Multivariate Quality Control Using Loss-Scaled Principal Components

Abstract

We consider a principal components based decomposition of the expected value of the multivariate quadratic loss function, i.e., MQL. The principal components are formed by scaling the original data by the contents of the loss constant matrix, which defines the economic penalty associated with specific variables being off their desired target values. We demonstrate the extent to which a subset of these ``loss-scaled principal components", i.e., LSPC, accounts for the two components of expected MQL, namely the trace-covariance term and the off-target vector product. We employ the LSPC to solve a robust design problem of full and reduced dimensionality with deterministic models that approximate the true solution and demonstrate comparable results in less computational time. We also employ the LSPC to construct a test statistic called loss-scaled T^2 for multivariate statistical process control. We show for one case how the proposed test statistic has faster detection than Hotelling's T^2 of shifts in location for variables with high weighting in the MQL. In addition we introduce a principal component based decomposition of Hotelling's T^2 to diagnose the variables responsible for driving the location and/or dispersion of a subgroup of multivariate observations out of statistical control. We demonstrate the accuracy of this diagnostic technique on a data set from the literature and show its potential for diagnosing the loss-scaled T^2 statistic as well.