FLORIDA THE STATE UNIVERSITY COLLEGE OF ENGINEERING ROBUST CHANGE DETECTION AND CHANGE POINT ESTIMATION FOR POISSON COUNT PROCESSES
By Marcus B. Perry
A Dissertation submitted to the Department of Industrial Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Summer Semester, 2004
Copyright 2004 Marcus B. Perry All Rights Reserved
The members of the Committee approve the dissertation of Marcus B. Perry defended on May 28, 2004. _______________________________ Joseph J. Pignatiello Jr. Professor Directing Dissertation _______________________________ Anuj Srivastava Outside Committee Member _______________________________ James R. Simpson Committee Member _______________________________ Chuck Zhang Committee Member Approved: _________________________________ Ben Wang, Chair, Department of Industrial and Manufacturing Engineering _________________________________ Ching-Jen Chen, Dean, College of Engineering
The Office of Graduate Studies has verified and approved the above named committee members.
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ACKNOWLEDGEMENTS
I would like to express my appreciation to Dr. Joseph Pignatiello and Dr. James Simpson for their excellent mentoring and contributions to this manuscript. Additionally, I would like to give a special thanks to Dr. Ben Wang for his support throughout my endeavors at The Florida State University.
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TABLE OF CONTENTS
List of Tables ....................................................................................................................vii List of Figures...................................................................................................................xiv Abstract.............................................................................................................................xix
1.0 INTRODUCTION..................................................................................................... 1 1.1 c-charts for Monitoring Poisson Counts ................................................................ 3 1.2 CUSUM for Monitoring Poisson Counts............................................................... 4 1.3 EWMA for Monitoring Poisson Counts ................................................................ 7 1.4 Choice of Control Chart........................................................................................ 8 1.5 Change Point Estimation....................................................................................... 9 1.6 Statement of the Problem.................................................................................... 11 2.0 ESTIMATION OF THE CHANGE POINT OF A POISSON RATE PARAMETER FOR SPC APPLICATIONS .......................................................................................... 13 2.1 Introduction ........................................................................................................ 13 2.2 Poisson Process Step Change Model and Derivation of the MLE ........................ 14 2.3 Poisson CUSUM Control Chart .......................................................................... 16 2.4 Poisson EWMA Control Chart............................................................................ 17 2.5 Comparison of Change Point Estimators............................................................. 18 2.5.1 False Alarms................................................................................................ 18 2.5.2 Change Point Estimators Used With Poisson CUSUM Control Charts ......... 19 2.5.3 Change Point Estimators Used With Poisson EWMA Control Charts........... 24 2.6 Confidence Sets Based on the Change Likelihood Function................................ 28 2.6.1 Confidence Sets for Process Change Point After a Signal from a c-chart ...... 36 2.6.2 Confidence Sets for the Process Change Point After a Signal from a Poisson CUSUM Control Chart .......................................................................................... 37 2.6.3 Confidence Sets for the Process Change Point After a Signal from a Poisson EWMA Control Chart ........................................................................................... 38 2.7 Choice of for Performance Evaluation of the Confidence Set Estimator......... 38 2.8 Summary ............................................................................................................ 47 3.0 A MAGNITUDE-ROBUST CONTROL CHART FOR MONITORING AND ESTIMATING STEP CHANGES IN A POISSON RATE PARAMETER .................... 50 3.1 Introduction ........................................................................................................ 50 3.2 Process Behavior Model and Associated Hypothesis Tests for the Poisson CUSUM .................................................................................................................... 50 3.3 Behavior Model for Poisson Process Rate Parameter .......................................... 53 3.4 Likelihood Ratio Test: Control Chart for a Poisson Process Step Change Model 53 3.5 Average Run Length Comparison ....................................................................... 57 3.5.1 Simulation Modeling of a Step Change ........................................................ 58 3.5.2 False Alarms................................................................................................ 58 3.5.3 ARL Calibration of Control Charts .............................................................. 59
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3.5.4 Initial ARL Performance Comparisons......................................................... 59 3.5.5 Steady State ARL Performance Comparisons............................................... 62 3.6 Implementation Issues and Illustration ................................................................ 69 3.7 Discussion .......................................................................................................... 71 4.0 ESTIMATING THE CHANGE POINT OF A POISSON RATE PARAMETER WITH A LINEAR TREND DISTURBANCE IN SPC APPLICATIONS ...................... 79 4.1 Introduction ........................................................................................................ 79 4.2 Poisson Process Linear Trend Change Model and Derivation of the MLE........... 80 4.3 Newtons Method for Finding Maximum Likelihood Estimate of Slope Parameter .................................................................................................................................. 82 4.4 Poisson CUSUM Control Chart .......................................................................... 83 4.5 Comparison of Change Point Estimators............................................................. 84 4.5.1 False Alarms................................................................................................ 85 4.5.2 Accuracy Performances of Change Point Estimators .................................... 85 4.5.3 Precision Performances of Change Point Estimators .................................... 86 4.6 Confidence Sets Based on the Change Likelihood Function for the Change Point of a Poisson Rate Parameter ...................................................................................... 90 4.6.1 Cardinality and Coverage Performances of Confidence Set Estimators ........ 94 4.7 Summary ............................................................................................................ 98 5.0 ESTIMATING THE CHANGE POINT OF A POISSON RATE PARAMETER WITH AN ISOTONIC CHANGE DISTURBANCE IN SPC APPLICATION ............ 100 5.1 Introduction ...................................................................................................... 100 5.2 Poisson Process Behavior Model and Derivation of the MLE ........................... 102 5.3 Poisson CUSUM Control Chart ........................................................................ 104 5.4 Comparison of Change Point Estimators........................................................... 105 5.4.1 False Alarms.............................................................................................. 105 5.4.2 Change Point Estimators with a Single Point Step Change Disturbance...... 106 5.4.3 Change Point Estimators with a Linear Trend Change Disturbance ............ 111 5.4.4 Change Point Estimators with a Multiple Step Change Disturbance ........... 113 5.5 Summary and Discussion.................................................................................. 118 6.0 CONTROL CHARTS FOR MONITORING AND ESTIMATING MONOTONIC CHANGES IN A POISSON RATE PARAMETER .................................................... 122 6.1 Introduction ...................................................................................................... 122 6.2 Process Behavior Model ................................................................................... 125 6.3 Control Chart for Isotonic Changes in a Poisson Rate: Likelihood Ratio Test ... 125 6.4 Average Run Length Comparison ..................................................................... 129 6.4.1 Simulation Modeling of Process Change.................................................... 129 6.4.2 False Alarms.............................................................................................. 130 6.4.3 ARL Calibration of Control Charts ............................................................ 130 6.4.4 ARL Performances for Step Change Case .................................................. 131 6.4.5 ARL Performances for Linear Trend Change Case..................................... 134 6.4.6 ARL Performances for Multiple Step Change Case.................................... 148 6.5 Implementation Issues and Illustration .............................................................. 150
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6.6 Discussion ........................................................................................................ 167 7.0 CONCLUSION .................................................................................................... 171 APPENDIX................................................................................................................. 178 REFERENCES ........................................................................................................... 181 BIOGRAPHICAL SKETCH ....................................................................................... 184
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LIST OF TABLES
1. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 20 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in , = 100 , h =23.19. ............................................................................................ 21 2. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 10 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in , = 100, h = 12.1. ............................................................................................ 22
3. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 5 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 3 in , = 100, h= 7.47...................................................................................................... 22 4. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM charts were designed for shifts of 4 in the Poisson rate parameter, = 100, 0 = 20 . Precision for CUSUM is shown in parentheses. ................................................. 25 5. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM charts were designed for shifts of 4 in the Poisson rate parameter, = 100, 0 = 10 . Precision for CUSUM is shown in parentheses. ................................................. 26
6. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM chart was designed for shifts of 3 in the Poisson rate parameter, = 100, 0 = 5 . Precision for CUSUM is shown in parentheses. ................................................... 27
7. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 20 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.67. .............. 30 8. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 20 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.20, = 100, A = 2.835. ............ 30
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9. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 10 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.60. .............. 31 10. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 5 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.42............................. 31 11. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 20 . Precision for EWMA is shown in parentheses. .................................... 32 12. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 10 . Precision for EWMA is shown in parentheses. Precision of MLE and
EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 5 . Precision for EWMA is shown in parentheses.......................................................................................... 33
13. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 5 . Precision for EWMA is shown in parentheses. ....................... 34 14. In-control ARL estimates for the magnitude robust for step changes (MR-SC) and three different CUSUM schemes for each in-control rate parameter value investigated. ARLs reflect those obtained from one-sided control charts for increases. ....................... 60 15. In-control ARL estimates for the magnitude robust for step changes (MR-SC) and three different CUSUM schemes for each in-control rate parameter value investigated. ARLs reflect those obtained from one-sided control charts for decreases. ...................... 61 16. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5......................................................................... 63 17. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5......................................................................... 64 18. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10 . ..................................................................... 64
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19. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10. ....................................................................... 65 20. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20 ....................................................................... 65 21. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20. ...................................................................... 66 22. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5. ........................................................................ 66 23. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5......................................................................... 67 24. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10....................................................................... 67 25. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10....................................................................... 68 26. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20....................................................................... 68 27. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20....................................................................... 69 28. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 7.0; k = 6.38). Rounded standard errors are shown in parentheses. 0 = 5, = 100 and N = 10,000 independently-seeded runs. ............................................................................................. 87
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29. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 9.80; k = 11.89). Rounded standard errors are shown in parentheses. 0 = 10, = 100 and N = 10,000 independently-seeded runs. ...............................................................................................88 30. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 14.4; k = 22.41). Rounded standard errors are shown in parentheses. 0 = 20, = 100 and N = 10,000 independently-seeded runs................................................................................................ 89 31. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 7.0; k = 6.38). Precision estimates for SC are shown in parentheses. 0 = 5 and N = 10,000 independently-seeded runs.................................................................................................................................... 91 32. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 9.80; k = 11.89). Precision estimates for SC are shown in parentheses. 0 = 10 and N = 10,000 independentlyseeded runs........................................................................................................................ 92 33. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 14.4; k = 22.41). Precision estimates for SC are shown in parentheses. 0 = 20 and N = 10,000 independentlyseeded runs........................................................................................................................ 93 34. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a single-point step change disturbance is present. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20, = 25 and N = 100,000 independentlyseeded runs...................................................................................................................... 107 35. Estimated precision performances over a range of a values for , sc and lt following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in parenthesis. 0 = 20 and N = 100,000 independently-seeded runs............................... 112 36. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a linear trend disturbance is present. Rounded standard errors are shown in parentheses. 0 = 20 and N = 100,000 independently-seeded runs.................................................................. 113
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37. Estimated precision performances over a range of values for , sc and lt following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in
parenthesis. 0 = 20 and N = 100,000 independently-seeded runs................................ 115 38. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a multiple step change disturbance is present. Rounded standard errors are shown in parentheses. 0 = 20 , 1 = 25 , 2 = 35 and N = 100,000 independently-seeded runs. ......................................................................................................................................... 117 39. Input factors and levels considered in the 2 3 experiment. ....................................... 120 40. Direction of influence on estimated change point for each factor considered. An empty space in the table indicates no significant influence. ........................................... 120
41. Estimated precision performances over a range of 1 , 2 combinations for , sc and lt following a genuine signal from a Poisson CUSUM control chart
( h = 13.96 ; k = 22.41 ). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in parenthesis. 0 = 20 , 1 = 25, 2 = 35 and N = 100,000 independently-seeded runs.............................................................................................. 121 42. ARL comparison for increasing rate case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses.......................................................................135 43. ARL comparison for decreasing single-point step change case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ....................................... 136 44. ARL comparison for increasing single-point step change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses ........................................ 137 45. ARL comparison for decreasing single-point step change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ....................................... 138 46. ARL comparison for increasing single-point step change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ....................................... 139
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47. ARL comparison for decreasing single-point step change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses.. ....................... 140 48. ARL comparison for increasing single-point step change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses ........................................ 141 49. ARL comparison for decreasing single-point step change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ....................................... 142 50. ARL comparison for increasing single-point step change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ........................ 143 51. ARL comparison for decreasing single-point step change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ........................ 144 52. ARL comparison for increasing single-point step change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ........................ 145 53. ARL comparison for decreasing single-point step change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ........................ 146 54. ARL comparison for linear trend change case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ..................................................................... 151 55. ARL comparison for linear trend change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ............................................................. 152 56. ARL comparison for linear trend change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ............................................................. 153 57. ARL comparison for linear trend change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ............................................................. 154
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58. ARL comparison for linear trend change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ................................................. 155 59. ARL comparison for linear trend change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses .................................................. 156 60. ARL comparison for increasing multiple step change case with 5 change points, 0 = 5 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 157 61. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 5 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 158 62. ARL comparison for increasing multiple step change case with 5 change points, 0 = 10 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 159 63. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 10 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 160 64. ARL comparison for increasing multiple step change case with 5 change points, 0 = 20 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 161 65. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 20 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ......... 162 66. ANOVA table for experimental design. Response is SC ..................................... 178 67. ANOVA table for experimental design. Response is LT ..................................... 179 68. ANOVA table for experimental design. Response is OR ..................................... 180
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LIST OF FIGURES
1. c-chart for monitoring count of nonconformities.CUSUM control chart for increasing rate case. ......................................................................................................................... 4 2. CUSUM control chart for increasing rate case. ........................................................... 6 3. CUSUM control chart for decreasing rate case. .......................................................... 7 4. EWMA control chart for Poisson counts..................................................................... 9 5. Graphical representation of process step change model............................................. 15 6. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 20 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in . = 100 , h =23.19, N = 100,000 independently-seeded runs. .................................... 21 7. Plot of likelihood values computed at each potential change point t. ......................... 35 8. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 20. .............................................................................. 37 9. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 20. ............................................................................... 39 10. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 10. ............................................................................... 39 11. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 10. ............................................................................... 40 12. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 5. ................................................................................ 40
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3. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 20. ..................................... 41 14. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 20. ..................................... 41 15. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 10. ..................................... 42 16. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 10. ..................................... 42 17. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 5. ....................................... 43 18. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 5. ....................................... 43 19. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 20. ................................................. 44 20. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 20. ................................................. 44 21. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 10. ................................................. 45 22. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 10. ................................................. 45
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23. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 5. ................................................... 46 24. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 5. ................................................... 46 25. Effect of change in the simulated change point on coverage probabilities and average cardinality of confidence sets. ....................................................................................... 49 26. Example C codes for computing the magnitude-robust control chart statistics and diagnostics .................................................................................................................... 73 27. Two realizations of the magnitude-robust control charts for data experiencing a shift in from 5 to 7 following the 50th sample count obtained. Realization 1 signals at subgroup T = 57, while realization 2 signals at subgroup T = 63. .................................. 74 28. CUSUM control charts with h = 5.800 and k = 6.534 for the same two realizations of data experiencing a shift in from 5 to 7 following the 50th sample count obtained. Realization 1 signals at subgroup T = 57, while realization 2 did not produce a signal. .. 75 29. Diagnostic plots for data realization 1. Figure 3-4(a): R versus potential change point. Large values correspond to most likely change points. Figure 3-4(b): Estimated out-of-control rate parameter versus subgroup number. ................................................. 76 30. In-control ARL function approximations for a range of B and 0 using ordinary least squares. 3-5a: MR-SC for increases. 3-5b: MR-SC for decreases................................. 77 31. Pseudo-code for estimating in-control ARLs for the magnitude-robust control chart. Inputs into the program include B and N, the control limit and total number of simulation runs, respectively. The value of RT(i) is computed as given in equation (3.13).............. 78 32. C code implementing Newtons method for estimating the slope parameter at each potential change point t. ................................................................................................ 84 33. Plot of log likelihood values versus possible change points t for a single realization. Those values of t yielding log likelihood values that plot above log e L( ) - D (i.e. dotted horizontal line in plot) are considered potential change point candidates and are included in the confidence set...................................................................................................... 95 34. Surface plot obtained from confidence set estimator for linear trends showing estimated relationships between set cardinality, coverage, slope parameter and reference value D. 0 = 5, N = 10,000, = 100. ........................................................... 96
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35. Surface plot obtained from confidence set estimator for step changes (dashed-lines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 5 , N = 10,000, = 100 ........................................................................................ 97 36. Surface plot obtained from confidence set estimator for step changes (dashed-lines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 10 , N = 10,000, = 100 ....................................................................................... 97 37. Surface plot obtained from confidence set estimator for step changes (dotted-lines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 20, N = 10,000, = 100 ....................................................................................... 98 38. Non-decreasing change-types investigated............................................................ 106 39. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a single-point step change disturbance is introduced following the 25th subgroup. Results are provided for values of a = 22, 23, 24 and 25. N = 500 independent runs were used to obtain the estimated mean vectors................................ 110 40. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a linear trend change disturbance is introduced following the 25th subgroup. Results are provided for values of = 0.01, 0.02, 0.03, 0.04 and 0.05. N = 500 independent runs were used to obtain the estimated mean vectors. ................ 114 41. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a multiple step change disturbance is introduced following the 25th subgroup obtained. 1 = 25, 2 = 35 , N = 500 independent runs were used to obtain the estimated mean vectors.......................................................................................... 120 42. A single realization of the order-restricted control chart for data experiencing a shift in from 5 to 7 following the 50th sample count obtained. This realization signals at subgroup T = 64, yielding a runlength of 14. ............................................................... 164 43. Diagnostic plots for data realization 1. 6-2(a): R versus potential change point. Large values correspond to most likely change points. 6-2(b): Estimated process profile versus subgroup number.............................................................................................. 164 44. A single realization of the order-restricted control chart for data experiencing a linear trend in of magnitude = 0.20 following the 50th sample count obtained. This realization signals at subgroup T = 64, yielding a runlength of 14................................ 165 45. Diagnostic plots for data realization 2. 6-4(a): R versus potential change point. Large values correspond to most likely change points. 6-4(b): Estimated process profile versus subgroup number.............................................................................................. 166
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46. A single realization of the order-restricted control chart for data experiencing multiple step changes in following the 50th sample count obtained. This realization signals at subgroup T = 68, yielding a runlength of 18. ................................................ 168 47. Diagnostic plots for data realization 3. 6-6(a): R versus potential change point. Large values correspond to most likely change points. 6-6(b): Estimated process profile versus subgroup number.............................................................................................. 168 48. Pseudo-code for estimating in-control ARLs for the order-restricted control chart. Inputs into the program include B and N, the control limit and total number of simulation runs, respectively. The value of RT(i) is computed as given in equation (6.9)............. 169 49. Estimated process profile suggesting a change-type consisting of a single change point at = 50 ............................................................................................................ 175 50. Estimated process profile suggesting a change-type consisting of multiple change points and an initial change point of = 50 . ............................................................... 176
51. Normal probability and interaction plots for 2 3 experiment. Response is SC ...... 178 52. Normal probability and interaction plots for 2 3 experiment. Response is LT ...... 179 53. Normal probability and interaction plots for 2 3 experiment. Response is OR ...... 180
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ABSTRACT
In general, Poisson count process are often used to model the number of occurrences over some interval unit. In an industrial quality control setting, these processes are often used to model the number of nonconformities per unit of product. Current methods used for monitoring and estimating changes in Poisson count processes require that the magnitude and type of change be known apriori. Since rarely in practice are these known, this dissertation reports on the development and evaluation of several methods for detecting and estimating change points when the magnitude and type of change are unknown. Instead, the only assumption requires that the type of change belongs to a family of monotonic change types. Results indicate that the methodologies proposed throughout this dissertation research provide robust detection and estimation capabilities (relative to current methods) with regard to the magnitude of change and the type of monotonic change that may be present.
xix
CHAPTER 1 INTRODUCTION
In many industries, statistical process control (SPC) techniques have become some of the most frequently used tools for improving quality. The use of control charts is the most important and commonly used SPC technique. Such charts are useful in In using identifying the presence of assignable causes in manufacturing processes.
control charts, samples are taken from the output of the process and sample statistics are computed and plotted against control limits. If any assignable causes are present, the sample statistics are likely to plot outside of the pre-defined control limits, which give an out-of-control signal. Operators and engineers then search for the assignable causes and make necessary adjustments to bring the process back into its in-control state. The focus of this dissertation is on the statistical monitoring of Poisson count processes. These types of processes are, in general, used to model the number of Often in an industrial quality control setting, the Poisson occurrences over some interval unit. The interval unit can be time, distance, area, volume or some similar unit. distribution is used to model the number of defects or nonconformities per unit of product. That is, the probability that a randomly selected unit of product contains x nonconformities is given by P( X = x ) =
x
x!
exp( )
for x 0
where > 0 denotes the mean rate of nonconformities. We can exploit our assumption that these data are well modeled by the Poisson distribution and develop statistical procedures for quickly detecting changes in the rate parameter . The sooner a change is detected, the sooner process engineers can initiate their search for the special cause affecting the process. If the type of change is increasing in nature, then early detection can help prevent any nonconforming items from reaching the final stages of the production process. On the other hand, if the type of change is decreasing in nature, quick detection could help to accelerate efforts to improve quality.
1
The control chart most commonly used to monitor the rate of nonconformities (per single inspection unit) is the c-chart. Its simplicity makes it an attractive choice for monitoring these data type. On the other hand, it has some undesirable properties in that it is not very sensitive to smaller changes in the rate parameter. Even further, when using a c-chart, the detection of decreases in becomes an issue. That is, if the acceptable rate of nonconformities, denoted as 0 , is such that 0 9, then it is not possible to detect improvements in quality using this control-charting procedure. Other procedures have since been proposed, such as the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, and are well known to provide better detection performance than that offered by the c-chart, for both the increasing and decreasing rate cases. Even further, they offer change point diagnostic information to aid in the search for the special cause affecting the process. Although the CUSUM and EWMA procedures perform well, each requires a parameter in their design that specifies the magnitude of change for which the procedure should be best at detecting. This is problematic, however, as the magnitude of change is not likely to be known apriori. In this research, a series of papers are presented that detail new robust controlcharting and change point estimation procedures for use with those count processes well modeled by the Poisson distribution. The robustness, as it applies to this research, is specific to the magnitude of change and type of change that a process could exhibit. That is, the CUSUM and EWMA procedures are designed for a specific change magnitude, and although perform well at that level of change, may not perform well for others. Additionally, an underlying assumption in the derivation of the CUSUM statistic is that the type of change present is a simple step change. Although step changes are one potential change type, other change types could also exist. For example, process quality could gradually deteriorate according to some linear rate. Therefore, the robustness of a control chart can be gauged by how well it performs across a range of change magnitudes and/or change types. The following sections of this chapter provide background information on the aforementioned control-charting procedures (i.e. c-chart, CUSUM and EWMA). In subsequent sections some discussion is provided regarding the choice of control chart and
2
the importance of change point estimation. Finally, the problem statement is formally introduced.
1.1 c-chart for Monitoring Poisson Counts A c-chart is used to monitor the number of nonconformities per single inspection unit. The underlying statistical principles of the c-chart are based on the Poisson distribution. That is, the count of nonconformities obtained from a single inspection unit follows a Poisson distribution with rate parameter . Let C be a Poisson random variable, then the expected value of C is , and the variance of C is . Therefore, a control chart for monitoring the rate of nonconformities, or a c-chart, can be constructed by the following UCL = 0 + L+ 0 and LCL = 0 L 0
where UCL is the upper control limit, LCL is the lower control limit and 0 is the incontrol rate parameter. The values of L used in computing the control limits represent the number of standard deviation units from the in-control rate and are typically chosen to be L+ = L = 3 (see Montgomery (2002)). To apply this procedure, a sample is taken periodically from the output of the
process and a sample rate is obtained. If this sample rate plots above or below the
upper or lower control limits, respectively, the implication is that there is significant evidence suggesting a change in has occurred. An example of a c-chart is illustrated in Figure 1-1. The observations plotted on the c-chart in Figure 1-1 were simulated from a Poisson distribution with rate parameter = 20. A change was simulated to occur following sample number 100, where the rate parameter changed from = 20 to 24. The number of samples required for a control chart to signal following a change is called the run length. Since the run length is itself a random variable, the average run length (ARL) provides the basis for evaluating any control charting procedure. A large ARL is desirable when the quality of the process is at an acceptable level (i.e., yielding a low rate
3
of false signals). When the quality of the process is operating at an unacceptable level, however, smaller ARLs are desirable.
c-Chart 40
35
UCL
30
25 Sample Count
20
15
10
5
LCL
0
20
40
60 80 Sample Number
100
120
140
Figure 1-1. c-chart for monitoring count of nonconformities.
1.2 CUSUM for Monitoring Poisson Counts Although the CUSUM chart was developed by Page (1954, 1961) using a sequential probability ratio test (SPRT) under normal model assumptions, Brook and Evans (1972) were the first to suggest using such a scheme for monitoring count data processes. The CUSUM test statistic for detecting an increase in the mean count rate is given by
+ + C T = max 0, X T k + + C T 1
{
}
(1.1)
+ where CT is the cumulative sum at time T. Evidence of an increase in the mean count + rate is indicated by a signal occurring if CT > h + . The values of h + and k + are selected
on the basis of desirable ARL properties and 0 . Brook and Evans (1972) propose a Markov chain modeling approach for obtaining the exact run length distribution of a particular sampling scheme, allowing for a simple means of evaluating and comparing any such schemes.
4
Similar to the normal distribution, the Poisson CUSUM control statistic can be derived from the SPRT, however, under Poisson model assumptions. That is, the null hypothesis H 0 : = 0 is tested against the simple alternative hypothesis H a : = a whenever a new sequence of subgroup counts is obtained. comparing the sequential probability ratio
X a e
i
The SPRT operates by
ST =
i =1
T
a
X i! T e 0 X! i =1 i
Xi 0
(1.2)
to an appropriate constant A at each new subgroup. If S T > A , then the test concludes in favor of H a after observing subgroup T. The value of 0 is the known in-control rate parameter and a is the pre-specified change that one wishes to detect. To derive the form of the statistic in (1.1) for the increasing rate case, we can take the natural log of (1.2) and simplify to obtain
+ + S = 0 a + X i log e a . i =1 0
+ T T
(
)
(1.3)
Rewriting this expression as
+ + S T = 0 + + X T log e a + S T+1 a 0
(
)
(1.4)
and then dividing by log e + log e ( 0 ) and rearranging, we obtain a CT+ = X T k + + CT+1 ,
( )
(1.5)
where
CT+ =
+ + 0 ST a and k + = . + + log e a log e ( 0 ) log e ( a ) log e ( 0 )
( )
The value of C 0+ is typically taken to be zero. Resetting the expression in (1.5) to zero
+ whenever CT < 0 yields the form of the statistic in (1.1), or + + C T = max{ 0, X T k + + C T 1 }.
(1.6a)
5
If the test statistic in (1.6a) exceeds a decision interval h + , then the chart signals that an increase in the mean count rate has occurred. For the decreasing rate case, the CUSUM statistic is given by
CT = max{ 0, k X T + C T 1
}
(1.6b)
where k =
0 a and C 0 = 0. If CT h , then the chart signals that a ( 0 ) log e ( ) log e a
decrease in the mean count rate has occurred. An example of a Poisson CUSUM control chart for detecting increases is given in Figure 1-2 while Figure 1-3 shows a CUSUM chart for the decreasing rate case. The CUSUM chart in Figure 1-2 was designed for a step change in from 20 to 24, where the CUSUM chart in Figure 1-3 was designed for a step change in from 20 to 16.
Count Data CUSUM Control Chart 25
20
Cumulative Sum
15
10
h = 10
5
0
0
20
40
60 Sample Number
80
100
120
Figure 1-2. CUSUM control chart for increasing rate case.
Hawkins and Olwell (1998) provide extensive detail pertaining to the theoretical foundation and construction of CUSUM control charts in general, including coverage of the Poisson CUSUM. Lucas (1985) provides a comprehensive study on the ARL
6
Count Data CUSUM Control Chart 14
12
10
h = 10
Cumulative Sum
8
6
4
2
0
0
20
40
60 Sample Number
80
100
120
Figure 1-3. CUSUM control chart for decreasing rate case.
performances of the Poisson CUSUM control chart. Lucas investigates these charts with and without fast-initial response (FIR) features (see Lucas and Crosier 1982) and provides tables to aid the user in selecting appropriate h and k values.
1.3 EWMA for Monitoring Poisson Counts
For monitoring the location of a normal distribution, Roberts (1959) defined the exponentially weighted moving average at subgroup i to be
Z i = rX i + (1 r ) Z i 1
where 0 < r 1 is a constant and the starting value is the process target. The EWMA can be used to monitor a Poisson count process by using Z 0 = 0 . The upper and lower control limits for the Poisson EWMA control chart are given as U i = 0 + A + Var ( Z i ) and Li = 0 A Var ( Z i ) , respectively, where Var ( Z i ) is the variance of the weighted
7
average at time i and A are constants. Trevanich and Bourke (1993), Borror, Champ and Rigdon (1998) and Gan (1990) have studied the application of EWMA methods to Poisson count data. An example of a EWMA control chart for Poisson counts with a weighting coefficient of r = 0.10 and A = 2.65 is shown in Figure 1-4. Note that the control limits for this procedure will reach their asymptotic values as the number of samples obtained gets large.
1.4 Choice of Control Chart
The choice of implementing one control-charting procedure versus another will likely depend on how quick the procedure detects situations where the quality level has changed. As mentioned earlier, the quick detection of changes in a process is a desirable property of any control chart. It is well known that the CUSUM and EWMA controlcharting techniques will detect small to medium changes much faster than the standard cchart. This is evidenced by the works of Lucas (1985), Gan (1990) and Borror, Champ and Rigdon (1998). Also, the choice of control chart may depend on how much information is known about the process prior to the start of monitoring. It should be noted that the assumption of known in-control process parameters provides a basis for the standard control charting methods (i.e. c-, CUSUM and EWMA procedures). If the in-control process parameters are unknown, then they must be estimated. In these cases, any errors in the estimation of these parameters results in the inability to fix the in-control average run lengths for the more sensitive CUSUM and EWMA procedures. With the standard c-chart, however, having only estimates of the in-control process parameters is not a significant source of error, and as such, the in-control ARL can be viewed as fixed. Therefore, in these cases, a c-chart may be the better choice. Once the process had been made stable and a reliable estimate of the in-control rate parameter has been obtained, a more sensitive CUSUM or EWMA procedure could then be employed to replace the existing c-chart.
8
Poisson EWMA Control Chart 22.5 22
21.5 UCL 21 20.5
Weighted Average
20 19.5
19 18.5 LCL
18 17.5
0
20
40
60 Sample Number
80
100
120
Figure 1-4. EWMA control chart for Poisson counts.
1.5 Change Point Estimation
Another important area of concern is the estimation of the time of process change following a control chart signal. Equipped with this additional knowledge, the search for the cause of change could be greatly enhanced. For example, if process engineers knew the time at which the process first changed, they could simply revert back to their logbooks and determine what changes were made to the process at that point in time. Appropriate action undertaken to establish statistical control of the process could then be performed much earlier than if the change point were unknown. When the process under study can be parameterized according to some known statistical distribution, the problem of estimating the unknown change point is often addressed using statistical models. In particular, the change likelihood function can be used to derive a procedure for estimating the first subgroup from the changed process (or the last subgroup from the in-control process). A significant amount of work has been done regarding the change point problem for continuous random variables, however, not much has been explored with regard to discrete random variables. For continuous variables, Samuel and Pignatiello (1998a),
9
Nedumaran, Pignatiello and Calvin (2000), and Pignatiello and Samuel (2001) have studied performance measures of the maximum likelihood estimate of the point of process change. Bissell (1984) proposed a method for estimating the slope of a linear trend under normal model assumptions, but no published works were found that suggest an approach to estimating the point at which the linear trend first manifested itself into the data. Page (1954) and Nishina (1992) suggest change point estimators for use with CUSUM and EWMA control charts, respectively. In general, Page (1954) suggests using the estimator CUSUM = max{i : S i = 0} when a CUSUM chart signals that a change in the process mean has occurred, where CUSUM is the last subgroup number for which the cumulative sum is zero. Furthermore, Nishina (1992) suggests an appropriate estimator for use with a EWMA control chart is EWMA = max{i : Z i 0 } (for increasing cases), where EWMA is the last subgroup number for which the exponentially weighted moving average is less than or equal to the in-control process mean. For decreasing cases, the author suggests EWMA = max{t : Z t 0 } . Samuel and Pignatiello (1998b) propose a change point estimator based on the change likelihood function of a Poisson random variable. They demonstrate its performance on a Shewhart c-chart, considering only step changes in the rate parameter, and provide accuracy and precision measures. No literature was found that made direct performance comparisons between this estimator and those suggested by Page (1954) and Nishina (1992). Finally, no literature was found that proposed an estimation procedure for use with Poisson-based control charts in multiple change point situations. This is needed as the estimators proposed by Page (1954) and Samuel and Pignatiello (1998) are derived under the assumption of a single point step change. That is, the rate parameter is assumed to suddenly shift from its known in-control value to some unknown out-ofcontrol value at a single unknown point in the process.
10
1.6 Statement of the Problem
The primary focus of this research is to develop and evaluate new robust statistical process control charting methodologies for monitoring Poisson count processes. This research is motivated by the works of Pignatiello and Simpson (2002) where the authors propose a magnitude-robust control chart for monitoring and estimating step changes in a normal process mean. The procedures proposed in this research are modified extensions of their work into the Poisson model. The desirable properties of the proposed control charts aim to include (1) robust detection capabilities with regard to the magnitude and type of change and (2) the ability to provide meaningful change point diagnostic statistics to aid in the search for the special cause. This is significant because it will provide improved change detection and point of change estimation capabilities over an entire range of possible change magnitudes and change types. The types of changes considered in this research are those that can be classified as monotonic in nature. The proposed procedures are needed because when a change occurs in the process, the true magnitude of the change is rarely known. Therefore, when using a CUSUM or EWMA control chart, the concern lies in determining what magnitude of change these charts should be designed for. Procedures designed for a specific change magnitude will perform well at that magnitude of change. However, as the true magnitude of change deviates from the magnitude for which the chart was designed, the performances of these control charts tend to deteriorate. Further concern lies in the fact that the exact type of change is rarely known. As mentioned earlier, the CUSUM control chart is derived under strict step change assumptions. As such, this research aims to provide a control-charting procedure that provides robust change point detection and estimation performances across a range of potential change magnitudes and/or monotonic change types. The remaining chapters of this dissertation consist of separate stand-alone articles detailing new methods developed for robust change detection and change point estimation. Each article contains its own literature review, problem statement, research
11
objective, methodology description, evaluation plan and summary section. manuscript also includes a final chapter stating conclusions and total results.
This
12
CHAPTER 2 ESTIMATION OF THE CHANGE POINT OF A POISSON RATE PARAMETER FOR SPC APPLICATIONS
2.1 Introduction
Control charts are used to monitor for changes in a process by distinguishing between special causes and common causes of variation. Once a change is detected, process engineers can begin their search for the special cause disturbing the process. If process engineers knew the time at which the change first manifested itself in the data, valuable time could be saved in the search to find the special disturbance. Upon signaling, control charts do not provide specific information regarding the cause of the process change nor when the process changed (the process change point); rather, they only suggest that a change has occurred. Process control procedures such as the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have built-in change point estimators that provide point estimates of the time of process change. In this paper we discuss the built-in change point estimators of the CUSUM and EWMA control charts for Poisson count data, and investigate a maximum likelihood estimator (MLE) of the process change point in place of the built-in change point estimators of the CUSUM and EWMA control charts. Pignatiello and Samuel (2001) performed a similar study involving changes in a normal process mean. A distinct advantage of using the MLE is that it allows for the construction of an approximate confidence set to cover the true value of the process change point. The MLE approach is also advantageous as it provides a search window of possible change points and a suggests search strategy for the process engineers. In this paper, confidence sets based on the likelihood function of a Poisson process are discussed and their performance evaluated. In the next section, we consider a model for a permanent step change in the rate of a Poisson process. A permanent step change in the rate parameter occurs when the rate suddenly shifts from its in-control value to some out-of-control value, and remains at that
13
out-of-control value until the appropriate process adjustments are made. model, a maximum likelihood estimator of the process change point is derived.
From this
2.2 Poisson Process Step Change Model and Derivation of the MLE
Consider a permanent step change model for the behavior of a Poisson process rate parameter . The model assumes that the process is initially in control with independent observations coming from a Poisson distribution with known rate 0 . After an unknown point in time, the rate parameter changes from an in-control state of = 0 to an unknown out-of-control state of = a , where 0 a . The model also assumes that once the step change in the process rate occurs, the process remains at the new rate of
a until the special cause has been identified and removed.
