Phase II monitoring of linear profiles with random explanatory variable under Bayesian framework

Highlights

• Bayesian schemes for profiles monitoring with random explanatory variables are proposed.

• The classical multivariate EWMA (MEWMA) chart is extended to Bayesian MEWMA chart.

• Bayesian approach efficiently handled the parametric uncertainty of processes.

• Conjugate and non-conjugate priors are incorporated to handle parametric uncertainty.

• Conjugate priors are efficient among all priors in this study.

Abstract

Linear profiles monitoring have been successfully implemented in many industrial applications. The design structures of control charts for profiles monitoring are mostly based on two major classifications namely Classical and Bayesian. This study investigates the novel Bayesian exponentially weighted moving average and multivariate exponentially weighted moving average control charts for the monitoring of linear profiles, when explanatory variable(s) are random. The informative priors of normal and inverse gamma; and Bramwell, Holdsworth, Pinton (BHP) and Levy distributions are considered as conjugate and non-conjugate priors respectively. The proposed Bayesian schemes are evaluated using different run length characteristics. The schemes are also validated with simulation study and real-world data sets. The outcomes demonstrate that the Bayesian methods perform effectively better than the competing methods. The specified values of hyper-parameters are selected carefully after elicitation and sensitivity analysis of hyper-parameters. It has been observed that careful consideration is required while selecting the priors and possible values of hyper-parameters. The selection of appropriate priors and corresponding hyper-parameters comes up with efficient control structures which provide tangible benefits.

Introduction

Statistical Process Control (SPC) community has been actively involved in developing up-to-date statistical structures to enhance the performance of process structures in industrial engineering. The more emphasis on better quality of final product requires more information about process parameters and continuous process monitoring. The timely identification of changes in process parameters improves the quality of final product. There are several tools for process monitoring and the most essential one are the control charts. The quality control literature consists enormous articles that reflect the significance of this toolkit (i.e., Ahmad et al., 2014, Gunay and Kula, 2016, Stoumbos et al., 2000, Wu et al., 2017; references therein).

Currently, the SPC community is using two diverse approaches to define the quality characteristics of interest. These are the probability distribution or by using profiles function. In profiles function the quality characteristic(s) are defined using regression model where one is independent variable and other as dependent variable. The control charts are effectively used in profiles monitoring to check for possible change in relationship over time. The application of simple linear profiles was first introduced by Stover and Brill (1998) and attempt to resolve the appropriate calibration rate of recurrence for distinctive IC system with censored conductivity identification. Later, Kang and Albin (2000) characterized pressure and amount of flow using profiles function.

Since then, lots of literature has been published on profiles monitoring. They had tried to explore new dimensions and applications where profiles functions served effectively. The issue of interpretability in Kang and Albin (2000) was well addressed by Kim, Mahmoud, and Woodall (2003) by using transformed explanatory variable that results in zero covariance between Y-intercept and slope. Noorossana, Amiri, Vaghefi, and Roghanian (2004) introduced Multivariate Cumulative Sum (MCUSUM)/R chart for the monitoring of linear profiles in phase II. To handle the case of general linear profiles Zou, Tsung, and Wang (2007) incorporated Multivariate Exponentially Weighted Moving Average (MEWMA) charts. The idea of linear profiles monitoring was further extended by Noorossana, Eyvazian, and Vaghefi (2010) for multivariate cases. Soleimani and Noorossana (2012) investigated the impact of autocorrelation for the case of multivariate linear profiles. Zhang, He, Zhang, and Woodall (2014) evaluated the case of within-profile correlation in Phase II method. Zhang, Shang, He, and Wang (2016) used CUSUM schemes for the monitoring of pre specified changes in linear profiles. Zhang, Shang, Gao, and Wang (2017) used run rule schemes to monitor pre specified changes in linear profiles. Wang and Huang (2017) considered the case of first-order auto correlation in error terms while monitoring the linear profiles and provided diagnosis.

The control charting schemes discussed above and reference therein mainly focused the classical set-up to designed control charts. In classical set-up of process monitoring, assumption of fixed parameters seems impractical where parametric uncertainty is involved. This parametric uncertainty evolves when technological changes occur on regular basis to enhance the efficiency of products or processes. This demand from Statisticians to design flexible process monitoring structure to handle parametric uncertainty, which can be achieved through Bayesian control charts. Moreover, the use of posterior probability under Bayesian framework provides more practical grounds to identify the state of the process. There are numerous literature available where authors showed advantages of Bayesian approach over classical approach for the detection of shifts in process parameters (i.e., Hassan et al., 2011, Makis, 2008, Nikolaidis and Tagaras, 2017, Pan and Rigdon, 2012, Riaz et al., 2017, Wang, 2012, Wang and Lee, 2015; references therein).

In the view point of Bayesian approach, the elegant way to handle uncertainty is probability. For the case of repeated process parametric uncertainty occurs due to randomness in process. This scenario allows considering the fuzziness of the uncertain parameters in probabilistic way. Under these situations it seems more attractive to amassed most reliable values rather one value of the parameter (as in case of classical setup) Zellner, Ando, Basturk, Hoogerheide, and Van Dijk (2014). This parametric uncertainty incorporated in the form of probability distribution named as prior distribution. In Bayesian the information available for further analysis about process (i.e., environment of process, repairing history, observation of the engineer about process) are combined with response data by using Bayes theorem, mathematically defined as,

$$P\left(\eta \right|data)=\frac{P\left(data|\eta \right)P\left(\eta \right)}{P\left(data\right)}\propto P\left(data\right|\eta )P\left(\eta \right),$$

where η is the under study parameter. $P\left(\eta |data\right)$ is the posterior distribution of under study parameter η. $P\left(data|\eta \right)$ defined the likelihood function of sampling distribution of process information. $P\left(\eta \right)$ represent the prior distribution of uncertain parameter of interest. $P\left(data\right)$ is normalizing constant independent of parameter of interest which scale the posterior distribution.

Most of the literature available on profiles monitoring considered both explanatory variables and under study parameters as fixed. Noorossana, Fatemi, and Zerehsaz (2015) considered the case of random explanatory variables by taking a real process situation. However, there are many situations in process structures where the process parameters may vary along with explanatory variables. In this study, we have investigated randomness in both the process parameters and independent variable using Bayesian approach. The parametric uncertainty of process parameters integrated via prior distribution. This study investigated the properties of aforesaid Bayesian set-up after the modification of control limits coefficients, choice of priors and selection of hyper-parameters, choice of sampling distribution for dependent and independent variables, choice of loss function and impact of log and natural scale while monitoring the process standard deviations. Posterior estimates (posterior means) are obtained after incorporating prior knowledge into sample data. Three separate Bayesian EWMA control charts are designed under log and natural scale to monitor the process parameters of profiles model. The sensitivity analyses of hyper-parameters are performed to obtain the optimum choices of hyper-parameters values. The simulative study showed that there are processes defined by linear profiles where Bayesian set-up perform effectively and comes up with tangible benefits. The remaining article is structured as follows: Section 2 explains the linear profile model and Bayesian estimation. Classical and Bayesian control charting structure provided in Section 3. Section 4 describes the performance measure and step for simulation. Section 5 presents elicitation and sensitivity analysis for hyper-parameters. Section 6 demonstrates the comparative study. Section 7 presents the case study and Section 8 concludes the finding.