A study of new and advanced control charts for two categories of time related processes

A STUDY OF NEW AND ADVANCED CONTROL CHARTS FOR TWO CATEGORIES OF TIME RELATED PROCESSES DENG PEIPEI NATIONAL UNIVERSITY OF SINGAPORE 2013 A STUDY OF NEW AND ADVANCED CONTROL CHARTS FOR TWO CATEGORIES OF TIME RELATED PROCESSES DENG PEIPEI B.Sc., University of Science and Technology of China A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL & SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sourced of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously _ Deng Peipei 08 October 2013 i ACKNOWLEDGEMENTS Foremost, I would like to express my deepest appreciation to my supervisors, Prof Goh Thong Ngee and Prof Xie Min, for their continuous support, invaluable advice, inspiring guidance and great patience throughout my Ph.D study and research Their enlightening advice helped me in all the way of my study and the growth in my life I am deeply grateful to my senior and my friend, Dr Xie Yujuan for her insightful suggestions and help throughout my Ph.D life Besides, I want to thank National University of Singapore for offering me the opportunity and providing me the Scholarship to study here I want to express my appreciation to all the faculty members at the Department of Industrial and Systems Engineering as well for their supports My special thanks also goes to my fellow labmates and friends, Zhou Yuan, Liu Hongmei, Zhou Min, Chao Ankuo, Zhong Tengyue, Yang Linchang, Tang Muchen, Chen Liangpeng, Sheng Xiaoming, Ji Yibo and Xiao Hui at the Department of Industrial and Systems Engineering, National University of Singapore, for the stimulating discussions and all the fun in the last four years Last but not the least, I would like to express my heartfelt thanks to my parents and all my family members for their continuous care and precious support, and my boyfriend Yibo, for his understanding and illuminating encouragement in this endeavour ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ii SUMMARY viii LIST OF TABLES xi LIST OF FIGURES xiv LIST OF SYMBOLS xvi CHAPTER INTRODUCTION 1.1 CONTROL CHARTS 1.2 TWO CATEGORIES OF TIME-RELATED PROCESSES 1.2.1 Periodic Processes Monitoring 1.2.1 Time-between-Events Monitoring 1.3 RESEARCH SCOPE AND STRUCTURE OF THE STUDY CHAPTER LITERATURE REVIEW 10 2.1 PERIODIC PROCESS MONITORING 10 2.1.1 Periodic Processes 10 2.1.2 Cyclic Pattern 13 2.1.3 Periodic Process Extremes 14 2.1.4 Multivariate Periodic Processes 17 2.2 TIME BETWEEN EVENTS DATA MONITORING 18 2.2.1 Attribute TBE Control Charts 19 2.2.2 Variable Control Charts for TBE Data 20 2.3 TRANSFORMATIONS AND CHARTING EVALUATION 22 2.3.1 Transformation Techniques 22 2.3.2 Performance Evaluation of Control Charts 24 2.4 INNOVATIVE CONTROL CHARTS DESIGN 25 CHAPTER A CIRCLE CHART FOR PERIODIC MEASUREMENTS MONITORING 27 iii 3.1 THE BASIC CIRCLE CHART 28 3.1.1 An Illustration Example 29 3.1.2 Procedure to Plot A Circle Chart 31 3.1.3 Some Remarks 32 3.2 SOME FURTHER DEVELOPMENTS OF THE CIRCLE CHARTS 33 3.2.1 Properties of Circle Chart 33 3.2.2 Adjusting for Periodicity 34 3.2.3 Circle Chart based on Probability Limits 35 3.2.4 Normalization Transformation for Circle Chart Implementation 38 3.2.5 An Illustrative Example for Normalization Transformation 40 3.2.6 Circle Chart for Multivariate Characteristic 45 3.3 Discussions 47 CHAPTER CIRCLE CHART FOR THE MONITORING OF MAXIMA IN PERIODIC PROCESSES 48 4.1 INTRODUCTION 48 4.2 CIRCLE CHART BASED ON EXTREME VALUE DISTRIBUTION 50 4.2.1 Extreme-value Distribution and Parameter Estimation 