The Poisson process step change model can be parameterized as follows. Each subgroup consists of a count of nonconformities obtained from an inspection unit. It is assumed that the size of the inspection unit does not change between subgroups. During the formulation of subgroups i = 1, 2, , , the process rate i is equal to its known incontrol value 0 . For subgroups i = + 1 , ..., T , the process rate i is equal to some
unknown rate a and T is the current time. The unknown change point represents the last subgroup taken from the in-control process. graphically. The step change model can be used to derive a maximum likelihood estimator (MLE) for the process change point. We will denote the MLE of the change point as Figure 2-1 illustrates the model
. Assuming a process change point at , the likelihood function is
X X La ( , a | x ) = exp( 0 ) 0 i X i ! exp( a ) a i X i ! i =1 i = +1
T
(2.1)
where X i is the sample count of nonconformities at time i. The MLE of is the value of that maximizes the likelihood function, or equivalently, its logarithm. Taking the logarithm of the likelihood function and reducing algebraically yields
14
log e L( , a | x ) = K 0 + log e 0 X i (T ) a + log e a
i =1
i = +1
X
T
i
(2.2)
where K is a constant. For any given value of , it is easy to show that the value of a which maximizes log e L( , a | x ) is a ( ) =
T 1 X . (T ) i1 i = +
(2.3)
Substituting a ( ) for a in the log likelihood function and reducing yields
( ) log e L( | x ) = K 0 (T ) a ( )1 log e a . 0
(2.4)
Last subgroup taken from in-control rate parameter
a 0
1, 2 , ...
+ 1, ...
T
Figure 2-1. Graphical representation of process step change model
Therefore, the maximum likelihood estimate (MLE) of the change point is
MLE = argmax t 0 (T t ) a (t )1 log e
0 t <T
a (t ) , 0
(2.5)
that is, MLE is the value of t that provides the maximum log likelihood value.
15
The MLE of the change point can be applied when any Poisson count control chart gives an out-of-control signal, including the CUSUM, EWMA, and Shewhart ccharts. Samuel and Pignatiello (1998) demonstrated the application of such an estimator for use with a Shewhart c-chart. c-chart. They also evaluated the accuracy and precision performances of the estimator when applied following a signal from a standard Shewhart
2.3 Poisson CUSUM Control Chart
A CUSUM chart to monitor Poisson count data was suggested by Brook and Evans (1972). This approach is more effective than the Shewhart charts when the detection of small persistent shifts are of interest. A CUSUM scheme cumulates the difference between an observed value X i and a reference value k . If this sum exceeds a decision interval h, the chart signals a disturbance is present. The CUSUM control
statistics for detecting increases and decreases in the count rate are given as
S i+ = max{0, X i k + + S i+ 1 }
and
S i = max{0, k X i + S i 1 } , respectively, where
k + = ( + 0 ) /(ln( + ) ln( 0 )) and k = ( 0 ) /(ln( 0 ) ln( )) . The values of a a a a
are the given out-of-control process rates for which to design the CUSUMs. If a
S i+ exceeds a specified decision interval h + , then the chart signals that an increase in the
mean count rate has occurred. Similarly, if S i exceeds h , the chart signals that a decrease in the mean count rate has occurred. Lucas (1985) provides a comprehensive study on count data CUSUM control charts. Hawkins and Olwell (1998) provide extensive detail pertaining to the theoretical foundation and construction of CUSUM control charts in general, and does well in covering the Poisson CUSUM. The CUSUM chart has its own built-in change point estimator, as suggested by Page (1954). For example, if the CUSUM chart signals that an increase in the mean count rate has occurred, then an estimate of the change point is given by
CUSUM = max{i : S i+ = 0}
(2.6)
16
where CUSUM is taken to be most recent subgroup number yielding a cumulative sum of zero. Similarly, if the CUSUM chart signals that a decrease in the mean count rate has occurred, then an estimate of the change point is given by
CUSUM = max{i : S i = 0}.
(2.7)
Below, we compare the performance of the maximum likelihood change point estimator, MLE , to that of the CUSUM change point estimator, CUSUM , following a signal from a Poisson CUSUM control chart.
2.4 Poisson EWMA Control Chart
For monitoring the location of a normal distribution, Roberts (1959) defined the exponentially weighted moving average at time i to be
Z i = rX i + (1 r ) Z i 1
(2.8)
where 0 < r 1 is a constant and the starting value is the process target. The EWMA can be used to monitor a Poisson count process by using Z 0 = 0 . The upper and lower control limits for the Poisson EWMA control chart are given as U i = 0 + A + Var ( Z i ) and Li = 0 A Var ( Z i ) , respectively, where Var ( Z i ) is the variance of the weighted average at time i and A are constants. Trevanich and Bourke (1993), Borror, Champ and Rigdon (1998) and Gan (1990) have studied the application of EWMA methods to Poisson count data. Nishina (1992) suggests an appropriate change point estimator to be used with an EWMA control chart, following a signal resulting from an increase in the mean count rate, is
EWMA = max{i : Z i 0 }
(2.9)
where EWMA is the most recent subgroup number for which the exponentially weighted moving average is less than or equal to the in-control rate parameter, 0 . Similarly,
17
following the detection of a decrease in the mean count rate, Nishina estimates the change point as
EWMA = max{i : Z i 0 } .
(2.10)
Below, we compare the performance of the maximum likelihood estimator, MLE , to that of EWMA following a genuine out-of-control signal from a Poisson EWMA control chart.
2.5 Comparison of Change Point Estimators
In this section, we use Monte Carlo simulation to make performance comparisons between the maximum likelihood estimator of the process change point, MLE , and the
CUSUM estimator following a signal from a Poisson CUSUM control chart. We then
compare performances of MLE to the EWMA estimator following a signal from a Poisson EWMA control chart. All control chart signals are assumed to be genuine. In the case of a Poisson model, no procedure exists to standardize the data. As such we investigate a range of values for 0 , specifically 0 = 5, 10 and 20.
2.5.1 False Alarms
The handling of false alarms which occur in the simulation studies needs to be carefully addressed. When > 0 and a control chart issues a signal at subgroup T where
T , then the signal is a false alarm since the signal was given before the simulated
process change could occur. When a false alarm was encountered in a simulation run, it was treated the same way that a false alarm would be treated on an actual process. Namely, if one determines that a signal is indeed a false alarm then one is affirming that the process is currently in-control and could restart their monitoring of the process. Thus, when a false alarm was encountered at time T, the control chart was restarted at subgroup number T + 1 while not altering the scheduled change point. For example, if the change point was = 100 and a false alarm was issued in a
simulation run at subgroup number 75, then the appropriate statistics would be zeroed out
18
and that simulation run would continue as if subgroup number 76 was the first one from an in-control process. Thus, if there were no more false alarms on this particular simulation run, there would have been 25 subgroups observed from the restarted incontrol process when the first subgroup from the simulated changed process is observed. With this approach the number of subgroups since the chart was started or restarted that have been observed from the in-control process at the time of the process change will not necessarily be fixed at but will instead be random and less than or equal to . This is the same approach used by Pignatiello and Samuel (2001).
2.5.2 Change Point Estimators Used with Poisson CUSUM Control Charts
In this section, CUSUM charts designed for shifts of 4 (for 0 = 10 and 20 ) and 3 (for 0 = 5 ) are considered. These magnitudes of change correspond to mid-size shifts in the mean count rate. Each chart is designed to have an in-control average run length (ARL) close to that of a standard Shewhart c-chart, given the value of 0 . Monte Carlo simulation was used to study the performance of the change point estimators. The process change point was simulated to occur at = 100 . Independent observations were randomly generated from a simulated Poisson process with rate parameter 0 for subgroups i = 1, 2, , 100. Following subgroup 100, independent observations were randomly generated from a simulated Poisson process with rate parameter a until the CUSUM chart produced a signal. The estimates of the process change point, MLE and
CUSUM , were then computed. This procedure was repeated a total of N = 100,000 times
for each a value investigated. Averages of the change point estimates obtained from the 100,000 simulation runs were computed ( MLE and CUSUM ) along with the standard error of the estimates ( seMLE and seCUSUM ). We first consider a Poisson process with an in-control rate of 0 = 20 . CUSUM charts designed to detect an increase or decrease of 4 counts in the mean count rate was employed. The performances of the MLE and CUSUM estimators are given in Table 2.1 and illustrated graphically in Figure 2-2. Table 2.1 also shows E[ T ] , the estimated
19
expected time at which the control chart signals a genuine alarm. These estimates were also obtained by simulation and computed as ARL + . Figure 2-2 suggests that except for those shift magnitudes for which the CUSUM charts were designed, the MLE outperforms the CUSUM change point estimator. For increases in , the MLE of and the CUSUM estimate of are essentially unbiased when a = 24, since, CUSUM = 100.36 and MLE = 100.25. That is, both estimates are very close to the true process change point of = 100 . Figure 2-2 further suggests that the standard error of MLE is greater than that provided by CUSUM . This is no surprise since MLE requires estimating a and CUSUM requires no such estimate. Over the entire range of change magnitudes from a = 22 to a = 35 , however, MLE yields the least amount of bias. As such, we conclude MLE outperforms CUSUM . Results are similar for decreases. The results shown in Figure 2-2 reflect those obtained for in-control rate parameter values of 0 = 10 and 0 = 5 . Simulation results for these in-control rate parameter values are provided in Tables 2.2 and 2.3. For 0 = 10 , the CUSUM charts were designed to detect shifts of 4 in the rate parameter, while at 0 = 5, the CUSUM charts were designed to detect shifts of 3 in the rate parameter. The observed frequency of which the change point estimate was within a given number of the actual change point was also investigated. Such a measure provides a means of comparing the precision of the two estimators. Table 2.4 shows precision performances of the two estimators for increases and decreases with 0 = 20 . For
example, Table 2.4 shows that if 0 = 20 and a CUSUM chart is designed to detect changes of + 4 in , then if increases to = 30 , the estimated probability that MLE will be within 2 of the actual change point is 0.901 while the corresponding probability for CUSUM is only 0.737.
20
Accuracy vs Shift change: Increases 125 130
Accuracy vs Shift change: Decreases
120
MLE CUSUM
125 MLE CUSUM 120
115 change point estimate change point estimate 24 26 28 30 out-of-control rate parameter 32 34 36
115
110
110
105 105
100
100
95 22
95
4
6
8
10 12 out-of-control rate parameter
14
16
18
Standard error vs Shift change: Increases 0.09 0.12
Standard error vs Shift change: Decreases
0.08 MLE CUSUM 0.07 standard error of change point estimate standard error of change point estimate 0.1 MLE CUSUM
0.06
0.08
0.05
0.06
0.04
0.03
0.04
0.02 0.02 0.01
0 22
24
26
28 30 out-of-control rate parameter
32
34
36
0
4
6
8
10 12 out-of-control rate parameter
14
16
18
Figure 2-2. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 20 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in . = 100 , h =23.19, N = 100,000 independently-seeded runs.
Table 2.1. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 20 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in , = 100 , h =23.19. a seCUSUM MLE CUSUM se MLE E[T] 5.00 102.07 98.15 0.011 99.90 0.003 10.00 103.20 98.16 0.011 99.62 0.009 12.00 104.14 98.21 0.011 99.41 0.013 14.00 106.01 98.45 0.011 99.17 0.020 * 16.00 111.26 100.06 0.016 99.43 0.032 18.00 145.35 129.82 0.120 107.59 0.079 22.00 133.44 120.66 0.089 109.72 0.076 * 24.00 110.42 100.36 0.019 100.25 0.032 26.00 105.87 98.46 0.012 99.44 0.021 28.00 104.14 98.08 0.012 99.37 0.017 30.00 103.24 97.95 0.011 99.47 0.012 35.00 102.18 97.90 0.011 99.69 0.008
21
Table 2.2. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 10 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 4 in , = 100, h = 12.1.
a
0.25 0.50 2.00 4.00 * 6.00 8.00 12.00 * 14.00 16.00 18.00 20.00 25.00
E[T] 101.95 101.97 102.44 103.53 106.82 136.33 118.92 105.95 103.43 102.45 101.95 101.34
CUSUM
seCUSUM
MLE
se MLE
99.22 99.23 99.22 99.28 99.97 125.69 111.43 100.20 99.20 98.99 98.94 98.94
0.005 0.005 0.005 0.005 0.008 0.097 0.049 0.010 0.007 0.006 0.006 0.006
99.95 99.93 99.82 99.54 99.23 101.79 103.87 99.61 99.41 99.51 99.64 99.83
0.002 0.003 0.005 0.012 0.020 0.047 0.046 0.024 0.017 0.012 0.009 0.005
Table 2.3. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson rate parameter with 0 = 5 after a genuine signal from a Poisson CUSUM control chart. CUSUM charts were designed for shifts of 3 in , = 100, h= 7.47.
a
0.25 0.50 1.00 * 2.00 3.00 7.00 * 8.00 9.00 10.00 15.00
E[T] 102.89 103.07 103.68 106.25 117.23 109.22 105.03 103.46 102.66 101.34
CUSUM
seCUSUM
MLE
se MLE
99.37 99.37 99.39 99.83 107.03 103.56 100.35 99.59 99.33 99.13
0.004 0.004 0.004 0.006 0.035 0.020 0.009 0.006 0.006 0.005
99.81 99.76 99.60 99.30 99.69 101.10 99.69 99.45 99.48 99.76
0.005 0.006 0.010 0.017 0.027 0.030 0.022 0.018 0.015 0.007
The results in Table 2.4 indicate MLE provides a more precise estimate than
CUSUM . The only exception occurs when the rate changes to one of the specific values
22
for which the CUSUM was designed. Over the entire range of change magnitudes, however, MLE outperforms CUSUM with regard to precision performance. Table 2.5 shows precision performances of the two estimators for increases and decreases with 0 = 10 . Again, results indicate that the MLE of the change point was observed to lie within the specified interval more frequently than the CUSUM change point estimator for all shift magnitudes, with the exception of the shifts for which the CUSUM chart was designed. Over the entire range of shift magnitudes, however, MLE provides a more precise estimate. Finally, Table 2.6 shows precision performances of the two estimators for increases and decreases with 0 = 5 . Results indicate that CUSUM performs slightly better than MLE when small increases in are considered. However, MLE provides more precision for larger increases in the rate parameter. Even further, when decreases are considered, MLE provides better precision for all change magnitudes considered, with the exception of a = 2 , which is the magnitude of change the CUSUM chart was designed to quickly detect. The results in Table 2.6 suggest that for smaller values of 0 , say, 0 < 10,
CUSUM can provide slightly better precision than MLE when small increases in are
considered. For other changes in , however, CUSUM does not perform as well as does
MLE .
In this study, the MLE was also applied to CUSUM charts designed to quickly detect shifts in the rate parameter of 2 for each 0 value investigated. Although not shown here, similar results were obtained. That is, MLE yields better overall
performance when compared to CUSUM , regardless of the size of 0 investigated. Therefore, we conclude that the MLE of the change point provides a better estimate of the true process change point when compared to the built-in change point estimator of the Poisson CUSUM control chart. We have also examined the average performance of these estimators for = 10, 50 and 200 . Although not reported here, we found that using a simulated change point different from =100 does not alter our conclusions.
23
2.5.3 Change Point Estimators Used with Poisson EWMA Control Charts
We now consider an EWMA control chart with parameter r = 0.10 and symmetrical control limits set to yield an in-control ARL close to that of a standard Shewhart c-chart given the in-control rate parameter value. As in the previous section, we use Monte Carlo simulation to compare the performance of the estimators, MLE and
EWMA . The process change point was simulated at = 100 . Independent observations
were randomly generated from a Poisson process with rate parameter 0 for subgroups i = 1, 2, , 100. Following subgroup 100, independent observations were randomly generated from a Poisson process with rate parameter a until the EWMA chart produced a signal. The estimates MLE and EWMA were then computed. This procedure produced a signal. The estimates MLE and EWMA were then computed. This procedure was repeated a total of N = 100,000 times for each a value investigated. Averages of the change point estimates obtained from the 100,000 simulation runs were computed ( MLE and EWMA ) along with the standard error of the estimates ( seMLE and se EWMA ). Table 2.7 shows the accuracy performance of the two estimators following a genuine signal from an EWMA control chart with r = 0.10 and 0 = 20 . As Table 2.7 shows, MLE yields a better estimate than EWMA . With the exception of very small shifts (either increases or decreases), EWMA significantly underestimates the true process change point. As with the CUSUM chart, the accuracy of the MLE improves as the magnitude of the shift increases. Table 2.8 shows performances of the two estimators following a genuine signal from an EWMA control chart with r = 0.20 and 0 = 20 . In this case, MLE yields better performance for all shift magnitudes investigated. This was found to be the case for all values of 0 investigated with an EWMA weighting coefficient of r = 0.20. We now consider an EWMA control chart with weighting coefficient r = 0.10 and 0 = 10 . Performance comparisons are shown in Table 2.9 and suggest MLE
provides far better estimates over the entire range of shift magnitudes when compared to
24
Table 2.4. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM charts were designed for shifts of 4 in the Poisson rate parameter, = 100, 0 = 20 . Precision for CUSUM is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
5 0.952 (0.524) 0.986 (0.681) 0.991 (0.768) 0.993 (0.824) 0.996 (0.867) 0.996 (0.894)
10 0.751 (0.516) 0.909 (0.680) 0.952 (0.769) 0.967 (0.824) 0.975 (0.864) 0.981 (0.895)
12 0.606 (0.488) 0.823 (0.678) 0.903 (0.770) 0.935 (0.826) 0.954 (0.867) 0.964 (0.896)
14 0.428 (0.414) 0.671 (0.644) 0.785 (0.754) 0.849 (0.820) 0.891 (0.863) 0.918 (0.895)
16 0.233 (0.258) 0.435 (0.482) 0.556 (0.617) 0.644 (0.707) 0.708 (0.771) 0.756 (0.820)
18 0.068 (0.050) 0.154 (0.112) 0.224 (0.165) 0.281 (0.205) 0.327 (0.239) 0.370 (0.270)
22 0.066 (0.061) 0.148 (0.138) 0.215 (0.197) 0.270 (0.249) 0.321 (0.293) 0.361 (0.329)
24 0.205 (0.229) 0.390 (0.438) 0.512 (0.570) 0.598 (0.661) 0.668 (0.733) 0.720 (0.785)
26 0.356 (0.344) 0.594 (0.582) 0.719 (0.706) 0.799 (0.787) 0.851 (0.839) 0.886 (0.877)
28 0.497 (0.412) 0.732 (0.627) 0.839 (0.735) 0.895 (0.802) 0.925 (0.847) 0.945 (0.882)
30 0.610 (0.441) 0.824 (0.636) 0.901 (0.737) 0.938 (0.800) 0.956 (0.848) 0.967 (0.882)
35 0.802 (0.466) 0.931 (0.637) 0.962 (0.737) 0.974 (0.800) 0.980 (0.846) 0.985 (0.879)
25
Table 2.5. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM charts were designed for shifts of 4 in the Poisson rate parameter, = 100, 0 = 10 . Precision for CUSUM is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
0.25 0.975 (0.679) 0.992 (0.828) 0.995 (0.898) 0.998 (0.936) 0.998 (0.960) 0.999 (0.975)
0.50 0.964 (0.677) 0.989 (0.826) 0.994 (0.898) 0.996 (0.936) 0.997 (0.960) 0.998 (0.975)
2 0.885 (0.678) 0.965 (0.828) 0.982 (0.897) 0.989 (0.937) 0.991 (0.960) 0.993 (0.975)
4 0.688 (0.641) 0.879 (0.823) 0.935 (0.895) 0.961 (0.937) 0.972 (0.960) 0.978 (0.976)
6 0.410 (0.446) 0.652 (0.703) 0.772 (0.826) 0.839 (0.892) 0.884 (0.931) 0.912 (0.955)
8 0.131 (0.066) 0.275 (0.139) 0.377 (0.189) 0.452 (0.229) 0.510 (0.262) 0.564 (0.294)
12 0.117 (0.103) 0.244 (0.216) 0.341 (0.298) 0.416 (0.363) 0.477 (0.413) 0.528 (0.457)
14 0.321 (0.351) 0.549 (0.603) 0.686 (0.741) 0.766 (0.822) 0.826 (0.877) 0.866 (0.916)
16 0.509 (0.492) 0.751 (0.737) 0.852 (0.845) 0.909 (0.905) 0.935 (0.939) 0.952 (0.960)
18 0.654 (0.549) 0.856 (0.762) 0.923 (0.855) 0.952 (0.907) 0.965 (0.939) 0.973 (0.959)
20 0.756 (0.572) 0.912 (0.768) 0.954 (0.853) 0.971 (0.905) 0.979 (0.938) 0.984 (0.959)
25 0.894 (0.585) 0.967 (0.766) 0.982 (0.854) 0.989 (0.905) 0.993 (0.939) 0.994 (0.959)
26
Table 2.6. Estimated precision of MLE and CUSUM when used with a CUSUM control chart. CUSUM chart was designed for shifts of 3 in the Poisson rate parameter, = 100, 0 = 5 . Precision for CUSUM is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
0.25 0.893 (0.679) 0.969 (0.858) 0.983 (0.926) 0.988 (0.957) 0.993 (0.977) 0.995 (0.986)
0.50 0.835 (0.677) 0.951 (0.859) 0.974 (0.928) 0.983 (0.958) 0.989 (0.976) 0.992 (0.986)
1 0.727 (0.665) 0.898 (0.858) 0.950 (0.928) 0.968 (0.957) 0.978 (0.977) 0.983 (0.986)
2 0.489 (0.530) 0.729 (0.777) 0.838 (0.884) 0.894 (0.934) 0.925 (0.963) 0.943 (0.978)
3 0.254 (0.190) 0.468 (0.349) 0.593 (0.448) 0.679 (0.515) 0.742 (0.567) 0.790 (0.612)
7 0.200 (0.206) 0.383 (0.397) 0.508 (0.519) 0.596 (0.606) 0.665 (0.673) 0.722 (0.727)
8 0.339 (0.368) 0.578 (0.627) 0.705 (0.760) 0.789 (0.841) 0.845 (0.894) 0.884 (0.928)
9 0.467 (0.483) 0.713 (0.737) 0.826 (0.853) 0.887 (0.913) 0.923 (0.947) 0.943 (0.968)
10 0.569 (0.547) 0.803 (0.787) 0.891 (0.878) 0.933 (0.925) 0.954 (0.953) 0.967 (0.972)
15 0.856 (0.641) 0.953 (0.798) 0.976 (0.882) 0.985 (0.925) 0.989 (0.953) 0.992 (0.971)
27
EWMA .
Finally, we consider an EWMA control chart with weighting coefficient
r = 0.10 and 0 = 5 . Table 2.10 shows the accuracy performance of the two estimators
and suggests that the MLE is again the superior estimator. The corresponding estimated precision probabilities of the two estimators (obtained through the use of an EWMA control chart with r = 0.10 and 0 = 20 ) are shown in Table 2.11 for increases and decreases. The table indicates that MLE yields better precision when compared to EWMA , with the exception of the very small shift magnitudes investigated (i.e. a = 22 and a = 18 ). For an EWMA control chart with weighting coefficient r = 0.20 and 0 = 20, results showed the MLE of the change point was the superior estimator for all shift magnitudes investigated. Precision performances in Table 2.12 were obtained by using an EWMA control chart with weighting coefficient r = 0.10 and an in-control rate parameter value of
0 = 10. Results show that the MLE of the change point yields better precision for all
shift magnitudes, except for very small increases and decreases (i.e. a = 12 and
a = 8 ). Table 2.13 provides precision measures of the two estimators using an EWMA
control chart with weighting coefficient r = 0.10 and an in-control rate parameter value of 0 = 5. Again, similar results were obtained. Therefore, we conclude that, overall,
MLE provides more precision than EWMA regardless of the value of the in-control rate
parameter investigated. Even further, our results indicate improvement in the accuracy and precision performances of the MLE as 0 gets small. EWMA charts with weighting coefficients r = 0.20 were also investigated for the three values of 0 and similar results obtained.
2.6. Confidence Sets Based on the Change Likelihood Function
We now consider constructing confidence sets on the process change point . Such a set would provide a window of possible change points within which the true
28
process change point lies, with some level of confidence. This is useful because it provides process engineers a set of change point candidates from which to begin their search for the special cause. Such information can greatly enhance special cause identification, as well as, accelerate efforts to improve quality. Box and Cox (1964) suggest constructing confidence sets on parameter estimates using the likelihood function. Applying such a method yields a confidence set of the form CS = {t : log e L(t ) > log e L( ) D} (2.11) where log e L( ) is the maximum of the log likelihood function evaluated over all possible change points t. If the value of the log likelihood function at t, log e L(t ) , exceeds the maximum of the log likelihood function, less a reference value D, then t is included in the confidence set. To illustrate the use of the proposed confidence set estimator following a signal from a Poisson CUSUM control chart, consider monitoring a count process with known in-control rate parameter 0 = 10. A CUSUM chart designed to detect an increase of three in the mean count rate will be used. Suppose that such an increase in occurs following the 100th subgroup obtained, then Figure 2-3 shows the likelihood values obtained from using MLE following a genuine signal from the CUSUM chart. The maximum likelihood for this realization is found when t = 100. The horizontal line shown in Figure 3 represents the value of the log likelihood function at t = 100 , less the critical value D = 1.75. This line, or threshold, serves as a classifier that discriminates between those potential change points that are contained in the confidence set and those that are not. That is, values of t corresponding to all likelihoods that exceed the threshold are included in the confidence set. We see from the figure that the cardinality of the confidence set for this realization is 15 and the true change point of = 100 is contained in this set.
29
Table 2.7. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 20 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.67.
a
5.00 10.00 12.00 14.00 16.00 18.00 22.00 24.00 26.00 28.00 30.00 35.00
E[T] 102.40 103.55 104.52 106.35 110.91 133.84 130.74 110.92 106.53 104.69 103.70 102.51
EWMA
94.96 95.13 95.15 95.37 96.07 107.07 108.30 97.06 96.15 95.80 95.65 95.56
se EWMA
MLE
99.90 99.51 99.26 99.00 99.37 108.04 109.82 100.19 99.25 99.21 99.35 99.64
seMLE 0.004 0.011 0.014 0.021 0.032 0.075 0.074 0.032 0.022 0.017 0.014 0.009
0.028 0.028 0.029 0.029 0.030 0.070 0.068 0.029 0.027 0.027 0.027 0.026
Table 2.8. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 20 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.20, = 100, A = 2.835.
a
5.00 10.00 12.00 14.00 16.00 18.00 22.00 24.00 26.00 28.00 30.00 35.00
E[T]
101.97 102.99 103.95 105.90 111.90 150.89 137.00 111.07 105.96 104.09 103.15 102.08
EWMA
96.91 96.96 97.06 97.21 98.25 130.41 121.59 99.39 97.79 97.51 97.37 97.28
se EWMA
MLE
99.87 99.49 99.25 99.08 99.67 108.60 110.65 100.49 99.29 99.21 99.32 99.62
seMLE 0.005 0.011 0.016 0.021 0.032 0.081 0.078 0.032 0.023 0.019 0.015 0.009
0.017 0.017 0.017 0.017 0.020 0.138 0.100 0.021 0.016 0.015 0.015 0.015
30
Table 2.9. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 10 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.60.
a
0.25 0.50 2.00 4.00 6.00 8.00 12.00 14.00 16.00 18.00 20.00 25.00
E[T] 102.50 102.56 103.02 104.06 106.57 117.78 116.98 106.81 104.32 103.21 102.60 101.84
EWMA
94.94 94.89 94.97 95.02 95.38 97.80 99.84 96.39 96.00 95.80 95.67 95.60
s.e. EWMA
MLE
99.95 99.93 99.73 99.36 98.94 101.47 103.18 99.40 99.28 99.43 99.56 99.82
s.e.MLE 0.002 0.003 0.007 0.013 0.021 0.045 0.046 0.023 0.016 0.012 0.010 0.005
0.028 0.029 0.029 0.029 0.029 0.035 0.036 0.027 0.026 0.025 0.026 0.025
Table 2.10. Average change point estimates and associated standard errors for two methods of estimating the change point of a Poisson process with 0 = 5 after a signal from a Poisson EWMA control chart, EWMA parameter r = 0.10, = 100, A = 2.42.
a
0.25 0.50 1.00 2.00 3.00 7.00 8.00 9.00 10.00 15.00
E[T]
103.30 103.49 103.96 105.42 109.34 109.53 105.88 104.32 103.44 101.85
EWMA
94.98 95.02 95.10 95.21 95.83 97.72 96.65 96.37 96.26 95.96
s.e. EWMA
MLE
99.70 99.62 99.44 99.02 99.25 100.68 99.51 99.35 99.42 99.76
s.e.MLE 0.007 0.008 0.011 0.018 0.029 0.029 0.021 0.016 0.013 0.006
0.028 0.028 0.028 0.029 0.030 0.027 0.025 0.024 0.023 0.024
31
Table 2.11. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 20 . Precision for EWMA is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
5 0.953 (0.442) 0.986 (0.566) 0.991 (0.615) 0.993 (0.658) 0.995 (0.690) 0.996 (0.723)
10 0.749 (0.352) 0.907 (0.546) 0.947 (0.616) 0.965 (0.657) 0.971 (0.694) 0.976 (0.722)
12 0.604 (0.298) 0.822 (0.520) 0.895 (0.609) 0.930 (0.658) 0.948 (0.692) 0.957 (0.720)
14 0.426 (0.227) 0.664 (0.452) 0.780 (0.579) 0.844 (0.647) 0.882 (0.690) 0.911 (0.724)
16 0.234 (0.156) 0.433 (0.348) 0.558 (0.482) 0.642 (0.579) 0.706 (0.650) 0.757 (0.697)
18 0.066 (0.060) 0.152 (0.153) 0.236 (0.220) 0.272 (0.303) 0.325 (0.364) 0.366 (0.420)
22 0.063 (0.062) 0.144 (0.149) 0.208 (0.226) 0.265 (0.293) 0.307 (0.353) 0.350 (0.404)
24 0.203 (0.151) 0.389 (0.335) 0.509 (0.470) 0.595 (0.569) 0.660 (0.642) 0.713 (0.696)
26 0.355 (0.223) 0.592 (0.444) 0.710 (0.573) 0.788 (0.652) 0.839 (0.702) 0.876 (0.738)
28 0.492 (0.286) 0.725 (0.506) 0.830 (0.614) 0.886 (0.673) 0.918 (0.712) 0.936 (0.742)
30 0.606 (0.340) 0.819 (0.547) 0.899 (0.632) 0.930 (0.675) 0.949 (0.711) 0.959 (0.743)
35 0.801 (0.428) 0.928 (0.581) 0.960 (0.633) 0.972 (0.676) 0.978 (0.714) 0.982 (0.740)
32
Table 2.12. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 10 . Precision for EWMA is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
0.25 0.979 (0.430) 0.993 (0.558) 0.997 (0.610) 0.998 (0.654) 0.999 (0.690) 0.999 (0.718)
0.50 0.968 (0.424) 0.990 (0.562) 0.994 (0.614) 0.996 (0.651) 0.997 (0.689) 0.998 (0.718)
2 0.881 (0.384) 0.963 (0.556) 0.978 (0.611) 0.983 (0.654) 0.987 (0.690) 0.989 (0.716)
4 0.684 (0.313) 0.869 (0.526) 0.927 (0.611) 0.950 (0.651) 0.963 (0.689) 0.968 (0.717)
6 0.412 (0.223) 0.649 (0.442) 0.767 (0.572) 0.835 (0.643) 0.876 (0.687) 0.906 (0.719)
8 0.133 (0.108) 0.275 (0.256) 0.372 (0.375) 0.454 (0.472) 0.517 (0.548) 0.567 (0.611)
12 0.114 (0.103) 0.242 (0.239) 0.338 (0.349) 0.412 (0.445) 0.472 (0.516) 0.523 (0.581)
14 0.317 (0.214) 0.548 (0.427) 0.677 (0.561) 0.760 (0.647) 0.814 (0.700) 0.854 (0.740)
16 0.503 (0.297) 0.738 (0.519) 0.843 (0.625) 0.892 (0.680) 0.922 (0.715) 0.940 (0.749)
18 0.648 (0.367) 0.851 (0.562) 0.914 (0.638) 0.941 (0.680) 0.957 (0.719) 0.965 (0.747)
20 0.749 (0.418) 0.907 (0.581) 0.947 (0.638) 0.962 (0.682) 0.971 (0.720) 0.977 (0.747)
25 0.897 (0.489) 0.977 (0.588) 0.980 (0.640) 0.986 (0.680) 0.990 (0.716) 0.992 (0.742)
33
Table 2.13. Precision of MLE and EWMA when used with an EWMA control chart; r = 0.10 , = 100, 0 = 5 . Precision for EWMA is shown in parentheses.
a
P(| |) = 0 P(| |) 1 P(| |) 2 P(| |) 3 P(| |) 4 P(| |) 5
0.25 0.883 (0.350) 0.958 (0.546) 0.973 (0.606) 0.980 (0.648) 0.985 (0.685) 0.987 (0.712)
0.50 0.827 (0.334) 0.939 (0.540) 0.963 (0.605) 0.973 (0.648) 0.980 (0.683) 0.983 (0.713)
2 0.720 (0.304) 0.889 (0.524) 0.937 (0.604) 0.955 (0.648) 0.965 (0.683) 0.973 (0.716)
4 0.491 (0.241) 0.724 (0.464) 0.828 (0.581) 0.883 (0.644) 0.914 (0.682) 0.935 (0.715)
6 0.257 (0.161) 0.466 (0.360) 0.588 (0.497) 0.676 (0.591) 0.739 (0.656) 0.785 (0.699)
8 0.197 (0.155) 0.378 (0.334) 0.495 (0.471) 0.586 (0.571) 0.653 (0.646) 0.710 (0.705)
12 0.333 (0.225) 0.565 (0.442) 0.694 (0.578) 0.776 (0.659) 0.831 (0.715) 0.869 (0.754)
14 0.458 (0.288) 0.697 (0.512) 0.811 (0.623) 0.871 (0.687) 0.905 (0.727) 0.931 (0.762)
16 0.561 (0.340) 0.789 (0.555) 0.877 (0.642) 0.919 (0.692) 0.940 (0.731) 0.955 (0.762)
18 0.857 (0.484) 0.951 (0.596) 0.972 (0.651) 0.980 (0.692) 0.986 (0.733) 0.989 (0.760)
34
Likelihoods at Potential Change Likelihood Values at Potential Change Point t
Likelihoods at time t 4
Point t
3.5
3
2.5 Likelihood
2
Threshold
1.5
1
0.5
0
0
20
40
60 possible changepoint t
80
100
Figure 2-3. Plot of likelihood values computed at each potential change point t.
We now investigate the coverage probabilities and expected cardinality of the confidence sets obtained using critical values of D between 1.353 and 2.970. These values correspond to those suggested by Box and Cox (1964) and Siegmund (1986), respectively. The expected cardinality of the confidence sets is then plotted against the coverage probabilities for each value of 0 investigated. For each control chart and magnitude of change studied, a step-change in the Poisson rate parameter was simulated to occur following the 100th subgroup collected. The confidence set estimator was applied after a genuine signal from a control chart. The cardinality of the confidence set was recorded as well as whether the confidence set covered the true process change point of = 100 . This procedure was repeated for a total of N = 10,000 simulation runs for each value of a considered. The average
35
cardinality of the confidence set was computed as well as the proportion of the 10,000 runs that covered the true process change point.
2.6.1 Confidence Sets for Process Change Point After a Signal from a c-chart
In this section, we consider applying the likelihood-based method to estimating confidence sets after a genuine signal from a Shewhart c-chart. Estimates of the coverage probabilities and expected confidence set cardinality are obtained by using several different critical values D. Specifically, we investigate D at 1.353, 1.50, 1.75, 2.00, 2.25, 2.50, 2.75, and 2.97. Figures 2-4 and 2-5 show plots of expected confidence set cardinality versus coverage probabilities for increases and decreases, respectively, when 0 = 20. Figures 2-6 and 2-7 show plots of expected confidence set cardinality versus coverage probabilities for increases and decreases, respectively, when 0 = 10. Finally, plots of estimated expected confidence set cardinality versus coverage probabilities for increases in the Poisson rate parameter when 0 = 5 is shown in Figure 2-8. The Shewhart c-chart cannot detect decreases in for values of 0 9 , therefore, decreases were not considered for 0 = 5 . The figures show that as the magnitude of change in the Poisson rate parameter increases, the coverage probability increases while the cardinality decreases. The figures also show that as the value of D increases, more coverage is obtained, however, at the expense of greater cardinality. These charts can be useful in determining an appropriate value for D that will yield acceptable levels of expected coverage and cardinality. For example, suppose 0 = 20 and one has a particular interest in detecting increases of + 4 units or greater in the Poisson rate parameter. If at least 90 percent coverage is desirable, and cardinality greater than 15 is unacceptable, then the appropriate choice for D should lie between 2.50 and 2.75. Any value outside of this range will yield cardinality greater than 15 or coverage less than 90 percent. For a given critical value D, we note that as 0 decreases, the coverage probability increases and the cardinality decreases. We further note that for decreases in the Poisson rate parameter, the coverage probabilities are slightly greater than those
36
obtained from increases of the same shift magnitude, however, this occurs with a slight increase in the cardinality of the confidence set.
Cardinality versus Coverage Probabilities: c-chart, increasing rate case 1 35 0.95 30 28 26 24 0.9 22 0.85 Coverage Probability 2.75 0.8 2.25 0.75 2.00 0.7 1.75 0.65 1.50 0.6 1.353 Critical Value D 2.50 2.97 Rate Parameter
0.55
0
5
10
15
20
25
30
35
40
45
50
Average Cardinality
Figure 2-4. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 20.
2.6.2 Confidence Sets for the Process Change Point After a Signal from a Poisson CUSUM Control Chart
We now consider confidence sets based on the likelihood function following a signal from a Poisson CUSUM control chart. Each CUSUM chart was designed to detect a change of 4 units in the Poisson rate parameter, except for 0 = 5 , where the CUSUM chart was designed to detect a change of 3 counts. The same plots were generated as described in the previous section and are shown in Figures 2-9 through 2-14 below. Their behavior is very similar to those obtained from the Shewhart c-chart. When comparing the confidence sets obtained from the CUSUM chart to those obtained from the Shewhart c-chart, the CUSUM chart yield slightly greater cardinality at
37
small magnitudes of change than does the Shewhart c-chart. It is well known that, comparatively, the CUSUM chart has better ARL performance than the Shewhart c-chart in detecting small persistent shifts in the mean count rate. Therefore, such a result is expected since more information is available to estimate the confidence set when the ARL is larger.
2.6.3 Confidence Sets for the Process Change Point After a Signal from a Poisson EWMA Control Chart
We now consider confidence sets based on the likelihood function following a signal from a Poisson EWMA control chart. Each EWMA control chart investigated was set to have a weighting coefficient of r = 0.10 . The same plots were generated as described in the previous sections and are shown in Figures 2-15 through 2-20 below. Their behavior is very similar to those obtained from the Shewhart c-chart and the Poisson CUSUM control chart. Although not shown here, as the EWMA weighting coefficient gets large, both cardinality and coverage percentage increase. This was primarily the case for small shift magnitudes, say, a change of 2 units or less in the Poisson rate parameter. For large shifts in the rate parameter, however, and for larger values of the EWMA weighting coefficient, a decrease in cardinality is witnessed along with an increase in coverage percentage.
2.7. Choice of for Performance Evaluation of the Confidence Set Estimator
Our study of the performance of the likelihood-based confidence set estimator assumed a change point of = 100. This section aims to show that the decision to simulate the process change point at = 100 was appropriate. We investigated the coverage probabilities and cardinality of sets obtained following a signal from a standard Shewhart c-chart for various magnitudes of change when the value of the change point was varied between 10 and 500. Tables 14 and 15 show the results of this study with
D = 2.97 . Figure 2-21 shows these results graphically.
38
Cardinality versus Coverage Probabilities: c-chart decreasing rate case 1 10 12 0.95 14 16 18 0.9 2.75 Coverage Probability 2.50 0.85 2.25 2.97 Rate Parameter
2.00 0.8 1.75 0.75 1.50 0.7 Critical Value D
1.353
0.65
0
5
10
15
20
25
30
35
40
45
50
Average Cardinality
Figure 2-5. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 20.
Cardinality versus Coverage Probabilities: c-chart, increasing rate case 1 25 0.95 20 0.9
18 2.75
2.97
Coverage Probability
0.85 16 0.8 2.00 0.75 14 0.7 Rate Parameter 1.50 0.65 12 1.353 1.75 Critical Value D 2.25 2.50
0.6
0
5
10
15
20
25
30
35
Average Cardinality
Figure 2-6. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 10. 39
Cardinality versus Coverage Probabilities: c-chart decreasing rate case 1 2 4 6 0.95 2.75 0.9 Coverage Probability 2.25 2.00 0.85 2.50 Rate Parameter 8 2.97
1.75 0.8 1.50 0.75 1.353
Critical Value D
0.7
0
5
10
15 Average Cardinality
20
25
30
Figure 2-7. D al Value Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 10.
Cardinality versus Coverage Probabilities: c-chart, increasing rate case 1
0.95
15 2.97
0.9 10 0.85 9 0.8 8 0.75 Rate Parameter 1.50 0.7 7 1.353 1.75 2.00 Critical Value D 2.25 2.50
2.75
Coverage Probability
0.65
0
5
10
15
20
25
Average Cardinality
Figure 2-8. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a c-chart using several critical values D; = 100 , 0 = 5. 40
Cardinality versus Coverage Probabilities: CUSUM, increasing rate case 1
0.95 35 0.9 30 2.75 28 0.8 26 0.75 2.00 24 0.7 1.75 Rate Parameter 0.65 1.50 0.6 22 1.353 0.55 0 5 10 15 20 25 30 35 40 45 50 Critical Value D 2.25 2.50 2.97
0.85 Coverage Probability
Average Cardinality
Figure 2-9. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 20.
Cardinality versus Coverage Probabilities: CUSUM decreasing rate case 1 5 0.95
0.9 10 2.97 0.85 Coverage Probability 12 2.50 0.8 14 2.25 2.75
0.75 16 0.7 Rate Parameter 0.65 18 1.50 1.353 1.75
2.00 Critical Value D
0.6
0.55
0
5
10
15
20 25 30 Average Cardinality
35
40
45
50
Figure 2-10. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 20.
41
Cardinality versus Coverage Probabilities: CUSUM, increasing rate case 1
0.95
0.9
20 18 2.75 2.50 16 2.25
2.97
Coverage Probability
0.85
0.8 2.00 0.75 14 1.75 0.7 Rate Parameter 1.50 0.65 12 0.6 1.353 10 15 20 25 30 35 Critical Value D
0
5
Average Cardinality
Figure 2-11. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 10.
Cardinality versus Coverage Probabilities: CUSUM decreasing rate case 1 2 4 6 0.95 8 0.9 2.75 Coverage Probability 0.85 2.25 0.8 2.00 Critical Value D 0.75 1.75 2.50 2.97 Rate Parameter
0.7
1.50 1.353
0.65
0
5
10
15 Average Cardinality
20
25
30
Figure 2-12. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 10.
42
Cardinality versus Coverage Probabilities: CUSUM, increasing rate case 1 20 0.95
15 2.75
2.97
0.9
2.50 2.25
Coverage Probability
0.85
10 9 2.00 Critical Value D 1.75
0.8 8 0.75 Rate Parameter 0.7 1.50 7 1.353
0.65
0
5
10
15 Average Cardinality
20
25
30
Figure 2-13. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 5.
Cardinality versus Coverage Probabilities: CUSUM decreasing rate case 1 1 2 0.95 3 Rate Parameter
0.9
4 2.97
Coverage Probability
0.85 2.50 0.8 2.25 0.75 2.00 Critical Value D 0.7 1.75
2.75
0.65
1.50 1.353
0.6
0
5
10
15
20
25
30
35
40
45
50
Average Cardinality
Figure 2-14. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson CUSUM control chart using several critical values D; = 100 , 0 = 5.
43
Cardinality versus Coverage Probabilities: EWMA, increasing rate case 1 35 30 28 Rate Parameter 24
26
0.95
0.9 2.75 2.50 0.8 2.25 2.00 Critical Value D 0.7 1.75
22 2.97
0.85 Coverage Probability
0.75
0.65 1.50 0.6 1.353
0.55
0
10
20
30 Average Cardinality
40
50
60
Figure 2-15. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 20.
Cardinality versus Coverage Probabilities: EWMA decreasing rate case 1 10 0.95 12 14 Rate Parameter
16
18 0.9 2.75 Coverage Probability 0.85 2.50 2.25 2.97
0.8 2.00 0.75 1.75 0.7
Critical Value D
0.65
1.50 1.353
0.6
0
10
20
30 Average Cardinality
40
50
60
Figure 2-16. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 20.
44
Cardinality versus Coverage Probabilities: EWMA, increasing rate case 1 25 20 18
16
Rate Parameter 14 12 2.97
0.95
0.9 2.50 0.85 2.25
2.75
Coverage Probability
0.8
2.00 Critical Value D 1.75
0.75
0.7
1.50
1.353 0.65 0 5 10 15 20 25 30 35 40 45
Average Cardinality
Figure 2-17. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 10.
Cardinality versus Coverage Probabilities: EWMA decreasing rate case 1 2 4 6 0.95 8 2.97 0.9 2.50 Coverage Probability 0.85 2.00 0.8 1.75 0.75 1.50 0.7 1.353 Critical Value D 2.25 2.75 Rate Parameter
0
5
10
15
20 Average Cardinality
25
30
35
40
Figure 2-18. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 10.
45
Cardinality versus Coverage Probabilities: EWMA, increasing rate case 1 15 10 0.95 2.75 0.9 Coverage Probability 2.25 0.85 2.50 9 8 Rate Parameter 7 2.97
2.00
0.8
1.75
Critical Value D
1.50 0.75 1.353 0.7 0 5 10 15 20 25 30 35
Average Cardinality
Figure 2-19. Plot of coverage probabilities versus estimated cardinality of confidence sets for increases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 5.
Cardinality versus Coverage Probabilities: EWMA decreasing rate case 1 0.25 0.5 1 2 3 0.95 2.75 2.50 Coverage Probability 0.9 2.25 2.97
Rate Parameter
2.00 0.85 Critical Value D 1.75
0.8 1.50
0.75
1.353 0 5 10 15 20 25 Average Cardinality
Figure 2-20. Plot of coverage probabilities versus estimated cardinality of confidence sets for decreases in the Poisson rate parameter following a signal from a Poisson EWMA control chart using several critical values D; = 100 , 0 = 5. 46
Figure 2-21 shows that as the change point is varied between 10 and 500, there are no significant changes in the coverage probabilities obtained. For the increasing rate case, the expected cardinality increases with , however, not by an appreciable amount. Furthermore, for the decreasing rate case, the expected cardinality remains constant. This was found to be characteristic of the Shewhart c-chart only. Similar results were obtained with 0 = 5, 10 and 20 for all control charts and shift magnitudes investigated in this study. Therefore, we conclude that our choice of = 100 was a reasonable point at which to simulate the process change point.
2.8. Summary
If process engineers could determine when the process changed, their search for the responsible special causes could be narrowed down to finding which aspect of the process (e.g., which process variables) changed at that time. This could allow them to identify the special cause more quickly, and to use that information to improve the quality of the process or product sooner. Also, knowing the time of the process change would help to minimize identification of the wrong process variables as the special cause. Consequently, being able to estimate the time of a process change would be useful to process engineers. A point estimate as well as a confidence set of the time of the process change would provide process engineers with useful starting points in their search for a special cause following a control chart signal. In this paper we have considered a maximum likelihood estimator of the time of a step change in a Poisson rate parameter. We compared its performance with built-in estimators of the CUSUM and EWMA control charts. We showed that the MLE of the change point provides a more accurate estimate of the true process change point, regardless of the value of 0 , over the broad range of magnitudes of change considered. We showed that the precision of the MLE is superior to the precision of the built-in estimators of the CUSUM and EWMA control charts. The only exception is the CUSUMs CUSUM estimator, which has slightly better precision performance for small
47
increases in and smaller values of 0 . We conclude, however, that the MLE is still the better overall choice since the magnitude of the change is generally not known apriori. In addition, confidence sets are easily established using the MLE estimator. Using a maximum likelihood approach to obtain a point estimate of the time of the process change allows the construction of a confidence set which covers the true process change point with a given level of certainty. We have studied the use of a likelihood approach to confidence set estimation for the change point that can be used after a control chart signal. We have examined the use of this estimator with a Shewhart c-chart, a Poisson CUSUM control chart and a Poisson EWMA control chart. We have shown that a value of D = 2.97 will provide confidence sets on the change point with near 90% coverage or more for all shift magnitudes and all control charts considered in this study. We have also provided plots of expected cardinality versus coverage probabilities for all 0 values considered to aid in selecting an appropriate critical value D.
48
Effect of Simulated Changepoint on Estimated Coverage of Confidence Sets 0.97
a = 12 rate
0.96 rate a = 28
Estimated Coverage Probability
0.95
a rate = 16
0.94
0.93
a rate = 24
0.92
0.91 10
50
100
150
200
250
300
350
400
450
500
Simulated Changepoint
Effect of Simulated Changepoint on Estimated Cardinality 25
20 Estimated Cardinality of Confidence Set
rate a = 24
15
rate a = 16
10
rate = 28
a
5
rate = 12
a
0 10
50
100
150
200 250 300 Simulated Changepoint
350
400
450
500
Figure 2-21. Effect of change in the simulated change point on coverage probabilities and average cardinality of confidence sets.
49
CHAPTER 3 A MAGNITUDE-ROBUST CONRTOL CHART FOR MONITORING AND ESTIMATING STEP CHANGES IN A POISSON RATE PARAMETER
3.1 Introduction
Statistical process control (SPC) charts are used to monitor for changes in a process by distinguishing between common causes and special causes of variation. It is desirable to detect the presence of special cause variation quickly so that appropriate action can be taken to return the process to a state of statistical control. The sooner a process change is detected by a control chart, the sooner process engineers can initiate their search for the special cause effecting the process. This paper focuses on the statistical monitoring of count data processes, particularly, those that can be modeled by the Poisson distribution. That is, we consider count processes where X i , the count at subgroup i, is assumed to follow a Poisson distribution with rate parameter = i . The process is said to be in-control when = 0 and out-of-control when either > 0 or < 0 . We investigate a control chart that quickly detects changes in for any magnitude. This magnitude-robust control chart also provides estimates of the time at which the process changed and the magnitude of the change. These estimates can aid process engineers in finding the special cause(s) associated with the process disturbance. We show that the magnitude-robust chart can be derived from a likelihood ratio test for a step change in the mean count rate of a Poisson process. We also show that the proposed control charting strategy has better overall average run length (ARL) performance than that of any one CUSUM scheme.
3.2. Process Behavior Model and Associated Hypothesis Tests for the Poisson CUSUM
50
Although the CUSUM chart was developed by Page (1954, 1961) using a sequential probability ratio test (SPRT) under normal model assumptions, Brook and Evans (1972) were the first to suggest using such a scheme for monitoring count data processes. The CUSUM test statistic for detecting an increase in the mean count rate is given by
+ + C T = max{ 0, X T k + + C T 1
}
(3.1)
where CT+ is the cumulative sum at time T. Evidence of an increase in the mean count
+ rate is indicated by a signal occurring if CT > h + . The values of h + and k + are selected
on the basis of desirable ARL properties and 0 . Brook and Evans (1972) propose a Markov chain modeling approach for obtaining the exact run length distribution of a particular sampling scheme, allowing for a simple means of evaluating and comparing any such schemes. Similar to the normal distribution, the Poisson CUSUM control statistic can be derived from the SPRT, however, under Poisson model assumptions. That is, the null hypothesis H 0 : = 0 is tested against the simple alternative hypothesis H a : = a whenever a new sequence of subgroup counts is obtained. comparing the sequential probability ratio
X a e
i
The SPRT operates by
ST =
i =1
T
a
X i! X T 0 i e 0 X! i =1 i
(3.2)
to an appropriate constant A at each new subgroup. If S T > A , then the test concludes in favor of H a after observing subgroup T. The value of 0 is the known in-control rate parameter and a is the pre-specified change that one wishes to detect. To derive the form of the statistic in (3.1) for the increasing rate case, we can take the natural log of (3.2) and simplify to obtain
T + + S T = ( 0 + ) + X i log e a . a i =1 0
(3.3)
Rewriting this expression as
51
+ + S T = 0 + + X T log e a + S T+1 a 0
(
)
(3.4)
and then dividing by log e + log e ( 0 ) and rearranging, we obtain a CT+ = X T k + + CT+1 , where
+ + 0 ST + a and k = . C = log e + log e ( 0 ) log e ( + ) log e ( 0 ) a a + T
( )
(3.5)
( )
The value of C 0+ is typically taken to be zero. Resetting the expression in (3.5) to zero
+ whenever CT < 0 yields the form of the statistic in (3.1), or + C T = max{ 0, X T k + + CT+1 }.
(3.6a)
If the test statistic in (3.6a) exceeds a decision interval h + , then the chart signals that an increase in the mean count rate has occurred. For the decreasing rate case, the CUSUM statistic is given by
CT = max{ 0, k X T + C T 1
}
(3.6b)
0 a where k = and C 0 = 0. If CT h , then the chart signals that a log e ( 0 ) log e ( a )
decrease in the mean count rate has occurred. Hawkins and Olwell (1998) provide extensive detail pertaining to the theoretical foundation and construction of CUSUM control charts in general, including coverage of the Poisson CUSUM. Lucas (1985) provides a comprehensive study on the ARL performances of the Poisson CUSUM control chart. Lucas investigates these charts with and without fast-initial response (FIR) features (see Lucas and Crosier 1982) and provides tables to aid the user in selecting appropriate h and k values. An alternative to the SPRT is to consider a change point model and a likelihood ratio test. The change point model assumes that the process is in control for a period of time before shifting to an out-of-control state. Pignatiello and Simpson (2002) consider such an alternative in the development of their magnitude-robust control chart for step changes in normal process means. The purpose of this paper is to evaluate the likelihood
52
ratio strategy for detecting and characterizing step changes in the mean count rate of a Poisson process.
3.3. Behavior Model for Poisson Process Rate Parameter
Consider a permanent step change model for the behavior of a Poisson process rate parameter. The model assumes that the process is initially in-control with independent observations coming from a Poisson distribution with known rate 0 . After an unknown point in time the mean count rate changes from its in-control state of 0 to an unknown out-of-control state a . The model also assumes that once a step change in the mean count rate occurs, the process remains at the new level a until the special cause has been identified and removed. The Poisson process step change model can be parameterized as follows. Each observation collected consists of a subgroup count. During the formulation of subgroups
i=
1, 2, , , the process rate i is equal to its known in-control value 0 .
For
subgroups i = + 1 , ..., T , the process rate i is equal to some unknown rate a and T is the current subgroup number. The unknown change point represents the last subgroup taken from the in-control process.
3.4. Likelihood Ratio Test: Control Chart for a Poisson Process Step Change Model
After observing T subgroups, the null hypothesis of interest is that the process has been and still is in-control. Thus, H 0 : i = 0 for 1 i T , where 0 is the known value for the mean count rate when the process is in-control. The alternative hypothesis states the process is initially in-control, then after some unknown subgroup , the mean count rate changes from 0 to a . Thus, the alternative hypothesis can be stated as
53
H a : i = 0 for 0 i and i = a for + 1 i T where both and a are unknown. The likelihood under the null hypothesis can be written as
X e 0 0 i L0 (x ) = X i! i =1 T
(3.7)
where X i is the subgroup count at subgroup i and T is the total number of subgroup counts collected. Assuming a process change point at , the likelihood under the
alternative hypothesis can be written as
X e 0 0 i La ( , a | x ) = X i! i =1
X e a a i X! . i = +1 i T
(3.8)
Therefore, the ratio of La to L0 is given by
L( , a | x ) =
i =1
X 0 e
i
0
X i!
i =1
T
e
i = +1 X i 0 0
T
X a e
i
a
X i!
.
(3.9)
X i!
If we define R( , a | x ) as the natural logarithm of L( , a | x ) and simplify we obtain
T R( , a | x ) = (T )( 0 a ) + X i (log e a log e 0 ) , i = +1
(3.10)
where sufficiently large values of R( , a | x ) would favor the alternative hypothesis. Since and a are unknown, the test is conducted by maximizing R( , a | x ) over all possible values of and a given the observed subgroup counts x' = [ X 1 , X 2 , K , X T ] . We denote this maximum value as RT . For any given value of , it is easy to show that the value of a which maximizes R( , a | x ) is the maximum likelihood estimate (MLE) of a , or a ( ) =
T 1 X , (T ) t 1 t = +
(3.11)
the average of the (T ) most recent subgroup counts. Substituting the expression in (3.11) into (3.10) and reducing yields the following expression
54
( ) 1 , R , a ( ) | x = (T ) 0 + a ( ) log e a 0
(
)
(3.12) Thus, for each
which is expressed solely as a function of given the observations, x . to obtain the maximum. Letting t represent a possible change point, then RT = max R t , a (t ) | x
0t <T
new T, the expression in (3.12) is evaluated over all possible integer change point values
[ (
)]
(3.13) If RT is
where RT is the maximum of the log likelihood ratios at subgroup T.
sufficiently large, say RT > B (where B is an appropriately chosen constant), then the test concludes in favor of the alternative hypothesis. This hypothesis test could be implemented as a control chart by simply computing the test statistic, RT , with each new subgroup count obtained. If the value of RT is greater than an appropriately chosen constant (appropriate in the sense that desirable ARL properties are obtained), then the control chart signals that a change has occurred. Furthermore, when this control chart signals,
= arg max (T t )( 0 + a (t )) log e
0t <T
a (t ) 1 0
(3.14)
and a ( ) =
T 1 1X t (T ) t = +
(3.15)
provide the MLEs of the change point and new process rate a , respectively. It should be noted that the CUSUM chart given in (3.1) is a special case of the magnitude-robust chart when a is a known fixed quantity. As such, the proposed
magnitude-robust chart requires no prior specification of the out-of-control rate parameter
a . This is advantageous since the out-of-control rate parameter is rarely known apriori
in practice. The procedure described above produces a two-sided control chart since it can detect both increases and decreases in . This procedure can also be implemented as a one-sided control chart to detect either increases or decreases in . For example, for some applications process engineers may only be interested in detecting an increase in
55
the rate of nonconformities. To implement a one-sided procedure for detecting increases,
the estimate of a used in (3.12) becomes + (t ) = max 0 , a (t ) . a
{
}
Therefore, a one-
sided control chart for detecting increases in can be given as
+ RT = max R + t , + (t ) | x . a
0t <T
(
)
(3.16)
For a one-sided control chart to detect decreases in the estimate of a becomes
(t ) = min 0 , a (t ) . Hence, a one-sided control chart for decreases can be given by a
{
}
RT = max R t , (t ) | x . a
0t <T
(
)
(3.17)
The point estimates for using the one-sided procedures can then be expressed as
= arg max (T t )( 0 + (t )) log e a
0 t <T
(t ) a 1 . 0
(3.18)
A distinct advantage of using a control chart derived from the likelihood ratio approach is that it provides valuable diagnostic tools which can help process engineers focus their search for the special cause. Along with the signal of a process change, the charts described above additionally provide process engineers with point estimates of both the magnitude of the change and when that change first manifested itself in the data. Pignatiello and Samuel (1998) evaluated the performance of the estimator in (3.14) when applied following a signal from a Shewhart c-chart. Perry, Pignatiello and Simpson (2004a) evaluated the performance of the estimator in (3.14) when applied following signals from CUSUM and EWMA schemes. Those papers indicate that the MLE of provides good performance regardless of the magnitude of change. Perry, Pignatiello and Simpson (2004) also compare the estimator in (3.14) to estimators suggested by Page (1954) and Nishina (1992) for use with CUSUM and EWMA schemes, respectively. Their results show the MLE provides better overall estimation performance. The proposed control charting strategy also permits the construction of confidence sets on the change point , the last subgroup from the in-control process. Such a set would provide a window of possible change points that cover the true change point with a given level of confidence. Such a set would be useful because it provides process engineers with change point candidates from which to begin their search for the special
56
cause. Such information can lead to enhanced special cause identification, as well as accelerated efforts to improve process quality. Box and Cox (1964) suggest constructing confidence sets on parameter estimates using the log likelihood function. Applying such a method on yields a confidence set of the form
CS = {t : R t , a (t ) | x > RT D}
(
)
(3.19)
where RT is the maximum of the log likelihood ratios at time T. If R t , a (t ) | x , the
(
)
value of the log likelihood ratio at t, exceeds the maximum of the log likelihood ratio,
RT , less a reference value D, then t is included in the confidence set.
Perry, Pignatiello and Simpson (2004a) used simulation to study the effect of changes in the magnitude of the step change on the performance of the confidence set estimator in (3.19). They showed that at least 90% confidence can be obtained (for the range of change magnitudes they investigated) using a reference value D = 2.97. Smaller change magnitudes resulted in greater confidence set cardinality with a corresponding reduction in the confidence level of the set. They provided surface plots showing estimated relationships between confidence set coverage, set cardinality and step change magnitudes to aid the user in selecting a reference value D that will yield acceptable set coverage and cardinality over a range of potential change magnitudes.
3.5. Average Run Length Comparison
We now focus attention on the proposed charts detection performance relative to the CUSUM alternative. Specifically, we compare the ARL performance of the proposed likelihood ratio based chart to that of three CUSUM charts which differ in their reference values k and corresponding decision intervals h. That is, each CUSUM chart is designed to detect different-sized shifts in the mean count rate. Unlike the normal distribution case, since there is no standardized Poisson distribution, we investigate in-control rate parameter values of 0 = 5, 10 and 20.
57
Furthermore, since the CUSUMs are one-sided procedures, we only investigate the one sided procedures for the proposed control-charting strategy in our comparisons. That is, we consider increases and decreases separately, using the statistics defined in (3.16) and (3.17), respectively. The performance comparison assumes that a step change in the mean count rate occurs following subgroup 0 . We use Monte Carlo simulation to evaluate the ARL performances of the control charts. The Markov chain modeling approach suggested by Brook and Evans (1972) was used to verify the ARL results for the Poisson CUSUM chart. The results from the Markov chain and Monte Carlo simulation were in In the next section, simulation modeling of the step change is described. Some issues related to the handling of false alarms when > 0 and making fair ARL comparisons are discussed. Finally, ARL results are presented and compared with those of some other control charts. agreement.
3.5.1 Simulation Modeling of a Step Change
Monte Carlo simulation was used to estimate the ARL performances of the various control charts. The simulation study was conducted as follows. Observations were generated from an in-control Poisson distribution for subgroups i = 1, 2, , . Starting with subgroup + 1 , observations were generated from a Poisson distribution with an out-of-control rate parameter a . Observations were collected until the control chart issued a signal at subgroup T. The length for that run was then recorded as T . This procedure was repeated for a total of N = 100,000 independently-seeded runs for each value of a investigated. The average of the 100,000 independent run lengths was then recorded as the estimated ARL.
3.5.2 False Alarms
When > 0 and a control chart issues a signal at subgroup T where T , then the signal is a false alarm since the signal was given before the simulated process change could occur. We handled false alarms in our simulation runs as was done in Pignatiello and Simpson (2002). That is, when a false alarm was encountered in a simulation run, it
58
was treated the same way that a false alarm would be treated on an actual process. Namely, if one determines that a signal is indeed a false alarm then one is affirming that the process is currently in-control and could restart their monitoring of the process. Thus, when a false alarm was encountered at subgroup T, the control chart was restarted at subgroup T + 1 while not altering the scheduled change point. For example, if the change point was = 100 and a false alarm was issued at subgroup 75, then the appropriate statistics would be zeroed out and that simulation run would continue as if subgroup 76 was the first one from an in-control process. Thus, if there were no more false alarms on this particular run, there would have been 25 subgroups formed from the restarted in-control process when the first subgroup from the changed process is formed. With this approach the number of subgroups since the chart was started or restarted that have been observed from the in-control process at the time of the process change will not necessarily be fixed at but will instead be random and less than or equal to .
3.5.3 ARL Calibration of Control Charts
Without loss of generality, each chart was calibrated so that when the process was in-control, the ARL was equal to approximately 100. We have found that the same relative comparisons and results are obtained for other values for the in-control ARL. We investigate in-control rate parameter values of 0 = 5, 10 and 20 and consider both, increasing and decreasing rate cases separately. Table 3.1 shows the control charting schemes used in the increasing rate case comparisons and corresponding parameters yielding in-control ARLs of 100. Table 3.2 provides the same information for the decreasing rate cases. Simulation was used to determine the control limit value B required for the likelihood ratio based approach. A total of N = 10,000 independentlyseeded runs were used to estimate the in-control ARLs. A Markov chain approach to ARL estimation (as suggested by Brook and Evans 1972) was used to determine the decision intervals for the CUSUM schemes.
3.5.4 Initial ARL Performance Comparisons
We first consider control chart ARL performances for processes that are out-ofcontrol when a chart is first applied, i.e. when = 0. 59 Tables 3.3 and 3.4 show, for
increases and decreases, respectively, the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of = 5. Tables 3.5
0
and 3.6 show the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of 0 = 10 while Tables 3.7 and 3.8 show these estimates for 0 = 20 .
Table 3.1. In-control ARL estimates for the magnitude robust for step changes (MR-SC) and three different CUSUM schemes for each in-control rate parameter value investigated. ARLs reflect those obtained from one-sided control charts for increases.
0
5 10 20 ARL0 100.03 99.84 99.26 MR-SC B 3.335 3.245 3.440 95% C.I. [98.11, 101.95] [97.95, 101.73] [97.36, 101.16]
CUSUM Schemes CUSUM designed for increasing shift of +1 standard deviation units in mean count rate 0 k h ARL0 5 100.61 6.049 7.532 10 101.59 11.509 10.010 20 99.47 22.161 13.680 CUSUM designed for increasing shift of +1.5 standard deviation units in mean count rate 0 k h ARL0 5 100.29 6.534 5.800 10 102.82 12.219 7.600 20 100.33 23.193 10.420 CUSUM designed for increasing shift of +2 standard deviation units in mean count rate 0 k h ARL0 5 94.64 6.999 5.000 10 100.05 12.905 6.150 20 100.95 24.197 7.900
60
Table 3.2. In-control ARL estimates for the magnitude robust for step changes (MR-SC) and three different CUSUM schemes for each in-control rate parameter value investigated. ARLs reflect those obtained from one-sided control charts for decreases.
MR-SC B 4.4000 3.6000 3.7402
0
5 10 20
ARL0 100.42 100.15 102.25
95% C.I. [98.46, 102.38] [98.21, 102.10] [100.27, 104.22]
CUSUM Schemes CUSUM designed for decreasing shift of 10% of the mean count rate 0 k h ARL0 5 100.16 4.746 13.200 10 99.14 9.491 16.830 20 100.37 18.982 20.070 CUSUM designed for decreasing shift of 25% of the mean count rate 0 k h ARL0 5 99.88 4.345 8.450 10 100.32 8.690 9.740 20 100.30 17.380 10.470 CUSUM designed for decreasing shift of 40% of the mean count rate 0 k h ARL0 5 97.44 3.915 5.7450 10 99.31 7.830 6.0000 20 100.84 15.661 6.3215
The results in Tables 3.3-3.8 indicate that the CUSUM charts perform well when detecting the shifts which are close to the ones for which they were specifically designed. For example, in Table 3.3, of the control charts we have investigated here the k = 6.049 CUSUM (designed for an increase of 1 standard deviation unit in , or a = 7.236 ) provides the lowest ARLs for step changes in the interval a < 8.0. For values of a in the interval 8.0 a 10.0 , the k = 6.534 CUSUM (designed for an increase of 1.5 61
standard deviation units in , or a = 8.354 ) provides the lowest ARLs. For step changes to a > 9, the k = 6.999 CUSUM (designed for an increase of 2 standard deviation units in , or a = 9.472 ) provides the lowest ARLs. Although each CUSUM chart yields the best ARL performance in those regions close to the value of a for which they were designed, no single CUSUM control chart is uniformly best. For example, it can be seen that although the k = 6.049 CUSUM performs well for small changes, it does not perform as well as any of the other charts at detecting larger changes. Conversely, the k = 6.999 CUSUM, which does well at detecting the large changes, does not perform particularly well for smaller changes. The k = 6.534 can be considered somewhat of a compromise in that it does not perform badly for small or large changes, and it does expectedly well for values of a close to 8.354. For quick detection of step changes regardless of the magnitude, the magnituderobust chart should be considered, since it has a strong ARL performance for changes of all magnitudes. It outperforms the k = 6.534 CUSUM in Table 3.3 for values of a that are less than 8.0 and greater than 10.0. Although it does not have the best ARL performance at any specific value of a , there is no range of a where it does not perform well and it performs nearly the best for all values of a . In this sense, the control chart is robust to uncertainty in the magnitude of the change in the mean count rate. These results reflect those obtained for all in-control rate parameter values investigated. We note, however, that for the decreasing rate case and for small values of
0 , there is little room for change in since cannot be negative. We see that in Table
3.4 the CUSUM charts all perform quite well for the entire range of a . These results are intuitive since any step change in these cases that could occur would be fairly close to the CUSUMs pre-specified value of a . As such, for small values of 0 , it may be better to use an appropriately designed CUSUM chart when quick detection of decreases is of interest.
3.5.5 Steady State ARL Performance Comparisons
62
The second ARL performance study considers control charts which are applied on processes that are initially in-control, but experience a step change disturbance following the formation of subgroup > 0. The results shown in Tables 3.9-3.14 are for a change point of = 50. From simulation experiments not reported here, the ARL performances of the control charts for = 20 were approximately the same as for larger values of , such as = 75 and 100. Thus, the results reported here are indicative of a wide range of values of the change point. Tables 9-14 show results similar to the = 0 case. That is, the CUSUM charts yield the best ARL performances near those values of a for which they were specifically designed. Again, there is no single best control chart for all values of a .
