132 Chapter 6. Comparing Homogeneous and Inhomogeneous Time Markov Chains for Modelling Deterioration in Sewer Pipe Networks 6.4 Findings 6.4.1 Comparison between cohorts For all cohorts, the inhomogeneous time Markov chains (modelled with Gompertz, Weibull, Log-logistic, and Log-normal functions) outperform the homogeneous time Markov chains (modelled with the Exponential function and DTMCs) by achieving the lowest values in all goodness-of-fit metrics in Table 6.1. Notice that a smaller RMSE in Table 6.1 suggests a closer alignment of the Markov chains with the Turnbull estimator. Also, the goodness-of-fit metrics for both homogeneous time Markov chains are identical, which is consistent with the theoretical mapping of one into the other. This is visually corroborated in Figure 6.2. 6.4.2 Transition probabilities over time For further clarification and illustrative purposes in understanding the behaviour within di!erent types of Markov chains, Figure 6.3 displays the transition probability variations among Markov chains in the CS cohort. The homogeneous time Markov chain, employing the Exponential distribution, maintains constant transition probabilities over time, reflecting its homogeneous and memoryless properties. Conversely, the inhomogeneous time Markov chains reveal diverse behaviours in their transition probabilities, depicting distinct temporal variations. Notice that there are also di!erences in the transition probabilities between inhomogeneous Markov chains, due to the di!erent assumptions on the density functions. 6.4.3 Overfitting All inhomogeneous Markov chains map well where data is available (up to around 70-year-old pipes, see grey dashed vertical lines in Figure 6.2), however, beyond this point, these models tend to move faster to worse conditions. This is likely related to the additional degrees of freedom that inhomogeneous Markov provides. This e!ect is less in homogeneous Markov chains because they have fewer degrees of freedom. Thus, future research should consider this aspect in the model calibration, to improve the predictive capabilities of inhomogeneous time Markov chains. 6.4.4 Comparing inhomogeneous Markov chains Upon closer examination of inhomogeneous Markov chains modelled with LogNormal, Log-Logistic, Weibull, and Gompertz density functions, Table 6.1 reveals that the Gompertz distribution consistently demonstrates good performance across all cohorts and goodness-of-fit metrics, followed by Weibull and Log-Logistic density functions. Notably, the Weibull distribution shows poor performance for cohort PMW, likely due to sub-optimal parameters resulting from convergence in local optima.
RkJQdWJsaXNoZXIy MjY0ODMw