A Separability Index for Distance-based Clustering and Classification Algorithms

We propose a separability index that quantifies the degree of difficulty in a hard clustering problem under assumptions of a multivariate Gaussian distribution for each cluster. A preliminary index is first defined and several of its properties are explored both theoretically and numerically. Adjustments are then made to this index so that the final refinement is also interpretable in terms of the Adjusted Rand Index between a true grouping and its hypothetical idealized clustering, taken as a surrogate of clustering complexity. Our derived index is used to develop a datasimulation algorithm that generates samples according to the prescribed value of the index. This algorithm is particularly useful for systematically generating datasets with varying degrees of clustering difficulty which can be used to evaluate performance of different clustering algorithms. The index is also shown to be useful in providing a summary of the distinctiveness of classes in grouped datasets.