# A Bayesian Control Chart for Monitoring Process Variance
## 1 Introduction  
Due to the rapid development of artificial intelligence, service automation is emerging as a new wave of industrial revolution. Standardization and consistency of service quality are important parts of the automation process. The quality control methods widely used in the manufacturing industry can provide service quality measurement and service process monitoring. In particular, online process monitoring of the control chart can be used to quickly detect signals that indicates when the service process is out of control. However, the control of a service process is more difficult than that of a manufacturing process because the variability of a service process comes from widespread and complex factors [1; 2; 3]. First of all, the distribution of the service process is usually non-normal or unknown. Moreover, the skewness of process distribution is time-varying, even if the process is in control. Because Shewhart variables control charts depend on the normality assumption, several studies have searched for effective methods to measure the performance of control charts with non-normal or unknown distribution data. Among them, most studies dealwith process location monitoring, while fewer studies deal with process variability monitoring. Readers interested in process location monitoring can refer to [PERSON] [4], [PERSON] and [PERSON] [5], [PERSON] et al. [6], [PERSON] [7; 8], [PERSON] [9; 10], [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [12], [PERSON] et al. [13], [PERSON] et al. [14], [PERSON] and [PERSON] [15], and [PERSON] et al. [16; 17]. As for the process variability monitoring, [PERSON] and [PERSON] [15] developed the nonparametric likelihood-ratio (NLE) exponential weighted moving average (EWMA) control chart and the combination of EWMA mean chart and EWMA variance chart (CEW) to monitor the changes in process variance. [PERSON] and [PERSON] [18] proposed a distribution-free Phase I scale chart, which can be used to define the in-control state of the process variability. [PERSON] and [PERSON] [19] developed two Shewhart-type nonparametric control charts based on run rules for monitoring the changes in the process variability. The above non-parametric approaches can deal with non-normal or unknown distributed data, but they are not easy to apply for practitioners. The practitioners, usually not statisticians, do not quite understand the proper way to implement the control scheme. To achieve ease of use, [PERSON] et al. [20] proposed a new Sign Chart for variables data to monitor the deviation of process measurements from the target without the assumption of a normal process distribution or a distribution of known form. Similarly, a new Mean Chart and an improved Mean Chart for variables data were proposed by [PERSON] et al. [21] and [PERSON] [22], respectively, to monitor the process mean also without the assumption of a normal process distribution or a distribution of known form. As for the process variability monitoring, [PERSON] and [PERSON] [23] proposed a new distribution-free variance chart for variables data to monitor the process variance. [PERSON] and [PERSON] [24] further developed the interquartile range control chart for detecting the out-of-control variance of a critical quality characteristic that exhibits a non-normal or unknown distribution. The above easy-to-use approaches can deal with non-normal or unknown distributed data under the assumption that the skewness of the distribution is fixed. However, the assumption of a fixed skewness is not true for a process with the mixture distribution. Hence, the question arises: 'How can one measure the performance of control charts for monitoring a process in which skewness of the distribution may vary randomly by time?' The Bayesian approach is usually applied to address the uncertainty about the parameter of interest. [PERSON] and [PERSON] [25] used the Bayesian approach to deal with a class of statistical quality control procedures and continuous inspection procedures.