Table 3.3. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5 .
a
6.00 7.00 8.00 9.00 10.0 12.0 14.0 16.00 18.00 20.00
MR-SC 19.27 (0.05) 7.97 (0.02) 4.66 (0.01) 3.19 (0.01) 2.43 1.65 1.30 1.13 1.06 1.02 B = 3.335
k = 6.049 18.58 (0.05) 7.60 (0.02) 4.59 (0.01) 3.31 (0.01) 2.61 1.89 1.51 1.29 1.14 1.07 h = 7.532
CUSUM k = 6.534 20.67 (0.06) 8.01 (0.02) 4.55 (0.01) 3.14 (0.01) 2.42 1.72 1.39 1.20 1.09 1.04 h = 5.800
k = 6.999 22.14 (0.07) 8.46 (0.02) 4.59 (0.01) 3.06 (0.01) 2.31 1.60 1.29 1.13 1.05 1.02 h = 5.000
63
Table 3.4. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5.
a
4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.25
MR-SC 45.33 (0.13) 21.65 (0.05) 12.08 (0.03) 7.58 (0.02) 5.12 (0.01) 3.62 (0.01) 2.65 1.97 1.46 1.23 B = 4.4000
k = 4.746 32.57 (0.07) 16.88 (0.03) 11.06 (0.01) 8.16 (0.01) 6.49 (0.01) 5.42 4.65 4.03 3.46 3.17 h = 13.2000
CUSUM k = 4.345 35.41 (0.09) 16.75 (0.04) 10.06 (0.02) 6.97 (0.01) 5.31 (0.01) 4.28 3.59 3.07 2.65 2.40 h = 8.4500
k = 3.915 39.19 (0.11) 18.26 (0.05) 10.10 (0.02) 6.46 (0.01) 4.61 (0.01) 3.50 2.81 2.36 2.08 2.01 h = 5.7450
Table 3.5. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10 .
a
11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 22.00 24.00 26.00 28.00 30.00 MR-SC 27.69 (0.07) 12.14 (0.03) 7.06 (0.02) 4.81 (0.01) 3.58 (0.01) 2.84 2.36 2.01 1.77 1.59 1.34 1.19 1.10 1.05 1.02 B = 3.245 k = 11.509 28.49 (0.08) 12.10 (0.03) 7.01 (0.01) 4.86 (0.01) 3.73 (0.01) 3.05 2.58 2.25 2.00 1.81 1.52 1.33 1.19 1.11 1.05 h = 10.010 CUSUM k = 12.219 31.41 (0.09) 13.32 (0.04) 7.27 (0.02) 4.80 (0.01) 3.54 (0.01) 2.82 2.34 2.01 1.77 1.59 1.34 1.19 1.10 1.05 1.02 h = 7.600 k = 12.905 33.19 (0.10) 14.39 (0.04) 7.79 (0.02) 4.94 (0.01) 3.56 (0.01) 2.78 (0.01) 2.28 1.96 1.74 1.56 1.33 1.19 1.10 1.05 1.02 h = 6.150
64
Table 3.6. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10.
a
9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00
MR-SC 28.70 (0.08) 11.88 (0.03) 6.42 (0.01) 4.07 (0.01) 2.84 2.12 1.66 1.33 1.08 B = 3.600
k = 9.491 24.20 (0.05) 11.48 (0.02) 7.38 (0.01) 5.45 (0.01) 4.33 3.63 3.14 2.78 2.32 h = 16.830
CUSUM k = 8.690 27.00 (0.07) 10.97 (0.02) 6.22 (0.01) 4.27 (0.01) 3.25 2.66 2.27 2.05 2.00 h = 9.740
k = 7.830 31.71 (0.09) 12.53 (0.03) 6.35 (0.01) 3.95 (0.01) 2.86 2.25 1.89 1.60 1.26 h = 6.000
Table 3.7. ARL comparison for increasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20
a
21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00 32.00 34.00 36.00 38.00 40.00
MR-SC 39.47 (0.11) 19.43 (0.05) 11.52 (0.03) 7.78 (0.02) 5.67 (0.01) 4.41 (0.01) 3.58 (0.01) 2.98 (0.01) 2.55 2.22 1.78 1.51 1.32 1.20 1.12 B = 3.440
k = 22.161 38.75 (0.11) 18.60 (0.05) 10.94 (0.03) 7.37 (0.02) 5.52 (0.01) 4.39 (0.01) 3.65 (0.01) 3.14 (0.01) 2.75 2.46 2.03 1.74 1.53 1.37 1.25 h = 13.680
CUSUM k = 23.193 41.70 (0.13) 20.90 (0.06) 11.95 (0.03) 7.77 (0.02) 5.55 (0.01) 4.29 (0.01) 3.46 (0.01) 2.91 (0.01) 2.51 2.22 1.83 1.56 1.37 1.25 1.15 h = 10.420
k = 24.197 45.95 (0.14) 23.50 (0.07) 13.47 (0.04) 8.58 (0.02) 5.96 (0.01) 4.47 (0.01) 3.52 (0.01) 2.92 (0.01) 2.48 2.17 1.75 1.49 1.31 1.20 1.12 h = 7.900
65
Table 3.8. ARL comparison for decreasing rate case, = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20.
a
19.00 18.00 17.00 16.00 15.00 14.00 13.00 12.00 11.00 10.00 8.00 6.00
MR-SC 42.23 (0.12) 19.97 (0.05) 11.51 (0.05) 7.50 (0.03) 5.29 (0.01) 3.95 (0.01) 3.08 (0.01) 2.50 2.07 1.75 1.31 1.08 B = 3.7402
k = 18.982 34.32 (0.09) 16.81 (0.03) 10.47 (0.02) 7.49 (0.01) 5.83 (0.01) 4.78 4.05 3.54 3.14 2.83 2.35 2.06 h = 20.0700
CUSUM k = 17.380 40.87 (0.12) 19.47 (0.05) 10.86 (0.03) 6.91 (0.01) 4.91 (0.01) 3.79 3.08 2.61 2.29 2.04 1.71 1.39 h = 10.4700
k = 15.661 47.57 (0.15) 25.16 (0.08) 14.02 (0.04) 8.47 (0.02) 5.56 (0.01) 3.91 (0.01) 2.95 (0.01) 2.35 1.94 1.66 1.30 1.09 h = 6.3215
Table 3.9. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5.
a
6.00 7.00 8.00 9.00 10.00 12.00 14.00 16.00 18.00 20.00
MR-SC 18.18 (0.05) 7.43 (0.02) 4.34 (0.01) 3.00 (0.01) 2.31 1.60 1.28 1.12 1.05 1.02 B = 3.335
k = 6.049 17.69 (0.05) 7.02 (0.02) 4.17 (0.01) 3.00 (0.01) 2.38 1.74 1.41 1.22 1.11 1.05 h = 7.532
CUSUM k = 6.534 20.05 (0.06) 7.63 (0.02) 4.29 (0.01) 2.96 (0.01) 2.29 1.64 1.33 1.16 1.07 1.03 h = 5.800
k = 6.999 21.58 (0.07) 8.17 (0.02) 4.46 (0.01) 2.96 (0.01) 2.23 1.56 1.26 1.12 1.05 1.02 h = 5.000
66
Table 3.10. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 5.
a
4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.25
MR-SC 43.67 (0.12) 20.26 (0.05) 11.24 (0.03) 7.08 (0.01) 4.81 (0.01) 3.43 (0.01) 2.53 1.91 1.43 1.22 B = 4.4000
k = 4.746 27.09 (0.07) 13.38 (0.03) 8.54 (0.02) 6.27 (0.01) 4.94 (0.01) 4.11 (0.01) 3.54 3.10 2.74 2.58 h = 13.2000
CUSUM k = 4.345 32.56 (0.09) 14.71 (0.04) 8.56 (0.02) 5.87 (0.01) 4.43 (0.01) 3.58 3.01 2.60 2.27 2.09 h = 8.4500
k = 3.915 37.50 (0.11) 17.09 (0.05) 9.22 (0.02) 5.86 (0.01) 4.14 (0.01) 3.15 2.54 2.14 1.89 1.81 h = 5.7450
Table 3.11. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10.
a
11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 22.00 24.00 26.00 28.00 30.00
MR-SC 25.95 (0.07) 11.25 (0.03) 6.54 (0.01) 4.46 (0.01) 3.33 (0.01) 2.66 2.22 1.92 1.69 1.53 1.30 1.16 1.09 1.04 1.02 B = 3.245
k = 11.509 27.05 (0.08) 11.30 (0.03) 6.47 (0.01) 4.44 (0.01) 3.39 (0.01) 2.76 2.34 2.05 1.84 1.66 1.43 1.26 1.15 1.08 1.04 h = 10.010
CUSUM k = 12.219 30.70 (0.09) 12.76 (0.04) 6.95 (0.02) 4.57 (0.01) 3.35 (0.01) 2.65 2.22 1.91 1.69 1.53 1.30 1.17 1.09 1.04 1.02 h = 7.600
k = 12.905 32.88 (0.10) 14.09 (0.04) 7.55 (0.02) 4.80 (0.01) 3.42 (0.01) 2.65 (0.01) 2.19 1.89 1.67 1.51 1.29 1.16 1.08 1.04 1.02 h = 6.150
67
Table 3.12. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 10.
a
9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00
MR-SC 27.03 (0.07) 10.89 (0.03) 5.90 (0.01) 3.74 (0.01) 2.65 1.99 1.58 1.29 1.07 B = 3.600
k = 9.491 20.56 (0.05) 9.32 (0.02) 5.88 (0.01) 4.32 (0.01) 3.43 2.88 2.49 2.21 1.94 h = 16.830
CUSUM k = 8.690 25.29 (0.07) 9.97 (0.02) 5.50 (0.01) 3.73 (0.01) 2.84 2.33 1.99 1.80 1.69 h = 9.740
k = 7.830 30.56 (0.09) 11.90 (0.03) 5.93 (0.01) 3.67 (0.01) 2.63 2.07 1.73 1.47 1.20 h = 6.000
Table 3.13. ARL comparison for increasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20.
a
21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00 32.00 34.00 36.00 38.00 40.00 MR-SC 37.88 (0.11) 18.15 (0.05) 10.68 (0.03) 7.21 (0.02) 5.28 (0.01) 4.13 (0.01) 3.35 (0.01) 2.81 (0.01) 2.42 2.13 1.71 1.46 1.29 1.18 1.11 B = 3.440 k = 22.161 37.01 (0.11) 17.55 (0.05) 10.14 (0.03) 6.77 (0.02) 5.03 (0.01) 3.99 (0.01) 3.32 (0.01) 2.85 2.51 2.24 1.87 1.61 1.43 1.30 1.19 h = 13.680 CUSUM k = 23.193 41.09 (0.13) 20.33 (0.06) 11.55 (0.03) 7.43 (0.02) 5.27 (0.01) 4.05 (0.01) 3.27 (0.01) 2.75 (0.01) 2.37 2.10 1.73 1.49 1.32 1.21 1.13 h = 10.420 k = 24.197 45.61 (0.14) 23.11 (0.07) 13.27 (0.04) 8.38 (0.02) 5.78 (0.01) 4.33 (0.01) 3.40 (0.01) 2.81 (0.01) 2.40 2.08 1.69 1.45 1.29 1.18 1.11 h = 7.900
68
Table 3.14. ARL comparison for decreasing rate case, = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20.
a
19.00 18.00 17.00 16.00 15.00 14.00 13.00 12.00 11.00 10.00 8.00 6.00
MR-SC 40.26 (0.12) 18.65 (0.05) 10.57 (0.02) 6.87 (0.01) 4.88 (0.01) 3.68 (0.01) 2.90 (0.01) 2.36 1.97 1.67 1.28 1.08 B = 3.7402
k = 18.982 30.91 (0.09) 14.49 (0.04) 8.83 (0.02) 6.22 (0.01) 4.81 (0.01) 3.93 (0.01) 3.34 2.91 2.60 2.35 1.99 1.77 h = 20.0700
CUSUM k = 17.380 39.94 (0.12) 18.73 (0.05) 10.19 (0.03) 6.43 (0.01) 4.53 (0.01) 3.47 (0.01) 2.81 2.37 2.07 1.85 1.54 1.28 h = 10.4700
k = 15.661 47.57 (0.15) 24.88 (0.08) 13.83 (0.04) 8.26 (0.02) 5.38 (0.01) 3.78 (0.01) 2.85 (0.01) 2.27 1.88 1.61 1.26 1.07 h = 6.3215
The control chart that yields the best ARL performance depends upon the magnitude of the change. Again, although the magnitude-robust chart does not have the best ARL performance for any specific value of a , it has nearly the best ARL performance for all values of a . Thus, unless the magnitude of the change is known a priori, we conclude that the magnitude-robust chart has better ARL performance over the entire range of a values.
3.6. Implementation Issues and Illustration
The magnitude-robust chart can be implemented in a similar manner to most standard control charts for count data.
corresponding maximums of all R , ( ) | x a
(
That is, the RT statistics, which are the
) values over 0 < T , can be plotted
on charts against corresponding control limits B . In general, determining RT requires T
69
calculations of R (equation 3.12), which increases the amount of computation over most standard control charts. However, a simple program can be written using a common programming language or a spreadsheet to perform the calculations. Computing RT for relatively large values of T presents no problem even on a modest personal computer.
Values of RT that exceed B are worthy of special cause investigation. When RT > B , estimates for the change point and magnitude of change (both point estimates
and confidence sets) are provided to aid the search for the special cause. Since point
estimates of the change point and magnitude of process change are arguments of the RT
statistics, they are available immediately. Another few lines of code (or cells in a spreadsheet) can be added to calculate the confidence set for the process change point. The example C codes in Figure 3-1 demonstrate the ease in computing the magnitude robust control chart statistics RT as well as diagnostic statistics for any given T.
The magnitude-robust chart will be illustrated using simulated data generated from known distributions. Initially, the data are drawn from a Poisson process with an in-control rate parameter of = 5. Starting with subgroup 51, the data are drawn from a Poisson process with a higher rate of = 7 . As a result, the example dataset is in-control for the first 50 subgroups and experiences an increase of 2 in the mean count rate beginning with subgroup 51. Based on an average run length of about 8 for the magnitude-robust control chart with B = 3.335, we would expect the chart to signal outof-control somewhere around subgroup 58, depending on the actual dataset used. Two realizations of the above-described data were generated and the resulting
+ control chart statistic RT was plotted against the upper control limit B+ (see Figure 3-2).
Both control charts signal but the time of signal varies. That is, for the first realization, the magnitude-robust control chart signaled at subgroup T = 57, while, for the second realization, a signal was issued at subgroup T = 63. For comparison, the same data was charted using a CUSUM with similar in-control properties. Specifically, a CUSUM with reference value of k = 6.534 (designed for a specified increase in the mean count rate of 1.5 standard deviation units) and decision interval of h = 5.80 is used (see Figure 3-3). The time of signal was the same as the magnitude-robust chart for the first realization. For the second realization, however, the CUSUM chart did not signal.
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Once the magnitude-robust control chart signals, diagnostics are immediately available regarding the change point and magnitude of change. Figure 3-4 shows diagnostic plots from the first realization dataset, where the magnitude-robust chart signals at subgroup T = 57. Figure 3-4(a) shows computed values for R at each potential change point t. The largest value for R indicates the most likely change point . A confidence set on the most likely change points can also be developed. Figure 3-4(b) shows the out-of-control rate parameter estimates, including those corresponding to the two most likely change points (identified by the arrows). To design the magnitude-robust control chart, one must select an upper control limit B to obtain a desired in-control ARL of ARL0. For Poisson data, this ARL is dependent upon the value of the in-control rate parameter as well as B. Simulation runs of the magnitude robust control chart were made with = 0 and N = 10,000 for incontrol rate parameter values 0.50 0 50 and control limit values 3.0 B 5.5. Applying ordinary least squares (OLS) to those simulation results yields the ARL approximation plots in Figure 3-5 for increasing and decreasing rate cases separately. These plots can aid the user in selecting a control limit that produces a desirable incontrol ARL for a given in-control rate parameter value. For example, Figure 3-5(a) shows an estimated in-control ARL of about 185 when B = 4 and 0 = 5. If desired, the user can choose to fine-tune the control chart design via simulation. Figure 3-6 provides pseudo-code for fine-tuning the chart design and is easily implemented using programming languages like C or MATLAB.
3.7. Discussion
The magnitude-robust control chart offers some significant advantages over existing control charts. In addition to having better overall ARL performance, the magnitude-robust chart also provides valuable information to process engineers concerning the time of the change and the magnitude of the change. Although the CUSUM chart also provides a change point estimator (but not a confidence set), Perry,
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Pignatiello and Simpson (2004) showed that it is inferior to the MLE that the magnitude-robust control chart provides. Identifying which combination of the process variables is responsible for a change in a process allows engineers to improve quality by preventing or avoiding changes in those variables which lead to poor quality and by perpetuating those changes and optimizing those variables which can lead to better quality. Knowing when a process has changed and by how much can aid in the search for the special cause. If the time of the change could be determined, process engineers would have a smaller search window within which to look for the special cause. Consequently, the special cause can be identified more quickly and the appropriate actions can be implemented sooner to improve quality.
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/* Compute MR-SC control chart statistic for increasing rate case */ RT+ = -999999.99; for (t = 0; t < T; t++) { sumX =0.0e0; for (i = T; i > t; i--) { sumX += X[i]; } lambda_a = sumX/(T-t); if(lambda_a < lambda_0) { lambda_a = lambda_0; } R = (T-t)*(lambda_0 + lambda_a*(log(lambda_a/lambda_0)-1)); if(R > RT+) { RT+ = R; tau_hat = t; newrate = lambda_a; }
}
/* Compute MR-SC control chart statistic for decreasing rate case */ RT- = -999999.99; for (t = 0; t < T; t++) { sumX =0.0e0; for (i = T; i > t; i--) { sumX += X[i]; } lambda_a = sumX/(T-t); if(lambda_a > lambda_0) { lambda_a = lambda_0; } R = (T-t)*(lambda_0 + lambda_a*(log(lambda_a/lambda_0)-1)); if(R > RT-) { RT- = R; tau_hat = t; newrate = lambda_a; }
}
Figure 3-1. Example C codes for computing the magnitude-robust control chart statistics and diagnostics
73
Data Realization 1 4
3.5
B = 3.335
B = 3.335
3
2.5
RT
RT
2
1.5
1
0.5
0
5
10
15
20
25 30 35 Sample No. Index
40
45
50
55
60
Data Realization 2 4
3.5
B = 3.335
3
2.5
RT
2
1.5
1
0.5
0
10
20
30 40 Subgroup No. Index
50
60
70
Figure 3-2. Two realizations of the magnitude-robust control charts for data experiencing a shift in from 5 to 7 following the 50th sample count obtained. Realization 1 signals at subgroup T = 57, while realization 2 signals at subgroup T = 63.
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Data Realization 1 7
6
h = 5.80
h = 5.80
5
Cumulative Sum
CT
4
3
2
1
0
0
10
20
30 Sample No. Index
40
50
60
Data Realization 2 7
6
h = 5.80
5
CT
4
3
2
1
0
10
20
30 40 Sample No. Index
50
60
70
Figure 3-3. CUSUM control charts with h = 5.800 and k = 6.534 for the same two realizations of data experiencing a shift in from 5 to 7 following the 50th sample count obtained. Realization 1 signals at subgroup T = 57, while realization 2 did not produce a signal.
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Data Realization 1 3.5
3
2.5
R
R
2
1.5
1
0.5
0
0
10
20
30 t
40
50
60
Figure 3 - 4(a)
Data Realization 1 11
10
a
a
Estimated Mean Count Rate
9
8
7
6
5
0
10
20
30 Sample No. Index
40
50
60
Figure 3 - 4(b)
Figure 3-4. Diagnostic plots for data realization 1. Figure 3-4(a): R versus potential change point. Large values correspond to most likely change points. Figure 3-4(b): Estimated out-of-control rate parameter versus subgroup number.
76
In-control ARL Function Approximations using Ordinary Least Squares
Magnitude-Robust Control Chart for Increasing Step-Changes in Poisson Rate Parameter
750 700 650 600 550 500 450 ARL 400 350 300 250 200 150 100 50 0 0 10 20 30 40 50 In-control Rate Parameter Value B = 3.000 B = 4.000 B = 4.500 B = 5.000 B = 5.250 B = 5.500
Figure 3-5(a)
In-control ARL Function Approximations using Ordinary Least Squares
Magnitude-Robust Control Chart for Decreasing Step-Changes in Poisson Rate Parameter
600 550 500 450 400 350 ARL 300 250 200 150 100 50 0 0 10 20 30 40 50 In-control Rate Parameter Value B = 3.000 B = 4.750 B = 5.250 B = 5.500
B = 4.250
Figure 3-5(b)
Figure 3-5. In-control ARL function approximations for a range of B and 0 using ordinary least squares. 3-5a: MR-SC for increases. 3-5b: MR-SC for decreases.
77
% Simulate MR-SC control chart to obtain in-control ARL for n=1:N i=0; in_control = 1; while in_control == 1 i=i+1; generate X(i); compute RT(i); if RT(i)>=B runlength(n)= i; ave_runlength(n)= mean(runlength); in_control = 0; clear X RT end end end Y = ave_runlength(n)
Figure 3-6. Pseudo-code for estimating in-control ARLs for the magnitude-robust control chart. Inputs into the program include B and N, the control limit and total number of simulation runs, respectively. The value of RT(i) is computed as given in equation (3.13).
78
CHAPTER 4 ESTIMATING THE CHANGE POINT OF A POISSON RATE PARAMETER WITH A LINEAR TREND DISTURBANCE IN SPC APPLICATIONS
4.1. Introduction
Control charts are used to monitor for changes in a process by distinguishing between special causes and common causes of variation. Once a change is detected, process engineers can begin their search for the special cause disturbing the process. If process engineers knew the time at which the disturbance first manifested itself into the data, valuable time could be saved in the search to find the cause of the disturbance. Upon signaling, control charts do not provide specific information regarding the cause of process change nor when the process changed; rather, they only suggest that a change has occurred. Even further, most control-charting procedures and corresponding diagnostic tools are designed primarily for step changes in the parameters of interest. Although step changes are one potential change type, linear trends can also exist. For example, step changes can occur as a result of tool breakage, while, linear trends can occur as a result of tool wear. In this paper we propose an estimator for the change point of a Poisson rate parameter where the type of change is a linear trend. For Poisson processes, much of the literature on change point estimation is directed towards estimating the change point when the assumed change type is a step change. Samuel and Pignatiello (1998) suggested a maximum likelihood estimator (MLE) for the change point and derived their estimator under step change assumptions. They demonstrated the use of their estimator on a Shewhart c-chart when a step change is present and concluded that their estimator provided good overall performance. The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts also offer change point estimators as suggested by Page (1954) and Nishina (1992), respectively. Perry, Pignatiello and Simpson (2004a) show the estimator suggested by Samuel and Pignatiello (1998) is in superior step change disturbance situations when compared to the estimators offered by the CUSUM and EWMA
79
procedures. In this paper, we derive a maximum likelihood estimator for the point of process change using the change likelihood function for a linear trend. We use simulation to compare performances of this estimator and the estimator suggested by Samuel and Pignatiello (1998) when a linear trend disturbance is present and following a signal from a Poisson CUSUM control chart. In the next section, we consider a model for a linear trend change in the rate parameter of a Poisson process. A linear trend occurs when the rate begins to change linearly from its in-control value over time. From this model, a maximum likelihood estimator for the change point is derived.
4.2. Poisson Process Linear Trend Change Model and Derivation of the MLE
Consider a linear trend change model for the behavior of a Poisson process rate parameter . The model assumes that the process is initially in-control with independent observations coming from a Poisson distribution with known rate 0 . After an unknown point in time the rate parameter changes from its in-control state of = 0 to an unknown out-of-control state where = i for i = + 1, K , where, i > 0 and the functional form of i is given as
i = 0 + (i )
where is the magnitude (or slope) of the linear trend disturbance.
(4.1)
The Poisson process linear trend change model can be parameterized as follows. Each observation consists of a count from a subgroup formed from the output of the process. During the formulation of subgroups i = 1, 2, K, , the process rate i is equal to its known in-control value 0 . For subgroups i = + 1, K , T , the process rate i is equal to some unknown rate i = 0 + (i ) where T is the most recent subgroup sample. The model assumes two unknowns in and , representing the last subgroup taken from the in-control process and the slope parameter of the linear trend, respectively.
80
This model can be used to derive a maximum likelihood estimator for the process change point. We will denote the MLE of the proposed change point estimator as . Assuming a process change point at , the likelihood function is
X L( , | x ) = exp( 0 ) 0 i X i ! exp( ( 0 + (i )))( 0 + (i )) i =1 i = +1
T
Xi
X i ! , (4.2)
where X i is the count corresponding to the ith subgroup. The MLE of is the value of
that maximizes the likelihood in (4.2), or equivalently, its logarithm. Taking the
logarithm of that in (4.2) and reducing algebraically yields log e L( , | x ) =
K
T , 1 (T )(T + 1 ) + log e (0 ) X i + X i log e (0 + (i )) 2 i =1 i = +1
(4.3)
where K is a constant. Since the slope parameter is unknown, an expression is
required for in terms of that provides the maximum in (4.3), or . For any given
value of , finding an expression for is not trivial. The partial derivative of (4.3) with respect to is given by
T X i (i ) log e L( , | x ) 1 , = (T )(T + 1 ) + 2 i = +1 0 + (i )
(4.4)
where there is no closed-form solution for . As such, we adopt Newtons method to solve for in (4.4) at each potential change point value. This provides an estimate of for each without requiring an explicit closed-form expression. Substituting such an
estimate, , for in (4.3) and evaluating (4.3) over all possible change points in
search of the maximum yields = arg max t (T t )(T + 1 t ) + log e ( 0 ) X i +
0t <T i =1
1 2
t
i = t +1
X
T
i
log e 0 + t (i t ) , (4.5)
(
)
where is the maximum likelihood estimate for the last subgroup number obtained from the in-control process. The MLE of the change point can be applied when any Poisson count control chart gives an out-of-control signal, including the CUSUM, EWMA and Shewhart ccharts. Below, we demonstrate the application of the estimator in (4.5) using the CUSUM control charting procedure for Poisson counts. The next section provides a brief 81
outline of Newtons method and demonstrates how this procedure is used to obtain the MLE of .
4.3. Newtons Method for Finding Maximum Likelihood Estimate of Slope Parameter
Newtons method is a derivative-based root finding algorithm that uses the linear approximation
f ( x + x ) f ( x ) + d f ( x )x , dx
(4.6)
where, x = x k +1 x k . If we set f (x + x ) to zero and rearrange we obtain
x k +1 = x k f (x k ) . f ' (x k )
(4.7)
Therefore, given an initial point x0 and an appropriate stopping scheme, the procedure in (4.7) should converge at the root of f . The value of x0 should be chosen smartly, that is, somewhere near the root of the function. The reader is referred to Rardin (2000) for more detail regarding Newtons method. If was known, Newtons method could be used to solve for in (4.4) . That
is, at the k + 1st iteration can be written explicitly as
T X i (i ) 1 (T )(T + 1 ) + 2 i = +1 0 + ,k (i ) T X i (i )2 2 i = +1 0 + ,k (i )
,k +1 = ,k
(
)
(
)
,
(4.8)
where , 0 = 0. Notice that the denominator in (4.8) is always negative, which is a necessary condition for (4.3) to have a maximum. Even further, since the Poisson rate parameter is strictly greater than zero, must be strictly greater than 0
( i )
for
any given i. The procedure in (4.8) will work well for the increasing rate case since has no upper bound. For decreases, however, Newtons method will fail since is no longer unconstrained. Furthermore, for decreasing rates, the linear decreasing trend 82
would eventually produce negative rates which are impossible. Thus, only increasing trends are considered here. Consequently, estimates for can be obtained at each potential change point value given the observations x' = [ X 1 , X 2 , K , X T ] using the procedure defined in (4.8). The procedure is then repeated T times, once for each potential value of . The stopping scheme used for (4.8) should terminate the procedure if ,k ,k 1 is sufficiently small. For example, one could terminate the procedure if the squared difference between the last two iterations is less than, say, 1.0e-7. Using the estimator in (4.8) then allows for the evaluation of the argument in (4.5). Some example C code implementing Newtons Method is illustrated in Figure 4-1 and shows the ease in programming such a procedure. In the next section, an overview of the CUSUM control chart for Poisson counts is provided. In subsequent sections, issues pertaining to the simulation modeling of the estimator in (4.5) are addressed and performance comparisons are made between the estimator and the step change model estimator suggested by Samuel and Pignatiello (1998) when a linear trend disturbance is present. Finally, confidence sets on the point of process change are discussed and their performances evaluated.
4.4. Poisson CUSUM Control Chart
Although developed by Page (1954) for normal process means, a CUSUM chart to monitor Poisson count data was suggested by Brook and Evans (1972). This procedure cumulates the difference between an observed value X i and a reference value
k . If this sum exceeds a decision interval h, the chart signals a disturbance is present.
The CUSUM control statistic for detecting increases in the mean count rate is given by
S i+ = max{0, X i k + + S i+1 }, where k + = ( + 0 ) /(ln( + ) ln( 0 )) and S 0+ = 0. The a a
value for a is the out-of-control process rate for which to design the CUSUM. If
S i+ exceeds a specified decision interval h + , then the chart signals that an increase in the
mean count rate has occurred. Lucas (1985) provides a comprehensive study on Poisson CUSUM control charts. Hawkins and Olwell (1998) provide extensive detail pertaining 83
to the theoretical foundation and construction of CUSUM control charts in general, including the Poisson CUSUM.
4.5. Comparison of Change Point Estimators
In this section, we use Monte Carlo simulation to make performance comparisons
between the estimator derived for linear trends, , and the estimator suggested by Samuel
and Pignatiello (1998) when a linear trend disturbance is present and following a signal from a Poisson CUSUM control chart. Since the estimator proposed by Samuel and Pignatiello (1998) is derived under step change assumptions, we will refer to it as SC . Unlike the normal distribution case, since there is no standardized Poisson distribution, we investigate in-control rate parameter values of 0 = 5, 10 and 20. Results for other values of 0 , although not reported here, provide similar results.
/* Newton's Method to compute MLE of slope
for (t=0; t<T; t++) { k = 1; beta[k] = 0; gx = 1.0e0; E = 1.0e-7; while ((beta[k]-gx)*(beta[k]-gx) >= E) { Q=0; P=0; for (q = t+1; q<=T; q++) { P = P + ((X[q]*(q-t)) / ((lambda0 + beta[k]*(q-t)))); Q = Q + ((X[q]*(q-t)*(q-t)) / ((lambda0 + beta[k]*(q-t))*(lambda0 + beta[k]*(q-t)))); } gx = beta[k]; fx[k] = t*T + 0.5*t - 0.5*(T*T) - 0.5*T - 0.5*(t*t) + P; fpx[k] = (-1)*(Q); beta[k+1] = beta[k] - (fx[k]/fpx[k]); k += 1;
}
MLE_beta[t+1] = beta[k]; }
Figure 4-1. C code implementing Newtons method for estimating the slope parameter at each potential change point t.
84
4.5.1 False Alarms
When > 0 and a control chart issues a signal at subgroup T where T , the signal is a false alarm since the signal was given before the simulated process change could occur. When a false alarm was encountered in a simulation run, it was treated the same way that a false alarm would be treated on an actual process. Namely, if one determines that a signal is indeed a false alarm then one is affirming that the process is currently in-control and could restart the monitoring of the process. Thus, when a false alarm was encountered at subgroup T, the control chart was restarted at subgroup T + 1 while not altering the scheduled change point. For example, if the change point was = 100 and a false alarm was issued in a simulation run at subgroup 75, then the appropriate statistics would be zeroed out and that simulation run would continue as if subgroup number 76 was the first one from an in-control process. Thus, if there were no more false alarms on this particular simulation run, there would have been 25 subgroups observed from the restarted in-control process when the first subgroup from the simulated changed process is observed. With this approach the number of subgroups since the chart was started or restarted that have been observed from the in-control process at the time of the process change will not necessarily be fixed at but will instead be random and less than or equal to . This is the same approach used by Pignatiello and Samuel (2001).
4.5.2 Accuracy Performances of Change Point Estimators
In this section, we use simulation to compare accuracy of and SC following a
signal from a Poisson CUSUM control chart. All CUSUM control charts were calibrated such that their in-control average run lengths were approximately 150. The process change point was simulated to occur at = 100. Independent observations were simulated from a Poisson process with rate parameter 0 for subgroups i = 1, 2, , 100. Following subgroup 100, observations were simulated from a Poisson process with rate parameter i = 0 + (i 100) , where > 0, until the CUSUM chart produced a signal. The two estimates of the process change point, and SC , were then computed. This procedure was repeated a total of N = 10,000 times for each value investigated.
85
Averages of the change point estimates obtained from the 10,000 runs, and SC , were
computed along with their corresponding estimated standard errors. We first investigate the 0 = 5 case using a CUSUM chart designed to quickly detect an increase of 3 in the mean count rate. That is, a = 8. Table 4.1 shows accuracy performances for the two estimators, as well as the estimated ARL of the CUSUM
procedure, over a range of values. Results show that the estimator outperforms the
SC estimator for all values of considered since is closer to = 100 than is SC . The SC estimator tends to overestimate the true change point value, even for larger values of the slope parameter . We note that as the magnitude of the slope parameter increases, both estimators improve in accuracy. Similar results are obtained using incontrol rate parameter values of 0 = 10 and 20 and are given in Tables 4.2 and 4.3,
respectively. Thus, we conclude from Tables 4.1-4.3 that the proposed estimator
outperforms the SC estimator when a linear trend disturbance is present.
4.5.3 Precision Performances of Change Point Estimators
In this section, we use simulation to study the precision of the proposed estimator following a signal from a CUSUM control chart. simulated to occur at = 100. The process change point was Independent observations were generated from a
simulated Poisson process with rate parameter 0 for subgroups i = 1, 2, , 100. Following subgroup 100, observations were generated from a simulated Poisson process with rate parameter i = 0 + (i 100) , where > 0, until the CUSUM chart
produced a signal. The estimates of the process change point, and SC , were then computed. To evaluate and compare the precision of and SC , we repeated this procedure for a total of N = 10,000 independent simulation runs and recorded (for each estimator) the proportion of the N = 10,000 simulation runs that the estimator was within a specified tolerance of the simulated change point value. Doing this for a range of
86
Table 4.1. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 7.0; k = 6.38). Rounded standard errors are shown in parentheses. 0 = 5, = 100 and N = 10,000 independently-seeded runs.
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00
AR L 25.8 13.8 10.2 8.4 7.2 6.5 5.9 5.4 5.1 4.7 4.6 3.6 3.1 2.7 2.5
113.5 (0.16) 104.9 (0.10) 102.7 (0.09) 101.8 (0.07) 101.1 (0.07) 100.9 (0.06) 100.6 (0.06) 100.4 (0.05) 100.3 (0.05) 100.1 (0.05) 100.0 (0.05) 99.8 (0.04) 99.8 (0.03) 99.7 (0.03) 99.6 (0.03)
SC
117.6 (0.14) 108.0 (0.09) 105.2 (0.08) 104.0 (0.07) 103.2 (0.07) 102.8 (0.06) 102.3 (0.06) 102.0 (0.05) 101.8 (0.05) 101.6 (0.05) 101.6 (0.04) 101.1 (0.04) 100.8 (0.03) 100.7 (0.03) 100.6 (0.03)
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Table 4.2. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 9.80; k = 11.89). Rounded standard errors are shown in parentheses. 0 = 10, = 100 and N = 10,000 independently-seeded runs.
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00
AR L 31.5 11.0 8.7 10.3 8.9 7.8 7.2 6.6 6.1 5.7 5.6 4.4 3.7 3.3 3.0
117.2 (0.19) 105.7 (0.12) 103.6 (0.11) 102.4 (0.09) 101.8 (0.08) 101.2 (0.08) 101.0 (0.07) 100.7 (0.07) 100.4 (0.06) 100.3 (0.06) 100.2 (0.06) 99.9 (0.05) 99.8 (0.05) 99.8 (0.04) 99.8 (0.03)
SC
121.9 (0.17) 109.9 (0.10) 106.8 (0.09) 105.1 (0.08) 104.1 (0.07) 103.5 (0.07) 103.0 (0.06) 102.6 (0.06) 102.4 (0.06) 102.2 (0.06) 102.1 (0.06) 101.4 (0.05) 101.2 (0.04) 101.0 (0.04) 100.8 (0.04)
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Table 4.3. Accuracy performances for two different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 14.4; k = 22.41). Rounded standard errors are shown in parentheses. 0 = 20, = 100 and N = 10,000 independently-seeded runs. SC ARL 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00 36.5 20.0 14.9 12.3 10.5 9.4 8.6 7.9 7.3 6.9 6.7 5.3 4.5 3.9 3.6 121.9 (0.22) 108.7 (0.13) 105.2 (0.11) 103.5 (0.10) 102.5 (0.09) 101.9 (0.09) 101.5 (0.08) 101.2 (0.07) 100.8 (0.07) 100.7 (0.07) 100.7 (0.07) 100.2 (0.06) 99.9 (0.05) 99.9 (0.04) 99.7 (0.04) 126.3 (0.20) 112.4 (0.12) 108.7 (0.10) 106.6 (0.09) 105.4 (0.08) 104.6 (0.07) 104.0 (0.07) 103.5 (0.07) 103.3 (0.06) 103.0 (0.06) 102.8 (0.06) 102.0 (0.05) 101.5 (0.05) 101.3 (0.04) 101.0 (0.04)
89
yields the results shown in Tables 4.4, 4.5 and 4.6 for in-control rate parameter values of
0 = 5, 10 and 20, respectively.
Results indicate the proposed estimator provides better precision performance when compared to SC . That is, yields less variability about . For example, Table
4.4 shows the estimated probability that is within 3 of the true change point is 0.733
when = 0.75. The SC estimator, however, only yields an estimated probability of 0.605 in this case. Similar results are obtained using in-control rate parameter values of
0 = 10 and 20. We note that for any given value of , the precision obtained from
either estimator depends upon the value of 0 (i.e. better precision is obtained at smaller values of 0 ). This is the case for both estimators. Based on these results, we conclude that outperforms SC with regard to precision when a linear trend disturbance is present.
4.6. Confidence Sets Based on the Change Likelihood Function for the Change Point of a Poisson Rate Parameter
We now consider constructing confidence sets on the process change point. Such a set would provide a window of possible change points that covers the true process change point with a given level of confidence. This is useful because it provides process engineers a set of change point candidates from which to begin their search for the special cause. Such information should greatly enhance special cause identification as well as accelerate efforts to improve quality. Box and Cox (1964) suggest constructing confidence sets on parameter estimates using the likelihood function. Applying such a method yields a confidence set of the form
CS = {t : log e L(t ) > log e L( ) D} ,
(4.16)
where log e L( ) is the maximum of the log likelihood function evaluated over all possible change points t. If the value of the log likelihood function at t, log e L(t ) , exceeds the maximum of the log likelihood function, less a reference value D, then t is included in the confidence set. 90
Table 4.4. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 7.0; k = 6.38). Precision estimates for SC are shown in parentheses. 0 = 5 and N = 10,000 independently-seeded runs.
