A Combination of Stochastic Dynamic Programming and Bayesian Inference Approaches for Fault Detection and Diagnosis in Multivariate Statistical Quality Control Environments

In many quality control settings, the product (process) under examination may have two or more correlated quality characteristics (variables) and one needs an appropriate approach to monitor all these characteristics simultaneously. This introduces the multivariate quality control problem, which many researches are devoted to solve. One way of interpreting out-of-control signals in multivariate quality control environments is to view the corresponding univariate charts of a multivariate process to determine which statistic is causing the assignable cause of variation. Although this is a simple and plausible way of out-of-control analysis, there are some concerns associated with the adaptability of the technique. First, when there are many variables being measured, this technique tends to get tedious since there would be many univariate charts to interpret. Second, in multivariate quality control, an out-of-control signal is usually not caused by one variable, but rather a function of several correlated variables due to their interdependent behavior. Therefore, in many circumstances, the respective univariate charts may show no signs of being out-of-control, while on the contrary, the multivariate chart give out-of-control signals. The user of this technique needs to understand that there are other effective interpretation techniques that could be used with this technique to perform a better analysis of the out-of-control signals and that the user should not be limited to this technique simply because it is an obvious and simple approach to the interpretation. Early research on multivariate Shewhart charts goes back to Hotelling (1947) who introduced the problem of correlation between the quality characteristics of a process and came up with the well-known statistic (or in case of known process parameters) to identify whether the whole process is out-of-control. A major advantage of Hotelling’s statistic is that it is the optimal test statistic for detecting a general shift in the process mean vector for an individual multivariate observation (Hawkin 1991). However, the technique has several practical drawbacks. One of the most important ones is that when the statistic indicates that a process is out of control, it does not provide information on which variable or set of variables are out of control. Moreover, it is difficult to distinguish location shifts from scale shifts since the statistic is sensitive to both types of process changes.