Extending the Yule process to model recurrent pipe failures in water supply networks

摘要

This paper shows how to construct, within the mathematical framework of counting process theory, a Linear Extension of the Yule Process (LEYP) to model the recurrent failures of pressure pipes in water supply networks. The choice of the counting process framework is motivated by an analysis of the advantages and shortcomings of the modelling approaches proposed in the literature over the last thirty years. The parametric nature of the LEYP model enables the prediction of future failures, accounting for the effect of previous failures, pipe ageing, and explanatory factors, assuming proportional hazards. The counting process is Markovian by definition and shown to follow a negative binomial distribution. This property leads to a useful and simple formula for computing conditional predictions given past observed failures. The likelihood of the parameters with respect to a sequence of observed failures is derived using product-integration. Maximum likelihood estimates of the model parameters can be calculated with left-truncated data, i.e. failure observations restricted to a time interval that possibly starts long after pipe commissioning. Procedures for testing the significance of the model parameters, for assessing the model goodness of fit, and for validating the model predictions are presented. The predictive performance of the LEYP model is finally illustrated with extensive failure data provided by a French water utility.