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 ? 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
