116 Chapter 5. Deterioration Modelling of Sewer Pipes via Discrete-Time Markov Chains: A Large-Scale Case Study in the Netherlands as the maximum damage class found during an inspection for the relevant damage codes. This conservative approach helps determine which pipes require repair in the near future. To create Table 5.2, we define a time interval ∆t, group pipes by age at the time of inspection, and count the pipes in each damage class per group, normalising by the total number of pipes in that group. Table 5.2: Discretised table ˆp (ˆn) k for cohort CMW, surface damage (BAF), with ∆t =3 years. Count (c) PipeAge (years) Time (t) Step (ˆn) ˆp (ˆn) k k =1 k =2 k =3 k =4 k =5 832 [0,3) 1.5 0 0.95 0.03 0.01 0.01 0.00 .. . .. . .. . .. . .. . .. . .. . .. . .. . 2,339 [48,51) 49.5 16 0.35 0.50 0.12 0.02 0.01 .. . .. . .. . .. . .. . .. . .. . .. . .. . 64 [75,78) 76.5 25 0.44 0.20 0.28 0.05 0.03 .. . .. . .. . .. . .. . .. . .. . .. . .. . For instance, in Table 5.2, for cohort CMW, damage code BAF (surface damage), and ∆t =3 years, there were 2,339 pipes with 48 ⇓PipeAge <51 years at the time of inspection. The count vector (c) shows the total number of pipes within a PipeAge interval, t is the mean value of the PipeAge interval, and ˆn represents the discretisation step. For ˆn=16, corresponding to the interval 48 ⇓PipeAge <51 at t =49.5 years, 35% of pipes were in State 1 'i.e., ˆp (ˆn=16) k=1 =0.35(, and 50% in State 2 'i.e., ˆp (ˆn=16) k=2 =0.50(. Thus, ˆp (ˆn) k forms a |ˆn| ↘| S| matrix representing the ground truth for calibrating the DTMCs. The sum of counts (c) gives the total number of pipes in the network, which varies across PipeAge intervals. We incorporate this variation by defining a weight vector (Eq. 5.3) in the calibration of the DTMCs. We disregard interval-censoring in our dataset, assuming that the sewer pipe has just reached the condition observed during the inspection, though it may have reached this state earlier. This topic is further discussed in Chapter 6. 5.3.4 Step 4: Calibration of the Discrete-Time Markov Chain To calibrate a DTMC, we optimise the parameters p(0) (Eq. 5.1) and Pij (Definition 11, page 99) to minimise the Root Mean Weighted Square Error (Err) (Eq. 5.4). First, we normalise the counts (c) (Table 5.2) to compute a weight vector ¯w as shown in Eq. 5.3: ¯w= c max(c) , (5.3)
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