A covariate-dependent Markov jump process with application to the propagation of rail defect severity

Rail defects pose a significant threat to railway safety and efficiency. Refined modeling of the propagation of rail defect severity has the potential of informing maintenance activities for circumvention of dangerous rail degradation. We consider discretely observed degradation trajectories for defects discovered on the Norwegian rail network with the impact from tonnage and line speed, as well as rail curvature, profile and grade. The propagation of defect severity is modeled using a continuous-time Markov chain regressed on covariates. We propose two estimation approaches: (1) direct maximization of the discrete data log-likelihood using analytical gradient information, and (2) Monte Carlo simulation of fully observed defect trajectories, which informs an Expectation-Maximization algorithm Both methodologies give rise to fast convergence of model estimates with similar estimates, indicating a favorable local optimum. The covariate parameters are statistically significant and align with their expected physical effects. Model checking is performed by cross-validation. Experiments with indicator variables demonstrate that the included exogenous information satisfactorily accounts for the structural variability between the different rail lines. We compute expected transition times for defects on selected rail lines and demonstrate how spatially varying track conditions affect defect propagation.