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

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 movers 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, 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 and also in terms of computational complexity. Emphasis is placed on the set of possibilistic partitions, specifically noise and co-incident clusters, important cases that have received little-to-no attention to date in the comparing clusterings 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 to existing transportation problem based approaches.