Comparing Fuzzy, Probabilistic, and Possibilistic Partitions Using the Earth Mover’s Distance

Abstract

A number of noteworthy techniques have been put forth recently in different research fields for comparing clusterings. Herein, we introduce a new method for comparing soft (fuzzy, probabilistic, and possibilistic) partitions based on the earth mover's distance (EMD) and the ordered weighted average (OWA). The proposed method is a metric, depending on the ground distance, for all but possibilistic partitions. It is extremely flexible due to its EMD formulation, OWA aggregation, and abstract concept of ground distance. In theory, our method is agnostic to the type (uncertainty) of soft partition, clustering algorithm, and distance measure used in the clustering algorithm(s), and it is applicable to the clustering of both object and relational data. Validation is performed theoretically, experimentally, as well as in terms of computational complexity. Emphasis is placed on the set of possibilistic partitions, specifically noise and coincident clusters, which are important cases that have received little to no attention to date in the comparing clustering literature. Improvements are reported in terms of metric properties and computational complexity over existing extended concordance/discordance (e.g., soft Rand and Jaccard) approaches and improved design and robustness in comparison with existing transportation problem-based approaches.