6 6.2. Methods and materials 125 - Exploring alternative distributions, such as Log-Logistic and Log-Normal functions, in sewer network deterioration modelling. - Provide comprehensive formal definitions of the deterioration models. Additionally, for calibration, we combine the Metropolis-Hastings (M-H) algorithm with the Sequential Least Squares Programming (SLSQP) algorithm for parameter inference in di!erent Markov chains, a novel approach in this field. - Our implementation is available at https://gitlab.utwente.nl/fmt/ degradation-models/ihctmc. Chapter outline. Section 6.2 describes the methods and materials. Section 6.3 details the experimental setup and results. Section 6.4 analyses the findings. Section 6.5 concludes the chapter and suggests future research directions. Related work. The literature on sewer main deterioration modelling identifies two primary types of Markov chains: homogeneous and inhomogeneous (Table II.7 on page 105). Homogeneous-time Markov Chains (HTMCs) have constant transition probabilities, meaning the probabilities of transitioning between states do not change over time. In contrast, Inhomogeneous-time Markov Chains (IHTMCs) feature time-variable transition probabilities, indicating that the likelihood of state transitions can vary. From the literature, we observe that HTMCs o!er simplicity and computational e"ciency, making them easier to analyse. However, they often cannot adequately model non-linear patterns found in stochastic degradation processes, where assuming constant transition probabilities may be overly simplistic. In contrast, IHTMCs can handle these complexities better by accommodating time-varying transition probabilities. Yet, these chains are computationally intensive and sometimes lack feasible closed-form solutions, complicating their application. For completeness, other studies use di!erent forms of Markov chains, such as semi-Markov chains, fuzzy Markov chains, and ordered logistic models (Kleiner, 2001; Kleiner, Sadiq, and Rajani, 2004; Lubini and Fuamba, 2011). These types of Markov chains are outside the scope of our analysis. 6.2 Methods and materials Deterioration models for sewer mains are typically developed using inspection data conforming to standards like EN 13508:1 and EN 13508:2, which guide the classification of damages observed through CCTV inspections into severity levels. This data situates these deterioration models within the domain of Multi-State Modelling (MSM), which captures deterioration through finite states with welldefined deterioration indicators, providing a granular view of the process (Compare, Baraldi, Bani, et al., 2017). Thus, stochastic deterioration modelling of sewer mains is conducted via Markov chains, with states corresponding to severity levels.
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