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 xi

1 Introduction and Revision of Some Statistical Ideas 1

1.1 Necessity for Process Control 1

1.2 SPC and EPC 1

1.3 Process Monitoring Without a Model 3

1.4 Detecting a Signal in Noise 4

1.5 Measurement Data 4

1.6 Two Important Characteristics of a Probability Distribution 5

1.7 Normal Distribution 6

1.8 Normal Distribution Defined by μ and σ 6

1.9 Probabilities Associated with Normal Distribution 7

1.10 Estimating Mean and Standard Deviation from Data 8

1.11 Combining Estimates of σ2 9

1.12 Data on Frequencies (Events): Poisson Distribution 10

1.13 Normal Approximation to Poisson Distribution 12

1.14 Data on Proportion Defective: Binomial Distribution 12

1.15 Normal Approximation to Binomial Distribution 14

Appendix 1A: Central Limit Effect 15

Problems 17

2 Standard Control Charts Under Ideal Conditions As a First Approximation 21

2.1 Control Charts for Process Monitoring 21

2.2 Control Chart for Measurement (Variables) Data 22

2.3 Shewhart Charts for Sample Average and Range 24

2.4 Shewhart Chart for Sample Range 26

2.5 Process Monitoring With Control Charts for Frequencies 29

2.6 Data on Frequencies (Counts): Poisson Distribution 30

2.7 Common Causes and Special Causes 34

2.8 For What Kinds of Data Has the c Chart Been Used? 36

2.9 Quality Control Charts for Proportions: p Chart 37

2.10 EWMA Chart 40

2.11 Process Monitoring Using Cumulative Sums 46

2.12 Specification Limits, Target Accuracy, and Process Capability 53

2.13 How Successful Process Monitoring can Improve Quality 56

Problems 57

3 What Can Go Wrong and What Can We Do About It? 61

3.1 Introduction 61

3.2 Measurement Charts 64

3.3 Need for Time Series Models 65

3.4 Types of Variation 65

3.5 Nonstationary Noise 66

3.6 Values for constants 71

3.7 Frequencies and Proportions 74

3.8 Illustration 76

3.9 Robustness of EWMA 78

Appendix 3A: Alternative Forms of Relationships for EWMAs 79

Questions 79

4 Introduction to Forecasting and Process Dynamics 81

4.1 Forecasting with an EWMA 81

4.2 Forecasting Sales of Dingles 82

4.3 Pete’s Rule 85

4.4 Effect of Changing Discount Factor 86

4.5 Estimating Best Discount Factor 87

4.6 Standard Deviation of Forecast Errors and Probability Limits for Forecasts 88

4.7 What to Do If You Do Not Have Enough Data to Estimate θ 89

4.8 Introduction to Process Dynamics and Transfer Function 89

4.9 Dynamic Systems and Transfer Funtions 90

4.10 Difference Equations to Represent Dynamic Relations 90

4.11 Representing Dynamics of Industrial Process 96

4.12 Transfer Function Models Using Difference Equations 97

4.13 Stable and Unstable Systems 98

Problems 100

5 Nonstationary Time Series Models for Process Disturbances 103

5.1 Reprise 103

5.2 Stationary Time Series Model in Which Successive Values are Correlated 104

5.3 Major Effects of Statistical Dependence: Illustration 105

5.4 Random Walk 106

5.5 How to Test a Forecasting Method 107

5.6 Qualification of EWMA as a Forecast 107

5.7 Understanding Time Series Behavior with Variogram 110

5.8 Sticky Innovation Generating Model for Nonstationary Noise 113

5.9 Robustness of EWMA for Signal Extraction 118

5.10 Signal Extraction for Disturbance Model Due to Barnard 118

Questions 122

Problems 122

6 Repeated-Feedback Adjustment 125

6.1 Introduction to Discrete-Feedback Control 125

6.2 Inadequacy of NIID Models and Other Stationary Models for Control: Reiteration 125

6.3 Three Approaches to Repeated-Feedback Adjustment that Lead to Identical Conclusions 126

6.4 Some History 130

6.5 Adjustment Chart 132

6.6 Insensitivity to Choice of G 134

6.7 Compromise Value for G 135

6.8 Using Smaller Value of G to Reduce Adjustment Variance σ2x 136

Appendix 6A: Robustness of Integral Control 137

Appendix 6B: Effect on Adjustment of Choosing G Different from λ0: Obtaining Equation (6.12) 139

