134 Chapter 6. Comparing Homogeneous and Inhomogeneous Time Markov Chains for Modelling Deterioration in Sewer Pipe Networks Table 6.1: Goodness-of-fit metrics computed on the training and testing sets for di!erent Markov chains. Blue and red colours indicate the best and worst scores, respectively. Training set Test set Cohort Type Func. |⇀| RMSE AIC BIC RMSE AIC BIC CMW IHTMC Gompertz 24 0.0250 57431.0 57601.8 0.0337 20436.0 20583.2 IHTMC Weibull 24 0.0233 57414.8 57585.5 0.0361 20478.0 20625.2 IHTMC Log-Logistic 24 0.0219 58544.4 58715.1 0.0391 20861.2 21008.4 IHTMC Log-Normal 24 0.0221 58553.6 58724.3 0.0385 20823.2 20970.3 CTMC Exponential 15 0.0312 59574.6 59681.3 0.0359 21142.1 21234.0 DTMC - 15 0.0312 59574.6 59681.3 0.0359 21142.1 21234.0 CS IHTMC Gompertz 24 0.0358 3532.8 3665.8 0.0468 1179.4 1290.9 IHTMC Weibull 24 0.0326 3850.0 3983.0 0.0423 1269.0 1380.5 IHTMC Log-Logistic 24 0.0313 4111.3 4244.3 0.0427 1345.9 1457.4 IHTMC Log-Normal 24 0.0330 4035.1 4168.1 0.0446 1324.4 1436.0 CTMC Exponential 15 0.0593 4006.7 4089.8 0.0583 1359.1 1428.8 DTMC - 15 0.0593 4006.7 4089.8 0.0583 1359.1 1428.8 PWM IHTMC Gompertz 24 0.0199 2349.3 2495.3 0.0172 989.7 1114.7 IHTMC Weibull 24 0.0153 5522.0 5668.0 0.0403 1822.1 1947.0 IHTMC Log-Logistic 24 0.0217 4699.7 4845.7 0.0178 1523.1 1648.0 IHTMC Log-Normal 24 0.0211 3588.2 3734.2 0.0179 1493.6 1618.6 CTMC Exponential 15 0.0297 2438.6 2529.9 0.0256 1002.8 1080.9 DTMC - 15 0.0297 2438.6 2529.9 0.0256 1002.8 1080.9 6.5 Conclusions and future research We examine the e!ectiveness of homogeneous and inhomogeneous Markov chains in modelling stochastic deterioration in sewer mains. We introduce four inhomogeneous Markov chain models parametrised with Log-Normal, Log-Logistic, Weibull, and Gompertz density functions, and compare them against a homogeneous Markov chain with an Exponential distribution and discrete-time Markov chains using the same dataset.
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