Statistical Control by Monitoring and Adjustment

GEORGE E. P. BOX, PHD, is Ronald Aylmer Fisher Professor Emeritus of Statistics and Industrial Engineering at the University of Wisconsin–Madison. He is a Fellow of the Royal Society of London and the American Academy of Arts and Sciences. He is an honorary Fellow and Shewart and Deming Medalist of the American Society for Quality and an honorary member of the International Statistical Institute. He is also the recipient of the Samuel S. Wilks Memorial Medal of the American Statistical Association and the Guy Medal in Gold of the Royal Statistical Society. Dr. Box is the coauthor of Statistics for Experimenters: Design, Innovation, and Discovery, Second Edition; Response Surfaces, Mixtures, and Ridge Analyses, Second Edition; Evolutionary Operation: A Statistical Method for Process Improvements; Improving Almost Anything: Ideas and Essays, Revised Edition; Time Series Analysis: Forecasting and Control, Fourth Edition; and Bayesian Interface and Statistical Analyses, all published by Wiley. ALBERTO LUCEÑO, PHD, is Professor in the ETS de Ingenieros de Caminos at the University of Cantabria, Spain. He is an Associate Editor of the Journal of Quality Technology and Statistical Modelling: An International Journal. MARÍA DEL CARMEN PANIAGUA-QUIÑONES, PHD, is Visiting Assistant Professor in the School of Industrial Engineering at Purdue University. Her research interests include experimental design, quality productivity, and engineering management and control process improvement. She was the recipient of the Brumbaugh Award of the American Society for Quality for Best Technical Paper in 2007.

• Preface xi1 Introduction and Revision of Some Statistical Ideas 11.1 Necessity for Process Control 11.2 SPC and EPC 11.3 Process Monitoring Without a Model 31.4 Detecting a Signal in Noise 41.5 Measurement Data 41.6 Two Important Characteristics of a Probability Distribution 51.7 Normal Distribution 61.8 Normal Distribution Defined by μ and σ 61.9 Probabilities Associated with Normal Distribution 71.10 Estimating Mean and Standard Deviation from Data 81.11 Combining Estimates of σ2 91.12 Data on Frequencies (Events): Poisson Distribution 101.13 Normal Approximation to Poisson Distribution 121.14 Data on Proportion Defective: Binomial Distribution 121.15 Normal Approximation to Binomial Distribution 14Appendix 1A: Central Limit Effect 15Problems 172 Standard Control Charts Under Ideal Conditions As a First Approximation 212.1 Control Charts for Process Monitoring 212.2 Control Chart for Measurement (Variables) Data 222.3 Shewhart Charts for Sample Average and Range 242.4 Shewhart Chart for Sample Range 262.5 Process Monitoring With Control Charts for Frequencies 292.6 Data on Frequencies (Counts): Poisson Distribution 302.7 Common Causes and Special Causes 342.8 For What Kinds of Data Has the c Chart Been Used? 362.9 Quality Control Charts for Proportions: p Chart 372.10 EWMA Chart 402.11 Process Monitoring Using Cumulative Sums 462.12 Specification Limits, Target Accuracy, and Process Capability 532.13 How Successful Process Monitoring can Improve Quality 56Problems 573 What Can Go Wrong and What Can We Do About It? 613.1 Introduction 613.2 Measurement Charts 643.3 Need for Time Series Models 653.4 Types of Variation 653.5 Nonstationary Noise 663.6 Values for constants 713.7 Frequencies and Proportions 743.8 Illustration 763.9 Robustness of EWMA 78Appendix 3A: Alternative Forms of Relationships for EWMAs 79Questions 794 Introduction to Forecasting and Process Dynamics 814.1 Forecasting with an EWMA 814.2 Forecasting Sales of Dingles 824.3 Pete’s Rule 854.4 Effect of Changing Discount Factor 864.5 Estimating Best Discount Factor 874.6 Standard Deviation of Forecast Errors and Probability Limits for Forecasts 884.7 What to Do If You Do Not Have Enough Data to Estimate θ 894.8 Introduction to Process Dynamics and Transfer Function 894.9 Dynamic Systems and Transfer Funtions 904.10 Difference Equations to Represent Dynamic Relations 904.11 Representing Dynamics of Industrial Process 964.12 Transfer Function Models Using Difference Equations 974.13 Stable and Unstable Systems 98Problems 1005 Nonstationary Time Series Models for Process Disturbances 1035.1 Reprise 1035.2 Stationary Time Series Model in Which Successive Values are Correlated 1045.3 Major Effects of Statistical Dependence: Illustration 1055.4 