Appendix 6C: Average Reduction in Mean-Square Error Due to Adjustment for Observations Generated by IMA Model 140

Questions 140

Problems 140

7 Periodic Adjustment 143

7.1 Introduction 143

7.2 Periodic Adjustment 143

7.3 Starting Scheme for Periodic Adjustment 146

7.4 Numerical Calculations for Bounded Adjustment 146

7.5 Simple Device for Facilitating Bounded Adjustment 150

7.6 Bounded Adjustment Seen as Process of Tracking 153

7.7 Combination of Adjustment and Monitoring 153

7.8 Bounded Adjustment for Series not Generated by IMA Model 155

Problems 160

8 Control of Process with Inertia 163

8.1 Adjustment Depending on Last Two Output Errors 163

8.2 Minimum Mean-Square Error Control of Process With First-Order Dynamics 167

8.3 Schemes with Constrained Adjustment 169

8.4 PI Schemes with Constrained Adjustment 170

8.5 Optimal and Near-Optimal Constrained PI Schemes: Choice of P 171

8.6 Choice of G For P = 0 and P = −0.25 172

8.7 PI Schemes for Process With Dead Time 178

8.8 Process Monitoring and Process Adjustment 181

8.9 Feedback Adjustment Applied to Process in Perfect State of Control 182

8.10 Using Shewhart Chart to Adjust Unstable Process 182

8.11 Feedforward Control 183

Appendix 8A: Equivalence of Equations for PI Control 184

Appendix 8B: Effect of Errors in Adjustment 184

Appendix 8C: Choices for G and P to Attain Optimal Constrained PI Control for Various Values of λ0 and δ0 with d0 = 0 and d0 = 1 185

Questions 191

Problems 191

9 Explicit Consideration of Monetary Cost 193

9.1 Introduction 193

9.2 How Often Should You Take Data? 197

9.3 Choosing Adjustment Schemes Directly in Terms of Costs 203

Appendix 9A: Functions h(L/λσa) and q(L/λσa) in Table 9.1 205

Appendix 9B: Calculation of Minimum-Cost Schemes 205

Problems 207

10 Cuscore Charts: Looking for Signals in Noise 209

10.1 Introduction 209

10.2 How Are Cuscore Statistics Obtained? 216

10.3 Efficient Monitoring Charts 219

10.4 Useful Method for Obtaining Detector When Looking for Signal in Noise Not Necessarily White Noise 221

10.5 Looking for Single Spike 223

10.6 Some Time Series Examples 224

Appendix 10A: Likelihood, Fisher’s Efficient Score, and Cuscore Statistics 227

Appendix 10B: Useful Procedure for Obtaining Appropriate Cuscore Statistic 230

Appendix 10C: Detector Series for IMA Model 231

Problems 231

11 Monitoring an Operating Feedback System 235

11.1 Looking for Spike in Disturbance zt Subjected to Integral Control 235

11.2 Looking for Exponential Signal in Disturbance Subject to Integral Control 237

11.3 Monitoring Process with Inertia Represented by First-Order Dynamics 238

11.4 Reconstructing Disturbance Pattern 240

Appendix 11A: Derivation of Equation (11.3) 240

Appendix 11B: Derivation of Equation (11.10) 242

Appendix 11C: Derivation of Equation (11.14) 243

12 Brief Review of Time Series Analysis 245

12.1 Serial Dependence: Autocorrelation Function and Variogram 245

12.2 Relation of Autocorrelation Function and Variogram 246

12.3 Some Time Series Models 247

12.4 Stationary Models 247

12.5 Autoregressive Moving-Average Models 250

12.6 Nonstationary Models 253

12.7 IMA [or ARIMA(0, 1, 1)] Model 253

12.8 Modeling Time Series Data 255

12.9 Model Identification, Model Fitting, and Diagnostic Checking 256

12.10 Forecasting 261

12.11 Estimation with Closed-Loop Data 266

12.12 Conclusion 269

Appendix 12A: Other Tools for Identification of Time Series Models 269

Appendix 12B: Estimation of Time Series Parameters 270

Solutions to Exercises and Problems 273

References and Further Reading 307

Appendix Three Time Series 321

Index 327