Practical Applications of Bayesian Reliability

Preface xi

Acknowledgments xv

About the Companion Website xvii

1 Basic Concepts of Reliability Engineering 1

1.1 Introduction 1

1.1.1 Reliability Definition 3

1.1.2 Design for Reliability and Design for Six Sigma 4

1.2 Basic Theory and Concepts of Reliability Statistics 5

1.2.1 Random Variables 5

1.2.2 Discrete Probability Distributions 6

1.2.3 Continuous Probability Distributions 6

1.2.4 Properties of Discrete and Continuous Random Variables 6

1.2.4.1 Probability Mass Function 6

1.2.4.2 Probability Density Function 7

1.2.4.3 Cumulative Distribution Function 8

1.2.4.4 Reliability or Survival Function 8

1.2.4.5 Hazard Rate or Instantaneous Failure Rate 9

1.2.4.6 Cumulative Hazard Function 10

1.2.4.7 The Average Failure Rate Over Time 10

1.2.4.8 Mean Time to Failure 10

1.2.4.9 Mean Number of Failures 11

1.2.5 Censored Data 11

1.2.6 Parametric Models of Time to Failure Data 13

1.2.7 Nonparametric Estimation of Survival 14

1.2.8 Accelerated Life Testing 16

1.3 Bayesian Approach to Reliability Inferences 18

1.3.1 Brief History of Bayes' Theorem and Bayesian Statistics 18

1.3.2 How Does Bayesian Statistics Relate to Other Advances in the Industry? 19

1.3.2.1 Advancement of Predictive Analytics 20

1.3.2.2 Cost Reduction 20

1.4 Component Reliability Estimation 20

1.5 System Reliability Estimation 20

1.6 Design Capability Prediction (Monte Carlo Simulations) 21

1.7 Summary 22

References 23

2 Basic Concepts of Bayesian Statistics and Models 25

2.1 Basic Idea of Bayesian Reasoning 25

2.2 Basic Probability Theory and Bayes' Theorem 26

2.3 Bayesian Inference (Point and Interval Estimation) 32

2.4 Selection of Prior Distributions 35

2.4.1 Conjugate Priors 35

2.4.2 Informative and Non-informative Priors 38

2.5 Bayesian Inference vs. Frequentist Inference 44

2.6 How Bayesian Inference Works with Monte Carlo Simulations 48

2.7 Bayes Factor and its Applications 50

2.8 Predictive Distribution 53

2.9 Summary 57

References 57

3 Bayesian Computation 59

3.1 Introduction 59

3.2 Discretization 60

3.3 Markov Chain Monte Carlo Algorithms 66

3.3.1 Markov Chains 67

3.3.1.1 Monte Carlo Error 67

3.3.2 Metropolis-Hastings Algorithm 68

3.3.3 Gibbs Sampling 80

3.4 Using BUGS/JAGS 85

3.4.1 Define a JAGS Model 86

3.4.2 Create, Compile, and Run the JAGS Model 89

3.4.3 MCMC Diagnostics and Output Analysis 91

3.4.3.1 Summary Statistics 91

3.4.3.2 Trace Plots 92

3.4.3.3 Autocorrelation Plots 93

3.4.3.4 Cross-Correlation 93

3.4.3.5 Gelman-Rubin Diagnostic and Plots 94

3.4.4 Sensitivity to the Prior Distributions 95

3.4.5 Model Comparison 96

3.5 Summary 98

References 98

4 Reliability Distributions (Bayesian Perspective) 101

4.1 Introduction 101

4.2 Discrete Probability Models 102

4.2.1 Binomial Distribution 102

4.2.2 Poisson Distribution 104

4.3 Continuous Models 108

4.3.1 Exponential Distribution 108

4.3.2 Gamma Distribution 113

4.3.3 Weibull Distribution 115

4.3.3.1 Fit Data to a Weibull Distribution 116

4.3.3.2 Demonstrating Reliability using Right-censored Data Only 120

4.3.4 Normal Distribution 135

4.3.5 Lognormal Distribution 139

4.4 Model and Convergence Diagnostics 143

References 143

5 Reliability Demonstration Testing 145

5.1 Classical Zero-failure Test Plans for Substantiation Testing 146

5.2 Classical Zero-failure Test Plans for Reliability Testing 147

5.3 Bayesian Zero-failure Test Plan for Substantiation Testing 149

5.4 Bayesian Zero-failure Test Plan for Reliability Testing 161

5.5 Summary 162

References 163

6 Capability and Design for Reliability 165

6.1 Introduction 165

6.2 Monte Caro Simulations with Parameter Point Estimates 166

6.2.1 Stress-strength Interference Example 166

6.2.2 Tolerance Stack-up Example 171

6.3 Nested Monte Carlo Simulations with Bayesian Parameter Estimation 174

6.3.1 Stress-strength Interference Example 175

6.3.2 Tolerance Stack-up Example 182

6.4 Summary 186

References 186

7 System Reliability Bayesian Model 187

7.1 Introduction 187

7.2 Reliability Block Diagram 188

7.3 Fault Tree 196

7.4 Bayesian Network 197

7.4.1 A Multiple-sensor System 199

7.4.2 Dependent Failure Modes 202

7.4.3 Case Study: Aggregating Different Sources of Imperfect Data 204

7.5 Summary 214

References 214

8 Bayesian Hierarchical Model 217

8.1 Introduction 217

8.2 Bayesian Hierarchical Binomial Model 221

8.2.1 Separate One-level Bayesian Models 221

8.2.2 Bayesian Hierarchical Model 222

8.3 Bayesian Hierarchical Weibull Model 228

8.4 Summary 238

References 238

9 Regression Models 239

9.1 Linear Regression 239

9.2 Binary Logistic Regression 246

9.3 Case Study: Defibrillation Efficacy Analysis 257

9.4 Summary 277

References 278

Appendix A Guidance for Installing R, R Studio, JAGS, and rjags 279

A.1 Install R 279

A.2 Install R Studio 279

A.3 Install JAGS 280

A.4 Install Package rjags 280

A.5 Set Working Directory 280

Appendix B Commonly Used R Commands 281

B.1 How to Run R Commands 281

B.2 General Commands 281

B.3 Generate Data 282

B.4 Variable Types 283

B.5 Calculations and Operations 285

B.6 Summarize Data 286

B.7 Read and Write Data 287

B.8 Plot Data 288

B.9 Loops and Conditional Statements 290

Appendix C Probability Distributions 291

C.1 Discrete Distributions 291

C.1.1 Binomial Distribution 291

C.1.2 Poisson Distribution 291

C.2 Continuous Distributions 292

C.2.1 Beta Distribution 292

C.2.2 Exponential Distribution 292

C.2.3 Gamma Distribution 292

C.2.4 Inverse Gamma Distribution 293

C.2.5 Lognormal Distribution 293

C.2.6 Normal Distribution 293

C.2.7 Uniform Distribution 294

C.2.8 Weibull Distribution 294

Appendix D Jeffreys Prior 295

Index 299