Some new nonparametric distribution free control charts based on rank statistics

SOME NEW NONPARAMETRIC DISTRIBUTION-FREE CONTROL CHARTS BASED ON RANK STATISTICS LI SUYI NATIONAL UNIVERSITY OF SINGAPORE 2011 SOME NEW NONPARAMETRIC DISTRIBUTION-FREE CONTROL CHARTS BASED ON RANK STATISTICS LI SUYI (B.Eng., Tianjin University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL & SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements ACKNOWLEDGEMENTS Firstly, I would like to thank my main supervisor Prof Tang Loon-Ching at ISE department, NUS Prof Tang gave me a precious opportunity to pursue my PhD degree in NUS, which might have changed my whole life His enthusiasm, patience, and support have kept me working on the right track Prof Tang also has given me many valuable comments and suggestions, which helped a lot in improving the quality of this research My deep appreciation also goes to Prof Ng Szu-Hui, my cosupervisor Prof Ng helped me reviewing and revising the dissertation for many rounds, and she never missed any tiny details Her dedication and carefulness to research have always been inspiring me I would like to thank other professors in the ISE department as well, especially Prof Ang Beng-Wah, Prof Goh Thong-Ngee, Prof Xie Min, Prof Lee Loo-Hay, Prof Tan Kay-Chuan, Prof Poh Kim-Leng, and Prof Chai Kah-Hin, I have been in their classes for various courses, and I really learnt a lot from them The ISE department officers and lab technicians are always professional and helpful, and here I want to thank Ms Ow Lai-Chun and Mr Lau Pak-Kai I am very grateful to my fellow labmates in the QRE Lab, and other friends in the ISE Department To name a few, Zhou Peng, Fan Liwei, Lin Jun, Qian Yanjun, Wang Qi, Wang Xiaoyang, Xin Yan, Chang Hongling, Awie, Joyce, Liu Xiao, Han Dongling, Han Yongbin, Pan Jie, Vijay, Henry, Tony, and many others I benefited a lot through discussion with them, and more importantly, we spent so many memorable times together, I will cherish the friendships forever i Acknowledgements Lastly, I will thank the most important persons in my life: my wife Wang Miao, my son Li Junkai, and my parents My wife and I have known each other for 15 years by now, without her continuous support, I could not possibly come so far I wish my son Li Junkai happy and healthy My parents raised me and supported me for so long, but never asked for any return With your love I will not walk along LI SUYI February 2011 ii Table of Contents TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS iii SUMMARY v LIST OF TABLES vii LIST OF FIGURES ix CHAPTER INTRODUCTION 1.1 Median and max/min charts 1.2 Group signed-rank charts 1.3 Sequential rank charts 10 1.4 Research gaps 12 1.5 Structure of the dissertation 13 CHAPTER NONPARAMETRIC CUSUM AND EWMA CONTROL CHARTS FOR DETECTING STEP SHIFTS IN PROCESS MEAN 15 2.1 Introduction 15 2.2 The Wilcoxon Rank-Sum (WRS) based CUSUM and EWMA control charts 17 2.3 Design of W-CUSUM and W-EWMA control charts 20 2.4 ARL performance comparison 29 2.5 A numerical example 45 2.6 Effect of reference sample size and subgroup size 51 2.7 Conclusion 57 CHAPTER NONPARAMETRIC CUSUM AND EWMA CONTROL CHARTS FOR DETECTING STEP SHIFTS IN PROCESS VARIANCE .58 3.1 Introduction 58 3.2 Siegel-Tukey test based nonparametric control charts 60 3.3 Design of ST-CUSUM and ST-EWMA control charts 64 iii Table of Contents 3.4 Comparison to parametric control charts 69 3.5 Integration with W-charts 76 3.6 Numerical examples 83 3.7 Conclusion 91 CHAPTER NONPARAMETRIC CHANGE-POINT CONTROL CHART FOR PHASE I ANALYSIS 92 4.1 Introduction 92 4.2 A change-point type nonparametric Phase I control chart 94 4.3 Derivation of joint distribution 100 4.4 Performance comparison 106 4.5 A numerical example 116 4.6 Diagnostic application 119 4.7 Discussion 124 4.8 Conclusion 127 CHAPTER NONPARAMETRIC CONTROL CHARTS FOR MONITORING LINEAR PROFILES .128 5.1 Introduction 128 5.2 Linear mixed model and parameter estimation 133 5.3 Distribution-free profile control charts 138 5.4 ARL performance comparison 144 5.5 Numerical examples 157 5.6 Monitor error terms and Phase I analysis 166 5.7 Conclusion 169 CHAPTER CONCLUSIONS AND FUTURE WORKS 170 References .176 iv Summary SUMMARY Parametric control charts, such as Shewhart X chart, Cumulative Sum (CUSUM) chart, Exponentially Weighted Moving Average (EWMA) chart, and their extensions, have been proven to perform satisfactory in many situations However, they are often constructed based on the assumption that the underlying process follows normal (or multi-normal, for multivariate control charts) distribution The performance of parametric control charts could be