Multivariate process variability monitoring for general sample design

Abstract

In multivariate setting, from time to time, process variability is summarized and numerically represented as a covariance matrix, say S. It is generally measured as a non-negative real valued function of S such that the more scattered the population, the larger the value of the function and vice versa. In the literature, the three most popular functions are total variance (TV), generalized variance (GV) and vector variance (VV). Algebraically, TV is the sum of all eigenvalues of S, GV is the product of those eigenvalues, and VV is their sum of squares. The last two measures are designed for large sample size. If GV is to detect large shift in covariance matrix, VV is for small shift. Even though these measures can also be used for small sample size but the control limits must be determined based on heuristic approach. In order for those control limits to be related to the probability of false alarm, a control charting procedure that monitors multivariate variability changes is introduced in this thesis. The methodology is by maximizing the summation of conditional variances for all possible permutations or order of variables. This chart can be used to monitor process variability regardless of the sample size. Under normality, its exact distribution is derived. For practical purposes, when the sample size is small, we introduce the use of Solomon and Stephen’s approximation to that distribution with adjusted-probability of false alarm. The advantage of the proposed chart is that it could detect very small magnitude of disturbance in correlation structure which cannot be detected by existing charts. Besides that, the order of variables will lead to better diagnostic features. The performance of the proposed chart in terms of average run length (ARL) is very promising. Some industrial application examples are presented to illustrate the advantages of the proposed chart.