50 4.2.2 Circle Chart Construction Procedure for Process Extremes Monitoring 51 4.2.3 An Illustrative Example 52 4.3 EXTENSIONS AND TRANSFORMATIONS 54 4.3.1 A Periodic Extreme-value distributed Model 54 4.3.2 Mean Normalization for Periodic Extreme-value Distributed Models 56 4.3.3 An Illustrative Example with Mean Normalization 57 4.4 EVALUATION OF THE PERFORMANCE OF CHARTING PROCEDURE 61 4.4.1 ACRL Analysis of the Mean Normalization 61 4.4.2 Comparison Study 63 4.4.3 Effect of Parameter Estimation and Distribution Skewness 66 4.5 DISCUSSIONS AND CONCLUSIONS 69 iv CHAPTER A CYCLIC T2 CHART FOR MULTIVARIATE PERIODIC MEASUREMENTS MONITORING 70 5.1 INTRODUCTION 70 5.2 A BASIC CYCLIC T2 CHART 72 5.2.1 The Chart Construction in Phase I 72 5.2.2 The Chart Construction in Phase II 73 5.2.3 Procedure to Plot a Cyclic T2 Chart 74 5.2.4 Illustrative Example of a Basic Cyclic T2 Chart 75 5.3 MODIFICATIONS AND EXTENSIONS 78 5.3.1 A Multivariate Periodic Model with Seasonality 78 5.3.2 A Sequential T2 Chart and Normalization Technique 81 5.3.3 An Illustrative Comparison Example of Several Charts 82 5.4 ACRL ANALYSIS OF CYCLIC T2 CHARTS 87 5.5 CONCLUDING REMARKS 94 CHAPTER A FULL MEWMA CHART FOR GUMBEL’S BIVARIATE EXPONENTIALLY DISTRIBUTED DATA 96 6.1 INTRODUCTION 96 6.2 MEWMA CHART FOR GUMBEL’S BIVARIATE EXPONENTIAL DISTRIBUTION 98 6.2.1 Bivariate Exponential Models 98 6.2.2 Inference Procedures for GBE Model 99 6.2.3 Traditional MEWMA chart for GBE data 100 6.3 MULTIVARIATE EWMA CHART WITH A FULL SMOOTHING MATRIX 102 6.3.1 The FMEWMA Chart 102 6.3.2 The Double Square-Root Transformation 104 6.3.3 The Covariance Matrix 105 6.3.4 Illustrative Example 106 6.4 PERFORMANCE OF THE FMEWMA CHART 110 v 6.4.1 Control Limit Selection in the Initial State and the Steady State Monitoring 110 6.4.2 The Chart Performance under Different Distribution Dependence Parameter 112 6.4.3 The Smoothing Parameters Selection 115 6.5 THEORETICAL ANALYSIS 116 6.5.1 Average Run Length Analysis 116 6.5.2 The Eigenvalue Analysis for Small Shifts 118 6.5.3 The Simulation Study 121 6.6 DISCUSSIONS AND CONCLUSIONS 137 CHAPTER A FULL MEWMA CHART FOR TRANSFORMED GUMBEL’S BIVARIATE EXPONENTIALLY DISTRIBUTED DATA 139 7.1 FMEWMA CHART FOR TRANSFORMED GBE DATA 141 7.1.1 The Charting Statistic 141 7.1.2 The Transformed GBE Data 143 7.1.3 Eigenvalue Table Analysis 144 7.2 GENERAL PERFORMANCE ANALYSIS AND SIMULATION STUDY 148 7.2.1 The Average Run Length Comparison and Analysis for Transformed GBE (1, 1, 0.1) 148 7.2.2 The Average Run Length Comparison and Analysis for Transformed GBE (1, 1, 0.9) 156 7.3 DISCUSSIONS AND CONCLUSIONS 163 CHAPTER CONCLUSIONS AND DISCUSSIONS 165 8.1 SUMMARY OF CONTRIBUTIONS AND FINDINGS 165 8.1.1 Circle Chart for Periodic Processes 166 8.1.2 FMEWMA Chart for GBE data 167 8.2 DISCUSSIONS AND FUTURE WORKS 170 REFERENCES 172 vi APPENDIX 195 vii SUMMARY Two categories of time-related processes monitoring are studied in this dissertation: periodic process and time between events (TBE) data monitoring Periodical processes are very common and important in practical industries Monitoring of such processes is useful and should be investigated carefully Due to the fact that limited works have considered such process monitoring with periodicity as an inherent property, a circle chart is proposed and studied for different types of periodic processes Individual procedure and transformation techniques are presented in this study The TBE control charts are proven to be very effective in high quality manufacturing processes monitoring This