Post Hotelling's T-square procedure to identify fault variables

Hotelling's (Formula presented.) (HT) control chart is popular in monitoring the multivariate statistical process's mean vector. HT is a global testing procedure which only tells the existence of some unknown change in the p-variate mean. When the HT control chart detects the change in the p-variate mean, the next question would be which part of the mean vector is changed. We call the procedure to answer this as post-HT procedure. The post-HT procedure finds out the p-variate mean sub-vector, which is the most abnormal (is changed the most) given that the global hypothesis is rejected. In this paper, we propose to search all sub-vectors of the p-variate mean and find the sub-vector having the smallest unconditional and conditional p-values. We propose a stochastic optimization algorithm based on the shotgun stochastic search and the parallel tempering algorithms to search the solution efficiently. We numerically show the proposed post HT procedure performs better than the existing forward (MTY) or backward (adaptive step-down, ASD) procedures and the lasso-based procedure in sensitivity (telling the changes for the variables whose means are changed). We further apply our proposal to monitoring the weekly counts of seven emotional words related to suicide collected from all blogs of the company DAUMSOFT from January 1, 2008, to December 31, 2010.