5 5.4. Experimental evaluation 117 Err is then calculated as the di!erence between the discretised table 'ˆp (ˆn) k ( and the DTMC predictions using the Chapman-Kolmogorov equation (Eq. 4.5, page 99) '¯p (ˆn) k (, Err = )ˆn,k '¯p (ˆn) k →ˆp (ˆn) k ( 2 ∈ ¯wˆn |ˆn| ↘|S| (5.4) Minimisation of Err is performed using the Sequential Least-Squares Programming (SLSQP) algorithm fromScipy (Virtanen, Gommers, Oliphant, et al., 2020) with default parameters. All optimisation parameters are bounded in[0,1] and initialised as follows: (i) in ¯p(0), ¯p (0) k=1 =1 and ¯p (0) k↔ =1 =0; (ii) Pij is the identity matrix. We apply the constraints outlined in Definition 6 (page 96) for both Markov chains. We calibrate the DTMCs by randomly selecting 50% of the available data per cohort using repeated half-sample bootstrap (Saigo, Shao, and Sitter, 2001). The calibration process outputs the p(0) and Pij with the smallest Err for a given ∆t. 5.4 Experimental evaluation 5.4.1 Case study The detailed description of the case study is provided in Section II.4.3 on page 100. Based on these data, as recommended by domain experts, we focus on the damage codes (d.c.): infiltration (BBF), surface damage (BAF), and cracks (BAB), which were observed in 44%, 35%, and 18% of the inspections, respectively. 5.4.2 Results We visually compare the cohorts and chains using Figure 5.2. To construct these figures, we train a thousand DTMCs using repeated half-sampled bootstrap (Section 5.3.4) to account for uncertainty. Figure 5.2 presents selected results, with the full analysis available on Zenodo†. Figures 5.2(a)-(d) show the probability of being in state k given PipeAge, cohort, chain, and damage code. Markers represent the discretised table ˆp (ˆn) k (with ∆t =3 years), where size visualises counts. Dashed lines indicate a 95% confidence interval based on the projections of a thousand calibrated DTMCs, while the solid line shows the median value. Figures for comparing cohorts and chains are in Zenodo† under /comparing_cohorts and /comparing_chains. Figures 5.2(e)-(g) display expectations computed using Eq. 5.2, corresponding to the expected severity class for a given damage class at a specific PipeAge. The dashed and solid lines represent the confidence interval and median value, †All comparative figures and scripts are available at zenodo.org/record/6535853
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