6 6.2. Methods and materials 129 The M-H algorithm executes 50,000 iterations, with the first 49,000 as the burn-in period, and the subsequent 1,000 samples used to compute mean values and the output ⇀M↗H . Post convergence, ⇀M↗H serves as the initial guess for SLSQP, with ⇀ parameters constrained between , and ¯,. SLSQP employs convergence tolerances of eps = 1E - 5 and ftol = 1E - 50, and runs for up to 300 iterations. Upon SLSQP convergence, ⇀SLSQP is derived and selected as the optimal set of hyper-parameters of the Markov chain. 6.2.3 Non-parametric modelling Non-parametric estimators are widely used in survival analysis to estimate survival probabilities directly from time-to-event data with possible censoring, without relying on assumptions about the underlying data distribution. Examples include the Kaplan-Meier estimator (Jason T. Rich, J. Gail Neely, Randal C. Paniello, et al., 2010), which accommodates left- and right-censored data, and the Turnbull estimator (Turnbull, 1976), which is appropriate for interval-censored data where exact event times are unknown but are constrained within specific time intervals. It is noteworthy that these non-parametric estimators may be sensitive to the proportion of censored data and often require large sample sizes to achieve accurate estimates. Algorithm 3 Calibrate Turnbull Estimator from lifelines import KaplanMeierFitter as KM Input: DataFrame df with columns pipe_age (age at inspection) andk (damage severity level), where k ↓S. procedure CalibrateTurnbullEstimator(df) T_models ′empty dictionary Binarization based on damage severity level s for all s ↓S do yp ′new empty DataFrame with columns: ‘left’ and ‘right’, matching dimensions of df The condition s>k is not observed: yp[‘left’][df[k] <s] ′df[[k] <s][pipe_age] yp[‘right’][df[k] <s] ′△ The condition s>k is observed: yp[‘left’][df[k] >s] ′0 yp[‘right’][df[k] >s] ′df[[k] >s][pipe_age] T_models[s] ′KM().fit_interval_censoring(yp[‘left’], yp[‘right’]) Output: T_models Given the interval-censored nature of the inspection dataset for sewer networks, resulting from the data collection process, we employ the Turnbull estimator. This serves as the ground truth and is thus used as a reference for evaluating model
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