On the Complexity of Additive Clustering Models.
Additive clustering provides a conceptually simple and potentially powerful approach to modeling the similarity relationships between stimuli. The ability of additive clustering models to accommodate similarity data, however, typically arises through the incorporation of large numbers of parameterized clusters. Accordingly, for the purposes of both model generation and model comparison, it is necessary to develop quantitative evaluative measures of additive clustering models that take into account both data-fit and complexity. Using a previously developed probabilistic formulation of additive clustering, the Bayesian Information Criterion is proposed for this role, and its application demonstrated. Limitations inherent in this approach, including the assumption that model complexity is equivalent to cluster cardinality, are discussed. These limitations are addressed by applying the Laplacian approximation of a marginal probability density, from which a measure of cluster structure complexity is derived. Using this measure, a preliminary investigation is made of the various properties of cluster structures that affect additive clustering model complexity. Among other things, these investigations show that, for a fixed number of clusters, a model with a strictly nested cluster structure is the least complicated, while a model with a partitioning cluster structure is the most complicated. Copyright 2001 Academic Press.