Characterization and evaluation of similarity measures for pairs of clusterings

In evaluating the results of cluster analysis, it is common practice to make use of a number of fixed heuristics rather than to compare a data clustering directly against an empirically derived standard, such as a clustering empirically obtained from human informants. Given the dearth of research into techniques to express the similarity between clusterings, there is broad scope for fundamental research in this area. In defining the comparative problem, we identify two types of worst-case matches between pairs of clusterings, characterised as independently codistributed clustering pairs and conjugate partition pairs. Desirable behaviour for a similarity measure in either of the two worst cases is discussed, giving rise to five test scenarios in which characteristics of one of a pair of clusterings was manipulated in order to compare and contrast the behaviour of different clustering similarity measures. This comparison is carried out for previously-proposed clustering similarity measures, as well as a number of established similarity measures that have not previously been applied to clustering comparison. We introduce a paradigm apparatus for the evaluation of clustering comparison techniques and distinguish between the goodness of clusterings and the similarity of clusterings by clarifying the degree to which different measures confuse the two. Accompanying this is the proposal of a novel clustering similarity measure, the Measure of Concordance (MoC). We show that only MoC, Powers's measure, Lopez and Rajski's measure and various forms of Normalised Mutual Information exhibit the desired behaviour under each of the test scenarios.