seriously affected if the normal assumption is violated, despite the effect of central limit theorem In this research, several distribution-free nonparametric control charts are proposed The proposed control charts not rely on normal assumption, and they can be used when the underlying process distribution is not well known The nonparametric control charts are developed to address some major topics in statistical process control (SPC), such as monitoring process mean, monitoring process variance, Phase I (retrospective) analysis of historical data sample, and monitoring linear profiles The nonparametric methods are often less favorable compared to parametric control charts, due to their lower power-of-the-test However, it is shown in the dissertation that, our proposed nonparametric control charts perform quit close to their parametric counterparts, if the process parameters are considered being estimated from reference sample The exact run-length distributions of the proposed control charts are derived, the average run-length (ARL) properties are investigated, and several numerical examples are presented for illustration purpose It has been found, parametric control charts generally have too short in-control ARLs under non-normal distributions, and v Summary the proposed nonparametric control charts perform consistently in terms of in-control ARL under all distribution scenarios A notable improvement of the proposed nonparametric control charts, over existing nonparametric control charts, is that they are still sensitive under normal distribution Therefore, they can be used in place of the traditional parametric control charts without losing much power vi List of Tables LIST OF TABLES Table 1.1 In-control ARLs for X and X (subgroup size 5) charts for data from χ2-distributions Table 1.2 Products from a statistical process control database Table 2.1 Distribution parameters for N(0,1), G(3,1), and t(5) 32 Table 2.2 Control limits and parameter settings for the 12 control charts used in the simulation comparison 32 Table 2.3 Performance comparison of W-CUSUM, W-EWMA with other control charts, under Normal(0,1) distribution 33 Table 2.4 Performance comparison of W-CUSUM, W-EWMA with other control charts, under Gamma(3,1) distribution 34 Table 2.5 Performance comparison of W-CUSUM, W-EWMA with other control charts, under t(5) distribution 35 Table 2.6 Percentage deviations of ARL values for W-CUSUM and X - CUSUM (adjust) charts 36 Table 2.7 Percentage deviations of ARL values for W-EWMA and some selected X - EWMA charts under Normal (0, 1) distribution 36 Table 2.8 Percentage deviations of ARL values for W-EWMA and some selected X - EWMA charts under Gamma (3, 1) distribution 36 Table 2.9 Percentage deviations of ARL values for W-EWMA and some selected X - EWMA charts under t (5) distribution 37 Table 2.10 W-CUSUM and W-EWMA charts ARL performance under various reference sample size and subgroup size combinations, t(5) distribution 52 Table 3.1 Performance comparison of ST-CUSUM, ST-EWMA with ln(S2)CUSUM and ln(S2)-EWMA charts, under Normal(0,1) distribution 71 Table 3.2 Performance comparison of ST-CUSUM, ST-EWMA with ln(S2)CUSUM and ln(S2)-EWMA charts, under Gamma(3,1) distribution 71 Table 3.3 Performance comparison of ST-CUSUM, ST-EWMA with ln(S2)CUSUM and ln(S2)-EWMA charts, under t(5) distribution 72 Table 3.4 Performance comparison of W-CUSUM, W-EWMA charts with STCUSUM, ST-EWMA charts, when process mean shifts 78 vii List of Tables Table 3.5 Performance comparison of W-CUSUM, W-EWMA charts with STCUSUM, ST-EWMA charts, when process variance shifts 78 Table 3.6 Performance comparison of W-CUSUM, W-EWMA charts with STCUSUM, ST-EWMA charts, when both process mean and variance shift 79 Table 4.1 The correlation coefficients of various t and s when sample size is 10 96 Table 4.2 The probability of detecting shift under different change-point location, underlying process follows t(5) distribution, N=50 110 Table 4.3 The probability of detecting shift of SW-chart, Shewhart chart, and lrt chart under Normal (0,1) distribution, N=50 110 Table 4.4 The probability of detecting shift of S-W, Shewhart chart, and lrt chart under Gamma (3,1) distribution, N=50 111 Table 4.5 The probability of detecting shift of SW-chart, Shewhart chart, and lrt chart under Student’s t(5) distribution, N=50 111 Table 4.6 The probability of detecting shift of SW-chart under different sample size, with FAR=0.05, t(5) distribution 115 Table 4.7 Probability of correctly indicating the change-point by using the SWchart under various combinations of total sample size and change-point location, when shift magnitude is 1σ / m 122 Table 4.8 Probability of correctly indicating the change-point by using the SWchart under various combinations of total sample size and change-point location, when shift magnitude is 2σ / m 123 Table 4.9 Probability of correctly indicating the change-point by using the SWchart under various combinations of total sample size and change-point location, when shift magnitude is 3σ / m 123 Table 5.1 Performance comparison of RT2-CUSUM, RT2-EWMA charts with T2CUSUM , T2-EWMA, and T2(Kang & Albin) charts under normal distribution scenario 149 Table 5.2 Performance comparison of RT2-CUSUM, RT2-EWMA charts with T2CUSUM , T2-EWMA charts under Gamma distribution scenario 150 Table 5.3 Performance comparison of RT2-CUSUM, RT2-EWMA charts with T2CUSUM , T2-EWMA charts under t(5) distribution scenario 151 Table 5.4 ARL & SDRL comparison of profile control charts under various combinations of reference sample size and number of measurements in each profile, when the process is normally distributed and the shift size is 156 viii Chapter CHAPTER CONCLUSIONS AND FUTURE WORKS As has been discussed, non-normality commonly exists in practice, and could affect the performance of parametric control charts seriously In this dissertation, some new nonparametric control charts are introduced These control charts are distribution-free, i.e., their in-control ARLs are not affected by the distribution of the underlying process This characteristic is useful when the underlying process distribution is not well known Different to existing nonparametric control charts, the proposed methods perform close to their parametric counterparts, even when the process follows normal distribution In the dissertation, we also cautioned that, for parametric control charts, the effect of using estimated parameters from a reference sample should be considered Our comparisons are all based on the assumption that the process parameters are not known a priori After a review of the development of nonparametric control charts, it is found that the existing nonparametric charts are less effective than parametric charts, and there is a lack of nonparametric control chart for monitoring process variance and nonparametric Phase I control chart These issues are addressed in the respective chapters of the dissertation We first proposed two control charts, W-CUSUM and W-EWMA charts, based on the Wilcoxon Rank-Sum statistic The run-length distribution is obtained by using conditioning method, and then the run-length distribution is used to determine the control limits of the proposed charts Through simulation study, the proposed Wcharts are found to be superior to their parametric counterparts under non-normal distributions, and still effective under normal distribution Most importantly, their incontrol ARLs remain unchanged under different distributions The effect of the 170 Chapter reference sample size and subgroup size is also investigated, and the results suggest that the W-charts perform well over a wide range of combinations of reference sample size and subgroup size Follow the similar idea of the W-CUSUM and W-EWMA charts, we then proposed the ST-CUSUM and ST-EWMA charts for monitoring process variation These charts are based on the Siegel-Tukey test, and they are designed in the same way as the W-charts Their performance is also compared to the parametric charts Like the W-charts, the ST-charts are superior to their parametric counterparts under non-normal distributions, and also effective under normal distribution We propose that the W-charts and the ST-charts can be used together to monitor the process mean and variance simultaneously The simulation results show that jointly using the Wcharts and ST-charts can help to identify whether the process mean or variance has shifted, since the W-charts will give out-of-control signal earlier if the process mean has shifted, and ST-charts will give out-of-control signal earlier if the process variance has shifted A nonparametric change-point type control chart, based on sequential WRS statistic, was proposed for Phase I application Since all the sequential WRS statistics obtained from a historical data set are correlated, they are considered together as a multi-dimensional vector The joint distribution of the vector is derived by conditional probability and theory of combinations, and the joint distribution is then used for the design of the nonparametric Phase I chart The proposed chart can effectively detect sustained mean shift under different distributions, and meanwhile the false alarm rate remains the same The sequential WRS test is also considered to be used as a 171 Chapter diagnostic