D-optimality criterion for weighting variables in K-means clustering

The aim of the study is how to achieve best K-means clustering structure so that k groups uncovered reveal more meaningful within-group coherence by assigning weights w1,…,wm to m clustering variables Z1,…,Zm. We propose Wilks' lambda as a criterion to be minimized with respect to variable weights w1,…,wm. This criterion, that is the ratio of the determinant of the within-cluster sums of squares and cross products matrix and that of the between clusters sums of squares and cross products matrix, is equivalent to the D-optimality criterion in the optimal design theory and related to minimization of the volume of the simultaneous confidence region of the cluster means. We will present the computing algorithm for such K-means clustering and numerical examples, among which one is simulated, two are real and the other one is the real data set augmented with additional simulated noise variables.