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00
P ( = 0) 0.022 (0.012) 0.039 (0.021) 0.054 (0.025) 0.075 (0.036) 0.096 (0.049) 0.107 (0.051) 0.122 (0.058) 0.146 (0.070) 0.145 (0.080) 0.165 (0.087) 0.179 (0.082) 0.234 (0.122) 0.313 (0.157) 0.357 (0.188) 0.413 (0.214)
P ( 1) 0.058 (0.036) 0.117 (0.063) 0.162 (0.088) 0.217 (0.120) 0.264 (0.146) 0.294 (0.170) 0.334 (0.199) 0.375 (0.227) 0.400 (0.262) 0.431 (0.285) 0.453 (0.291) 0.583 (0.419) 0.677 (0.539) 0.747 (0.632) 0.801 (0.709)
P( 2) 0.091 (0.056) 0.186 (0.105) 0.271 (0.159) 0.351 (0.214) 0.411 (0.271) 0.461 (0.313) 0.515 (0.375) 0.579 (0.420) 0.605 (0.466) 0.646 (0.511) 0.662 (0.532) 0.790 (0.708) 0.860 (0.822) 0.904 (0.887) 0.920 (0.928)
P( 3) 0.124 (0.074) 0.261 (0.154) 0.371 (0.232) 0.471 (0.321) 0.550 (0.401) 0.604 (0.464) 0.672 (0.546) 0.733 (0.605) 0.760 (0.666) 0.799 (0.713) 0.803 (0.728) 0.899 (0.886) 0.930 (0.949) 0.945 (0.968) 0.951 (0.973)
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Table 4.5. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 9.80; k = 11.89). Precision estimates for SC are shown in parentheses. 0 = 10 and N = 10,000 independently-seeded runs.
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00
P ( = 0) 0.015 (0.009) 0.031 (0.018) 0.045 (0.023) 0.057 (0.030) 0.068 (0.035) 0.080 (0.040) 0.085 (0.046) 0.105 (0.048) 0.114 (0.055) 0.128 (0.061) 0.138 (0.060) 0.189 (0.087) 0.236 (0.110) 0.282 (0.139) 0.329 (0.150)
P ( 1) 0.045 (0.030) 0.087 (0.051) 0.129 (0.074) 0.169 (0.095) 0.200 (0.107) 0.237 (0.132) 0.261 (0.144) 0.297 (0.166) 0.331 (0.187) 0.353 (0.206) 0.370 (0.211) 0.489 (0.306) 0.582 (0.401) 0.650 (0.486) 0.714 (0.561)
P( 2) 0.071 (0.047) 0.140 (0.086) 0.214 (0.126) 0.276 (0.162) 0.325 (0.194) 0.384 (0.235) 0.424 (0.267) 0.460 (0.303) 0.518 (0.354) 0.548 (0.378) 0.561 (0.393) 0.702 (0.557) 0.794 (0.687) 0.845 (0.782) 0.884 (0.852)
P( 3) 0.096 (0.065) 0.198 (0.126) 0.288 (0.181) 0.373 (0.239) 0.442 (0.293) 0.515 (0.355) 0.562 (0.405) 0.608 (0.453) 0.659 (0.522) 0.700 (0.561) 0.713 (0.579) 0.838 (0.769) 0.900 (0.874) 0.921 (0.933) 0.936 (0.959)
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Table 4.6. Estimated precision performances over a range of values for and SC following a genuine signal from a Poisson CUSUM control chart (h = 14.4; k = 22.41). Precision estimates for SC are shown in parentheses. 0 = 20 and N = 10,000 independently-seeded runs.
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.00 1.50 2.00 2.50 3.00
P ( = 0) 0.013 (0.011) 0.027 (0.018) 0.034 (0.018) 0.049 (0.024) 0.058 (0.029) 0.061 (0.033) 0.078 (0.036) 0.083 (0.037) 0.095 (0.045) 0.104 (0.050) 0.104 (0.050) 0.148 (0.073) 0.187 (0.082) 0.226 (0.102) 0.259 (0.109)
P ( 1) 0.036 (0.029) 0.077 (0.046) 0.103 (0.056) 0.137 (0.073) 0.162 (0.086) 0.186 (0.098) 0.218 (0.115) 0.240 (0.131) 0.267 (0.147) 0.285 (0.159) 0.294 (0.165) 0.395 (0.237) 0.479 (0.296) 0.554 (0.371) 0.612 (0.417)
P( 2) 0.058 (0.045) 0.123 (0.072) 0.170 (0.098) 0.223 (0.128) 0.263 (0.150) 0.305 (0.174) 0.343 (0.207) 0.382 (0.237) 0.419 (0.266) 0.450 (0.286) 0.465 (0.302) 0.600 (0.440) 0.694 (0.545) 0.764 (0.645) 0.819 (0.724)
P( 3) 0.078 (0.060) 0.169 (0.103) 0.233 (0.142) 0.311 (0.188) 0.362 (0.224) 0.416 (0.264) 0.466 (0.308) 0.512 (0.358) 0.561 (0.400) 0.595 (0.438) 0.611 (0.451) 0.756 (0.633) 0.836 (0.757) 0.878 (0.847) 0.912 (0.903)
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To illustrate the use of the confidence set estimator in (4.16), we consider a Poisson count process with in-control rate parameter 0 = 5. Following the 50th subgroup obtained, a linear trend disturbance with slope parameter = 0.75 (as described by the function in (4.1)) is introduced to the process. Subgroups are formulated from these distributions until the control chart issues a genuine signal at subgroup T. Figure 4-2 plots log e L(t ) versus t obtained from a single realization of the above described data set. The plot in Figure 4-2 indicates the control chart used in this example signaled at subgroup T = 57. Applying the confidence set estimator in (4.16) yields a confidence set cardinality of 18 using a reference value of D = 1.75. That is, there are 18 values of t that yield log likelihood values greater than log e L( ) 1.75 . We note that in this example, the true change point of = 50 is included in this set. Perry, Pignatiello and Simpson (2004a) evaluated the performance of the confidence set estimator in (4.16) derived under step change assumptions following signals from various Poisson control charting procedures. Their study, however, does not consider how the estimator in (4.16) derived for step changes performs under the presence of a linear trend. Therefore, given that a linear trend disturbance is present, we compare the estimator in (4.16) when derived under step change assumptions to one that is derived under linear trend assumptions following a signal from a Poisson CUSUM control chart.
4.6.1 Cardinality and Coverage Performances of Confidence Set Estimators
In this section, we use simulation to evaluate two forms of the confidence set estimator in (4.16) when a linear trend is present and following a signal from a Poisson CUSUM control chart. Specifically, we investigate an estimator derived for step changes, and another derived for linear trends. Set cardinality and coverage measures are used to evaluate the performance of the confidence set estimators and are obtained using critical values of D between 1.50 and 3.00. For each magnitude of change studied, the change was simulated to occur following the 100th subgroup collected. The confidence set estimators were applied after a genuine signal from the control chart. The cardinality
94
Log Likelihoods Versus t 5
4.5
4
3.5
3 L(t) 2.5
2
1.5
1 0.5
0
T = 57
0 10 20 30 t (potential change point) 40 50 60
Figure 4-2. Plot of log likelihood values versus possible change points t for a single realization. Those values of t yielding log likelihood values that plot above log e L( ) - D (i.e. dotted horizontal line in plot) are considered potential change point candidates and are included in the confidence set.
of the confidence set was recorded as well as whether the confidence set covered the true process change point of = 100 . This procedure was repeated for a total of N = 10,000 independent simulation runs for each value of considered. The average cardinality was computed over the 10,000 runs, as well as the proportion of the 10,000 runs that included the true change point in the estimated set. Figure 4-3 provides a surface plot for the 0 = 5 case showing estimated relationships between cardinality, coverage, and D for the confidence set estimator derived under linear trend change assumptions. Figure 4-4 shows the surface obtained from the confidence set estimator derived for step changes (dashed surface) superimposed on the surface shown in Figure 4-3. The results in Figures 4.3 and 4.4 show that more coverage can be obtained at any given value of D using the confidence set estimator derived for linear trends. In general, 95
the confidence set estimator derived for step changes will yield confidence sets of smaller set cardinality and less coverage than the confidence set estimator derived for linear trends. Similar results are obtained for in-control rate parameter values of 0 = 10 and 20, and are shown in Figures 4-5 and 4-6, respectively. The axes for and D in Figures 4.5 and 4.6 are the same scale as is given in Figure 4-3. Figures 4-4 through 4-6 indicate that at least 80% coverage can be obtained using the confidence set estimator derived for linear trends and a reference value of D = 3.0 for all values of considered in this study. Furthermore, for any given value of D, the confidence set estimator derived for linear trends will provide more coverage than that offered by the estimator derived for step changes. This advantage comes at the expense of a slight increase in the cardinality of the set. Thus, if a linear trend is present and step change model methods are used, process engineers may be misled as to when the process change occurred.
1 0.95
0.9
3.00
0.85
3.00 1.00 2.75 2.50 2.25 0.15
0.8 Coverage
0.75 0.7
0.35 0.25
0.65
2.00
D
1.75
0.05
0.6
.
0.55
1.50
0.5
0
10
20
30
40 50 Cardinality
60
70
80
90
Figure 4-3. Surface plot obtained from confidence set estimator for linear trends showing estimated relationships between set cardinality, coverage, slope parameter and reference value D. 0 = 5, N = 10,000, = 100.
96
1
0.9
0.8
0.7 Coverage
0.6
0.5
D
0.4
0.3
0
10
20
30
40 Cardinality
50
60
70
80
Figure 4-4. Surface plot obtained from confidence set estimator for step changes (dashed-lines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 5, N = 10,000, = 100.
1
0.9
0.8
0.7 Coverage
0.6
0.5
D
0.4
0.3
0.2
0
10
20
30
40 50 Cardinality
60
70
80
90
Figure 4-5. Surface plot obtained from confidence set estimator for step changes (dashed-lines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 10, N = 10,000, = 100.
97
1
0.9
0.8
0.7 Coverage
0.6
0.5
0.4
D
0.3
0.2
0
10
20
30
40
50 Cardinality
60
70
80
90
100
Figure 4-6. Surface plot obtained from confidence set estimator for step changes (dottedlines) superimposed on the surface obtained from confidence set estimator for linear trends. 0 = 20, N = 10,000, = 100.
4.7. Summary
If process engineers could determine when the process changed, their search could be narrowed to finding which aspect of the process (e.g., which process variable) changed at that time. This could allow them to identify the special cause more quickly, and to use that information to improve the quality of the process or product sooner. Also, knowing the time of the process change would help to minimize identification of the wrong process variables as the special cause and reduce the costs of unnecessary experiments. Consequently, being able to estimate the time of a process change would be useful to process engineers. A point estimate as well as a confidence set on the process change point would provide process engineers with useful starting points in their search for a special cause following a control chart signal. In this paper a maximum likelihood estimator for the time of linear trend change in a Poisson rate parameter was derived and evaluated. We showed that Newtons method can be used to estimate the slope parameter at each potential change point
98
value, allowing for an estimate of the change point to be obtained. We consider only the increasing rate case due to the constrained nature of in the decreasing direction. We also showed that the proposed estimator provides good overall accuracy and precision performances when a linear trend disturbance is present. We compared these results with those obtained using the MLE derived for step changes and found that the step change estimator tends to overestimate the true change point value. Confidence sets on the change point were also investigated. We showed that the confidence set estimator derived for linear trends will provide better coverage when a linear trend is present than that offered by the confidence set estimator derived for step changes for any given value of D. This added coverage results in a slight increase in set cardinality. Finally, we found that a reference value of D = 3.0 will provide at least 80% coverage for all slope parameter magnitudes and in-control rate parameter values investigated in this study.
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CHAPTER 5 ESTIMATING THE CHANGE POINT OF A POISSON RATE PARAMETER WITH AN ISOTONIC CHANGE DISTURBANCE IN SPC APPLICATION
5.1. Introduction
Poisson count processes, in general, are used to model the number of occurrences over some interval unit. The interval unit can be time, distance, area, volume or some similar unit. Often in an industrial quality control setting, the Poisson distribution is used to model the number of defects or nonconformities per unit of product. That is, the probability that a randomly selected unit of product contains x nonconformities is given by
P( X = x ) =
x
x!
e
for x 0
where > 0 denotes the mean rate of nonconformities. (SPC) chart.
Furthermore, the on-line
monitoring of is typically accomplished through the use of a statistical process control In general, SPC charts are used to detect changes in a process by distinguishing between special causes and common causes of process variation. For monitoring Poisson rates, the c-, CUSUM and EWMA control charting procedures are most commonly used (see Ryan (2000) for more detail on these procedures). Once a control chart detects a change, process engineers can begin their search for the special cause disturbing the process. If process engineers knew the time at which the change first manifested itself into the data, then valuable time could be saved in the search to find the root cause of process change. Upon signaling, the CUSUM and EWMA control-charting procedures can provide estimates of the process change point as described by Page (1961) and Nishina (1992), respectively. Even further, Samuel and Pignatiello (1998) proposed a maximum likelihood estimator (MLE) for the process change point using the step change likelihood function for a Poisson random variable. They evaluated its performance following a cchart signal and conclude good overall accuracy and precision performance. Perry, 100
Pignatiello and Simpson (2004a) compared the estimator suggested by Samuel and Pignatiello (1998) to those suggested by Page (1961) and Nishina (1992) following signals from CUSUM and EWMA control charts, respectively. Considering only step changes, they concluded that the estimator suggested by Samuel and Pignatiello (1998) outperforms those offered by the CUSUM and EWMA procedures across a wide range of change magnitudes. Although a step change is certainly one possible change-type, a process may just as likely experience, say, a linear trend. Such a change type can occur as a result of, say, tool wear. Thus, Perry, Pignatiello and Simpson (2004b) derived the MLE for the time of linear trend change in a Poisson rate parameter. They used simulation to study and compare performances between their estimator and the estimator suggested by Samuel and Pignatiello (1998) following a CUSUM control chart signal. They concluded that the MLE of the change point derived from a linear trend change model outperforms the MLE of the change point derived from a step change model when a linear trend is present. They note, however, that due to the nature of their methods, their procedure is not valid for decreasing . The change point estimators suggested by Samuel and Pignatiello (1998) and Perry, Pignatiello and Simpson (2004b) are derived under strict change-type assumptions (e.g., step changes and linear trends). Unfortunately, in practice, rarely is the type of change known apriori, and thus, any deviations in the true change-type from the given change-type is likely to affect the performance of the estimator. Therefore, in this paper we investigate a change point estimator derived from the change likelihood function for a Poisson random variable, however, without assuming prior knowledge of the exact change-type. The only assumption is that the anticipated change-type is one from a family of isotonic change-types. We then compare performances between our estimator and those suggested by Samuel and Pignatiello (1998) and Perry, Pignatiello and Simpson (2004b) following a genuine CUSUM control chart signal. We do this for a range of potential change-types and change magnitudes. In the next section we consider a model for the behavior of a Poisson rate parameter . The model assumes that the type of behavior can be described by one of a
101
countably infinite number of non-decreasing change-types. From this model, a maximum likelihood estimator for the time of isotonic change is derived.
5.2. Poisson Process Behavior Model and Derivation of the MLE
Consider a model for the behavior of a Poisson process rate parameter where the change in can be described as belonging to a family of isotonic change-types. The set of isotonic changes is the set of all possible change types that are non-decreasing in nature. We note that this set includes the step and linear trend change-types, as well as, a countably infinite number of other non-decreasing change-types. The model assumes that the process is operating in a state of statistical control for the first subgroups with
= 0 . Following an unknown subgroup , the process changes to an unknown out of
control state such that the behavior in can be described by +1 > 0 and i i 1 for
i = + 2, K , T , where T denotes the most recent subgroup.
This model can be used to derive the MLE of , assuming the type of change is non-decreasing. We will denote the MLE of the proposed change point estimator as . Assuming a process change point at , the likelihood function is given by
L( , T | x T , 0 ) =
i =1
x 0 e
i
0
xi !
i = +1
T
ix e
i
i
xi !
(5.1)
where x T denotes the given observation vector, 0 denotes the known in-control rate parameter value and T denotes an unknown time series of out-of-control rate parameter values. The MLE of is the value of that maximizes the likelihood in (5.1), or equivalently, its logarithm. Taking the natural log of L and reducing we obtain log e L( , T | x T , 0 ) = K 0 + log e (0 ) xi +
i =1
i = +1
(x
T
i
log e (i ) i ) , (5.2)
where K is a constant. Since and T are unknown, they must be estimated from x . This involves finding estimates for and T that provide the maximum in (5.2), or the
102
MLEs. Assuming the MLE of T can be expressed in terms of and x (denoted as
T ), then the MLE of the change point can be found by replacing T in (5.2) with T and evaluating (5.2) over the set of all possible integer change points t in search of
its maximum, or
= arg max t 0 + log e ( 0 ) xi +
0 t <T
t
i =1
i = t +1
(x
T
i
log e i i
( )
)
(5.3)
From (5.3) we see that in order to find , we must first find T t for each t.
Recall from our process behavior model that the type of change is only known to be nondecreasing. Thus, for any given t, we could obtain an initial estimate for T t (denoted as
~ T t ) given by
i = i 0
~
x
if x i > 0 if x i 0
for i = t + 1, K , T
,
~ where 0 is the known in-control rate parameter value. Now, for any given T t , we can obtain T t as the solution to the following convex program
T ~ ~ maximize x i log e i i i = t +1
[
( ) ]
(5.4)
subject to
i i 1
~
~
for i = t + 2, K , T
where the objective function is taken directly from the log likelihood function in (5.2) ~ ~ and the decision variables are taken to be T t . The initial values of the elements in T t provide a starting point in the solution to (4.4).
We can solve for T t in (5.4) using a generalized reduced gradient algorithm;
however, Robertson, Wright and Dykstra (1988, p. 34) provide a theorem that suggests ~ T t is simply found by fitting an isotonic regression function to the random vector T t . That is, they show that since the Poisson mass function belongs to the exponential family of densities, then the MLE of T t obtained under simple order restriction is given by the
103
~ ~ isotonic regression of T t . Therefore, for any given t and T t , the MLE of T t can be
obtained from
~ T t = I T t ~ where I returns the isotonic regression of T t .
(
)
(5.5)
Robertson et al (1988) further suggest some computation algorithms for performing the isotonic regression, the simplest of these being the pooled-adjacency violators (PAV) algorithm (see Ayer et al (1955) and Best et al (1990)). For details on the computational complexity of the PAV algorithm, see Grotzinger and Witzgall (1984). In the next section we use Monte Carlo simulation to evaluate the performance of the estimator in (5.3) relative to the procedures suggested by Samuel and Pignatiello (1998) and Perry, Pignatiello and Simpson (2004b) for a variety of isotonic change-types following signals from a CUSUM control chart. We use the active set formulation of the
PAV algorithm as described by Best and Chakravarti (1990) to obtain T t for a given
0t <T
.
5.3. Poisson CUSUM Control Chart
Although developed by Page (1961) for normal process means, a CUSUM chart to monitor Poisson count data was suggested by Brook and Evans (1972). This procedure cumulates the difference between an observed value X i and a reference value
k . If this sum exceeds a decision interval h, the chart signals a disturbance is present.
The CUSUM control statistic for detecting increases in the mean count rate is given by S i+ = max{0, X i k + + S i+1 }, where k + = ( + 0 ) /(ln( + ) ln( 0 )) and S 0+ = 0. The a a value for a is the out-of-control process rate for which to design the CUSUM. If S i+ exceeds a specified decision interval h + , then the chart signals that an increase in the mean count rate has occurred. Lucas (1985) provides a comprehensive study on Poisson CUSUM control charts. Hawkins and Olwell (1998) provide extensive detail pertaining
104
to the theoretical foundation and construction of CUSUM control charts in general, including the Poisson CUSUM.
5.4. Comparison of Change Point Estimators
In this section, we use Monte Carlo simulation to study the accuracy and precision
performance (i.e. bias and variability about , respectively) of . Specifically, we compare performances between and the change point estimators suggested by Samuel
and Pignatiello (1998) and Perry et al (2004b), denoted as sc and lt , respectively, following a signal from a Poisson CUSUM control chart. We investigate three distinct non-decreasing change-types, each of which are illustrated graphically in Figure 5-1. These include (1) a single point step change, (2) a linear trend and (3) multiple point step changes. Each of these change-types is described in further detail in subsequent sections.
5.4.1 False Alarms
This section addresses the handling of false alarms in the simulation model. When > 0 and a control chart issues a signal at subgroup T where T , the signal is a false alarm since the signal was given before the simulated process change could occur. When a false alarm was encountered in a simulation run, it was treated the same way that a false alarm would be treated on an actual process. Namely, if one determines that a signal is indeed a false alarm then one is affirming that the process is currently in-control and could restart the monitoring of the process. Thus, when a false alarm was encountered at subgroup T, the control chart was restarted at subgroup T + 1 while not altering the scheduled change point. This is the same approach used by Samuel and Pignatiello (1998).
105
Step Change
Linear Trend
0
1 2
...
0
Subgroup
+1
...
T
1
2
...
Subgroup
+1
...
T
Multiple Step Changes
0
Subgroup
1
2
...
+1
...
T
Figure 5-1. Non-decreasing change-types investigated.
5.4.2 Change Point Estimators with a Single Point Step Change Disturbance We first consider the single point step change case where the process behavior can be described by a sudden and persistent shift in the mean count rate, . This type of change can occur when a single influential input factor suddenly changes (or, multiple input factors change at the same time) causing a shift in the output of the process. For example, a change in material may cause an increase in the number of nonconformities found on a machined part. The process change point was simulated to occur at = 25 Independent observations were simulated from a Poisson process with rate parameter 0 for subgroups i = 1, 2, , 25. Following subgroup 25, observations were simulated from a Poisson process with rate parameter a , where a 0 , until the CUSUM chart
produced a signal. The three estimates of the process change point, , sc and lt , were
then computed. This procedure was repeated a total of N = 100,000 times for each a value investigated. Averages of the change point estimates obtained from the 100,000
runs (i.e., , sc and lt ) were computed along with their corresponding estimated
standard errors.
106
We consider the 0 = 20 case using a one-sided CUSUM chart designed to quickly detect a 25% increase in 0 with an in-control average run length (ARL) of approximately 132. Such a CUSUM procedure yields a reference value and decision interval of k = 22.41 and h = 13.96, respectively. Table 5.1 shows accuracy performances for the three estimators, as well as the estimated ARL (i.e., E [ T ] ) of the CUSUM procedure, over a range of a values. In general, the results provided in Table 5.1 show the proposed estimator underestimates the true process change point when a single point step change occurs. Furthermore, the sc estimator provides the best overall relative performance across the range of step change magnitudes investigated. The lt estimator also performs relatively well, but also seems to slightly underestimate the true process change point as the magnitude of step change gets large.
Table 5.1. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a single-point step change disturbance is present. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. 0 = 20, = 25 and N = 100,000 independently-seeded runs.
a
22 24 26 28 30 35 40
E[ T 20.6 7.4 4.2 3.0 2.3 1.6 1.2
]
29.7 (0.05) 22.0 (0.02) 21.1 (0.02) 21.0 (0.02) 21.0 (0.02) 21.0 (0.02) 21.0 (0.02)
sc
36.7 (0.05) 27.5 (0.02) 25.8 (0.01) 25.4 (0.01) 25.2 (0.01) 25.1 25.1
lt
32.8 (0.06) 24.6 (0.02) 23.8 (0.01) 23.9 (0.01) 24.1 (0.01) 24.4 24.7
107
Although a point estimate of may be better obtained by using the procedures suggested by Samuel and Pignatiello (1998) and Perry et al. (2004b) in this case, these methods assume that the behavior of the rate parameter is well modeled by a known function (e.g., single step or linear trend-type functions). Rarely is the case, however, that the behavior of a random process is known apriori. Thus, one advantage to using the proposed change point estimation procedure is that, along with a point estimate for ,
T
provides the maximum likelihood estimate for the unknown process profile
following . Since T is a random vector, we use Monte Carlo simulation to study the average performance of T with regard to how well it estimates the true underlying
process behavior. The simulation model is described below. The process change point was simulated to occur at = 25. Independent observations were simulated from a Poisson process with rate parameter 0 = 20 for subgroups i = 1, 2, , 25. Following subgroup 25, observations were simulated from a Poisson process with rate parameter a , where a 0 , until the CUSUM chart
produced a signal. The estimates, and T , were then computed by the equations
given in (5.3) and (5.5), respectively. We used these estimates to compute the overall profile of the process, or T , whose total elements are given by
1 i , i = 0 i < i T
where T denotes the length of T (or the time of signal), denotes the estimated change point value, and 1 i T denotes the subgroup index. This procedure was repeated for a total of N = 500 independent simulation runs for each value of a considered. The average of the profile estimates was then obtained from the 500 simulation runs, denoted
as , where T is the estimated average number of runs from the start of the process
T
until the CUSUM chart produces a signal. Figure 5-2 shows estimated mean profiles for values of a = 22, 23, 24 and 25 superimposed on the same plot. The abscissa of the plot is the subgroup index, while the
108
ordinate axis is the estimated value of . We see that, on average, T does well in estimating the true process behavior when a single-point step change disturbance is present. Such a diagnostic tool is quite powerful since it can provide process engineers with good insight into the correlation between time and the quality of the process. Finally, we use Monte Carlo simulation to investigate the precision obtained using the proposed change point estimator . That is, we study the variability of about its target, . We then compute precision estimates obtained using the sc and lt estimators and evaluate the performances of all three estimators relative to each other. performances. The process change point was simulated to occur at = 25. Independent observations were generated from a simulated Poisson process with rate parameter The following explains the simulation model used in obtaining the estimated precision
0 = 20 for subgroups i = 1, 2, , 25. Following subgroup 25, observations were
generated from a simulated Poisson process with rate parameter a , where a 0 ,
until the CUSUM chart produced a signal. The estimates of the process change point, ,
sc and lt were then computed. To evaluate and compare the precision of , sc and
lt , we repeated this procedure for a total of N = 100,000 independent simulation runs
and recorded (for each estimator) the proportion of the N = 100,000 simulation runs that the estimator was within a specified tolerance of the simulated change point value. Doing this for a range of a yields the results shown in Table 5.2. The results provided in Table 5.2 indicate that sc provides better relative precision across the range of a values considered. This is not surprising since the sc estimator is derived for the single-point step change scenario. The estimator provides as good or better precision (relative to the other procedures) for smaller a , however, as the magnitude of the step change gets large, the sc and lt estimators begin to dominate. We note, however, that the precision provided by does not worsen as a increases, rather, as a , the precision appears to approach a limiting value noticeably less than those offered by sc and lt . For example, using the proposed estimator, the estimated
109
limiting value of the probability, P ( 3 ) , is 0.64 as a , while, the estimated limiting value for the same probability using the sc estimator is nearly 1. We find in this section that if a step change disturbance is present, the sc estimator will provide the better point estimate for . This is an expected result since sc is derived specifically for this change-type. As noted earlier, rarely is the type of change known apriori. Thus, in these situations, the proposed estimator should be considered. We find that, on average, the estimator does fair in providing a point estimate for in this case. That is, it performs well, considering it is not derived specifically for step changes. More importantly, however, the procedure used in computing offers the maximum likelihood estimate of the unknown process profile, of which, provides a fairly accurate description of the process behavior. This knowledge could be critical for process engineers.
Mean Process Profiles 25
a = 25
24.5 24
a = 24 a = 23
23.5
mean count rate
23
22.5
22
a = 22
21.5
21
20.5
= 25
20 5 10 15 20 25 30 subgroup index 35 40 45 50
Figure 5-2. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a single-point step change disturbance is introduced following the 25th subgroup. Results are provided for values of a = 22, 23, 24 and 25. N = 500 independent runs were used to obtain the estimated mean vectors.
110
5.4.3 Change Point Estimators with a Linear Trend Change Disturbance
We now consider the linear trend change case where the process behavior can be described by a constant drift in the mean count rate, . This type of change can occur when, say, a single influential input factor is changing according to some constant rate
. For example, tool wear is a likely cause of this type of disturbance. In this section,
we use Monte Carlo simulation to study the accuracy performance of relative to sc and lt when a linear trend change disturbance is present. The process change point was simulated to occur at = 25. Independent observations were simulated from a Poisson process with rate parameter 0 = 20 for subgroups i = 1, 2, , 25. Following subgroup 25, observations were simulated from a Poisson process with rate parameter i = 20 + (i 25) , where > 0, until the
CUSUM chart produced a signal at time T. At this point, , sc and lt , were then computed. We repeated the procedure for a total of N = 100,000 independent simulation
runs and recorded the average values (i.e., , sc and lt ) obtained over the 100,000 runs.
Results are shown in Table 5.3. and suggest that, if a linear trend is present, the proposed
estimator will provide the better point estimate for across the range of considered.
The lt estimator yields the next best performance, followed by sc . We also study the accuracy performance of the profile estimate T obtained when a linear trend disturbance is present for a range of . Figure 5-3 provides estimated mean profiles using the proposed procedure (obtained from N = 500 independent simulation runs) for values of = 0.10, 0.20, 0.30, 0.40 and 0.50 . Results suggest that the proposed estimation procedure will, on average, provide a good estimate of the true process behavior when the type of disturbance present is a linear trend. Finally, Table 5.4 shows precision estimates for the three MLEs when the changetype present is a linear trend. Results indicate that the proposed provides the better relative precision across the range of values considered. Thus, we conclude that
111
provides the best overall performance (relative to sc and lt ) when a linear trend disturbance is present.
Table 5.2. Estimated precision performances over a range of a values for , sc and lt following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in parenthesis. 0 = 20 and N = 100,000 independently-seeded runs.
a
22 24 26 28 30 35 40
P ( 0)
P ( 1)
P ( 2)
P ( 3)
0.09 [0.06] (0.07) 0.20 [0.16] (0.18) 0.27 [0.27] (0.30) 0.30 [0.37] (0.41) 0.32 [0.47] (0.51) 0.33 [0.68] (0.69) 0.33 [0.83] (0.80)
0.21 [0.15] (0.16) 0.39 [0.38] (0.38) 0.45 [0.60] (0.56) 0.47 [0.75] (0.69) 0.47 [0.85] (0.78) 0.47 [0.95] (0.89) 0.47 [0.98] (0.94)
0.30 [0.23] (0.24) 0.52 [0.51] (0.51) 0.56 [0.73] (0.70) 0.57 [0.86] (0.82) 0.57 [0.93] (0.88) 0.57 [0.98] (0.95) 0.57 [0.99] (0.97)
0.38 [0.28] (0.30) 0.61 [0.61] (0.61) 0.64 [0.82] (0.79) 0.64 [0.92] (0.88) 0.64 [0.96] (0.92) 0.64 [0.99] (0.97) 0.64 [0.99] (0.99)
112
Table 5.3. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a linear trend disturbance is present. Rounded standard errors are shown in parentheses. 0 = 20, = 25 and N = 100,000 independently-seeded runs.
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
E[ T
24.3 16.4 13.1 11.1 9.8 8.8 8.0 7.4 6.9 6.5
]
34.7 (0.04) 28.8 (0.03) 26.5 (0.02) 25.3 (0.02) 24.5 (0.02) 23.4 (0.02) 23.5 (0.02) 23.2 (0.02) 22.9 (0.02) 22.7 (0.02)
sc
41.9 (0.04) 35.6 (0.03) 32.9 (0.02) 31.4 (0.02) 30.4 (0.02) 29.6 (0.01) 29.1 (0.01) 28.7 (0.01) 28.4 (0.01) 28.1 (0.01)
lt
38.1 (0.04) 32.3 (0.03) 29.9 (0.02) 28.6 (0.02) 27.8 (0.02) 27.2 (0.02) 26.8 (0.02) 26.5 (0.02) 26.3 (0.02) 26.0 (0.01)
5.4.4 Change Point Estimators with a Multiple Step Change Disturbance
In this section, we study the relative performances of , sc and lt when
multiple step changes are introduced to the process.
The multiple step changes
considered in this section are of an isotonic nature. This type of process behavior could likely be caused by several influential input factors changing, however, at different times. For example, consider a the manufacture of a machined part whose quality is dependent upon, say, the raw material supplier and the skill of the machinist. A multiple step change pattern could potentially develop if, say, a change in supplier occurred at 1 and a change in machinist occurred at 2 . If the new suppliers raw material resulted in a
113
sudden increase in the mean count of nonconformities, and the new machinist is a less skilled operator than his predecessor (causing additional nonconformities), then a multiple step change of an isotonic nature could potentially develop.
Mean Process Profiles 26
= 0.50 = 0.40
25
= 0.30
24 mean count rate
= 0.20
23
= 0.10
22
21
20
= 25
5 10 15 20 25 30 subgroup index 35 40 45 50
Figure 5-3. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a linear trend change disturbance is introduced following the 25th subgroup. Results are provided for values of = 0.01, 0.02, 0.03, 0.04 and 0.05. N = 500 independent runs were used to obtain the estimated mean vectors.
In this section, we use Monte Carlo simulation to study the accuracy performance of relative to sc and lt when a multiple step change disturbance is present. We assume two change points in our simulation model, 1 and 2 . Thus, independent observations were simulated from a Poisson process with rate parameter 0 = 20 for subgroups i = 1, 2, , 1 . For subgroups i = 1 + 1 , , 2 , observations were simulated
114
Table 5.4. Estimated precision performances over a range of values for , sc and lt following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in parenthesis. 0 = 20 and N = 100,000 independently-seeded runs.
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
P ( 0)
P ( 1)
P ( 2 )
P ( 3)
0.02 [0.01] (0.02) 0.04 [0.02] (0.03) 0.05 [0.02] (0.04) 0.07 [0.03] (0.05) 0.08 [0.03] (0.06) 0.09 [0.04] (0.07) 0.10 [0.04] (0.08) 0.11 [0.04] (0.09) 0.11 [0.05] (0.10) 0.12 [0.05] (0.11)
0.07 [0.04] (0.06) 0.12 [0.05] (0.09) 0.16 [0.07] (0.12) 0.20 [0.08] (0.15) 0.23 [0.10] (0.18) 0.25 [0.11] (0.21) 0.28 [0.13] (0.23) 0.29 [0.14] (0.25) 0.31 [0.16] (0.27) 0.33 [0.17] (0.30)
0.12 [0.06] (0.09) 0.20 [0.09] (0.15) 0.26 [0.12] (0.20) 0.32 [0.15] (0.25) 0.37 [0.17] (0.29) 0.40 [0.20] (0.33) 0.43 [0.23] (0.36) 0.46 [0.25] (0.40) 0.48 [0.28] (0.43) 0.50 [0.31] (0.47)
0.16 [0.09] (0.13) 0.28 [0.13] (0.21) 0.36 [0.17] (0.27) 0.43 [0.21] (0.34) 0.49 [0.26] (0.39) 0.53 [0.30] (0.45) 0.57 [0.34] (0.49) 0.59 [0.38] (0.54) 0.61 [0.42] (0.58) 0.63 [0.46] (0.61)
from a Poisson process with rate parameter i = 1 , where 1 > 0 . Following subgroup
2 , observations were generated from a Poisson process with rate parameter i = 2 ,
115
where 2 > 1 until the CUSUM chart produced a signal at time T. At this point in the simulation run, , sc and lt , are computed, each of which estimate 1 (or the last subgroup from the in-control process). We repeated this procedure for a total of
N = 100,000 independent simulation runs and recorded the average values (i.e.,
, sc and lt ) obtained over the 100,000 runs. The values of 1 and 2 were simulated at
25 and 35, respectively. Accuracy results are shown in Table 5.5 and indicate that the proposed estimator provides the better point estimate for this change-type scenario. We also study the accuracy performance of the profile estimate T obtained when a multiple step change disturbance is present for several isotonic combinations of 1 and 2 . Figure 5-4 provides estimated mean profiles using the proposed procedure (obtained from
N = 500 independent simulation runs) for several
(1 , 2 )
combinations and with
1 = 25 and 2 = 35. Results suggest that the proposed estimation procedure will, on
average, provide a good estimate of the true process behavior when the type of disturbance present is a multiple step change. Since the values of 1 , 2 , 1 and 2 would not be fixed in practice, we need to evaluate the performances of the estimation procedures, , sc and lt , while considering the randomness associated with these parameters. To do this, we study the robustness of the estimation procedures (with regard to changes in the relative locations of 1 and 2 , as well as, changes in 1 and 2 ) using a 2 3 full factorial designed experiment in two replicates. The factors and their input levels considered in the experiments are given in Table 5.6, while the detailed statistical findings from the analysis are included in the Appendix. The responses analyzed include , sc and lt . Results of the designed experiment indicate that, in general, for the sc and lt estimators, there are two significant 2-factor interaction effects. The first of these involves ( 2 1 ) and 1 , while the second involves 1 and 2 . This suggests that the robustness of these procedures are affected by ( 2 1 ) , 1 and 2 . For the proposed estimator, however, the only significant 2-factor interaction effect involves ( 2 1 ) and
116
1 , suggesting that the robustness of is only affected by changes in these parameters.
