Recurrent Event Modeling Based on the Yule Process: Application to Water Network Asset Management

This book presents research work into the reliability of drinking water pipes. The infrastructure of water pipes is susceptible to routine failures, namely leakage or breakage, which occur in an aggregative manner in pipeline networks. Creating strategies for infrastructure asset management requires accurate modeling tools and first-hand experience of what repeated failures can mean in terms of socio-economic and environmental consequences. Devoted to the counting process framework when dealing with this issue, the author presents preliminary basic concepts, particularly the process intensity, as well as basic tools (classical distributions and processes). The introductory material precedes the discussion of several constructs, namely the non-homogeneous birth process, and further as a special case, the linearly extended Yule process (LEYP), and its adaptation to account for selective survival. The practical usefulness of the theoretical results is illustrated with actual water pipe failure data.

Table of Contents:
Preface ix Chapter 1. Introduction  1 1.1. Notation 2 1.2. General theoretical framework  4 1.2.1. The concept of a counting process  4 1.2.2. The intensity function of a counting process  5 1.3. The non-homogeneous Poisson process 6 1.4. The Eisenbeis model  7 1.5. Other approaches for water pipe failure modeling 8 1.6. Why mobilize the Yule process? 9 1.7. Structure of the book 10 Chapter 2. Preliminaries  13 2.1. The Yule process and the negative binomial distribution  13 2.2. Gamma-mixture of NHPP 17 2.3. The negative binomial power series 19 2.4. The negative multinomial distribution  19 2.5. The negative multinomial power series 22 Chapter 3. Non-homogeneous Birth Process  23 3.1. NHBP intensity  24 3.2. Conditional distribution of the counting process  24 Chapter 4. Linear Extension of the Yule Process  33 4.1. LEYP intensity 33 4.2. Conditional distribution of the LEYP  34 4.2.1. Distribution of N(b) − N(a) | N(a−) 34 4.2.2. Marginal distribution of N(t) 36 4.2.3. Marginal distribution of N(b) − N(a)  36 4.2.4. Conditional distribution of N(a−) given N(b) − N(a) 37 4.2.5. Conditional distribution of N(c) − N(b) given N(b−) − N(a) 38 4.2.6. Distribution of N(b−) − N(a) given N(c) − N(b)  39 4.2.7. Distribution of N(d) − N(c) given N(b) − N(a) 40 4.3. Limiting distribution when α tends to 0+  42 4.4. Partition of an interval 44 4.5. Generalization to any subset of a partition  46 4.6. Discontinuous observation interval  49 Chapter 5. LEYP Likelihood and Inference  51 5.1. LEYP likelihood  51 5.2. LEYP parameter estimation  55 5.2.1. Maximum likelihood estimator  55 5.2.2. Null hypothesis of parameter estimates 56 5.2.3. The Yule–Weibull–Cox intensity 56 5.2.4. Null hypothesis test implemented for the Yule-Weibull-Cox intensity 57 5.2.5. Parameter estimation algorithm 58 5.3. Validation of the estimation procedure  58 5.3.1. Conditional distribution of the inter-event time 59 5.3.2. LEYP event simulation  59 5.4. LEYP model goodness of fit 60 5.5. Validating LEYP model predictions 62 5.5.1. Lorenz curve  63 5.5.2. Prediction bias checking 65 Chapter 6. Selective Survival 67 6.1. Left-truncation, right-censoring and decommissioning decisions  67 6.2. Coupling failure and decommissioning processes: LEYP2s model 68 6.3. LEYP2s discretization scheme  69 6.4. Failure and decommissioning probabilities 71 6.4.1. Probability of no decommissioning 71 6.4.2. Distribution of N(b) − N(a) given R(a−) = 0  73 6.4.3. Conditional probability of R(a−) = 0 given N(b) − N(a) 75 6.4.4. Conditional distribution of N(c) − N(b) given N(b) − N(a) and R(a−) = 0 77 6.4.5. Conditional distribution of N(d) − N(c) given N(b) − N(a) and R(a−) = 0 78 6.4.6. Conditional distribution of N(a−) given N(b) − N(a) and R(a−) = 0 79 Chapter 7. LEYP2s Likelihood and Inference  83 7.1. Validation of the estimation procedure for LEYP2s  88 7.1.1. Constrained and selective decommissioning survival functions 88 7.1.2. Random failure and decommissioning data generation  89 7.1.3. Checking parameter estimate accuracy 93 7.1.4. Checking log-likelihood convexity  94 Chapter 8. Case Study Application of the LEYP2s Model  97 8.1. Lausanne water utility 97 8.2. Lausanne water supply network 97 8.3. Lausanne network segment failure and decommissioning data 99 8.4. Model parameter estimates  100 8.5. Model goodness of fit assessment  104 8.6. Model validation  105 8.7. Service lifetime  108 Chapter 9. Conclusion and Outlook  111 9.1. Software implementation: Casses  111 9.2. Model enhancement needs  112 9.2.1. More flexible analytical form for the failure intensity function 112 9.2.2. Time-dependent covariates  112 9.3. LEYP2s model as element of IAM decision helping  114 9.3.1. Accounting for vulnerability to failures: toward a risk approach  115 Appendices  117 Appendix A  119 Appendix B  121 Bibliography 123 Index  127