Self-Starting Multivariate Exponentially Weighted Moving Average Control Charting

Abstract

Multivariate control charts are valuable tools for industrial quality control. The conventional discussion of them rests on the presumption that the in-control process parameters are known a priori. The more common reality is that practitioners plug in parameter estimates gathered from a special phase I sample to establish parameter values for the charts. But no sample will establish the exact process parameters, and quite small random errors translate into serious distortions of the run behavior, particularly of sensitive charts, and can affect chart performance. So-called “self-starting” methods can begin the control of the process right after startup without the preliminary step of a large phase I sample. Univariate self-starting methods for converting the unknown-parameter stream of process readings into a known-parameter sequence have been available for some time now. This article develops a multivariate equivalent by providing a way to transform the process readings into a stream of vectors following an exact known-parameter distribution. Although our approach is far from being the first proposal for self-starting charting of multivariate data, we believe it is the first that does so by transforming the unknown-parameter process vectors into known-parameter vectors of the same dimensionality. This stream of vectors has many potential uses. In particular, it may be used to construct any multivariate control chart, such as Hotelling'sT2, or any of the multivariate cusum methods. We illustrate using the transformed stream to set up a multivariate exponentially weighted moving average chart. With the self-starting front end, this (or any other) chart will have the same in-control properties as if the true process mean and covariance matrix were known exactly, thereby allowing multivariate control charting to proceed without a large and costly phase I data-gathering exercise.