Distances in evidence theory: Comprehensive survey and generalizations

AbstractThe 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 Dempster’s 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

Clustering of discretely observed diffusion processes

In this paper a new dissimilarity measure to identify groups of assets dynamics is proposed. The underlying generating process is assumed to be a diffusion process solution of stochastic differential equations and observed at discrete time. The mesh of observations is not required to shrink to zero. As distance between two observed paths, the quadratic distance of the corresponding estimated Markov operators is considered. Analysis of both synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence that this distance seems capable to catch differences in both the drift and diffusion coefficients contrary to other commonly used metrics

Comparison and contrast in perceptual categorization

People categorized pairs of perceptual stimuli that varied in both category membership and pairwise similarity. Experiments 1 and 2 showed categorization of 1 color of a pair to be reliably contrasted from that of the other. This similarity-based contrast effect occurred only when the context stimulus was relevant for the categorization of the target (Experiment 3). The effect was not simply owing to perceptual color contrast (Experiment 4), and it extended to pictures from common semantic categories (Experiment 5). Results were consistent with a sign-and-magnitude version of N. Stewart and G. D. A. Brown's (2005) similarity-dissimilarity generalized context model, in which categorization is affected by both similarity to and difference from target categories. The data are also modeled with criterion setting theory (M. Treisman & T. C. Williams, 1984), in which the decision criterion is systematically shifted toward the mean of the current stimuli