between two BBAs m 1 (·) and m 2 (·) defined on the powerset of a given FoD Θ = {θ 1 , . .…”

“…Theoretically any other strict distance metric, for instance Jousselme's distance [27][28][29], could be used instead of d BI (·, ·). We have chosen d BI distance because of its ability to provide good and reasonable behavior [23] as will be shown. When there exists a tie between multiple decisions {X j , j > 1}, then the prudent decision corresponding to their disjunctionX = ∪ jXj should be preferred (if allowed), otherwise the final decisionX is made by a random selection of elementsX j .…”

Inter-Criteria Analysis Based on Belief Functions for GPS Surveying Problems

“…We use the d BI (., .) distance presented in [33] for measuring the distance d(m 1 , m 2 ) between the two BBAs 8 m 1 (·) and m 2 (·) over the same FoD. It is defined by…”