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Robust Change Detection and Change Point Estimation for Poisson Count Processes Marcus B Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2004 Robust Change Detection and Change Point Estimation for Poisson Count Processes Marcus B. Perry Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA 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. ii 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. iii 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 τ 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 iv 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 Newton’s 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 ...........................................................
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