Phase II monitoring of linear profiles with random explanatory variable under Bayesian framework
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,where η is the under study parameter. is the posterior distribution of under study parameter η. defined the likelihood function of sampling distribution of process information. represent the prior distribution of uncertain parameter of interest. 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.
Section snippets
Linear profiles model and estimation under Bayesian framework
This section provides detail estimation procedure of simple linear profiles model under classical and Bayesian setup. The simple linear profiles model can be defined as:where and are the specified process parameters under an in-control scenario for the jth profile. The least square estimates are and with respective variances as: and The
Phase II classical and Bayesian control charting structure
This section presents control charting structure based on classical and Bayesian framework using phase II method with random X for linear profiles monitoring. The SPC community designed control charting structures either in Phase I or in Phase II. These have different goals and demonstration when applied. Further we demonstrate the competing and proposed control charts in Phase II after incorporating the posterior estimates.
Performance measures and simulation setting
The control charts mostly evaluated on the basis of run length distribution properties. The ARL values are the widely acknowledged measure of evaluation. It is interpreted as the average number of values required by a control structure to identify first out-of-control signal or signal false alarm. There are several methods available in literature to compute ARL values, like the integral equation method of Page, 1954, Brook and Evans, 1972 scheme based on MCMC techniques, and the method based on
Elicitation and sensitivity analysis
This section describes the hyper-parameter’s elicitation and corresponding sensitivity analysis to obtain the best choice of hyper-parameters. The hyper-parameter values contribute significantly for the designed control structures using Bayesian setups. So it is quite important to elicitate these hyper-parameter values to conduct further Bayesian analysis. A method proposed by Garthwaitea, Al-Awadhib, Elfadalya, and Jenkinsonc (2013) is an effective way of elicitating the hyper-parameters. We
Comparison of proposed method
This section provides comparative analysis of proposed Bayesian scheme for the monitoring of linear profiles with the existing classical approach. The evaluations of the proposed and existing control charts are based on different individual and overall measures. The smoothing constant value of 0.2 used by Noorossana et al. (2015) also used for Bayesian analyses. The EWMA control charts under classical setup represented by EWMA-Cx, while the proposed Bayesian EWMA charts for non-conjugate and
Real data applications
This section presents and discusses case studies based on real data to justify simulative work findings. The in-control and out-of-control scenarios are designed based on real data set to validate the effectiveness of control charts under Bayesian setup for linear profiles monitoring with random explanatory variable. The following steps are involved to estimate Bayesian EWMA charts.
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Check the linearity and normality assumption of each profile.
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Elicitation of hyper-parameters based on real data.
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Conclusion and recommendations
There are situations in manufacturing industry where the process parameters are random then Bayesian approach is a wiser choice to handle such processes. This study has investigated linear profiles under random explanatory variables using Bayesian EWMA and MEWMA control charts in phase II. It is to be mentioned that the priors and the corresponding hyper-parameters should be selected carefully for efficient monitoring of linear profiles. From the run length comparison it is observed that the
Acknowledgments
The authors would like to acknowledge the Editor and Referees for their constructive comments that lead to significant improvements in the manuscript. The authors would like to acknowledge their parent Universities for providing excellent research facilities. Moreover, Dr. Muhammad Riaz would like to acknowledge Deanship of Scientific Research for supporting the study under Project Number IN171016.
References (40)
- et al.
On monitoring of linear profiles using Bayesian methods
Computers and Industrial Engineering
(2016) - et al.
On efficient use of auxiliary information for control charting in SPC
Computers and Industrial Engineering
(2014) - et al.
The S chart with variable charting statistic to control Bi and trivariate processes
Computers & Industrial Engineering
(2017) - et al.
Integration of production quantity and control chart design in automotive manufacturing
Computers & Industrial Engineering
(2016) - et al.
Monitoring simple linear profiles in multistage processes by a Maxewma control chart
Computers and Industrial Engineering
(2016) - et al.
New indices for the evaluation of the statistical properties of Bayesian Xbar control charts for short runs
European Journal of Operational Research
(2017) - et al.
Phase Ii monitoring of multivariate simple linear profiles
Computers & Industrial Engineering
(2010) - et al.
A comparison study of effectiveness and robustness of a control charts for monitoring process mean
International Journal of Production Economics
(2012) - et al.
Monitoring the performance of Bayesian EWMA control chart using loss functions
Computers & Industrial Engineering
(2017) - et al.
Combined Shewhart CUSUM charts using auxiliary variable
Computers & Industrial Engineering
(2017)
Statistical quality control applied to ion chromatography calibrations
Journal of Chromatography A
A simulation-based multivariate Bayesian control chart for real time condition-based maintenance of complex systems
European Journal of Operational Research
Phase II monitoring and diagnosis of autocorrelated simple linear profiles
Computers & Industrial Engineering
Properties and enhancements of robust likelihood cusum control chart
Computers & Industrial Engineering
On the prior selection for variance of the normal distribution
Journal of Stats
Universality of rare fluctuations in turbulence and critical phenomena
Nature
An approach to the probability distribution of cusum run length
Biometrika
Prior distribution elicitation for generalized linear and piecewise-linear models
Journal of Applied Statistics
Bayesian estimation of the time of a linear trend in risk-adjusted control charts
International Journal of Computer Science
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