5 5.2. Discrete-Time Markov Chains 113 As a first step, we decided to use Discrete-Time Markov Chains (DTMCs) because these proved to be a straightforward approach to model deterioration patterns associated with sewer networks. Moreover, we are interested in two typical types of DTMCs (see Figure 5.1) that we call Chains ‘Multi’ and ‘Single’, where the former contains additional transitions compared to the latter, and we are interested in evaluating which of them best suits our case study. Similar to Caradot, Riechel, Fesneau, et al., 2018, our goal with these DTMCs is to predict the probability for a group of pipes to be in a severity class for a certain type of damage, based on the pipe age and a set of numerical or categorical variables (called covariates) organised in 6 cohorts (i.e., group of pipes with the same characteristics) of interest. Our research questions are both application-oriented (RQ1) and methodological (RQ2): RQ1 how do the predefined cohorts compare in terms of deterioration rate? RQ2 how can DTMCs assist in getting this insight, and how do Chains ‘Multi’ and ‘Single’ compare in terms of performance? The experimental evaluation is based on a large-scale case study in the city of Breda in the Netherlands (see Section II.4.3), where we have information on sewer mains built since the 1920s which contains information on di!erent covariates. We focus on three typical types of sewer mains damages namely infiltration, surface damage, and cracks. Each damage has an associated severity index ranging from 1 to 5. Our main contribution is to demonstrate the application of existing deterioration models in a large-scale case study. The present work is a valuable step toward the development of an evidence-based asset management framework. The scripts and comparative figures can be found at zenodo.org/record/6535853. The structure of this chapter is as follows. Section 5.2 provides the theoretical background on DTMCs. Section 5.3 presents our methodology. In Section 5.4 we preset the case study, the experimental evaluation and the main results. We discuss and conclude in Section 5.5. 5.2 Discrete-Time Markov Chains Refer to Definition 11 on page 99 for the formal definition of Discrete-Time Markov Chains (DTMCs). Our DTMCs are defined by five states, S =↔1,2,3,4,5↗, with the index of the state denoted as k ↓S. The values pij with i, j ↓S in Figure 5.1 are discussed in Section 5.4.2. The initial probability distribution p(0) indicates the probability that the sewer main is in the state k at the step n=0, p(0) =↔p (0) 1 , p (0) 2 , p (0) 3 , p (0) 4 , p (0) 5 ↗ T (5.1)
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