II.4.2 Case studies in sewer networks: overview 99 Definition 11 (Discrete-Time Markov Chain). ADiscrete-Time Markov Chain (DTMC) is defined by the tuple M=↔S, p(0), Pij↗; see S, p (0), and Pij in Definition 6. Here: - Pij : S↘S ↑[0,1] represents the transition probability matrix. Each entry pij of the matrix Pij specifies the probability of transitioning from state i to state j over the time step ∆t, where i, j ↓S. - The matrix Pij is time-invariant, characteristic of a time-homogeneous Markov process. The uniform time step ∆t ensures it applies to transitions at regular intervals, simplifying temporal modelling. This form is the simplest and most common type of Markov chains, where the state probabilities can be calculated with the Chapman-Kolmogorov equation: p(n) =p(0)Pn ij (4.5) Here, p(n) represents the state probability distribution at the nth step, and Pn ij is the n-th power of the transition probability matrix. II.4.2 Case studies in sewer networks: overview Validating degradation models through real-world case studies is a common practice. Depending on the model, di!erent types of data are collected from sewer networks. For instance, Machine Learning models—used e.g., for anomaly detection or damage classification—often utilise datasets containing images‡ or videos§. Examples include Haurum and Moeslund, 2021. We are interested in condition data collected through inspections using Closed Circuit Television (CCTV), which include reports of various damage types and their classification with a severity index, following guidelines such as EN 13508:1. Table II.1 presents a (non-exhaustive) overview of research utilising similar data to develop models for condition assessment and degradation modelling. For details on the datasets, such as construction year and population age, we recommend checking the sources. Table II.1 presents the reference, city, and country of each case study; the total sewer network length and the portion analysed (e.g., after data cleaning); the number of pipes studied (considered as the population of pipes); and the availability of the data. N.A. indicates that the reference does not explicitly mention data availability, while U.R. denotes data available upon request. From Table II.1, we conclude that, to our knowledge, the only publicly available dataset for degradation assessment is that provided by Jimenez-Roa, Heskes, Tinga, et al., 2022, which is part of the results of this dissertation. ‡https://paperswithcode.com/dataset/sewer-ml §https://videopipe.github.io/cctvpipe/index.html
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