Causation-Based Monitoring and Diagnosis for Multivariate Categorical Processes With Ordinal Information

Abstract

The monitoring and diagnosis of multivariate categorical processes (MCPs) have drawn increasing attention lately, as categorical variables have been frequently involved in modern quality control applications. In these applications, there may exist causal relationships among multiple categorical variables, where the attribute level of a cause variable influences that of its effect variable. In such a case, shifts occurring in a cause variable will propagate to its effect variable based on the causal structure. Furthermore, there usually exists natural order among the attribute levels of some categorical variables such as good, neutral, and bad for measuring the product quality. By assuming a latent continuous variable, the attribute levels of an ordinal categorical variable can be determined by classifying the value of the latent variable based on thresholds. In this paper, we leverage Bayesian networks (BNs) to characterize MCPs with a causal structure, where the categorical variables can be either nominal, ordinal or a combination of both. We develop one general control chart and one directional control chart, both of which fully exploit the causal relationships and the ordinal information for better process monitoring and diagnosis. Numerical simulations have demonstrated the superiority and robustness of our method in detecting and diagnosing the conditional probability shifts of nominal factors as well as the conditional latent location shifts of ordinal factors. Note to Practitioners —This paper aims at addressing the challenges in monitoring and diagnosing MCPs when there are causal relationships among the categorical variables. The developed method is greatly beneficial, especially when there are nominal and ordinal variables involved in the MCP. Specifically, a BN is employed to characterize the dependence structure of the variables involved in the process, and a latent continuous variable is utilized to model the orders of attribute levels of the ordinal variables. Then, a novel method is proposed to detect the probability shift in the nominal factors and the location shift on the latent variables of the ordinal variables based on the likelihood ratio test. A general monitoring control chart as well as a directional version which also facilitates diagnosis is proposed. In this paper, although our method is based on the assumption that the latent continuous variables of the ordinal variables follow logistic distributions, the proposed charts are demonstrated to perform efficiently and robustly in various cases as shown in the simulations and case studies.