tool, in order to indicate the change-point location after the Phase II chart gives out-of-control signal The proposed nonparametric control charts were applied to monitor linear profile data We consider the situation that the profile data can be represented well by the linear mixed model, and distribution-free estimation technique is used to obtain the estimators of the model parameters We focus on the Phase II monitoring of the random effect term, and introduce the concept for monitoring error terms and Phase I analysis The nonparametric profile charts perform well under different distributions This method is flexible and can be easily extended to deal with more situations In this dissertation, several new nonparametric control charts are introduced The topics cover the Phase II charts for process mean, Phase II charts for process variance, Phase I chart and diagnostic tool for process mean, and control charts for linear profile data Although these methods have constituted a system of nonparametric control charts, some issues still need to be addressed or further explained We list our plan for future research here: In Chapter 3, we have considered that the mean and variance have shifted the same magnitude More study is needed to consider other scenarios, such as mean and variance shift in different magnitude Further study will be conducted on the using W-charts and ST-charts simultaneously In section 3.5, Chapter 3, ARL was used to measure the performance of the W-charts and ST-charts Other performance index, however, could also be useful For instance, the probability of Pr{RLST-CUSUM – RLW-CUSUM >d} (d is a small number of steps) can better reflect whether the integration of the two charts still performs 172 Chapter well for detecting a mean shift only, i.e., the later ST-CUSUM’s alarms will not affect the monitoring decision for detecting the process change with a mean shift only The nonparametric Phase I control chart for monitoring process variance is not covered in this dissertation, although some ideas have come up Some preliminary works on this topic have been conducted by the author Further study could be done to investigate the performance of the SWchart for diagnostic purpose In Chapter 4, we only considered limited scenarios of simulation settings, and a more comprehensive study could better promote the usage of the approach The proposed control charts, especially the W-CUSUM, W-EWMA, ST-CUSUM, ST-EWMA, and SW-chart, can be easily applied by using the Microsoft Office Excel spreadsheet We have done some preliminary works to use the proposed charts in Excel spreadsheet, and we will try to develop it into some package or add-ins The methods for profile monitoring in Chapter can be used as multivariate distribution-free control charts as well Hence, we will study their performance as nonparametric multivariate control chart and compare them to the parametric multivariate CUSUM and EWMA control charts and some existing nonparametric multivariate control charts, such as the data-depth control charts (Liu (1995)) We will also investigate how to extend the proposed methods to polynomial mixed model and nonlinear mixed model situations 173 Chapter The nonparametric control charts for monitoring profile error terms could be compared to existing methods, although we believe the results will be similar to what has been shown in Chapter Phase I analysis for profile data could be further developed Some important issues, such as detecting outliers and accuracy of the estimation techniques could be discussed in detail And the nonparametric Phase I profile chart could be compared to the parametric chart proposed by Jensen et al (2008) For the Phase II nonparametric profile control charts, more study could be done to consider the unbalanced data and missing data scenarios, although it is expected that the proposed method will still perform well 10 The nonparametric profile charts could be 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Journal of Quality Technology, 39(4): 364-375 184 ... possibility of developing nonparametric control charts based on two-sample tests The objective is to develop some new nonparametric control charts, which are not only distribution- free but also effective... process is in -control Therefore, the control charts based on Siegel-Tukey statistic have the same in -control run-length distribution to the similar control charts based on Wilcoxon rank- sum statistic... X-EWMA charts, and also to some existing nonparametric control charts Among existing nonparametric control charts (see the Introduction section), the Median chart and the methods based on group