Table 5.7 quickly summarizes the analysis results of the designed experiment. Specifically, it shows the estimated direction of influence for each factor and response considered. These results suggest that by implementing , process engineers can be assured that the robustness of is not significantly affected by any lack of prior knowledge of 2 .
Table 5.5. Accuracy performances for three different MLEs of the change point following a genuine signal from a Poisson CUSUM control chart (h = 13.96; k = 22.41) when a multiple step change disturbance is present. Rounded standard errors are shown in parentheses. 0 = 20, 1 = 25, 2 = 35 and N = 100,000 independently-seeded runs.
1 , 2
21, 22 21, 23 21, 24 21, 25 21, 26 21, 27 21, 28 21, 29 21, 30 21, 35
E[ T
25.6 18.3 15.1 13.5 12.6 12.0 11.6 11.3 11.1 10.4
]
34.2 (0.05) 29.2 (0.03) 27.9 (0.03) 27.5 (0.02) 27.3 (0.02) 27.2 (0.02) 27.2 (0.02) 27.1 (0.02) 27.1 (0.02) 27.1 (0.02)
sc
41.4 (0.05) 36.1 (0.03) 34.4 (0.02) 33.7 (0.02) 33.4 (0.02) 33.3 (0.02) 33.2 (0.02) 33.2 (0.01) 33.3 (0.02) 33.4 (0.01)
lt
37.2 (0.06) 32.3 (0.04) 31.1 (0.03) 30.7 (0.02) 31.0 (0.02) 31.1 (0.02) 31.3 (0.02) 31.5 (0.02) 31.7 (0.02) 32.3 (0.02)
Finally, Table 5.8 shows precision estimates for the three MLEs when the changetype present is a multiple step change. Results indicate that the proposed provides the better relative precision across the combinations of 1 , 2 values considered. Thus, we conclude that provides the best overall performance (relative to sc and lt ) when a multiple step change disturbance is present. 117 We also studied multiple step change
scenarios consisting of more than two change points, and although not shown here, similar results were obtained.
5.5. Summary and Discussion
The results provided in this paper suggest that unless the type of change is known apriori, the proposed estimator should be considered since the only required assumption of is that the anticipated change-type be a non-decreasing function. Specifically, we showed that performs well across the change-types considered in this paper, and outperforms sc and lt when a linear trend or multiple step change disturbance is present. The advantages to implementing in place of the estimators suggested by Samuel and Pignatiello (1998) and Perry, Pignatiello and Simpson (2004b) are significant. First, the performance of is more robust (or insensitive) to changes in the type of isotonic change that may be present. This is important for process engineers since rarely is the type of change known apriori. Second, an estimate of the profile of the process could provide process engineers with insight into the true behavior of the process over time. Such a diagnostic report contains information pertaining to the times of change, magnitudes of change and the correlation between these quantities. This could help process engineers to more quickly and precisely pin-point any special causes that may be affecting the process. Third, process engineers can be assured that the robustness of is not significantly affected by the magnitudes of the changes that occur beyond the initial point of change. We found that this was not the case when using the procedures suggested by Samuel and Pignatiello (1998) and Perry et al (2004b). The procedure outlined in this paper could just as easily be used with the set of all non-increasing functions (or the antitonic set). The only difference lies in how we estimate T . That is, if the process behavior can be described as non-increasing, then
i = i 0
~
x
if x i < 0 for i = t + 1, K , T if x i 0
provides a starting point in the solution to the following convex program 118
maximize
i = +1
[x
T
i
~ ~ log e i i
( ) ]
(5.6)
subject to
i i 1
~
~
for i = + 2, K , T
where the difference here lies in the direction of the order constraints. In this case, the antitonic regression provides the solution to (5.6). That is, the estimate of T for the non-increasing case is given by
~ T = A T
(
)
~ where the function A returns the antitonic regression of T . We studied the relative
performances of and sc across non-increasing single and multiple step change types, and although not reported here, the results obtained are similar to those obtained for the non-decreasing case. In this paper, results are reported for a single value of the in-control rate parameter 0 and performance is demonstrated when applied to a Poisson CUSUM control chart. We note that, although not shown here, we conducted additional simulation studies and found that the same relative performances are obtained regardless of the value of , 0 or the type of control chart used. Thus, we conclude that the proposed estimator is a good alternative to sc and lt when the true behavior of the process is only known to be monotonic.
119
Mean Process Profiles 25
1 , 2 = 21,26 1 , 2 = 21,25
24.5
24 23.5 estimated count rate
1 , 2 = 21,24
23 22.5
1 , 2 = 21,23
22
1 , 2 = 21,22
21.5 21
20.5
1 = 25
20 0 10 20
2 = 35
40 50 60
30 subgroup index
Figure 5-4. Plots of estimated mean process profiles obtained from the proposed change point estimation procedure when a multiple step change disturbance is introduced following the 25th subgroup obtained. 1 = 25, 2 = 35 , N = 500 independent runs were used to obtain the estimated mean vectors.
Table 5.6.
Input factors and levels considered in the 2 3 experiment. Input Levels Factor high (+) Low (-) A: ( 2 1 ) 20 10 B: 1 21 22 C: 2 23 24
Table 5.7. Direction of influence on estimated change point for each factor considered. An empty space in the table indicates no significant influence. Influence Factors sc lt + A: ( 2 1 ) + + B: 1 C: 2 AB: ( 2 1 ) 1 + AC: 1 2 +
120
Table 5.8. Estimated precision performances over a range of 1 , 2 combinations for , sc and lt following a genuine signal from a Poisson CUSUM control chart ( h = 13.96 ; k = 22.41 ). Precision estimates for sc are shown in brackets; precision estimates for lt are shown in parenthesis. 0 = 20 , 1 = 25, 2 = 35 and N = 100,000 independently-seeded runs.
1 , 2
21, 22 21, 23 21, 24 21, 25 21, 26 21, 27 21, 28 21, 29 21, 30 21, 35
P ( 0)
P ( 1)
P ( 2 )
P ( 3)
0.05 [0.03] (0.03) 0.05 [0.03] (0.04) 0.05 [0.03] (0.04) 0.06 [0.03] (0.04) 0.06 [0.03] (0.03) 0.06 [0.02] (0.03) 0.06 [0.02] (0.03) 0.06 [0.02] (0.03) 0.06 [0.02] (0.03) 0.06 [0.02] (0.02)
0.12 [0.08] (0.09) 0.13 [0.08] (0.10) 0.14 [0.07] (0.10) 0.15 [0.07] (0.10) 0.15 [0.07] (0.09) 0.15 [0.07] (0.09) 0.15 [0.07] (0.08) 0.15 [0.06] (0.08) 0.15 [0.06] (0.07) 0.15 [0.06] (0.06)
0.18 [0.11] (0.14) 0.21 [0.12] (0.15) 0.22 [0.11] (0.15) 0.23 [0.11] (0.15) 0.23 [0.11] (0.14) 0.23 [0.10] (0.13) 0.23 [0.10] (0.12) 0.23 [0.10] (0.12) 0.23 [0.09] (0.11) 0.23 [0.09] (0.09)
0.24 [0.15] (0.18) 0.27 [0.15] (0.20) 0.29 [0.15] (0.20) 0.30 [0.15] (0.20) 0.30 [0.14] (0.18) 0.30 [0.13] (0.17) 0.30 [0.13] (0.16) 0.30 [0.13] (0.16) 0.30 [0.12] (0.15) 0.30 [0.11] (0.12)
121
CHAPTER 6 CONTROL CHARTS FOR MONITORING AND ESTIMATING MONOTONIC CHANGES IN A POISSON RATE PARAMETER
6.1. Introduction
Poisson count processes, in general, are used to model the number of occurrences over some interval unit. The interval unit can be time, distance, area, volume or some similar unit. Often in an industrial quality control setting, the Poisson distribution is used to model the number of defects or nonconformities per unit of product. That is, the probability that a randomly selected unit of product contains x nonconformities is given by P( X = x ) =
x
x!
e
for x 0
where > 0 denotes the mean rate of nonconformities. The on-line monitoring of is typically accomplished through the use of a statistical process control (SPC) chart. In general, SPC charts are used to detect changes in a process by distinguishing between special causes and common causes of process variation. It is desirable to detect the presence of special cause variation quickly so that appropriate action can be taken to return the process to a state of statistical control. The sooner a process change is detected by a control chart; the sooner process engineers can initiate their search for the special cause affecting the process. This paper focuses on the statistical monitoring of count data processes, particularly, those that can be modeled by the Poisson distribution. That is, we consider count processes where X i , the count at subgroup i, is assumed to follow a Poisson distribution with rate parameter = i . The process is said to be in-control when = 0 and out-of-control when either > 0 or
< 0 .
122
Quick detection of changes in is often accomplished by way of a cumulative sum (CUSUM) control chart. Although the CUSUM chart was developed by Page (1961) for normal process means using a sequential probability ratio test (SPRT) under normal model assumptions, Brook and Evans (1972) were the first to suggest using such a procedure for monitoring count data processes. The Poisson CUSUM control chart can be derived from the SPRT, however, under Poisson model assumptions. The null hypothesis H 0 : = 0 is tested against the simple alternative hypothesis H a : = a whenever a new sequence of subgroup counts is obtained, where a denotes the mean count rate from the out-of-control process. In general, the SPRT operates by comparing the sequential probability ratio
X a e
i
ST =
i =1
T
a
X i! T e 0 X! i =1 i
Xi 0
(6.1)
to an appropriate constant A at each new subgroup. If S T > A , then the test concludes in favor of H a . It turns out that a CUSUM procedure designed to detect a change in from = 0 to = a will provide the best ARL performance when the magnitude of the change is such that the true value of the rate parameter is a (see Moustakides (1986)). As the true value of the rate parameter deviates from a , however, the ARLs obtained from this procedure will divert from optimality. An alternative to the CUSUM control chart is to consider a change point model and a likelihood ratio test. Perry, Pignatiello and Simpson (2004c) suggest such a procedure and show that its performance is more robust to uncertainty in the magnitude of change relative to any one CUSUM chart. Termed a magnitude-robust control chart, it is derived from the likelihood ratio test for homogeneity of Poisson means with a simple alternative. In formulating the hypotheses for this procedure, one assumption is that the process behavior beyond the unknown point of change is known apriori. Specifically, the magnitude-robust procedure models process behavior using a single-point step change function. Many times, however, this assumption cannot be validated. In fact, rarely in practice is the type of change known apriori, and thus, the ARL performance of the 123
magnitude-robust procedure may suffer as the true change-type deviates from the assumed step change behavior. The CUSUM procedure mentioned earlier is also derived under this specific condition. A major advantage to using the magnitude-robust procedure over the CUSUM is that, upon signaling, useful diagnostic statistics are immediately provided in order to aid process engineers in their search for the special cause affecting the process (i.e. change point, magnitude of change, etc.). Perry, Pignatiello and Simpson (2004d), however, show that the performances of the diagnostic statistics obtained from the magnituderobust procedure are not insensitive to any uncertainty in the type of change that may be present. Thus, if the type of change present is one other than a step change, process engineers can easily be mislead into changing process input variables that do not require adjustment. In this paper, we investigate a control chart that quickly detects changes in , however, without requiring exact knowledge of the change-type apriori. Instead, we assume that the type of change present is one from a family of monotonic change-types. We show that the proposed control charts can be derived from the likelihood ratio test for homogeneity of Poisson means with an order restricted alternative. We also show that the proposed control chart has better overall ARL performance relative to the Poisson CUSUM and magnitude-robust control-charting procedures suggested by Page (1961) and Perry, Pignatiello and Simpson (2004b), respectively, across a range of monotonic change-types. Also, like the magnitude-robust procedure, the proposed control charts offer useful diagnostic information. Perry et al (2004d) show that the diagnostic information obtained from the proposed control charting procedure outperforms that obtained from the magnitude-robust procedure across a range of monotonic change-types. In the next section, we define the process behavior model used in deriving the proposed control chart. In subsequent sections we describe the test and derive the form of the control chart statistic. Finally, we compare ARL performances of the proposed strategy with those obtained from the Poisson CUSUM and magnitude-robust control charts.
124
6.2. Process Behavior Model
Consider a model for the behavior of a Poisson process rate parameter where the change in can be described as belonging to a family of, say, isotonic change-types. The set of isotonic changes is the set of all possible change types that are non-decreasing in nature. We note that this set includes the step change-type, as well as, a countably infinite number of other non-decreasing change-types. The model assumes that the process is operating in a state of statistical control for the first subgroups with = 0 . Following an unknown subgroup , the process changes to an unknown out of control state such that the behavior in can be described by +1 > 0 and i i 1 i = + 2, K , T , where T denotes the most recent subgroup. for
6.3. Control Chart for Isotonic Changes in a Poisson Rate: Likelihood Ratio Test
The proposed control-charting strategy is developed from the likelihood ratio test for homogeneity of Poisson means with a restricted alternative. hypotheses considered in its development are given by H 0 : i = 0 H a : i = 0 for i = 1, 2, K , T for i = 1, 2, K , , for i = + 2, K , T (6.2) Specifically, the
+1 > 0
and
i i 1
model.
where the order-restricted alternative is taken directly from the assumed process behavior Assuming a change point at , the likelihood functions under H 0 and H a are given as
l0 =
i =1 T X 0 e
i 0
X!
(6.3)
and
125
la =
i =1
X 0 e
i
0
X i!
i = +1
T
iX e
i
i
X i!
,
(6.4)
respectively. Thus, by taking the ratio of la to lo and simplifying we obtain the log likelihood ratio given by R( , T | x ) = 0 (T ) +
i = +1
[X
T
i
log e ( i 0 ) i ] ,
(6.5)
where sufficiently large values of this statistic would favor the alternative hypothesis. Since and T are unknown, the test is conducted by maximizing R( , T | x ) over all possible values of and non-decreasing T given the observed subgroup counts x' = [ X 1 , X 2 , K , X T ] . We denote this maximum value as RT . If we knew , it can be shown that the vector T that maximizes R( , T | x ) is the maximum likelihood estimate (MLE) of T under simple order restriction. Recall from our process behavior model that the type of change is only known to be nondecreasing. Thus, for any given , we could obtain an initial estimate for T (denoted
~ as T ) given by
i = i 0
~
x
if x i > 0 if x i 0
for i = + 1, K , T
,
~ where 0 is the known in-control rate parameter value. Now, for any given T , we can obtain T (or the MLE) as the solution to the following convex program
maximize
i = +1
[x
T
i
~ ~ log e i 0 i
(
) ]
(6.6)
subject to
i i 1
~
~
for i = + 2, K, T
where the objective function is taken directly from the log likelihood function in (6.5) ~ and the decision variables are taken to be T . The initial values of the elements in
~ T provide a starting point in the solution to (6.6).
126
We can solve for T in (6.6) using a generalized reduced gradient algorithm;
however, Robertson, Wright and Dykstra (1988, p. 34) provide a theorem that suggests
T is simply found by fitting an isotonic regression function to the random vector ~ T . That is, they show that since the Poisson mass function belongs to the exponential
family of densities, then the MLE of T obtained under simple order restriction can be
~ ~ found by fitting the isotonic regression to T . Therefore, for any given and T ,
the MLE of T can be obtained from
~ T = I T
(
)
(6.7)
~ where the function I returns the isotonic regression of T .
Robertson et al (1988) further suggest some computation algorithms for performing the isotonic regression, the simplest of these being the pooled-adjacency violators (PAV) algorithm (see Ayer et al (1955) and Best et al (1990)). For details on the computational complexity of the PAV algorithm, see Grotzinger and Witzgall (1984).
Substituting T into (6.5) then yields the following expression
R , T | x = 0 (T ) +
(
)
i = +1
[X
T
i
log e i 0 i ,
(
) ]
(6.8)
which is expressed solely as a function of given the observations, x . Thus, for each new T, the expression in (6.8) is evaluated over all possible integer change point values to obtain the maximum. Letting t represent a possible change point, then RT = max R t , T t | x
0t <T
[ (
)]
(6.9) If RT is
where RT is the maximum of the log likelihood ratios at subgroup T.
sufficiently large, say RT > B (where B is an appropriately chosen constant), then the test concludes in favor of the alternative hypothesis. This hypothesis test could be implemented as a control chart by simply computing the test statistic, RT , with each new subgroup count obtained. If the value of RT is greater than an appropriately chosen constant (appropriate in the sense that desirable ARL properties are obtained), then the control chart signals that a change has occurred. Furthermore, when this control chart signals, 127
= arg max 0 (T ) +
0 t <T
i = +1
[X
T
i
log e i 0 i
(
) ]
(6.10)
and
~ T = I T
(
)
(6.11)
provide the MLEs of the change point and unknown process profile, respectively. The procedure outlined above could just as easily be used with the set of all nonincreasing functions (or the antitonic set). The only difference lies in how we estimate
T . That is, if the process behavior can be described as non-increasing, then
i = i 0
~
x
if xi < 0 for i = + 1, K , T if xi 0
provides a starting point in the solution to the following convex program maximize
i = +1
[x log ( ) ]
T
~
~
i
e
i
i
subject to
(6.12) for i = + 2, K, T
i i 1
~
~
where the difference here lies in the direction of the order constraints. In this case, the antitonic regression provides the solution to (6.12). That is, the estimate of T for the non-increasing case is given by
~ T = A T
(
)
~ where the function A returns the antitonic regression of T .
A distinct advantage of using a control chart derived from the likelihood ratio approach is that it provides valuable diagnostic tools which can help process engineers focus their search for the special cause. Along with the signal of a process change, the control charts described above additionally provide process engineers with estimates for
and T .
Perry, Pignatiello and Simpson (2004d) evaluated the performances of the estimators in (6.10) and (6.11) when applied following signals from a Poisson CUSUM control chart. They considered three distinct isotonic change-types in their study, 128
namely, single-point step changes, linear trends, and multiple-point step changes. They compared accuracy and precision performances between in (6.10) and the estimators suggested by Samuel and Pignatiello (1998) and Perry, Pignatiello and Simpson (2004b). Results showed that the performance of is more robust (or insensitive) to uncertainty in the type of non-decreasing change-type that may be present. Furthermore, they showed that, on average, the estimator in (6.11) does well in describing the true nature of the outof-control process.
6.4 Average Run Length Comparison
We now focus attention on the proposed charts detection performance relative to the CUSUM and magnitude-robust for step changes alternatives. monotonic change-types are considered in our evaluation. Three distinct We consider two one-sided
CUSUM charts for detecting increases in , each of which differ by their reference values k and corresponding decision intervals h. Specifically, CUSUM charts designed to detect 30% and 45% changes in the mean count rate are considered. The magnituderobust procedures considered are the one-sided charts. Unlike the normal distribution case, since there is no standardized Poisson distribution, we investigate small and large values of the in-control rate parameter, namely, 0 = 5, 10 and 20.
6.4.1 Simulation Modeling of Process Change
Monte Carlo simulation was used to estimate the ARL performances of the various control charts. Specifically, we investigate the performance of the proposed control charts relative to the others under the presence of (1) a single-point step change, (2) a linear trend and (3) multiple step changes. The simulation model for the singlepoint step change assumes i = 0 for i = 1, 2, , and i = a for i = +1, , T, where a is the new value of the mean count rate and is the unknown change point. For the linear trend, the simulation model assumes i = 0 for i = 1, 2, , and
i = 0 + (i ) for i = +1, , T, where is the slope of the linear trend. Finally,
129
the multiple step change simulation model assumes five change points. That is, the rate parameter is in-control for the first 1 subgroups, but then i = 0 + j for
i = j + 1, K, j +1 where j = 1, 2, K, 4 and i = 0 + 5 for i = 5 + 1, K , T . In this case,
1 , 2 , K , 5 are the magnitudes of the step changes at 1 , 2 , K, 5 , respectively. Also, 1 2 L 5 for the isotonic case and 1 2 L 5 for the antitonic case.
The simulation study was conducted as follows. Observations were generated from an in-control Poisson distribution for subgroups i = 1, 2, , (or 1 for the multiple step change case). Starting with subgroup + 1 , observations were generated from an out-of-control Poisson distribution where the rate parameter behavior can be described by one of the change-types mentioned above. Observations were collected until the control chart issued a signal at subgroup T. The length for that run was then recorded as T (or T 1 for the multiple step change case). This procedure was repeated for a total of N = 100,000 independently-seeded runs for each change-type and at several magnitudes of change. The average of the 100,000 independent run lengths was then recorded as the estimated ARL.
6.4.2 False Alarms
When > 0 and a control chart issues a signal at subgroup T where T , then the signal is a false alarm since the signal was given before the simulated process change could occur. We handled false alarms in our simulation runs as was done in Perry, Pignatiello and Simpson (2004b). That is, when a false alarm was encountered in a simulation run, it was treated the same way that a false alarm would be treated on an actual process. Namely, if one determines that a signal is indeed a false alarm then one is affirming that the process is currently in-control and could restart their monitoring of the process. Thus, when a false alarm was encountered at subgroup T, the control chart was restarted at subgroup T + 1 while not altering the scheduled change point.
6.4.3 ARL Calibration of Control Charts
Without loss of generality, for the isotonic case, each control chart was calibrated so that when the process was in-control, the ARL was approximately equal to the in-
130
control ARLs obtained from the proposed order-restricted procedure with a control limit of B = 4 , given 0 . For the antitonic case, each control chart was calibrated so that when the process was in-control, the ARL was approximately equal to the in-control ARLs obtained from the proposed order-restricted procedure with a control limit of
B = 5 . We have found that the same relative comparisons and results are obtained for
other values for the in-control ARL and values of 0 .
A total of N = 100, 000
independently-seeded runs were used to estimate the in-control ARLs for the orderrestricted, magnitude-robust and CUSUM procedures. A Markov chain approach to ARL estimation (as suggested by Brook and Evans (1972)) was used to verify the decision intervals obtained for the CUSUM schemes.
6.4.4 ARL Performances for Step Change Case
We first consider control chart ARL performances for processes that are out-ofcontrol when a chart is first applied, i.e. when = 0. Tables 6.1 and 6.2 show, for increases and decreases, respectively, the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of 0 = 5. Tables 6.3 and 6.4 show the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of 0 = 10 while Tables 6.5 and 6.6 show these estimates for 0 = 20. Results indicate that ARL performances of the CUSUM procedures depend on the magnitude of change exhibited by the process. That is, the CUSUM procedure will perform well for those magnitudes of change for which the CUSUM was designed, however, performance deteriorates as the true change magnitude deviates from the CUSUMs pre-specified value. Although each CUSUM chart yields the best ARL performance in those regions close to the value of a for which they were designed, no single CUSUM control chart is uniformly best. For example, it can be seen that although the 30% CUSUM performs well for small changes, it does not perform as well as the 45% CUSUM at detecting larger changes. Conversely, the 45% CUSUM, which does well at detecting the large changes, does not perform particularly well for smaller changes.
131
To obtain a measure of the robustness of the control-charting procedures, we count the number of times each control chart yields an ARL performance that is either first or second best amongst the four control charts considered in this study. For example, in Table 6.1, the 30% CUSUM was either first or second best (relative to the other procedures) for all out-of-control rate parameter values less than 7.5, yielding a count of 4. Thus, of the 15 levels of change magnitudes considered in Table 6.1, the 30% CUSUM provides either the first or second best relative ARL performances at only 4 distinct levels of change (e.g., a = 5.5, 6.0, 6.5, and 7.0). The 45% CUSUM was either first or second best (relative to the other procedures) for all values of a in the interval
[6.5, 9.0] , yielding a count of 6.
Thus, of the 15 levels of change magnitudes considered
in Table 6.1, the 45% CUSUM provides either the first or second best relative ARL performances at 6 distinct levels of change (e.g., a = 6.5, 7.0, 7.5, 8.0, 8.5, and 9.0). Both, the magnitude-robust and order-restricted control charts provide either the first or second best ARL performances at 11 distinct levels of change, implying these procedures are more robust to uncertainty in the change magnitude than is the CUSUM procedure. Using the same procedure outlined above, and for the 0 = 5 case (however, this time considering both increases and decreases in ), of the 25 total distinct levels of changes studied (see Tables 6.1 and 6.2), the 30% CUSUM provides either the first or second best relative ARL performances at 9 distinct levels of change, while the 45% CUSUM provides either the first or second best at 13 distinct levels of change. For comparison, the magnitude-robust control chart provides the first or second best relative ARL performances at 16 distinct levels of change, while the order-restricted procedure provides the first or second best at 15 distinct levels of change. The implication here is that, in the initial state, the magnitude-robust procedure is least sensitive (relative to the other procedures) to any uncertainty in the magnitude of step change a process may exhibit, followed by the order-restricted procedure. Similar results are obtained for other values of the in-control rate parameter 0 . Thus, for quick detection of step changes regardless of the magnitude, the magnitude-robust or order-restricted control charting procedures should be considered, since they have strong ARL performances for step changes of all magnitudes. We note
132
that, although the proposed order-restricted control charting procedure performs similar to the magnitude-robust control chart for this change-type, it also appears to provide more power than the magnitude-robust procedure when smaller step changes in the rate parameter occur. The second ARL performance study considers control charts which are applied on processes that are initially in-control, but experience a step change disturbance following the formation of subgroup > 0. The results shown in Tables 6.7-6.12 are for a change point of = 50. From simulation experiments not reported here, the ARL performances of the control charts for = 20 were approximately the same as for larger values of , such as = 75 and 100. Thus, the results reported here are indicative of a wide range of values of the change point. Tables 6.7-6.12 show results similar to the = 0 case. That is, the CUSUM charts yield the best ARL performances near those values of a for which they were specifically designed. Again, there is no single best control chart for all values of a . The control chart that yields the best ARL performance depends upon the magnitude of the change. Although the likelihood-based control charts will not provide the best ARL performance for any specific value of a , they have nearly the best ARL performance for all values of a . Thus, unless the magnitude of the change is known apriori, we conclude that the likelihood-based control charts have better ARL performance over the entire range of a values. To obtain a measure of the robustness of the control-charting procedures in the steady state, we, again, count the number of times each control chart yields an ARL performance that is either first or second best amongst the four control charts considered. For the 0 = 5 case (and considering both increases and decreases), the magnitude-robust control chart provides the first or second best relative ARL performances at 15 distinct levels of change, while the order-restricted provides the first or second best at 21 distinct levels of change. This indicates that, in the steady state, the order-restricted procedure is more robust to uncertainty in the magnitude of change than is the magnitude-robust control chart (or any of the CUSUM procedures) for this change-type. Similar results are obtained for other values of the in-control rate parameter 0 . 133
We note that, in many of these cases, the ARL results obtained using the orderrestricted procedure are only marginally different when compared to the CUSUM or magnitude-robust procedures. Thus, the real advantage to using the proposed orderrestricted control charting procedure lies in the diagnostic statistics that are immediately available upon signaling. Along with an estimate for the process change point, process engineers are also provided the maximum likelihood estimate of the process profile. These are important diagnostic statistics since it is not likely that the change point, type of change or magnitude of change will be known apriori.
6.4.5 ARL Performances for Linear Trend Change Case
We now consider control chart ARL performances for processes that experience an increasing linear trend when a chart is first applied, i.e. when = 0. Table 6.13 shows the corresponding estimated ARL values and their associated standard errors for an incontrol rate parameter value of 0 = 5. Table 6.14 shows the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of
0 = 10 while Table 6.15 shows these estimates for 0 = 20 . We do not consider linear
trends for decreasing rates since the linear decreasing trend would eventually produce negative rates, which are impossible. In Tables 6.13-6.15, results indicate that the detection performance of the CUSUM control chart is dependent upon the magnitude of the slope parameter . That is, in general, for smaller values of , the 30% CUSUM will provide better detection performance than the 45% CUSUM. On the other hand, for mid-to-large values of the slope parameter, the 45% CUSUM will provide more power. Thus, similar to the step change case, no single CUSUM control chart is uniformly best across all values of . To obtain a measure of the robustness of the control-charting procedures for this change-type, we count the number of times each control chart yields an ARL performance that is either first or second best amongst the four control charts considered. For the 0 = 5 case, the 45% CUSUM provides either the first or second best relative ARL performances at 14 distinct levels of change (out of a total of 17 levels considered),
134
Table 6.1. ARL comparison for increasing rate case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 140 h = 10.27 * = 6.50 a 30.0% ARLcusum 44.0 (0.12) 20.3 (0.05) 12.2 (0.03) 8.6 (0.02) 6.6 (0.01) 5.3 (0.01) 4.5 (0.01) 3.9 (0.01) 3.5 3.1 2.6 2.2 2.0 1.8 1.6 h = 7.89 * = 7.25 a 45.0% ARLcusum 47.7 (0.14) 21.7 (0.06) 12.5 (0.03) 8.5 (0.02) 6.3 (0.01) 5.0 (0.01) 4.2 (0.01) 3.6 (0.01) 3.1 2.8 2.3 2.0 1.7 1.5 1.4
B = 4.00 ARLOR 46.5 (0.13) 21.7 (0.05) 12.9 (0.03) 8.8 (0.02) 6.6 (0.01) 5.1 (0.01) 4.2 (0.01) 3.6 (0.01) 3.1 2.7 2.2 1.8 1.6 1.4 1.3
B = 3.527 ARLMR 47.1 (0.13) 21.7 (0.05) 12.8 (0.03) 8.7 (0.02) 6.4 (0.01) 5.0 (0.01) 4.1 (0.01) 3.5 (0.01) 3.0 (0.01) 2.6 2.1 1.8 1.6 1.4 1.3
a
5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 11 12 13 14 15
135
Table 6.2. ARL comparison for decreasing single-point step change case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 138
a
B = 5.00 ARLOR 50.8 (0.14) 22.7 (0.05) 12.7 (0.02) 8.2 (0.01) 5.8 (0.01) 4.3 (0.01) 3.3 2.7 2.2 2.1
B = 4.93* ARLMR 54.6 (0.15) 25.0 (0.06) 13.6 (0.03) 8.5 (0.02) 5.7 (0.01) 4.1 (0.01) 3.0 2.2 1.6 1.3
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.25
h = 8.234 * = 3.50 a 30.0% ARLcusum 45.6 (0.13) 19.6 (0.05) 11.1 (0.02) 7.4 (0.01) 5.5 (0.01) 4.4 3.6 3.1 2.6 2.4
h = 5.56 * = 2.75 a 45.0% ARLcusum 51.5 (0.15) 22.4 (0.06) 11.7 (0.03) 7.2 (0.01) 5.1 (0.01) 3.9 3.2 2.7 2.3 2.1
*The magnitude-robust control chart could not be calibrated to 138 in this case, as such, its in-control ARL was set to the next closest value less than 138 (or 123).
