On the comparison and representation of fuzzy partitions

A fuzzy partition assigns to each among n objects a distribution over a categories. Elementary linear algebraic methods permit to introduce and investigate concepts and properties such as a) variance and inertia decomposition; b) coarseand fine-graining (nestedness); c) iteration of fuzzy partitions; d) stability of a group in regard to another partition; e) (euclidean embeddable) dissimilarities between objects; f) (euclidean embeddable) dissimilarities between partitions. Unweighted (R) or weighted (T , P ) object similarities are further investigated, and found to be related to the chi-square as well as to the indices of Gini, variety and Mirkin-Cherny-Rand. Weighted versions T and P differ for fuzzy partitions, allowing various non-equivalent constructions characterizing differing aspects of fuzzy partitions and possessing no formal analog at the crisp level.