A proof for the positive definiteness of the Jaccard index matrix

Authors:
Mathieu Bouchard;Anne-Laure Jousselme;Pierre-Emmanuel Doré
Affiliations:
Université Laval, Quebec, QC, Canada G1K 7P4;Defence R&D Canada -- Valcartier Quebec, QC, Canada G3J 1X5;IRISA -- Université de Rennes 1, 22302 Lannion Cedex, France
Venue:
International Journal of Approximate Reasoning
Year:
2013

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Abstract

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.