136
Table 6.3. ARL comparison for increasing single-point step change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. h = 11.27 * = 13.00 a 30.0% ARLcusum 32.0 (0.09) 13.1 (0.03) 7.5 (0.01) 5.2 (0.01) 4.0 (0.01) 3.3 2.8 2.4 2.1 1.9 1.8 1.6 1.5 1.4 1.3 h = 8.67 * = 14.50 a 45.0% ARLcusum 37.2 (0.11) 14.8 (0.04) 7.9 (0.02) 5.2 (0.01) 3.8 (0.01) 3.0 (0.01) 2.5 2.1 1.9 1.7 1.5 1.4 1.3 1.3 1.2
ARL0 132
B = 4.00 ARLOR 31.7 (0.08) 13.5 (0.03) 7.9 (0.02) 5.4 (0.01) 4.0 (0.01) 3.1 (0.01) 2.6 2.2 1.9 1.7 1.5 1.4 1.3 1.2 1.2
B = 3.662 ARLMR 32.6 (0.09) 13.7 (0.03) 7.8 (0.02) 5.2 (0.01) 3.8 (0.01) 3.0 (0.01) 2.5 2.1 1.8 1.6 1.5 1.4 1.3 1.2 1.2
a
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
137
Table 6.4. ARL comparison for decreasing single-point step change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. h = 10.057 * = 7.00 a 30.0% ARLcusum 42.9 (0.12) 14.6 (0.03) 7.5 (0.01) 4.9 (0.01) 3.6 2.9 2.4 2.1 2.0 h = 6.51 * = 5.50 a 45.0% ARLcusum 53.7 (0.16) 18.2 (0.05) 8.1 (0.02) 4.7 (0.01) 3.2 2.4 2.0 1.6 1.3
ARL0 197
B = 5.00 ARLOR 40.2 (0.10) 15.1 (0.03) 8.1 (0.01) 5.1 (0.01) 3.6 2.6 2.0 1.6 1.3
B = 4.67 ARLMR 43.6 (0.11) 15.9 (0.03) 8.2 (0.02) 5.0 (0.01) 3.4 (0.01) 2.5 1.8 1.4 1.1
a
9 8 7 6 5 4 3 2 1
138
Table 6.5. ARL comparison for increasing single-point step change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. h = 12.17 * B = 3.56 a = 26 30.0% ARLMR ARLcusum 45.0 (0.13) 21.1 (0.05) 12.3 (0.03) 8.2 (0.02) 6.0 (0.01) 4.7 (0.01) 3.8 (0.01) 3.2 2.7 2.4 2.1 1.9 1.7 1.6 1.5 48.5 (0.15) 22.8 (0.06) 12.8 (0.03) 8.3 (0.02) 5.9 (0.01) 4.5 (0.01) 3.7 (0.01) 3.1 2.7 2.4 2.2 2.0 1.8 1.7 1.6 h = 8.77 * = 29 a 45.0% ARLcusum 54.7 (0.17) 27.2 (0.08) 15.3 (0.04) 9.5 (0.03) 6.5 (0.02) 4.8 (0.01) 3.7 (0.01) 3.1 (0.01) 2.6 2.2 2.0 1.8 1.6 1.5 1.4
ARL0 126
B = 4.00 ARLOR 44.1 (0.12) 20.6 (0.05) 12.2 (0.03) 8.3 (0.02) 6.1 (0.01) 4.8 (0.01) 3.9 (0.01) 3.2 2.7 2.4 2.1 1.9 1.7 1.6 1.5
a
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
139
Table 6.6. ARL comparison for decreasing single-point step change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. h = 10.64 * = 14.0 a 30.0% ARLcusum 74.4 (0.23) 31.5 (0.09) 15.6 (0.04) 9.0 (0.02) 6.0 (0.01) 4.4 (0.01) 3.5 (0.01) 2.8 2.4 2.1 1.9 1.7 1.6 1.4 1.2 h = 6.16 * = 11.0 a 45.0% ARLcusum 93.4 (0.29) 43.5 (0.13) 21.9 (0.06) 12.0 (0.03) 7.3 (0.02) 4.9 (0.01) 3.6 (0.01) 2.8 2.3 1.9 1.6 1.4 1.3 1.2 1.1
ARL0 210
B = 5.00 ARLOR 63.0 (0.17) 25.7 (0.06) 14.2 (0.03) 9.1 (0.02) 6.5 (0.01) 4.9 (0.01) 3.8 (0.01) 3.1 2.6 2.2 1.9 1.6 1.4 1.3 1.1
B = 4.65 ARLMR 66.1 (0.56) 27.6 (0.06) 14.8 (0.03) 9.3 (0.02) 6.4 (0.01) 4.7 (0.01) 3.7 (0.01) 2.9 2.4 2.0 1.7 1.5 1.3 1.2 1.1
a
19 18 17 16 15 14 13 12 11 10 9 8 7 6 5
140
Table 6.7. ARL comparison for increasing single-point step change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 140 h = 10.27 * = 6.50 a 30.0% ARLcusum 41.4 (0.12) 18.6 (0.05) 10.9 (0.03) 7.6 (0.02) 5.8 (0.01) 4.7 (0.01) 4.0 (0.01) 3.4 (0.01) 3.0 2.7 2.3 2.0 1.8 1.6 1.5 h = 7.89 * = 7.25 a 45.0% ARLcusum 45.8 (0.14) 20.5 (0.06) 11.7 (0.03) 7.8 (0.02) 5.8 (0.01) 4.6 (0.01) 3.8 (0.01) 3.3 (0.01) 2.9 2.6 2.1 1.8 1.6 1.5 1.3
B = 4.00 ARLOR 44.3 (0.13) 19.9 (0.05) 11.7 (0.03) 7.9 (0.02) 5.9 (0.01) 4.7 (0.01) 3.8 (0.01) 3.2 (0.01) 2.8 2.5 2.0 1.7 1.5 1.3 1.2
B = 3.527 ARLMR 44.6 (0.13) 20.4 (0.05) 11.8 (0.03) 8.0 (0.02) 5.9 (0.01) 4.6 (0.01) 3.8 (0.01) 3.2 (0.01) 2.8 2.5 2.0 1.7 1.5 1.4 1.3
a
5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 11 12 13 14 15
141
Table 6.8. ARL comparison for decreasing single-point step change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 138 h = 8.234 * = 3.50 a 30.0% ARLcusum 42.8 (0.13) 17.8 (0.05) 9.8 (0.02) 6.4 (0.01) 4.8 (0.01) 3.8 3.1 2.7 2.3 2.1 h = 5.56 * = 2.75 a 45.0% ARLcusum 49.7 (0.15) 21.1 (0.06) 10.8 (0.03) 6.5 (0.01) 4.6 (0.01) 3.5 2.8 2.4 2.1 1.9
B = 5.00 ARLOR 47.9 (0.14) 20.6 (0.05) 11.3 (0.02) 7.2 (0.01) 5.0 (0.01) 3.7 (0.01) 2.9 2.2 1.7 1.5
B = 4.93* ARLMR 53.0 (0.15) 23.5 (0.06) 12.7 (0.03) 7.9 (0.02) 5.4 (0.01) 3.8 (0.01) 2.9 2.1 1.5 1.3
a
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.50 0.25
142
Table 6.9. ARL comparison for increasing single-point step change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 132 h = 11.27 * = 13.00 a 30.0% ARLcusum 30.5 (0.09) 12.1 (0.03) 6.9 (0.01) 4.7 (0.01) 3.6 (0.01) 3.0 2.5 2.2 2.0 1.8 1.6 1.5 1.4 1.3 1.3 h = 8.67 * = 14.50 a 45.0% ARLcusum 36.4 (0.11) 14.3 (0.04) 7.5 (0.02) 4.9 (0.01) 3.6 (0.01) 2.8 (0.01) 2.3 2.0 1.8 1.6 1.5 1.4 1.3 1.2 1.2
B = 4.00 ARLOR 29.7 (0.08) 12.4 (0.03) 7.1 (0.02) 4.9 (0.01) 3.6 (0.01) 2.8 (0.01) 2.3 2.0 1.7 1.6 1.4 1.3 1.2 1.2 1.1
B = 3.662 ARLMR 31.0 (0.09) 12.8 (0.03) 7.2 (0.02) 4.9 (0.01) 3.6 (0.01) 2.8 (0.01) 2.4 2.0 1.8 1.6 1.4 1.3 1.2 1.2 1.1
a
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
143
Table 6.10. ARL comparison for decreasing single-point step change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 197 h = 10.057 * = 7.00 a 30.0% ARLcusum 41.3 (0.12) 13.6 (0.03) 6.8 (0.01) 4.4 (0.01) 3.2 2.6 2.2 1.9 1.8 h = 6.51 * = 5.50 a 45.0% ARLcusum 52.8 (0.16) 17.7 (0.05) 7.8 (0.02) 4.4 (0.01) 3.0 2.3 1.8 1.5 1.2
B = 5.00 ARLOR 37.2 (0.10) 13.5 (0.03) 7.2 (0.01) 4.5 (0.01) 3.1 2.3 1.8 1.4 1.1
B = 4.67 ARLMR 41.2 (0.11) 14.7 (0.03) 7.6 (0.01) 4.7 (0.01) 3.2 (0.01) 2.4 1.8 1.3 1.1
a
9 8 7 6 5 4 3 2 1
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Table 6.11. ARL comparison for increasing single-point step change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 126 h = 12.17 * B = 3.56 a = 26 30.0% ARLMR ARLcusum 43.1 (0.12) 19.7 (0.05) 11.3 (0.03) 7.6 (0.02) 5.5 (0.01) 4.3 (0.01) 3.5 (0.01) 3.0 2.6 2.3 2.0 1.8 1.6 1.5 1.4 48.0 (0.15) 22.3 (0.07) 12.2 (0.03) 7.8 (0.02) 5.5 (0.01) 4.3 (0.01) 3.5 (0.01) 2.9 (0.01) 2.5 2.3 2.0 1.9 1.7 1.6 1.5 h = 8.77 * = 29 a 45.0% ARLcusum 54.3 (0.17) 27.1 (0.08) 15.0 (0.04) 9.3 (0.03) 6.3 (0.02) 4.7 (0.01) 3.6 (0.01) 3.0 (0.01) 2.5 2.2 1.9 1.7 1.6 1.5 1.4
B = 4.00 ARLOR 41.7 (0.12) 19.1 (0.05) 11.1 (0.03) 7.5 (0.02) 5.5 (0.01) 4.3 (0.01) 3.5 (0.01) 3.0 2.5 2.2 2.0 1.8 1.6 1.5 1.4
a
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
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Table 6.12. ARL comparison for decreasing single-point step change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 210 h = 10.64 * = 14.0 a 30.0% ARLcusum 73.4 (0.23) 30.8 (0.09) 15.0 (0.04) 8.6 (0.02) 5.7 (0.01) 4.1 (0.01) 3.2 (0.01) 2.7 2.3 2.0 1.8 1.6 1.5 1.3 1.2 h = 6.16 * = 11.0 a 45.0% ARLcusum 91.9 (0.29) 43.1 (0.13) 21.8 (0.06) 11.8 (0.03) 7.2 (0.02) 4.8 (0.01) 3.4 (0.01) 2.7 2.2 1.9 1.6 1.4 1.3 1.1 1.0
B = 5.00 ARLOR 59.7 (0.17) 23.7 (0.06) 12.8 (0.03) 8.2 (0.02) 5.7 (0.01) 4.3 (0.01) 3.4 (0.01) 2.8 2.3 1.9 1.7 1.5 1.3 1.2 1.1
B = 4.65 ARLMR 64.2 (0.18) 25.7 (0.06) 13.7 (0.03) 8.6 (0.02) 6.0 (0.01) 4.4 (0.01) 3.4 (0.01) 2.8 2.3 1.9 1.6 1.4 1.3 1.1 1.1
a
19 18 17 16 15 14 13 12 11 10 9 8 7 6 5
146
while the order-restricted procedure provides either the first or second best relative ARL performances at 13 distinct levels of change. CUSUM. For the 0 = 10 case, the order-restricted procedure and the 30% CUSUM provide either the first or second best relative ARL performances at 15 distinct levels of change, followed by the magnitude-robust and 45% CUSUM procedures, respectively. Finally, for the 0 = 20 case, the order-restricted procedure is again the least sensitive to changes in , followed by the magnitude-robust, 30% CUSUM and 45% CUSUM procedures, respectively. These results indicate that the order-restricted procedure is more robust to uncertainty in the magnitude of the slope parameter than is the magnitude-robust control chart, or any of the CUSUM procedures, for this change-type. Thus, for quick detection of linear trends regardless of the magnitude of , the order-restricted control charting procedure should be considered, since it has strong ARL performances for linear trends of all magnitudes. Again, we note that, in general, the proposed order-restricted procedure will provide more power than the magnitude-robust procedure for smaller magnitudes of the slope parameter . However, as the magnitude of becomes large, the magnitude-robust procedure slightly outperforms the proposed order-restricted procedure. The second ARL performance study considers control charts which are applied on processes that are initially in-control, but experience a linear trend disturbance following the formation of subgroup > 0. The results shown in Tables 6.16-6.18 are for a change point of = 50. Again, from simulation experiments not reported here, the ARL performances of the control charts for = 20 were approximately the same as for larger values of , such as = 75 and 100. Thus, the results reported here are indicative of a wide range of values of the change point. Tables 6.16-6.18 show results similar to the = 0 case. Again, there is no single best control chart for all values of and the control chart that yields the best ARL performance depends upon the magnitude of the change. To obtain a measure of the The magnitude-robust control chart provides the next best performance (with regard to robustness), followed by the 30%
147
robustness of the control-charting procedures in the steady state, we, again, count the number of times each control chart yields an ARL performance that is either first or second best amongst the four control charts considered. For the 0 = 5 case, the order-restricted control chart provides either the first or second best relative ARL performances at 15 distinct levels of change (out of a total of 17 distinct levels of change considered), while the 30% and 45% CUSUMs provide either the first or second best at 12 distinct levels of change. The magnitude-robust procedure provides either the first or second best at 9 distinct levels of change in this case. This indicates that, even in the steady state, the order-restricted procedure is more robust to uncertainty in the magnitude of than is any of the other control charting procedures for this change-type. Similar results are obtained for the other values of the in-control rate parameter 0 . Thus, unless the magnitude of is known apriori, we conclude that the proposed order-restricted control chart provides better overall ARL performance than the magnitude-robust or CUSUM procedures for this change-type. We note that, again, in many of these cases, the ARL results obtained using the order-restricted procedure are marginally different when compared to the CUSUM or magnitude-robust procedures. Thus, the real advantage to using the proposed orderrestricted control charting procedure in this case lies in the diagnostic statistics that are immediately available upon signaling. Furthermore, Perry, Pignatiello and Simpson (2004d) showed that the diagnostic statistics obtained from the proposed order-restricted procedure outperform those obtained from the magnitude-robust procedure when linear trend changes are present.
6.4.6 ARL Performances for Multiple Step Change Case
In this section, we consider control chart ARL performances for processes that experience multiple step changes. Tables 6.19 and 6.20 show the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of 0 = 5 for increases and decreases, respectively. Tables 6.21 and 6.22 show the corresponding estimated ARL values and their associated standard errors for an in-control rate parameter value of 0 = 10 for increases and decreases,
148
respectively, while Tables 6.23 and 6.24 show these estimates for 0 = 20. The j ' s and
j ' s used in the simulation were chosen arbitrarily, with the only constraint being that
the resulting pattern be monotonic in nature. General results indicate that a CUSUM procedure designed for small changes in
(say, 10-15%) could be used to provide better relative detection performance when
multiple step change occurrences (of small magnitudes) are present. This is evidenced by the 0 = 5 case. Such a scheme, however, will not necessarily provide good detection performance for, say, medium-to-large change magnitude occurrences. In this sense, the CUSUM procedure is not robust to the type of change that may be present. For example, we see from Tables 6.19 and 6.20 that with 0 = 5 , a 30% CUSUM chart (i.e. a CUSUM designed for an increase of 1.5 in the rate parameter) will provide better relative detection performance for this change type than that offered by the other control-charting procedures considered in this paper. On the other hand, detection performance suffers (relative to the other procedures) when mid-to-large magnitude single-point step changes in occur. This is evidenced by the results provided in Tables 6.1 and 6.2. For the 0 = 10 and 0 = 20 cases, the proposed order-restricted control-charting procedure provides the overall better ARLs for this change-type. A CUSUM chart could be designed for smaller changes in (say, a 15% change) and would likely perform better than the order-restricted procedure in these cases, however, its ARL performance would suffer for larger magnitude change occurrences. Thus, we conclude that the CUSUM procedure is more sensitive to the magnitude of change (and hence, the type of change) that may be present than is the proposed order-restricted procedure. Therefore, unless the exact type of change is known apriori, the proposed order-restricted procedure should be considered. We note that there are an infinite number of monotonic change-types that a process could exhibit, and thus, by no means can every change-type be evaluated. The change-types evaluated in this section were chosen arbitrarily and fall into the quadratic trend category, however, with a slope parameter that does not remain constant between subgroups. Since the CUSUM procedure is sensitive to the magnitude of change exhibited by a process, it is also sensitive to the type of change. This is because the type
149
of change present depends upon the magnitudes of change and the relative locations of the change points (i.e. j ' s and j ' s ). In comparing the proposed procedure with the magnitude-robust control chart, Tables 6.19-6.24 indicate the proposed method outperforms the magnitude-robust procedure for nearly all multiple step change scenarios considered in this paper. The only exceptions are when the j ' s are of a large magnitude, at which the magnituderobust control chart will perform as well as the order-restricted procedure. Thus, we conclude that when a process exhibits multiple step changes, the proposed orderrestricted procedure should be used in place of the magnitude-robust.
6.5 Implementation Issues and Illustration
The order-restricted control chart can be implemented in a similar manner to most standard control charts for count data.
corresponding maximum of all R , T | x values over 0 < T , can be plotted on a
(
)
That is, the RT statistic, which is the
chart against a control limit B . In general, determining RT requires T calculations of R (equation 6.8), which increases the amount of computation over most standard control charts. However, a simple program can be written using a common programming language or a spreadsheet to perform the calculations. Computing RT for relatively large values of T presents no problem even on a modest personal computer. Values of RT that exceed B are worthy of special cause investigation. When
RT > B , estimates for the initial change point (both a point estimate and confidence set)
and unknown process profile are provided to aid in the search for the special cause. Since estimates of the change point and process profile are arguments of the RT statistic, they are available immediately. Another few lines of code (or cells in a spreadsheet) can be added to calculate the confidence set for the initial process change point.
150
Table 6.13. ARL comparison for linear trend change case, 0 = 5 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 140
B = 4.00 ARLOR 54.2 (0.09) 39.7 (0.06) 32.5 (0.05) 28.1 (0.03) 25.1 (0.03) 17.3 (0.02) 11.9 (0.01) 9.5 (0.01) 8.1 (0.01) 7.1 (0.01) 6.4 (0.01) 5.9 (0.01) 5.4 5.1 4.8 3.2 2.5
B = 3.527 ARLMR 54.4 (0.10) 39.9 (0.06) 32.7 (0.05) 28.2 (0.04) 25.1 (0.03) 17.4 (0.02) 11.8 (0.01) 9.4 (0.01) 8.0 (0.01) 7.0 (0.01) 6.3 (0.01) 5.8 (0.01) 5.4 (0.01) 5.0 4.7 3.2 2.5
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
h = 10.27 * = 6.50 a 30.0% ARLcusum 53.0 (0.09) 38.7 (0.06) 31.7 (0.04) 27.4 (0.04) 24.5 (0.03) 17.0 (0.02) 11.7 (0.01) 9.5 (0.01) 8.1 (0.01) 7.2 (0.01) 6.6 (0.01) 6.1 5.7 5.3 5.0 3.5 2.9
h = 7.89 * = 7.25 a 45.0% ARLcusum 55.0 (0.10) 39.8 (0.06) 32.7 (0.05) 28.1 (0.04) 25.0 (0.03) 17.1 (0.02) 11.6 (0.01) 9.3 (0.01) 7.9 (0.01) 7.0 (0.01) 6.4 (0.01) 5.8 (0.01) 5.4 5.1 4.8 3.3 2.7
151
Table 6.14. ARL comparison for linear trend change case, 0 = 10 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 132 h = 11.27 * = 13.00 a 30.0% ARLcusum 61.5 (0.12) 45.8 (0.08) 37.9 (0.06) 32.9 (0.05) 29.3 (0.04) 20.3 (0.03) 13.9 (0.02) 11.1 (0.01) 9.4 (0.01) 8.3 (0.01) 7.6 (0.01) 6.9 (0.01) 6.4 (0.01) 6.1 (0.01) 5.7 3.9 3.2 h = 8.67 * = 14.50 a 45.0% ARLcusum 64.9 (0.13) 48.7 (0.09) 40.2 (0.07) 34.9 (0.05) 31.1 (0.05) 21.2 (0.03) 14.3 (0.02) 11.3 (0.01) 9.5 (0.01) 8.3 (0.01) 7.5 (0.01) 6.9 (0.01) 6.4 (0.01) 5.9 (0.01) 5.6 (0.01) 3.7 3.0
B = 4.00 ARLOR 61.0 (0.11) 45.7 (0.08) 37.9 (0.06) 33.0 (0.05) 29.5 (0.04) 20.6 (0.03) 14.1 (0.02) 11.3 (0.01) 9.6 (0.01) 8.5 (0.01) 7.6 (0.01) 7.0 (0.01) 6.5 (0.01) 6.1 (0.01) 5.7 (0.01) 3.8 3.0
B = 3.662 ARLMR 61.5 (0.12) 46.0 (0.08) 38.4 (0.06) 33.4 (0.05) 29.8 (0.04) 20.8 (0.03) 14.2 (0.02) 11.3 (0.01) 9.6 (0.01) 8.5 (0.01) 7.6 (0.01) 7.0 (0.01) 6.4 (0.01) 6.0 (0.01) 5.6 (0.01) 3.8 3.0
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
152
Table 6.15. ARL comparison for linear trend change case, 0 = 20 , = 0 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 126 h = 12.17 * = 26 a 30.0% ARLcusum 70.5 (0.15) 54.5 (0.10) 45.8 (0.08) 40.0 (0.07) 36.0 (0.06) 25.1 (0.04) 17.1 (0.02) 13.5 (0.02) 11.4 (0.01) 10.0 (0.01) 9.0 (0.01) 8.2 (0.01) 7.6 (0.01) 7.1 (0.01) 6.7 (0.01) 4.5 3.6 h = 8.77 * = 29 a 45.0% ARLcusum 73.2 (0.16) 57.5 (0.11) 48.5 (0.09) 42.4 (0.07) 38.3 (0.06) 26.8 (0.04) 18.1 (0.02) 14.2 (0.02) 11.9 (0.01) 10.4 (0.01) 9.3 (0.01) 8.5 (0.01) 7.8 (0.01) 7.3 (0.01) 6.8 (0.01) 4.5 3.5
B = 4.00 ARLOR 67.5 (0.14) 51.9 (0.09) 43.7 (0.07) 38.5 (0.06) 34.6 (0.05) 24.4 (0.03) 16.9 (0.02) 13.5 (0.02) 11.5 (0.01) 10.2 (0.01) 9.2 (0.01) 8.4 (0.01) 7.8 (0.01) 7.3 (0.01) 6.9 (0.01) 4.6 3.6
B = 3.56 ARLMR 68.8 (0.14) 52.8 (0.09) 44.4 (0.07) 38.9 (0.06) 35.1 (0.05) 24.6 (0.03) 17.0 (0.02) 13.6 (0.02) 11.5 (0.01) 10.2 (0.01) 9.2 (0.01) 8.4 (0.01) 7.8 (0.01) 7.3 (0.01) 6.8 (0.01) 4.6 3.6
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
153
Table 6.16. ARL comparison for linear trend change case, 0 = 5 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 140 h = 10.27 * = 6.50 a 30.0% ARLcusum 51.2 (0.09) 37.3 (0.06) 30.7 (0.05) 26.5 (0.04) 23.6 (0.03) 16.3 (0.02) 11.2 (0.01) 9.0 (0.01) 7.7 (0.01) 6.8 (0.01) 6.2 (0.01) 5.7 (0.01) 5.3 (0.01) 5.0 4.7 3.3 2.6 h = 7.89 * = 7.25 a 45.0% ARLcusum 53.7 (0.10) 39.1 (0.06) 31.8 (0.05) 27.5 (0.04) 24.4 (0.03) 16.7 (0.02) 11.3 (0.01) 9.0 (0.01) 7.6 (0.01) 6.8 (0.01) 6.1 (0.01) 5.6 (0.01) 5.2 (0.01) 4.9 4.6 3.1 2.5
B = 4.00 ARLOR 52.8 (0.09) 38.5 (0.06) 31.6 (0.05) 27.3 (0.03) 24.3 (0.03) 16.7 (0.02) 11.4 (0.01) 9.1 (0.01) 7.7 (0.01) 6.8 (0.01) 6.1 (0.01) 5.6 (0.01) 5.2 4.8 4.6 3.1 2.4
B = 3.527 ARLMR 53.2 (0.10) 38.8 (0.06) 32.0 (0.05) 27.5 (0.04) 24.5 (0.03) 16.8 (0.02) 11.4 (0.01) 9.1 (0.01) 7.7 (0.01) 6.8 (0.01) 6.1 (0.01) 5.6 (0.01) 5.2 (0.01) 4.9 (0.01) 4.6 3.1 2.4
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
154
Table 6.17. ARL comparison for linear trend change case, 0 = 10 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 132 h = 11.27 * = 13.00 a 30.0% ARLcusum 59.9 (0.12) 44.8 (0.08) 37.0 (0.06) 32.1 (0.05) 28.7 (0.04) 19.8 (0.03) 13.4 (0.02) 10.7 (0.01) 9.1 (0.01) 8.0 (0.01) 7.3 (0.01) 6.7 (0.01) 6.2 (0.01) 5.8 (0.01) 5.4 (0.01) 3.7 3.0 h = 8.67 * = 14.50 a 45.0% ARLcusum 64.7 (0.13) 48.1 (0.09) 39.7 (0.07) 34.4 (0.06) 30.7 (0.05) 21.0 (0.03) 14.1 (0.02) 11.1 (0.01) 9.4 (0.01) 8.2 (0.01) 7.4 (0.01) 6.7 (0.01) 6.2 (0.01) 5.8 (0.01) 5.5 (0.01) 3.6 2.9
B = 4.00 ARLOR 59.2 (0.11) 44.5 (0.08) 36.8 (0.06) 31.8 (0.05) 28.6 (0.04) 19.8 (0.03) 13.6 (0.02) 10.8 (0.01) 9.2 (0.01) 8.1 (0.01) 7.3 (0.01) 6.7 (0.01) 6.2 (0.01) 5.8 (0.01) 5.4 (0.01) 3.6 2.8
B = 3.662 ARLMR 60.3 (0.12) 45.1 (0.08) 37.5 (0.06) 32.6 (0.05) 29.2 (0.04) 20.3 (0.03) 13.9 (0.02) 11.0 (0.01) 9.3 (0.01) 8.2 (0.01) 7.4 (0.01) 6.8 (0.01) 6.3 (0.01) 5.8 (0.01) 5.5 (0.01) 3.7 2.9
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
155
Table 6.18. ARL comparison for linear trend change case, 0 = 20 , = 50 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 126 h = 12.17 * = 26 a 30.0% ARLcusum 69.5 (0.15) 53.7 (0.10) 45.0 (0.08) 39.5 (0.07) 35.5 (0.06) 24.7 (0.04) 16.8 (0.02) 13.3 (0.02) 11.2 (0.01) 9.9 (0.01) 8.8 (0.01) 8.1 (0.01) 7.5 (0.01) 6.9 (0.01) 6.5 (0.01) 4.4 3.5 h = 8.77 * = 29 a 45.0% ARLcusum 73.1 (0.16) 57.0 (0.11) 48.1 (0.09) 42.5 (0.08) 38.1 (0.06) 26.6 (0.04) 18.0 (0.02) 14.1 (0.02) 11.9 (0.01) 10.4 (0.01) 9.3 (0.01) 8.4 (0.01) 7.8 (0.01) 7.2 (0.01) 6.7 (0.01) 4.4 3.4
B = 4.00 ARLOR 66.0 (0.14) 50.7 (0.09) 42.7 (0.07) 37.2 (0.06) 33.6 (0.05) 23.7 (0.03) 16.4 (0.02) 13.0 (0.02) 11.1 (0.01) 9.8 (0.01) 8.8 (0.01) 8.1 (0.01) 7.5 (0.01) 7.0 (0.01) 6.6 (0.01) 4.4 3.5
B = 3.56 ARLMR 67.1 (0.14) 51.6 (0.10) 43.4 (0.07) 37.9 (0.06) 34.2 (0.05) 24.0 (0.03) 16.6 (0.02) 13.2 (0.02) 11.2 (0.01) 9.9 (0.01) 8.9 (0.01) 8.1 (0.01) 7.5 (0.01) 7.0 (0.01) 6.6 (0.01) 4.4 3.5
0.01 0.02 0.03 0.04 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00
156
Table 6.19. ARL comparison for increasing multiple step change case with 5 change points, 0 = 5 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 140
1 , 2 , 3 , 4 , 5
0.10, 0.13, 0.17, 0.25, 0.50 0.10, 0.20, 0.30, 0.40, 0.50 0.05, 0.10, 0.35, 0.50, 1.00 0.23, 0.36, 0.46, 0.72, 0.75 0.05, 0.35, 0.70, 3.72, 4.11 0.20, 0.30, 0.50, 0.80, 1.25 0.30 0.50, 0.60, 1.60, 2.10 0.35, 0.65, 0.70, 0.75, 3.75 0.10, 0.40, 0.90, 1.20, 1.45 0.65, 0.94, 1.12, 8.35, 9.22 0.10, 0.25, 2.27, 2.54, 2.62 0.05, 1.10, 1.15, 1.75, 2.50 0.35, 0.65, 6.35, 6.75, 7.12 1.12, 1.69, 2.22, 9.67, 9.89 2.50, 2.60, 2.75, 2.80, 3.00 2.75, 3.15, 3.62, 3.77, 3.79 3.00, 3.50, 3.75, 4.20, 4.37
B = 4.00 ARLOR
92.4 (0.28) 69.5 (0.20) 65.9 (0.19) 54.3 (0.14) 52.4 (0.10) 47.0 (0.13) 39.3 (0.11) 34.1 (0.09) 29.7 (0.06) 26.1 (0.06) 25.3 (0.03) 25.2 (0.04) 17.7 (0.02) 14.9 (0.03) 5.6 (0.01) 4.8 (0.01) 3.9 (0.01)
B = 3.527 ARLMR
93.0 (0.28) 71.8 (0.21) 66.4 (0.18) 55.1 (0.14) 52.8 (0.10) 48.0 (0.13) 39.8 (0.11) 34.5 (0.09) 30.3 (0.06) 26.4 (0.06) 25.3 (0.03) 25.5 (0.05) 17.8 (0.02) 15.0 (0.03) 5.6 (0.01) 4.8 (0.01) 3.9 (0.01)
1, 2 , 3 4 , 5
32, 44, 48, 56, 65 10, 15, 19, 22, 25 36, 45, 49, 53, 65 44, 60, 68, 73, 77 25, 55, 60, 63, 65 15, 19, 23, 30, 35 21, 26, 30, 34, 41 46, 52, 62, 65, 68 11, 16, 22, 28, 31 42, 62, 75, 82, 84 27, 44, 49, 51, 55 17, 27, 31, 33, 35 23, 34, 42, 45, 48 62, 81, 86, 94, 97 14, 16, 20, 22, 23 32, 36, 38, 41, 42 19, 20, 22, 24, 25
h = 10.27 * = 6.50 a 30.0% ARLcusum
88.7 (0.27) 65.9 (0.19) 62.5 (0.17) 51.5 (0.14) 50.8 (0.10) 44.1 (0.12) 36.8 (0.10) 32.0 (0.08) 28.1 (0.06) 24.6 (0.05) 25.0 (0.03) 24.0 (0.04) 17.7 (0.02) 14.1 (0.03) 5.6 (0.01) 4.9 (0.01) 4.0 (0.01)
h = 7.89 * = 7.25 a 45.0% ARLcusum
94.5 (0.29) 71.4 (0.21) 67.8 (0.19) 56.3 (0.16) 53.9 (0.11) 48.8 (0.14) 40.8 (0.12) 35.5 (0.10) 30.5 (0.07) 26.6 (0.06) 25.2 (0.03) 25.4 (0.05) 17.8 (0.02) 14.8 (0.03) 5.5 (0.01) 4.8 (0.01) 3.9 (0.01)
157
Table 6.20. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 5 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 138
1 , 2 , 3 , 4 , 5
0.25, 0.75, 1.25, 1.45, 1.50 0.50, 0.75, 1.00, 2.00, 2.50 2.00, 2.10, 2.15, 2.30, 2.35 2.50, 2.55, 2.75, 3.00, 3.10 0.50, 1.20, 2.20, 2.50, 2.75 0.25, 0.65, 2.75, 2.95, 3.15 0.15, 0.35, 0.65, 0.85, 3.25 0.75, 3.35, 3.85, 3.95, 4.10 0.25, 0.55, 3.35, 4.00, 4.15 2.25, 2.75, 2.85, 2.95, 3.25 0.10, 0.80, 0.95, 2.20, 2.65 0.25, 0.75, 3.35, 3.95, 4.75 1.35, 1.65, 3.75, 3.95, 4.35
B = 5.00 ARLOR
24.5 (0.04) 26.1 (0.05) 6.7 (0.01) 4.6 (0.01) 15.6 (0.02) 16.9 (0.01) 39.8 (0.10) 7.1 (0.01) 11.9 (0.01) 4.7 (0.01) 30.5 (0.06) 9.2 (0.01) 6.0 (0.01)
B = 4.93* ARLMR
26.5 (0.04) 28.9 (0.06) 7.3 (0.01) 4.9 (0.01) 16.4 (0.02) 17.3 (0.02) 43.9 (0.11) 7.3 (0.01) 12.1 (0.01) 5.0 (0.01) 33.1 (0.07) 9.4 (0.01) 6.3 (0.01)
1, 2 , 3 4 , 5
10, 20, 25, 30, 35 25, 35, 37, 39, 40 10, 12, 13, 16, 18 15, 16, 18, 19, 20 15, 25, 28, 32, 35 20, 30, 35, 37, 40 25, 30, 32, 35, 40 25, 30, 32, 33, 35 35, 42, 45, 47, 49 50, 52, 54, 56, 57 16, 26, 30, 33, 37 20, 23, 27, 32, 35 25, 27, 30, 33, 35
h =8.234 * = 3.50 a 30.0% ARLcusum
22.4 (0.03) 23.5 (0.05) 6.0 (0.01) 4.5 (0.01) 14.9 (0.02) 16.7 (0.01) 35.5 (0.09) 7.3 (0.01) 12.2 (0.01) 4.6 (0.01) 27.6 (0.05) 9.3 (0.01) 6.1 (0.01)
h =5.56 * = 2.75 a 45.0% ARLcusum
24.6 (0.04) 26.8 (0.06) 6.0 (0.01) 4.3 (0.01) 15.1 (0.02) 16.7 (0.01) 41.5 (0.11) 7.1 (0.01) 12.0 (0.01) 4.4 (0.01) 31.0 (0.07) 9.1 (0.01) 6.0 (0.01)
*The magnitude-robust control chart could not be calibrated to 138 in this case, as such, its in-control ARL was set to the next closest value less than 138 (or 123).
158
Table 6.21. ARL comparison for increasing multiple step change case with 5 change points, 0 = 10 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 132
1 , 2 , 3 , 4 , 5
0.20, 0.30, 0.50, 0.80, 1.25 0.30, 0.50, 0.60, 1.60, 2.10 0.10, 0.40, 0.90, 1.20, 1.45 0.40, 0.50, 1.10, 1.30, 1.50 0.05, 1.10, 1.15, 1.75, 2.50 0.10, 0.25, 2.27, 2.54, 2.62 0.30, 0.90, 1.60, 1.70, 2.20 1.00, 1.30, 1.70, 1.90, 2.30 0.90, 1.50, 2.10, 2.50, 3.00 2.50, 2.60, 2.75, 2.80, 3.00 2.75, 3.15, 3.62, 3.77, 3.79 3.00, 3.50, 3.75, 4.20, 4.37 6.00, 6.15, 6.35, 6.75, 7.12 6.00, 7.25, 7.36, 8.12, 8.50
B = 4.00
1, 2 , 3 4 , 5
B = 3.662 ARLMR
61.7 (0.18) 53.6 (0.15) 41.0 (0.10) 36.1 (0.08) 34.0 (0.07) 28.6 (0.04) 25.0 (0.05) 19.9 (0.04) 18.2 (0.03) 8.8 (0.02) 7.0 (0.01) 5.8 (0.01) 2.8 2.8
ARLOR
59.9 (0.18) 51.9 (0.15) 39.5 (0.10) 34.9 (0.08) 32.8 (0.07) 28.3 (0.03) 24.2 (0.05) 19.2 (0.04) 17.6 (0.03) 8.5 (0.02) 6.8 (0.01) 5.7 (0.01) 2.8 2.8
h = 11.27 * = 13.00 a 30.0% ARLcusum
60.5 (0.18) 52.9 (0.16) 40.3 (0.10) 35.2 (0.08) 33.5 (0.08) 28.1 (0.03) 24.5 (0.05) 19.3 (0.04) 17.6 (0.03) 8.2 (0.02) 6.7 (0.01) 5.5 (0.01) 2.9 2.9
h = 8.67 * = 14.50 a 45.0% ARLcusum
68.6 (0.21) 60.3 (0.18) 46.3 (0.13) 40.3 (0.10) 38.1 (0.09) 29.7 (0.04) 27.6 (0.06) 22.3 (0.05) 19.8 (0.04) 9.2 (0.02) 7.1 (0.01) 5.8 (0.01) 2.8 2.8
15, 19, 23, 30, 35 21, 26, 30, 34, 41 11, 16, 22, 28, 31 5, 12, 21, 26, 36 17, 27, 31, 33, 35 27, 44, 49, 51, 55 42, 49, 55, 62, 65 18, 26, 28, 33, 35 16, 22, 32, 35, 40 14, 16, 20, 22, 23 32, 36, 38, 41, 42 19, 20, 22, 24, 25 23, 25, 29, 31, 32 42, 46, 52, 54, 58
159
Table 6.22. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 10 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 197
1 , 2 , 3 , 4 , 5
0.20, 0.30, 0.50, 0.80, 1.25 0.30, 0.50, 0.60, 1.60, 2.10 0.10, 0.40, 0.90, 1.20, 1.45 0.40, 0.50, 1.10, 1.30, 1.50 0.05, 1.10, 1.15, 1.75, 2.50 0.10, 0.25, 2.27, 2.54, 2.62 0.30, 0.90, 1.60, 1.70, 2.20 1.00, 1.30, 1.70, 1.90, 2.30 0.90, 1.50, 2.10, 2.50, 3.00 2.50, 2.60, 2.75, 2.80, 3.00 2.75, 3.15, 3.62, 3.77, 3.79 3.00, 3.50, 3.75, 4.20, 4.37 6.00, 6.15, 6.35, 6.75, 7.12 6.00, 7.25, 7.36, 8.12, 8.50
B = 5.00
1, 2 , 3 4 , 5
B = 4.67 ARLMR
90.1 (0.26) 76.0 (0.21) 53.8 (0.13) 45.5 (0.10) 42.2 (0.09) 31.0 (0.03) 29.2 (0.05) 23.5 (0.05) 20.9 (0.03) 9.4 (0.02) 7.2 (0.01) 5.7 (0.01) 2.3 2.3
ARLOR
83.1 (0.24) 70.3 (0.20) 49.5 (0.12) 43.3 (0.09) 39.0 (0.08) 30.1 (0.03) 27.2 (0.05) 21.6 (0.04) 19.5 (0.03) 8.8 (0.02) 6.9 (0.01) 5.5 (0.01) 2.3 2.3
h =10.057 * = 7.00 a 30.0% ARLcusum
88.2 (0.27) 76.0 (0.23) 53.7 (0.14) 45.2 (0.11) 42.0 (0.10) 30.1 (0.03) 28.5 (0.06) 22.4 (0.05) 19.8 (0.04) 8.4 (0.02) 6.6 (0.01) 5.3 (0.01) 2.6 2.6
h = 6.51 * = 5.50 a 45.0% ARLcusum
100.8 (0.31) 88.9 (0.27) 65.1 (0.18) 54.9 (0.14) 51.6 (0.14) 32.6 (0.04) 34.9 (0.08) 28.4 (0.07) 23.7 (0.05) 9.9 (0.02) 7.2 (0.01) 5.5 (0.01) 2.3 2.3
15, 19, 23, 30, 35 21, 26, 30, 34, 41 11, 16, 22, 28, 31 5, 12, 21, 26, 36 17, 27, 31, 33, 35 27, 44, 49, 51, 55 42, 49, 55, 62, 65 18, 26, 28, 33, 35 16, 22, 32, 35, 40 14, 16, 20, 22, 23 32, 36, 38, 41, 42 19, 20, 22, 24, 25 23, 25, 29, 31, 32 42, 46, 52, 54, 58
160
Table 6.23. ARL comparison for increasing multiple step change case with 5 change points, 0 = 20 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 126
1 , 2 , 3 , 4 , 5
0.20, 0.30, 0.50, 0.80, 1.25 0.30, 0.50, 0.60, 1.60, 2.10 0.10, 0.40, 0.90, 1.20, 1.45 0.40, 0.50, 1.10, 1.30, 1.50 0.05, 1.10, 1.15, 1.75, 2.50 0.10, 0.25, 2.27, 2.54, 2.62 0.30, 0.90, 1.60, 1.70, 2.20 1.00, 1.30, 1.70, 1.90, 2.30 0.90, 1.50, 2.10, 2.50, 3.00 2.50, 2.60, 2.75, 2.80, 3.00 2.75, 3.15, 3.62, 3.77, 3.79 3.00, 3.50, 3.75, 4.20, 4.37 6.00, 6.15, 6.35, 6.75, 7.12 6.00, 7.25, 7.36, 8.12, 8.50
B = 4.00
1, 2 , 3 4 , 5
B = 3.56 ARLMR
73.2 (0.22) 65.9 (0.20) 52.6 (0.14) 46.4 (0.11) 44.2 (0.11) 33.5 (0.05) 33.0 (0.07) 27.7 (0.07) 24.5 (0.05) 13.4 (0.03) 10.1 (0.02) 8.7 (0.02) 4.2 (0.01) 4.1 (0.01)
ARLOR
70.9 (0.21) 63.5 (0.19) 50.8 (0.14) 45.3 (0.11) 43.2 (0.11) 32.9 (0.05) 32.2 (0.07) 27.1 (0.07) 23.9 (0.05) 13.1 (0.03) 10.0 (0.02) 8.6 (0.02) 4.2 (0.01) 4.1 (0.01)
h = 12.17 * = 26 a 30.0% ARLcusum
77.1 (0.23) 69.7 (0.21) 56.8 (0.16) 50.0 (0.14) 48.3 (0.13) 35.1 (0.06) 36.2 (0.09) 31.0 (0.08) 26.7 (0.06) 14.4 (0.04) 10.6 (0.02) 9.0 (0.02) 4.2 (0.01) 4.0 (0.01)
h = 8.77 * = 29 a 45.0% ARLcusum
81.7 (0.25) 75.4 (0.23) 62.6 (0.18) 56.0 (0.16) 54.0 (0.15) 38.6 (0.07) 41.7 (0.11) 36.2 (0.10) 31.1 (0.08) 17.7 (0.05) 12.6 (0.03) 10.8 (0.03) 4.5 (0.01) 4.2 (0.01)
15, 19, 23, 30, 35 21, 26, 30, 34, 41 11, 16, 22, 28, 31 5, 12, 21, 26, 36 17, 27, 31, 33, 35 27, 44, 49, 51, 55 42, 49, 55, 62, 65 18, 26, 28, 33, 35 16, 22, 32, 35, 40 14, 16, 20, 22, 23 32, 36, 38, 41, 42 19, 20, 22, 24, 25 23, 25, 29, 31, 32 42, 46, 52, 54, 58
161
Table 6.24. ARL comparison for decreasing multiple step change case with 5 change points, 0 = 20 . Each estimated ARL is based on N = 100,000 independently-seeded runs. Rounded standard errors greater than or equal to 0.01 are shown in parentheses. ARL0 210 h = 10.64 * = 14.0 a 30.0% ARLcusum
49.5 (0.14) 68.1 (0.20) 31.9 (0.08) 19.6 (0.05) 36.4 (0.08) 35.2 (0.08) 83.5 (0.25) 16.5 (0.04) 35.9 (0.09) 5.0 (0.01) 5.9 (0.01) 5.9 (0.01) 8.2 (0.02) 41.7 (0.12)
B = 5.00 ARLOR
39.0 (0.10) 54.9 (0.14) 25.8 (0.05) 17.0 (0.03) 30.5 (0.05) 29.8 (0.05) 69.9 (0.20) 14.7 (0.03) 29.3 (0.06) 5.2 (0.01) 6.0 (0.01) 6.1 (0.01) 8.0 (0.01) 32.2 (0.08)
B = 4.65 ARLMR
42.3 (0.10) 59.1 (0.15) 27.8 (0.06) 18.1 (0.04) 32.5 (0.06) 31.8 (0.05) 75.1 (0.20) 15.6 (0.03) 31.4 (0.06) 5.3 (0.01) 6.2 (0.01) 6.2 (0.01) 8.3 (0.02) 35.1 (0.09)
1 , 2 , 3 , 4 , 5
0.50, 1.25, 1.50, 1.75, 2.25 0.25, 0.75, 1.15, 1.35, 4.25 1.15, 1.65, 2.13, 2.26, 3.75 2.15, 2.65, 2.85, 3.00, 3.25 0.65, 0.90, 2.15, 2.65, 3.70 1.00, 2.00, 2.30, 2.35, 2.50 0.60, 0.70, 0.90, 4.67, 5.35 2.10, 3.10, 3.30, 3.60, 7.25 0.96, 1.35, 2.10, 2.25, 8.25 5.00, 5.25, 6.85, 6.90, 7.12 4.41, 5.26, 5.85, 6.87, 9.35 1.65, 1.85, 6.80, 6.90, 6.95 3.62, 4.15, 4.25, 5.69, 6.12 1.43, 1.53, 1.66, 9.54, 9.95
1 , 2 , 3 4 , 5
15, 19, 23, 30, 35 21, 26, 30, 34, 41 11, 16, 22, 32, 39 5, 12, 21, 26, 36 17, 27, 31, 33, 35 27, 44, 49, 51, 55 42, 49, 55, 62, 65 18, 26, 28, 33, 35 16, 22, 32, 35, 40 14, 16, 20, 22, 23 32, 36, 38, 41, 42 19, 20, 22, 24, 25 23, 25, 29, 31, 32 42, 46, 52, 54, 58
h = 6.16 * = 11.0 a 45.0% ARLcusum
65.4 (0.19) 85.6 (0.26) 43.4 (0.12) 27.1 (0.07) 47.3 (0.12) 45.0 (0.11) 101.6 (0.31) 22.2 (0.05) 47.3 (0.13) 5.6 (0.01) 6.9 (0.01) 6.3 (0.01) 11.0 (0.03) 57.0 (0.17)
162
Three illustrations of the order-restricted control charting procedure will be performed using simulated data generated from known distributions. parameter of = 5. For the first illustration, the data are initially drawn from a Poisson process with an in-control rate Starting with subgroup 51, the data are drawn from a Poisson process with a higher rate of = 7 . As a result, the example dataset is in-control for the first 50 subgroups and experiences a step change such that an increase of 2 in the mean count rate occurs beginning with subgroup 51. For the second illustration, the data are initially drawn from a Poisson process with an in-control rate parameter of = 5. Starting with subgroup 51, however, the data are drawn from a Poisson process with a rate of = 5 + 0.20(i 50 ) , where 51 i T represents the subgroup index. This change-type is recognized as that of a linear trend with a constant slope of = 0.20 . Finally, for the third illustration, the data are initially drawn from a Poisson process with an in-control rate parameter of = 5. However, three step changes in the process occur with magnitudes 1 = 0.45, 2 = 0.70 and 3 = 1.10 and at corresponding change points 1 = 50, 2 = 61 and 3 = 65 . Single realizations of the above described data sets were generated and the resulting control chart statistic RT was plotted against the upper control limit B = 4.00. Figure 6-1 shows results of the order-restricted control chart for the first illustration (i.e., step change in from 5 to 7). The control chart signaled at subgroup 64 in this case, yielding a run length of 14 for this realization. Once the order-restricted control chart signals, diagnostics are immediately available regarding the change point and the unknown process profile. Figure 6-2 shows the diagnostics for the realization shown in Figure 6-1, where the order-restricted control chart signals at subgroup T = 64. Figure 6-2(a) shows computed values for R at each potential change point t. The largest value for R indicates the most likely change point . A confidence set on the most likely change points can also be developed. Figure 6-2(b) shows the maximum likelihood estimate for the unknown process profile.
