k-plus Anticlustering: An Improved k-means Criterion for Maximizing Between-Group Similarity

Anticlustering refers to the process of partitioning elements into disjoint groups with the goal of obtaining high between-group similarity and high within-group heterogeneity. Anticlustering thereby reverses the logic of its better known twin—cluster analysis—and is usually approached by maximizing instead of minimizing a clustering objective function. This paper presents k-plus, an extension of the classical k-means objective to maximize between-group similarity in anticlustering applications. K-plus represents between-group similarity as discrepancy in distribution moments (means, variance, and higher order moments), whereas the k-means criterion only reflects groups differences with regard to means. While constituting a new criterion for anticlustering, it is shown that k-plus anticlustering can be implemented by optimizing the original k-means criterion after the input data has been augmented with additional variables. A computer simulation and practical examples show that k-plus anticlustering achieves high between-group similarity with regard to multiple objectives. In particular, optimizing between-group similarity with regard to variances usually does not compromise similarity with regard to means; the k-plus extension is therefore generally preferred over classical k-means anticlustering. Examples are given on how k-plus anticlustering can be applied to real norming data using the open source R package anticlust, which is freely available via CRAN.