Factorial and reduced K-means reconsidered

Factorial K-means analysis (FKM) and Reduced K-means analysis (RKM) are clustering methods that aim at simultaneously achieving a clustering of the objects and a dimension reduction of the variables. Because a comprehensive comparison between FKM and RKM is lacking in the literature so far, a theoretical and simulation-based comparison between FKM and RKM is provided. It is shown theoretically how FKM's versus RKM's performances are affected by the presence of residuals within the clustering subspace and/or within its orthocomplement in the observed data. The simulation study confirmed that for both FKM and RKM, the cluster membership recovery generally deteriorates with increasing amount of overlap between clusters. Furthermore, the conjectures were confirmed that for FKM the subspace recovery deteriorates with increasing relative sizes of subspace residuals compared to the complement residuals, and that the reverse holds for RKM. As such, FKM and RKM complement each other. When the majority of the variables reflect the clustering structure, and/or standardized variables are being analyzed, RKM can be expected to perform reasonably well. However, because both RKM and FKM may suffer from subspace and membership recovery problems, it is essential to critically evaluate their solutions on the basis of the content of the clustering problem at hand.