Weighting variables in K-means clustering

The aim of this study is to assign weights w1, . . . , wm to m clustering variables Z1, . . . , Zm, 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 w1, . . . , wm.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.