Concordance indices for comparing fuzzy, possibilistic, rough and grey partitions

Many indices have been proposed in literature for the comparison of two crisp data partitions, as resulting from two different classifications attempts, two different clustering solutions or the comparison of a predicted vs. a true labelling. Crisp partitions however cannot model ambiguity, vagueness or uncertainty in class definition and thus are not suitable to model all cases where information lacks, terms definitions are intrinsically imprecise or the classification results from a human expert knowledge representation. In presence of vagueness, it is not obvious how to quantify overlap or agreement of two different partitions of the same data and many facets of vagueness have emerged in literature through complimentary theories. The aim of the paper is to give simple numerical indices to quantify partitions agreement in the fuzzy, possibilistic, rough and grey frameworks. We propose a method based on pseudo counts, intuitive in the meaning and simple to implement that is very general and allows comparing fuzzy, possibilistic, rough and grey partitions, even with a different number of classes. The proposed method has just one free parameter used to model sensitivity to higher values of membership.