We consider measuring the number of clusters (cluster size) in the finite mixture models for interpreting their structures. Many existing information criteria have been applied for this issue by regarding it as the same as the number of mixture components (mixture size); however, this may not be valid in the presence of overlaps or weight biases. In this study, we argue that the cluster size should be measured as a continuous value and propose a new criterion called mixture complexity (MC) to formulate it. It is formally defined from the viewpoint of information theory and can be seen as a natural extension of the cluster size considering overlap and weight bias. Subsequently, we apply MC to the issue of gradual clustering change detection. Conventionally, clustering changes have been regarded as abrupt, induced by the changes in the mixture size or cluster size. Meanwhile, we consider the clustering changes to be gradual in terms of MC; it has the benefits of finding the changes earlier and discerning the significant and insignificant changes. We further demonstrate that the MC can be decomposed according to the hierarchical structures of the mixture models; it helps us to analyze the detail of substructures.
Keywords: change detection; clustering; finite mixture model; gradual change; information theory.