130 Chapter 6. Comparing Homogeneous and Inhomogeneous Time Markov Chains for Modelling Deterioration in Sewer Pipe Networks performance. Algorithm 3 outlines the methodology used to calibrate the estimators, with the results presented in Figure 6.2. The non-parametric Turnbull estimator is computed using the lifelines toolbox (Davidson-Pilon, 2019) implemented in Python. 6.2.4 Goodness-of-fit metrics Markov chains performance is evaluated via likelihood-based metrics: Akaike Information Criterion (AIC) (Akaike, 1998) (Eq. 6.6a) and Bayesian Information Criterion (BIC) (Schwarz, 1978) (Eq. 6.6b), which aid in model selection. Moreover, the Root Mean Squared Error (RMSE) (Eq. 6.6c) quantifies the Euclidean distance between the predictions of the Markov chains and the Turnbull estimator. AIC=2|⇀| →2⇁ (6.6a) BIC=ln(|y|)|⇀| →2⇁ (6.6b) RMSE= 1 |y| ↘|S| t→y k→Ω' pk(t) →ˆpk(t)( 2 (6.6c) Both AIC and BIC include |⇀|, the number of parameters in the model, with BIC additionally considering ln(|y|), the natural logarithm of the sample size. RMSE involves pk(t) and ˆpk(t), which denote the probabilities of being in state k at pipe age t, obtained from the Markov chains and the Turnbull estimator, respectively. 6.3 Experimental setup and evaluation 6.3.1 Case study The case study is detailed in Section II.4.3 on page 100. Mohammadi, Najafi, Kermanshachi, et al., 2020; Salihu, Hussein, Mohandes, et al., 2022 identify age, material, and content as the primary factors a!ecting sewer pipe condition. Based on these, we categorise pipes into three cohorts and examine the BAF damage code, indicating infiltration. Cohort CMW: Concrete pipes for mixed and waste content, Length: 469 km, Pipes: 11,942. Cohort CS: Concrete pipes for stormwater, Length: 172 km, Pipes: 4,701. Cohort PMW: PVC pipes for mixed and waste content, Length: 294 km, Pipes: 10,777. 6.3.2 Experimental setup Our experiment aims to assess the e"cacy of homogeneous and inhomogeneous Markov chains in predicting stochastic deterioration of sewer mains using the same dataset. Employing cross-validation, 70% of the sewer mains from the case study
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