163
Order-restricted Control Chart 4.5
4
3.5
3
2.5 RT
2
1.5
1
0.5
0
10
20
30
40
50
60
Figure 6-1. A single realization of the order-restricted control chart for data experiencing a shift in from 5 to 7 following the 50th sample count obtained. This realization signals at subgroup T = 64, yielding a runlength of 14.
Subgroup Index
Rv ersus t 4.5 8
Estimated Process Profile
4
= 50
7.5
3.5
3 Magnitude of Change Likelihood ratio
7
2.5
6.5
2
1.5
6
1 5.5 0.5
0 0
10
20
30 40 potential change point t
50
60
70
5
10
20
30
40
50
60
Subgroup Index
Figure 6-2(a)
Figure 6-2(b)
Figure 6-2. Diagnostic plots for data realization 1. 6-2(a): R versus potential change point. Large values correspond to most likely change points. 6-2(b): Estimated process profile versus subgroup number.
164
The diagnostic plots in Figures 6.2(a,b) indicate that the process change point occurred at = 50 , while the estimated process profile gives good indication that the type of change present is a step change from = 5 to = 7 following subgroup 50. These estimates are quite accurate since the true process change was indeed a step change in from = 5 to = 7 and did occur following the 50th subgroup. Figure 6-3 shows results of the order-restricted control chart for the second illustration (i.e., linear trend in of magnitude = 0.20 ). The control chart signaled at subgroup 64 in this case, yielding a run length of 14 for this realization.
Order-restricted Control Chart 5
4.5 4
3.5 3
RT
2.5 2
1.5 1
0.5 0
10
20
30
40
50
60
Subgroup Index
Figure 6-3. A single realization of the order-restricted control chart for data experiencing a linear trend in of magnitude = 0.20 following the 50th sample count obtained. This realization signals at subgroup T = 64, yielding a runlength of 14.
165
Figure 6-4 shows the diagnostics for the realization shown in Figure 6-3, where the order-restricted control chart signals at subgroup T = 64. Figure 6-4(a) shows computed values for R at each potential change point t. The largest value for R indicates the most likely change point . A confidence set on the most likely change points can also be developed. Figure 6-4(b) shows the maximum likelihood estimate for the unknown process profile. The diagnostic plot in Figure 6-4(a) indicates that the process change point occurred at = 50 , while the estimated process profile in Figure 6-4(b) gives good indication that the type of change is one involving multiple change points following subgroup 50. These estimates are again quite accurate since the true process change was one involving multiple change points (i.e. linear trend) following the 50th subgroup.
Rv ersus t 5 11
Estimated Process Profile
4.5
= 50
10
4 9 Magnitude of Change 3.5 Likelihood ratio
3
8
2.5
7 2 6 1.5
1 0
10
20
30 40 potential change point t
50
60
70
5
10
20
30
40
50
60
Subgroup Index
Figure 6-4(a)
Figure 6-4(b)
Figure 6-4. Diagnostic plots for data realization 2. 6-4(a): R versus potential change point. Large values correspond to most likely change points. 6-4(b): Estimated process profile versus subgroup number.
Finally, Figure 6-5 shows results of the order-restricted control chart for the third illustration (i.e., multiple step changes). The control chart signaled at subgroup 68 in this case, yielding a run length of 18 for this realization.
166
Figure 6-6 shows the diagnostics for the realization shown in Figure 6-5, where the order-restricted control chart signals at subgroup T = 68. Figure 6-6(a) shows computed values for R at each potential change point t. The largest value for R indicates the most likely change point . A confidence set on the most likely change points can also be developed. Figure 6-6(b) shows the maximum likelihood estimate for the unknown process profile. The diagnostic plot in Figure 6-6(a) indicates that the process change point occurred at = 50 , while the estimated process profile in Figure 6-6(b) gives good indication that the type of change is one involving multiple change points following subgroup 50. Again, these estimates are quite accurate since the true process change was one involving multiple change points (i.e. multiple step changes) following the 50th subgroup. To design the order-restricted control chart, one must select an upper control limit B to obtain a desired in-control ARL of ARL0. For Poisson data, this ARL is dependent upon the value of the in-control rate parameter as well as B. To obtain a value for B, simulation must be used. Thus, Figure 6-7 provides pseudo-code for obtaining the control limit B and is easily implemented using a programming language like C.
6.6. Discussion
The order-restricted control chart offers some significant advantages over existing control charts. In addition to having more robust ARL performance (that is, across monotonic change types and corresponding change magnitudes), the order-restricted control chart also provides valuable information to process engineers concerning the time of the change, type of monotonic change and magnitude of change at any given change point. Although the magnitude-robust control chart also provides an estimate of the process change point, Perry, Pignatiello and Simpson (2004d) showed that it is inferior to the estimator offered by the order-restricted control chart when multiple change points exist. This assumes that the initial change is not one of an extreme magnitude. This is
167
Order-restricted Control Chart 6
5
4
RT
3
2
1
0
10
20
30
40
50
60
70
Subgroup Index
Figure 6-5. A single realization of the order-restricted control chart for data experiencing multiple step changes in following the 50th sample count obtained. This realization signals at subgroup T = 68, yielding a runlength of 18.
Rv ersus t 5.5 5 11
Estimated Process Profile
= 50
10
4.5 4 9 Likelihood ratio 3.5 3 Magnitude of Change
8
2.5 2
7
1.5 6 1
0.5 0
10
20
30 40 potential change point t
50
60
70
5
10
20
30
40
50
60
70
Subgroup Index
Figure 6-6(a)
Figure 6-6(b)
Figure 6-6. Diagnostic plots for data realization 3. 6-6(a): R versus potential change point. Large values correspond to most likely change points. 6-6(b): Estimated process profile versus subgroup number.
168
% Simulate OR control chart to obtain in-control ARL for n=1:N i=0; in_control = 1; while in_control == 1 i=i+1; generate X(i); compute RT(i); if RT(i)>=B runlength(n)= i; ave_runlength(n)= mean(runlength); in_control = 0; clear X RT end end end Y = ave_runlength(n)
Figure 6-7. Pseudo-code for estimating in-control ARLs for the order-restricted control chart. Inputs into the program include B and N, the control limit and total number of simulation runs, respectively. The value of RT(i) is computed as given in equation (6.9). primarily due to the fact that the magnitude-robust procedure is derived under singlepoint step change assumptions. The CUSUM procedure is also derived under the assumption that a single-point step change is present. Furthermore, this procedure requires a user-specified level of change in its design. Since rarely in practice is this value a known quantity, designing these procedures can become problematic. Even further, since the type of change is rarely known exactly, it becomes even less evident as to which level of change these procedures should be designed for. Finally, the CUSUM procedure also offers an estimate of the unknown process change point, however, confidence sets on the point of process change are not readily available using this method. Identifying which combination of the process variables is responsible for a change in a process allows engineers to improve quality by preventing or avoiding changes in those variables which lead to poor quality and by perpetuating those changes and optimizing those variables which can lead to better quality. Knowing when a process has
169
changed, how it has changed and by how much can aid in the search for the special cause. If the time of the change and type of change could be determined, process engineers would have a smaller search window within which to look for the special cause(s). Consequently, the special cause(s) can be identified more quickly and the appropriate actions can be implemented sooner to improve quality. When using the magnitude-robust or CUSUM procedures, it is assumed that the type of process disturbance is a steady and persistent step change. In these cases, if the type of change present is one other than a step change, process engineers can easily be mislead into changing process input variables that do not require adjustment. Thus, when the change-type is unknown (however, still monotonic in nature), implementation of the proposed order-restricted procedure can reduce or eliminate any unnecessary adjustments to process variables by providing more information regarding the exact nature of the underlying change.
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CHAPTER 7 CONCLUSION
The primary focus of this research was to develop and evaluate new robust statistical process control-charting methodologies for use with Poisson count processes. General results indicate that the methods proposed throughout this dissertation provide robust detection and estimation performances with regard to the magnitude of change and the type of monotonic change that may be present. Along with quick detection performance, implementation of the methodologies introduced throughout this research can provide process engineers with valuable diagnostic statistics to aid in the search for the special cause(s) affecting the process. For example, Chapter 2 developed the maximum likelihood estimator (MLE) for the point of process change and evaluated the quality of this estimator when applied to a variety of well documented control-charting procedures (i.e., c-chart, CUSUM and EWMA). It was concluded in this chapter that the MLE of the process change point outperforms the change point estimators offered by the CUSUM and EWMA controlcharting procedures over a range of possible change magnitudes. Furthermore, using the MLE of the process change point allows for a confidence set on the true change point to be constructed, providing process engineers with a window of potential change point candidates from which to begin their search for the special cause(s) affecting the process. Even further, along with an estimate of the change point, an estimate of the magnitude of change is readily available uising the proposed procedure. Knowing the magnitude of change can further help to reduce the cost of locating the cause of process change. In Chapter 3, a magnitude-robust control chart for Poisson count process was developed and evaluated. Results of this chapter show that unless the magnitude of process change is known apriori, the ARL performances of this procedure outperform those provided by any one CUSUM procedure. Although the magnitude-robust control chart will not provide the best ARL performance for any one distinct level of change, it will provide nearly the best ARL performance for all levels of change. Furthermore, the
171
diagnostic statistics provided by the magnitude-robust procedure are superior to those offered by the CUSUM procedure, and are readily available upon signaling. For example, when the magnitude-robust control chart issues a signal, the maximum likelihood estimates of the process change point and out-of-control rate parameter value are immediately provided to process engineers. Additionally, this procedure supports the construction of confidence sets on the process change point, as well as, confidence intervals on the unknown out-of-control rate parameter value. The change point estimation methodology described in Chapter 2 assumes that the type of change present is a steady and persistent shift in the underlying distribution of the process. It is possible, however, that a Poisson process could exhibit a linear trend disturbance, say, as a result of tool wear. Thus, in Chapter 4 we developed and evaluated a maximum likelihood estimator designed for Poisson count processes that experience increasing linear trends. We made direct performance comparisons between the MLE designed for step changes and the MLE designed for linear trends when a linear trend disturbance is present. Results showed that the MLE for linear trends outperforms the MLE for step changes in the sense that the MLE for step changes tends to overestimate the true process change point. Such a result can be misleading to process engineers, resulting in unnecessary adjustments being made to the process. We also evaluated the confidence set estimators using the two MLEs and found that when a linear trend disturbance is present, better confidence sets (i.e., increased coverage with reduced cardinality) are obtained using the MLE designed for linear trends. Both of the change point estimation methods described in Chapters 2 and 4 assume that the underlying process change is known exactly. For example, the MLE developed in Chapter 2 assumes that the underlying change-type is a single-point step change, while the MLE developed in Chapter 4 assumes the change-type is a linear trend. Rarely in practice, however, is the exact type of change known apriori. Instead, process engineers may only know that the type of change present is monotonic in nature. As such, Chapter 5 developes and evaluates a maximum likelihood estimator for the change point that does not require knowledge of the exact change-type exhibited by the process. Rather, the only assumption is that the change-type present can be described as belonging
172
to a family of monotonic change-types (i.e., either isotonic or antitonic). Such a family includes the single-point step and linear trend change-types, as well as, an infinite number of other monotonic change-types. Direct performance comparisons were made between the MLEs designed for step changes and linear trends and the MLE designed for monotonic changes. General results indicate that the MLE designed for monotonic changes outperforms the MLEs designed under strict change-type assumptions when change-types consisting of multiple change points are present. That is, unless the initial magnitude of change is large, the MLEs for step changes and linear trends tend to overestimate the true process change point. Furthermore, along with an estimate for the process change point, the MLE designed for monotonic changes also offers process engineers the maximum likelihood estimate of the unknown process profile. This is significant as such a tool can provide valuable information pertaining to the correlation between the magnitude of changes, and the times at which the changes occurred. This additional information can help process engineers to better pin-point the underlying cause(s) of process change. Monte Carlo simulation was used to evaluate the average performance of the estimated process profile. Results suggest that the MLE of the process profile does well at estimating the unknown process behavior following the initial point of change. Finally, in Chapter 6, control charts for monotonic changes in Poisson count processes were developed and evaluated. The motivation for this chapter lies in the fact that the Poisson CUSUM and magnitude-robust control-charting procedures are designed for detecting single-point step changes in a Poisson rate parameter. Since in Chapter 2 it was determined that the performance of the CUSUM procedure is sensitive to changes in the magnitude of change that a process may exhibit, the implication is that its performance is sensitive to the type of change that a process may exhibit. This is because the type of change present depends upon the relative locations of the change points and the magnitudes of change that correspond to these change points. In this chapter, ARL performance comparisons were made between the Poisson CUSUM, magnitude-robust and the order-restricted control chart (i.e., control chart for montonic chnges) across three distinct change-types. Namely, single-point step changes, linear trends and multiple step change scenarios. General results indicate that, although
173
marginal, the order-restricted procedure does provide the better ARL performances across the various change-types, as well as, the various magnitudes of change associated with these change-types. It is apperant that a CUSUM procedure could be designed to perform best for a given range of change magnitudes (that is, given a particular changetype), however, its performance will suffer for other regions of change magnitudes. The order-restricted procedure will not provide the best ARL performance for a given changetype and corresponding change magnitude, however, it will provide nearly the best ARL performance for all monotonic change-types and corresponding change magnitudes. In this sense, the order-restricted procedure is robust to the type of monotonic change (as well as the magnitude of that change) that a process may be exhibiting. significant as rarely is the type of change known apriori. The ARL performance of the order-restricted control charting procedure outlined in Chapter 6 performs very similar to the magnitude-robust control chart outlined in Chapter 2. The exception is that for smaller change magnitudes (regardless of the change-type), the order-restricted procedure will provide more power than the magnitude-robust. Additionally, the order-restricted procdure will, in general, outperform the magnitude-robust procedure when multiple step changes are present. On the other hand, when the magnitude of change is large, the magnitude-robust will perform as good, or in some cases, marginally better than the order-restricted procedure. Thus, the real advantage to using the order-restricted procedure as an alternative to the CUSUM or magnitude-robust control charts lies in the diagnostic information provided upon signaling. That is, when the order-resticted control chart signals, the MLEs of the process change point and unknown process profile are immediately provided to aid process engineers in diagnosing the cause(s) of process change. Analysis of the profile estimate can give a subjective indication as to whether the type of change is one consisting of single or multiple change points. For example, Figure 7-1 shows an estimated process profile suggesting the type of change present is one consisting of a single change point. This is the same diagnostic plot that was given in Chapter 6 for a step change in from = 5 to = 7 . We see that, except for the last subgroup, the estimated process profile suggests that the process was in control for the first 50 subgroups, but then experienced a step change beginning This is
174
with subgroup 51. Although subject to some error, this type of pattern in the process profile was found to be typical of processes that experience a single-point step change. On the other hand, through analysis of the process profile shown in Figure 7-2, we have subjective indication that the process has experienced a change indicative of multiple change points. Figure 7-2 is the same diagnostic plot that was given in Chapter 6 for a linear trend change in with slope parameter = 0.20 . Again, although subject to error, this type of pattern in the process profile was found to be typical of change-types consisting of multiple change points. Further research could be conducted in attempts to remove the subjective nature of the analysis of the profile estimate in attempts to classify the type of change as either one consisting of a single change point or multiple change points. For example, a trained artificial neural network may warrant investigation here.
Estimated Process Profile 8
7.5
7 Magnitude of Change
6.5
6
5.5
5
10
20
30
40
50
60
Subgroup Index
Figure 7-1. Estimated process profile suggesting a change-type consisting of a single change point at = 50 .
175
Estimated Process Profile 11
10
9 Magnitude of Change
8
7
6
5
10
20
30
40
50
60
Subgroup Index
Figure 7-2. Estimated process profile suggesting a change-type consisting of multiple change points and an initial change point of = 50 .
Future research could also include the development of a control-charting procedure for Poisson counts that quickly detects sporatic changes in . For example, a process could suddenly behave in a chaotic manner, fluctuating between the in-control and out-of-control states, and in both the increasing and decreasing directions. In these situations, the procedures outlined in this research will not likely perform well due to an averaging effect in our estimate of the out-of-control rate parameter value. One possible solution is to apply some level of constraint to the estimate of the process behavior that is indicative of the sporatic behavior one wishes to detect. At this point, however, more investigation is needed in order to define the appropriate constraint (or constraints) that should be applied in these situations.
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The methodologies proposed throughout this dissertation research are applicable to Poisson count processes and have been shown to provide robust detection and estimation performances relative to other procedures. Thus, in general, we conclude that unless the type and magnitude of process change are known apriori, the procedures outlined in this dissertation should be considered. Furthermore, the control-charting procedures developed offer process engineers valuable diagnostic information to aid in the search for the special cause(s) affecting the process. This is significant as current procedures only provide insight into whether or not a process has changed. Knowing when a process changed, how a process changed, and by how much can help to reduce or eliminate the costs associated with locating the cause of process change. Additionally, when a control chart signals, the methods proposed throughout this research can help to reduce or eliminate any unnecessary adjustments being made to the process.
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APPENDIX
Table A1. ANOVA table for experimental design. Response is SC
ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Model 146.39 5 29.28 72.27 A 43.89 1 43.89 108.34 B 75.26 1 75.26 185.76 C 3.33 1 3.33 8.22 AB 18.28 1 18.28 45.11 BC 5.64 1 5.64 13.92 Residual 4.05 10 0.41 Lack of Fit 0.20 2 0.098 0.20 Pure Error 3.85 8 0.48 Cor Total 150.44 15 Std. Dev. Mean C.V. 0.64 35.78 1.78 R-Squared Adj R-Squared Pred R-Squared 0.9731 0.9596 0.9311
Prob > F < 0.0001 < 0.0001 < 0.0001 0.0167 < 0.0001 0.0039 0.8199
significant
not significant
PRESS
10.37
Adeq Precision
22.674
D SIG -E E T P E N XP R lot Tau-hat-sc A: tau2-tau1 B: lam bda1 C lam : bda2
N al plot orm
99
95
A BC
Norm % probability al
90 80 70 50 30 20 10 5
C A B B
1
-4.34
-2.42
-0.51
1.40
3.31
Effect
D SIG -E E T P E N XP R lot Tau-hat-sc X = A: tau2-tau1 Y = B: lam bda1 B- 21.000 B+ 22.000 Actual F actor C lam : bda2 = 23.50
39.55 42
Interaction G raph
B: lambda1
D SIG -E E T P E N XP R lot Tau-hat-sc X = B: lam bda1 Y = C lam : bda2 C 23.000 C 24.000 + Actual F actor A: tau2-tau1 = 15.00
39.55 42
Interaction G raph
C: lambda2
Tau-hat-sc
Tau-hat-sc
37.1
37.1
34.65
34.65
32.2
32.2
10.00
12.50
15.00
17.50
20.00
21.00
21.25
21.50
21.75
22.00
A tau2-tau1 :
B: lambda1
Figure A1. Normal probability and interaction plots for 2 3 experiment. Response is SC .
178
Table A2. ANOVA table for experimental design. Response is LT
ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Model 142.50 5 28.50 84.51 A 42.25 1 42.25 125.28 B 76.56 1 76.56 227.02 C 1.00 1 1.00 2.97 AB 18.06 1 18.06 53.56 BC 4.62 1 4.62 13.71 Residual 3.37 10 0.34 Lack of Fit 0.45 2 0.23 0.62 Pure Error 2.92 8 0.37 Cor Total 145.87 15 Std. Dev. Mean C.V. 0.58 32.02 1.81 R-Squared Adj R-Squared Pred R-Squared 0.9769 0.9653 0.9408
Prob > F < 0.0001 < 0.0001 < 0.0001 0.1158 < 0.0001 0.0041 0.5620
significant
not significant
PRESS
8.63
Adeq Precision
24.4640
D SIG -E E T P E N XP R lot Tau-hat-lt A: tau2-tau1 B: lam bda1 C lam : bda2
N al plot orm
99
95
A BC
Norm % probability al
90 80 70 50 30 20 10 5
C A B B
1
-4.38
-2.47
-0.56
1.34
3.25
Effect
D SIG -E E T P E N XP R lot Tau-hat-lt X = A: tau2-tau1 Y = B: lam bda1 B- 21.000 B+ 22.000 Actual Factor C lam : bda2 = 23.50
35.625 38
Interaction G raph
B: lambda1
D SIG -E E T P E N XP R lot Tau-hat-lt X = B: lam bda1 Y = C lam : bda2 C 23.000 C 24.000 + Actual Factor A: tau2-tau1 = 15.00
35.625 38
Interaction G raph
C: lambda2
Tau-hat-lt
33.25
Tau-hat-lt
33.25
30.875
30.875
28.5
28.5
10.00
12.50
15.00
17.50
20.00
21.00
21.25
21.50
21.75
22.00
A tau2-tau1 :
B: lambda1
Figure A2. Normal probability and interaction plots for 2 3 experiment. Response is LT .
179
Table A3. ANOVA table for experimental design. Response is OR
ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Model 174.81 4 43.70 125.78 A 78.77 1 78.77 226.70 B 81.45 1 81.45 234.43 C 4.52 1 4.52 13.00 AB 10.08 1 10.08 29.01 Residual 3.82 11 0.35 Lack of Fit 0.89 3 0.30 0.81 Pure Error 2.93 8 0.37 Cor Total 178.63 15 Std. Dev. Mean C.V. PRESS 0.59 29.13 2.02 8.09 R-Squared Adj R-Squared Pred R-Squared Adeq Precision 0.9786 0.9708 0.9547 30.3860
Prob > F < 0.0001 < 0.0001 < 0.0001 0.0041 0.0002 0.5251
significant
not significant
D SIG -E E T P E N XP R lot tau-hat-or A: tau2-tau1 B: lam bda1 C lam : bda2
N al plot orm
99
D SIG -E E T P E N XP R lot tau-hat-or X = A: tau2-tau1 Y = B: lam bda1 B- 21.000 B+ 22.000 Actual Factor C lam : bda2 = 23.50
32.35 34.9
Interaction G raph
B: lambda1
95
A
Norm % probability al
90 80 70 50 30 20 10 5
tau-hat-or
29.8
C A B B
27.25
1
24.7
-4.51
-2.27
-0.04
2.20
4.44
10.00
12.50
15.00
17.50
20.00
Effect
A tau2-tau1 :
Figure A3. Normal probability and interaction plots for 2 experiment. Response is OR .
3
180
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1. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T., and Silverman, E. (1955). An Empirical Distribution Function for Sampling with Incomplete Information, Annals of Mathematical Statistics 26, pp. 641-647. 2. Best, M. J. and Chakravarti, N. (1990). Active Set Algorithms for Isotonic Regression; A Unifying Framework, Mathematical Programming 47, pp. 425439. 3. Borror, C. M.; Champ, C. W.; and Rigdon, S. E. (1998). Poisson EWMA Control Charts. Journal of Quality Technology 30(4), pp.352-361. 4. Box, G. E. P. and Cox, D. R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society B26, pp. 211-243. 5. Brook, D. and Evans, D. A. (1972). An Approach to the Probability Distribution of CUSUM Run Length. Biometrika 59, pp. 539-549. 6. Gan, F. F. (1990). Monitoring Poisson Observations Using Modified Exponentially Weighted Moving Average Control Charts. Communications in Statistics Simulation and Computation 19(1), pp.103-124. 7. Grotzinger, S. J. and Witzgall, C. (1984). Projections onto Order Simplexes, Applied Mathematics and Optimization 12, pp. 247-270. 8. Hawkins, D. M., and Olwell, D. H., (1998). Cumulative Sum Charts and Charting for Quality Improvement, Springer Verlag, New York. 9. Lucas, J. M. (1985). Counted Data CUSUMs. Technometrics 27(2), pp.129144. 10. Lucas and Crosier (1982). Fast Initial Response for CUSUM Quality-Control Schemes: Give Your CUSUM a Headstart, Technometrics 24, pp. 199-205. 11. Lucas, J. M. and Saccucci, M. S. (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics 32(1), pp. 1-12. 12. Montgomery, D. C. (2001). Introduction to Statistical Quality Control, 4th ed. John Wiley & Sons, New York, NY.
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13. Moustakides, G. V. (1986). Optimal Stopping Times for Detecting Changes in Distributions, The Annals of Statistics 14(4), pp. 1379-1387. 14. Nishina, K (1992). A Comparison of Control Charts from the Viewpoint of Change point Estimation. Quality and Reliability Engineering International 8, pp. 537-541. 15. Page, E. S. (1954). Continuous Inspection Schemes. Biometrika 41, pp. 1-9. 16. Page, E. S. (1961). Cumulative Sum Control Charts, Technometrics 3, pp. 1-9. 17. Pignatiello, J. J. Jr. and Samuel, T. R. (2001). Estimation of the Change point of a Normal Process Mean in SPC Applications. Journal of Quality Technology 33(1), pp.82-95. 18. Pignatiello, J. J., Jr. and Simpson J. R. (2002). A Magnitude-Robust Control Chart for Monitoring and Estimating Step Changes for Normal Process Means, Quality and Reliability Engineering International 18, pp. 1-13. 19. Perry, M. B., Pignatiello, J. J., Jr. and Simpson, J. R.. (2004a) Estimation of the Change Point of a Poisson Rate Parameter for SPC Applications, IIE Transactions (Submitted for publication). 20. Perry, M. B., Pignatiello, J. J., Jr. and Simpson, J. R.. (2004b) A MagnitudeRobust Control Chart for Monitoring and Estimating Step Changes in a Poisson Rate Parameter, IIE Transactions (Under Review). 21. Perry, M. B., Pignatiello, J. J., Jr. and Simpson, J. R.. (2004c) Estimating the Change Point of a Poisson Rate Parameter with a Linear Trend Disturbance in SPC Application, Quality and Reliability Engineering International (Under Review). 22. Perry, M. B., Pignatiello, J. J., Jr. and Simpson, J. R.. (2004d) Estimating the Change Point of a Poisson Rate parameter with an Isotonic Change Disturbance in SPC Application, (Submitted for publication). 23. Rardin, R. L. (2000). Optimization in Operations Research. Prentice Hall Upper Saddle River, New Jersey. 24. Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics 1, pp. 239-250. 25. Robertson, T., Wright, F.T., and Dykstra, R.L. (1988). Order Restricted Statistical Inference. John Wiley & Sons, New York, NY.
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26. Samuel, T. R. and Pignatiello, J. J. Jr. (1998). Identifying the Time of a Change in a Poisson Rate Parameter. Quality Engineering 10(4), pp. 673-681. 27. Siegmund, D. (1986). Boundary Crossing Probabilities and Statistical Applications. Annals of Statistics 14, pp. 361-404. 28. Trevanich, A and Bourke, P (1993). EWMA Control Charts Using Attributes Data. The Statistician 42, p. 215.
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BIOGRAPHICAL SKETCH
Educational Background
Degree Doctor of Philosophy in In du st r ia l E ngineer in g Emphasis: Qu a lit y E ngineer in g/Applied St a t ist ics Thesis advisor: Dr. J oseph J . P ign a t iello J r . Master of Science in Ma nu fa ctu r in g Syst em s Thesis advisor: Dr. J u lie K. McBr ide Month/Year Awarded Institution F lor ida St a t e Un iver sit y Ta lla h a ssee, F L
06/04
08/00
Sou t h er n Illin ois Un iver sit y Ca r bon da le, IL Sou t h er n Illin ois Un iver sit y Ca r bon da le, IL
Bachelor of Science in In du st r ia l Tech nology
05/98
Teaching Appointments
Position Assist a n t P r ofessor Dates Employer Air F or ce In st it u t e of Tech n ology WP AF B OH Courses Respon se Sur fa ce Met h odology Design of Experiments Statistical Process Control P r oba bilit y a n d St a t ist ics for E n gin eer s St a t ist ica l P r ocess Con t rol E n gin eer in g Ma n a gem en t In t egr a t ed Pr oduct ion Syst em s Sen ior Design P r oject
06/04 pr esen t
Adju n ct In st r uct or
01/04 05/04
F lor ida St a t e Un iver sit y Ta lla h a ssee, F L
Tea ch in g Assist a n t
08/01 12/03
F lor ida St a t e Un iver sit y Ta lla h a ssee, F L
Adju n ct In st r uct or
07/02 08/02 (su m m er t er m )
Sou t h er n Illin ois Un iver sit y Ca r bon da le, IL Sou t h ea st Missour i St a t e Ca pe Gir a r dea u , MO
F a cilit ies P la n n in g F a cilit ies P la n n in g E n gin eer in g Gr a ph ics
In st r uct or
08/00 05/01
184
Research Appointments
Client Un it ed St a t es Ar m y (ARL-H RE D) Dates Employer F lor ida A&M Un iver sit y 05/02 05/04 Su pervisor : Dr. J a m es R. Sim pson Responsibilities P r ovide ou t pu t an a lysis en h an cem en t s t o ARL-H RE D soldier wor kloa d discret e even t sim u la t ion softwa r e (IMP RINT)
Industrial Experience
Position P la n t E ngin eer Dates 06/96 08/00 Employer Ma t er ia l Ser vice Cor p. At h en s, IL 62613 Su pervisor : J a ck Br own Responsibilities In volved in th e developm en t a n d im plem en t a t ion of effect ive qu a lit y a ssu r an ce syst em s. Assist ed in t h e design a n d im plem en t a t ion of efficien t pla n t la you t a n d m a t er ia l h a n dling syst em s. Develop a nd m a in t a in pla n t dr a wings th r ou gh ba sic CAD syst em s. Act ively involved in qu a lit y a ssu r a n ce t hr ough im plem en t a t ion of SP C pr ocess m on it or ing. Developm en t a n d im plem en t a t ion of fa cilit y wide m a n a gem en t in for m a t ion syst em s.
Ma ch in e Sh op Assist a n t
06/99 - 06/00
Ma yt a g La un dr y P r oduct s H er r in , IL 62948 Su pervisor : Rick Wilson
185