study further considers advanced control charts for small shifts detection: a multivariate exponentially weighted moving average (MEWMA) control chart with a full smoothing matrix The chart is designed for a Gumbel’s bivariate exponential (GBE) distribution commonly used to model practical TBE data The principle of the chart design is to enlarge the test statistics through the selection of off-diagonal elements under certain shifts for the charting efficiency improvement The thesis consists of four parts: Chapter 1-2; Chapter 3-5; Chapter 6-7, and a concluding Chapter Chapter introduces several basic concepts as well as the foundation and motivation of this study The research line and target is presented in this chapter Chapter reviews many related work and shows the gap between the existing literature and our concerned problems Periodic process with cyclebased signals e.g a stamping process or a forging process, is useful and should be monitored carefully Direct construction of traditional control charts to monitor such signals would either raise lots of false alarms or reduce the charting sensitivity Comparison between successive periods is not convenient as well Circle chart is proposed under such circumstances On the other hand, study towards the TBE data monitoring has drawn lots of attention and been developed viii McCool J.I and Joyner-Motley T (1998) ‘Control charts applicable when the fraction nonconforming is small’, Journal of Quality Technology, 30:3, 240-247 Memar A.O and Niaki S.T.A (2011) ‘Multivariate Variability Monitoring Using EWMA Control Charts Based on Squared Deviation of Observations from Target’, Quality and Reliability Engineering International, 27, 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‘Cycle-based signal monitoring using a directionally variant multivariate control chart system’, IIE Transactions, 37:11, 971–982 Zhou S.Y and Jin J (2005b) ‘Automatic feature selection for unsupervised clustering of cycle-based signals in manufacturing processes’, IIE Transactions, 37:6, 569-584 Zhou S.Y., Sun B and Shi J (2006) ‘An SPC monitoring system for cycle-based waveform signals using Haar transform’, IEEE Transactions on Automation Science and Engineering, 3:1, 60-72 194 APPENDIX The appendix presents a discussion towards the section θt in the Chapter Previous analysis assumes equal section θt In this appendix, we consider the situation with unequal ones and discuss two approaches to construct corresponding circle charts Refer to Chapter 4, the periodic extreme value distributed model is assumed to have equal sectional θt representing the ratio of the location parameter to the scale parameter θt = ut / bt The situation with different sectional θt is discussed as follows Refer to (4.11), the CDF for the transformed Yt with accurate parameter estimation is, F ( yt ) = exp( − exp( − yt − θt / (θt + γ ) )) / (θt + γ ) (A.1) Probability control limits become following express with different sectional θt: ICLα /2 = θt − α log( − log( )), θt + γ θt + γ θt α OCL1−α /2 = − log( − log(1 − )) θt + γ θt + γ (A.2) Direction expression of control limits becomes section dependent Note that the circle chart can be drawn with time-dependent control limits However, the circle chart could lose its clarity in this manner Without loss of generality, we still assume a periodicity of two with d = Following study present one alternative approach to consider a unified extreme-value distribution assumption with certain | θ1 − θ2 | 195 Notice that standard deviation of transformed section variable