Random Walk 1065.5 How to Test a Forecasting Method 1075.6 Qualification of EWMA as a Forecast 1075.7 Understanding Time Series Behavior with Variogram 1105.8 Sticky Innovation Generating Model for Nonstationary Noise 1135.9 Robustness of EWMA for Signal Extraction 1185.10 Signal Extraction for Disturbance Model Due to Barnard 118Questions 122Problems 1226 Repeated-Feedback Adjustment 1256.1 Introduction to Discrete-Feedback Control 1256.2 Inadequacy of NIID Models and Other Stationary Models for Control: Reiteration 1256.3 Three Approaches to Repeated-Feedback Adjustment that Lead to Identical Conclusions 1266.4 Some History 1306.5 Adjustment Chart 1326.6 Insensitivity to Choice of G 1346.7 Compromise Value for G 1356.8 Using Smaller Value of G to Reduce Adjustment Variance σ2x 136Appendix 6A: Robustness of Integral Control 137Appendix 6B: Effect on Adjustment of Choosing G Different from λ0: Obtaining Equation (6.12) 139Appendix 6C: Average Reduction in Mean-Square Error Due to Adjustment for Observations Generated by IMA Model 140Questions 140Problems 1407 Periodic Adjustment 1437.1 Introduction 1437.2 Periodic Adjustment 1437.3 Starting Scheme for Periodic Adjustment 1467.4 Numerical Calculations for Bounded Adjustment 1467.5 Simple Device for Facilitating Bounded Adjustment 1507.6 Bounded Adjustment Seen as Process of Tracking 1537.7 Combination of Adjustment and Monitoring 1537.8 Bounded Adjustment for Series not Generated by IMA Model 155Problems 1608 Control of Process with Inertia 1638.1 Adjustment Depending on Last Two Output Errors 1638.2 Minimum Mean-Square Error Control of Process With First-Order Dynamics 1678.3 Schemes with Constrained Adjustment 1698.4 PI Schemes with Constrained Adjustment 1708.5 Optimal and Near-Optimal Constrained PI Schemes: Choice of P 1718.6 Choice of G For P = 0 and P = −0.25 1728.7 PI Schemes for Process With Dead Time 1788.8 Process Monitoring and Process Adjustment 1818.9 Feedback Adjustment Applied to Process in Perfect State of Control 1828.10 Using Shewhart Chart to Adjust Unstable Process 1828.11 Feedforward Control 183Appendix 8A: Equivalence of Equations for PI Control 184Appendix 8B: Effect of Errors in Adjustment 184Appendix 8C: Choices for G and P to Attain Optimal Constrained PI Control for Various Values of λ0 and δ0 with d0 = 0 and d0 = 1 185Questions 191Problems 1919 Explicit Consideration of Monetary Cost 1939.1 Introduction 1939.2 How Often Should You Take Data? 1979.3 Choosing Adjustment Schemes Directly in Terms of Costs 203Appendix 9A: Functions h(L/λσa) and q(L/λσa) in Table 9.1 205Appendix 9B: Calculation of Minimum-Cost Schemes 205Problems 20710 Cuscore Charts: Looking for Signals in Noise 20910.1 Introduction 20910.2 How Are Cuscore Statistics Obtained? 21610.3 Efficient Monitoring Charts 21910.4 Useful Method for Obtaining Detector When Looking for Signal in Noise Not Necessarily White Noise 22110.5 Looking for Single Spike 22310.6 Some Time Series Examples 224Appendix 10A: Likelihood, Fisher’s Efficient Score, and Cuscore Statistics 227Appendix 10B: Useful Procedure for Obtaining Appropriate Cuscore Statistic 230Appendix 10C: Detector Series for IMA Model 231Problems 23111 Monitoring an Operating Feedback System 23511.1 Looking for Spike in Disturbance zt Subjected to Integral Control 23511.2 Looking for Exponential Signal in Disturbance Subject to Integral Control 23711.3 Monitoring Process with Inertia Represented by First-Order Dynamics 23811.4 Reconstructing Disturbance Pattern 240Appendix 11A: Derivation of Equation (11.3) 240Appendix 11B: Derivation of Equation (11.10) 242Appendix 11C: Derivation of Equation (11.14) 24312 Brief Review of Time Series Analysis 24512.1 Serial Dependence: Autocorrelation Function and Variogram 24512.2 Relation of Autocorrelation Function and Variogram 24612.3 Some Time Series Models 24712.4 Stationary Models 24712.5 Autoregressive Moving-Average Models 25012.6 Nonstationary Models 25312.7 IMA [or ARIMA(0, 1, 1)] Model 25312.8 Modeling Time Series Data 25512.9 Model Identification, Model Fitting, and Diagnostic Checking 25612.10 Forecasting 26112.11 Estimation with Closed-Loop Data 26612.12 Conclusion 269Appendix 12A: Other Tools for Identification of Time Series Models 269Appendix 12B: Estimation of Time Series Parameters 270Solutions to Exercises and Problems 273References and Further Reading 307Appendix Three Time Series 321Index 327