Practical Applications of Bayesian Reliability

YAN LIU, PHD, is Principal Reliability Engineer at Medtronic PLC, (USA). She is a certified Master Black Belt at Medtronic and has 12 years of working and consulting experience on reliability engineering and design for Six Sigma. ATHULA I. ABEYRATNE, PHD, is Senior Principal Statistician and a certified DRM Black Belt at Medtronic PLC, (USA), where he has provided statistical consulting, training, data analyses, and modelling for 27 years.

• Preface xiAcknowledgments xvAbout the Companion Website xvii1 Basic Concepts of Reliability Engineering 11.1 Introduction 11.1.1 Reliability Definition 31.1.2 Design for Reliability and Design for Six Sigma 41.2 Basic Theory and Concepts of Reliability Statistics 51.2.1 Random Variables 51.2.2 Discrete Probability Distributions 61.2.3 Continuous Probability Distributions 61.2.4 Properties of Discrete and Continuous Random Variables 61.2.4.1 Probability Mass Function 61.2.4.2 Probability Density Function 71.2.4.3 Cumulative Distribution Function 81.2.4.4 Reliability or Survival Function 81.2.4.5 Hazard Rate or Instantaneous Failure Rate 91.2.4.6 Cumulative Hazard Function 101.2.4.7 The Average Failure Rate Over Time 101.2.4.8 Mean Time to Failure 101.2.4.9 Mean Number of Failures 111.2.5 Censored Data 111.2.6 Parametric Models of Time to Failure Data 131.2.7 Nonparametric Estimation of Survival 141.2.8 Accelerated Life Testing 161.3 Bayesian Approach to Reliability Inferences 181.3.1 Brief History of Bayes’ Theorem and Bayesian Statistics 181.3.2 How Does Bayesian Statistics Relate to Other Advances in the Industry? 191.3.2.1 Advancement of Predictive Analytics 201.3.2.2 Cost Reduction 201.4 Component Reliability Estimation 201.5 System Reliability Estimation 201.6 Design Capability Prediction (Monte Carlo Simulations) 211.7 Summary 22References 232 Basic Concepts of Bayesian Statistics and Models 252.1 Basic Idea of Bayesian Reasoning 252.2 Basic Probability Theory and Bayes’ Theorem 262.3 Bayesian Inference (Point and Interval Estimation) 322.4 Selection of Prior Distributions 352.4.1 Conjugate Priors 352.4.2 Informative and Non-informative Priors 382.5 Bayesian Inference vs. Frequentist Inference 442.6 How Bayesian Inference Works with Monte Carlo Simulations 482.7 Bayes Factor and its Applications 502.8 Predictive Distribution 532.9 Summary 57References 573 Bayesian Computation 593.1 Introduction 593.2 Discretization 603.3 Markov Chain Monte Carlo Algorithms 663.3.1 Markov Chains 673.3.1.1 Monte Carlo Error 673.3.2 Metropolis–Hastings Algorithm 683.3.3 Gibbs Sampling 803.4 Using BUGS/JAGS 853.4.1 Define a JAGS Model 863.4.2 Create, Compile, and Run the JAGS Model 893.4.3 MCMC Diagnostics and Output Analysis 913.4.3.1 Summary Statistics 913.4.3.2 Trace Plots 923.4.3.3 Autocorrelation Plots 933.4.3.4 Cross-Correlation 933.4.3.5 Gelman–Rubin Diagnostic and Plots 943.4.4 Sensitivity to the Prior Distributions 953.4.5 Model Comparison 963.5 Summary 98References 984 Reliability Distributions (Bayesian Perspective) 1014.1 Introduction 1014.2 Discrete Probability Models 1024.2.1 Binomial Distribution 1024.2.2 Poisson Distribution 1044.3 Continuous Models 1084.3.1 Exponential Distribution 1084.3.2 Gamma Distribution 1134.3.3 Weibull Distribution 1154.3.3.1 Fit Data to a Weibull Distribution 1164.3.3.2 Demonstrating Reliability using Right-censored Data Only 1204.3.4 Normal Distribution 1354.3.5 Lognormal Distribution 1394.4 Model and Convergence Diagnostics 143References 1435 Reliability Demonstration Testing 1455.1 Classical Zero-failure Test Plans for Substantiation Testing 1465.2 Classical Zero-failure Test Plans for Reliability Testing 1475.3 Bayesian Zero-failure Test Plan for Substantiation Testing 1495.4 Bayesian Zero-failure Test Plan for Reliability Testing 1615.5 Summary 162References 1636 Capability and Design for Reliability 1656.1 Introduction 1656.2 Monte Caro Simulations with Parameter Point Estimates 1666.2.1 Stress-strength Interference Example 1666.2.2 Tolerance Stack-up Example 1716.3 Nested Monte Carlo Simulations with Bayesian Parameter Estimation 1746.3.1 Stress-strength Interference Example 1756.3.2 Tolerance Stack-up Example 1826.4 Summary 186References 1867 System Reliability Bayesian Model 1877.1 Introduction 1877.2 Reliability Block Diagram 1887.3 Fault Tree 1967.4 Bayesian Network 1977.4.1 A Multiple-sensor System 1997.4.2 Dependent Failure Modes 2027.4.3 Case Study: Aggregating Different Sources of Imperfect Data 2047.5 Summary 214References 2148 Bayesian Hierarchical Model 2178.1 Introduction 2178.2 Bayesian Hierarchical Binomial Model 2218.2.1 Separate One-level Bayesian Models 2218.2.2 Bayesian Hierarchical Model 2228.3 Bayesian Hierarchical Weibull Model 2288.4 Summary 238References 2389 Regression Models 2399.1 Linear Regression 2399.2 Binary Logistic Regression 2469.3 Case Study: Defibrillation Efficacy Analysis 2579.4 Summary 277References 278Appendix A Guidance for Installing R, R Studio, JAGS, and rjags 279A.1 Install R 279A.2 Install R Studio 279A.3 Install JAGS 280A.4 Install Package rjags 280A.5 Set Working Directory 280Appendix B Commonly Used R Commands 281B.1 How to Run R Commands 281B.2 General Commands 281B.3 Generate Data 282B.4 Variable Types 283B.5 Calculations and Operations 285B.6 Summarize Data 286B.7 Read and Write Data 287B.8 Plot Data 288B.9 Loops and Conditional Statements 290Appendix C Probability Distributions 291C.1 Discrete Distributions 291C.1.1 Binomial Distribution 291C.1.2 Poisson Distribution 291C.2 Continuous Distributions 292C.2.1 Beta Distribution 292C.2.2 Exponential Distribution 292C.2.3 Gamma Distribution 292C.2.4 Inverse Gamma Distribution 293C.2.5 Lognormal Distribution 293C.2.6 Normal Distribution 293C.2.7 Uniform Distribution 294C.2.8 Weibull Distribution 294Appendix D Jeffreys Prior 295Index 299