depends only on location-scale ratio θt In the unified distribution assumption, probability limits from lower tail of the unified distribution shall fall into the interval of corresponding percentiles from two respective transformed distributions Situation is similar with percentile from upper tail of the distribution Corresponding lower tail intervals and upper tail intervals are studied in Fig A.1 for standardization 10 Inner Control Limit Outer Control Limit Control Limits -2 θt 10 Fig A.1: Control Limits under Mean Transformation for Different θ t Fig A.1 shows control limits difference under mean normalization when location-scale parameter ratio θt is of different values It can be observed that control limits difference decreases a lot after θt ≥ Difference is extremely large as θt approaches to zero Theoretical analysis is expressed as follows, CL1 − CL2 = θ1 − θ2 − log( − log( p)) 1 + log( − log( p)) θ1 + γ θ2 + γ θ1 + γ θ2 + γ (θ1 − θ ) = [γ + log( − log( p ))] (θ1 + γ )(θ2 + γ ) When θ1 − θ is set to certain value and p is set to 0.025 and 0.975, we have, 196 (A.3) | ICL1 − ICL2 |≤ 0.1471⋅ | θ1 − θ | and | OCL1 − OCL2 |≤ 0.2422⋅ | θ1 − θ |, for both θt ≥ In the same parameter settings, for both θt ≥ , we have, | ICL1 − ICL2 |≤ 0.0605⋅ | θ1 − θ | and | OCL1 − OCL2 |≤ 0.0996⋅ | θ1 − θ | Following expressions hold for a small difference between two sectional θt , | OCL1 − OCL2 |≤ 0.0996 , if | θ1 − θ2 |= , for θt ≥ (A.4) The unified control limits would be very near the true control limits in this manner Following figures show the difference between several distributions 1.8 TRANSFORMED EV(3,1) TRANSFORMED EV(4,1) UNIFIED 1.6 1.4 Probability Density 1.2 0.8 0.6 0.4 0.2 0 0.5 1.5 (a) 197 2.5 TRANSFORMED EV(9,1) TRANSFORMED EV(10,1) UNIFIED 3.5 Probability Density 2.5 1.5 0.5 0.4 0.6 0.8 1.2 1.4 1.6 1.8 (b) 2.5 TRANSFORMED EV(3,1) TRANSFORMED EV(5,1) UNIFIED Probability Density 1.5 0.5 0 0.5 1.5 (c) 198 2.5 TRANSFORMED EV(8,1) TRANSFORMED EV(10,1) UNIFIED 3.5 Probability Density 2.5 1.5 0.5 0.5 1.5 (d) TRANSFORMED EV(3,1) TRANSFORMED EV(10,1) UNIFIED 3.5 Probability Density 2.5 1.5 0.5 0 0.5 1.5 (e) 199 2.5 TRANSFORMED EV(5,1) TRANSFORMED EV(12,1) UNIFIED 4.5 Probability Density 3.5 2.5 1.5 0.5 0 0.5 1.5 2.5 (f) Fig A.2: The Transformed Distribution Comparison with Different (θ1 , θ ) Fig A.2 presents a visualizing difference between two transformed distributions and the unified distribution The case in Fig A.2(b) validates the expression (A.4) towards the difference between the unified control limits and the true control limits The upper tail and lower tail of three distributions almost coincides Under such circumstances, the unified control limits can be considered for the circle chart construction instead of section dependent ones This appendix only provides a discussion towards the case with different section θt, detailed numerical study can be conducted to further present the charting performance and assumption effect 200 ... mean and range normalization Baxter (1995) applied and studied standardization and transformation in 23 principal component analysis Normalization techniques for microarray data are studied and. . .A STUDY OF NEW AND ADVANCED CONTROL CHARTS FOR TWO CATEGORIES OF TIME RELATED PROCESSES DENG PEIPEI B.Sc., University of Science and Technology of China A THESIS SUBMITTED FOR THE DEGREE OF. .. Phase II control charts Common univariate Shewhart control charts, i.e X chart, and R chart or S chart, are for variable measurements monitoring Many Shewhart control charts for attributes are