II.4 Preliminaries 95 ML models for sewer network condition assessment are made by Nguyen and Seidu, 2022; El Morer, Wittek, and Rausch, 2024. Probabilistic models, grounded in probability theory, treat factors related to sewer network deterioration as random variables. Barraud, Bosco, Clemens-Meyer, et al., 2024 provides a comprehensive and updated overview of di!erent approaches used for deterioration modelling in sewer mains. These approaches have inherent limitations. Physics-based approaches become too extensive when capturing the complex deterioration behaviours present in large-scale systems with varying contexts such as sewer networks. ML-based and probabilistic models are only as e!ective as the quality and completeness of the data they use, a known challenge in sewer network systems (Noshahri, olde Scholtenhuis, Doree, et al., 2021). Additionally, despite the widespread application of ML-based techniques for diagnostic purposes, such as anomaly detection and condition or defect classification, they may be unsuitable for generating reliable, monotonous deterioration curves. This limitation, highlighted by Rokstad and Ugarelli, 2015; Caradot, Rouault, Clemens, et al., 2018; Kantidakis, Putter, Litière, et al., 2023, constrains their e!ectiveness in long-term maintenance planning. This research focuses on Markov chains, a probabilistic model used to predict the future distribution of the deterioration states. Markov chains are crucial in Multi-State Modelling (MSM) due to their ability to model state transitions. Tran, Setunge, and Shi, 2021 highlight the suitability of state-based Markov chains for sewer main deterioration modelling, especially when only a single inspection record is available. The key advantages of Markov chains are: (i) converting condition data into ordinal numbers, such as severity levels, commonly used in infrastructure asset rating (Tran, Lokuge, Setunge, et al., 2022); (ii) capturing the stochastic behaviour of sewer main deterioration; and (iii) providing outputs that estimate condition over a group of pipes, essential for optimising maintenance planning. II.4 Preliminaries II.4.1 Markov chains AMarkov chain models states and the transitions between them. As an example, Figure II.1(a) illustrates a Markov chain with three states: Sunny, Cloudy, and Rainy. The arrows represent transitions, with the arrow’s start showing the origin state and the end showing the destination state. The numbers on the arrows indicate the probability of moving from one state to another. Using the Markov chain in Figure II.1(a), we can calculate metrics such as the state probability, which is the probability of being in a particular state over time, given the current state. For instance, if the current state is Sunny, the Markov chain can determine the probabilities of being in each of the three states over time (i.e., days), as shown in Figure II.1(b). Here, the probability of staying
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