Model Selection Strategies for Determining the Optimal Number of Overlapping Clusters in Additive Overlapping Partitional Clustering

Abstract

In various scientific fields, researchers make use of partitioning methods (e.g., K-means) to disclose the structural mechanisms underlying object by variable data. In some instances, however, a grouping of objects into clusters that are allowed to overlap (i.e., assigning objects to multiple clusters) might lead to a better representation of the underlying clustering structure. To obtain an overlapping object clustering from object by variable data, Mirkin’s ADditive PROfile CLUStering (ADPROCLUS) model may be used. A major challenge when performing ADPROCLUS is to determine the optimal number of overlapping clusters underlying the data, which pertains to a model selection problem. Up to now, however, this problem has not been systematically investigated and almost no guidelines can be found in the literature regarding appropriate model selection strategies for ADPROCLUS. Therefore, in this paper, several existing model selection strategies for K-means (a.o., CHull, the Caliński-Harabasz, Krzanowski-Lai, Average Silhouette Width and Dunn Index and information-theoretic measures like AIC and BIC) and two cross-validation based strategies are tailored towards an ADPROCLUS context and are compared to each other in an extensive simulation study. The results demonstrate that CHull outperforms all other model selection strategies and this especially when the negative log-likelihood, which is associated with a minimal stochastic extension of ADPROCLUS, is used as (mis)fit measure. The analysis of a post hoc AIC-based model selection strategy revealed that better performance may be obtained when a different—more appropriate—definition of model complexity for ADPROCLUS is used.

A white paper on good research practices in benchmarking: The case of cluster analysis

“…Rossbroich et al (2022) addressed the (underinvestigated) issue of selecting the number of clusters in an overlapping clustering model (ADPROCLUS: Depril et al, 2008; Mirkin, 1987). For this purpose, they proposed and compared 13 model selection strategies, 11 of which were (minor or major) adaptations of similar strategies for partitioning models, and 2 of which were crossvalidation‐based.…”

“…For the actual simulations, either own code or existing data generators can be used. Regarding the latter, over the past decades quite a few generators have been proposed, including the Milligan (1985) algorithm for generating artificial test clusters, OCLUS (Steinley & Henson, 2005), the Qiu and Joe (2006) random cluster generation algorithm, and MixSim (Melnykov et al, 2012). In the justification of these generators, quite some emphasis has been put on the aspects of separability and overlap, where overlap refers to intensional rather than to extensional overlap, that is to say, overlap in terms of variables or component distributions, with all generated clusterings being partitions.…”

“…As a way out, one necessarily had to turn to benchmarking studies, with instances of seminal benchmarking attempts being reported by, for example, Baker (1974), Hubert (1974), and especially Milligan (Milligan, 1980, 1985; Milligan et al, 1983; Milligan & Cooper, 1985; for overviews of earlier benchmarking work in the area, see, e.g., Jain & Dubes, 1988; Milligan, 1981a, 1981b). Later on, there has been some follow‐up to this seminal work (e.g., Anderlucci & Hennig, 2014; Arbelaitz et al, 2013; Costa et al, 2022; Hennig, 2022; Rossbroich et al, 2022; Schepers et al, 2006; Shireman et al, 2017; Steinley, 2003; Steinley & Brusco, 2008; Šulc & Řezanková, 2019; Wilderjans et al, 2013). Nevertheless, there is much less of a benchmarking tradition in the clustering area than in the field of supervised learning.…”