Weighting variables in K-means clustering

The aim of this study is to assign weights w ₁, …, w _m to m clustering variables Z ₁, …, Z _m , so that k groups were uncovered to reveal more meaningful within-group coherence. We propose a new criterion to be minimized, which is the sum of the weighted within-cluster sums of squares and the penalty for the heterogeneity in variable weights w ₁, …, w _m . We will present the computing algorithm for such k-means clustering, a working procedure to determine a suitable value of penalty constant and numerical examples, among which one is simulated and the other two are real.