Hierarchical Machine Learning Model Improves Regional Multi-Leg Train Delay Predictions

A hierarchical machine learning framework that couples segment-specific predictive models with corridor-level meta-learning significantly improves arrival delay forecasting for multi-leg passenger rail journeys. Developed by researchers Elham Ahmadi, André Ludwig, and Henrik Leopold, the analytical architecture addresses a persistent operational challenge in regional rail transportation: standard performance metrics and machine learning predictors typically evaluate individual train runs in isolation, failing to capture the discontinuous, network-wide disruptions that occur when passengers must transfer between connecting services. By explicitly modeling intra-modal transfers and stacking out-of-fold delay forecasts from consecutive service segments, the framework bridges the operational gap between isolated train punctuality and end-to-end journey reliability. Empirical evaluation across seven years of operational data from central Germany demonstrates that this hierarchical structure successfully predicts destination delays while identifying the downstream consequences of maintained versus broken passenger connections.

Regional rail transportation networks form the backbone of passenger mobility by linking urban centers with surrounding communities, yet passenger itineraries across these corridors routinely require one or more transfers. Under conventional operating regimes, railway punctuality is evaluated through train-based benchmarks such as average delay durations or arrival thresholds at designated monitoring points. However, these metrics fail to reflect passenger experience, which is evaluated at the final journey destination rather than at intermediate stations. When an itinerary requires a connection, punctuality ceases to be a single-vehicle characteristic and becomes the property of an interdependent operational chain. A minor arrival disturbance on an incoming feeder train can consume scheduled interchange margins, causing passengers to miss downstream connections and face substantial, discrete delays while awaiting subsequent departures. Traditional machine learning models cannot easily capture these non-linear transitions when built as monolithic systems across heterogeneous routes.

To overcome these limitations, the research team designed a two-tiered supervised learning system tailored to the physical topology of connected railway corridors. Rather than fitting a single predictive model across an entire multi-stage itinerary, the architecture establishes separate base models for each distinct travel leg and combines their estimates through a corridor-level meta-learner. The base tier uses Extreme Gradient Boosting, commonly known as XGBoost, to predict next-station arrival delays by learning complex, nonlinear relationships among operational schedules, infrastructure attributes, temporal patterns, and environmental conditions. Station-level records are sequentially ordered along physical track distances, allowing the algorithms to track sequential delay propagation along each train run. By decomposing the network into its physical constituents, the system accommodates varying operational regimes, rolling-stock circulations, and track dynamics across disparate corridor segments.

Preventing data contamination in sequential forecasting requires rigorous preprocessing protocols, particularly when stacking algorithmic predictions. The researchers eliminated precomputed lag and cumulative delay fields from the historical logs, reconstructing strictly past-only delay variables within each individual trip trajectory. These leakage-safe predictors include one-step lagged arrival deviations, lagged departure delays, and cumulative past delays along the trip. To generate input features for the secondary meta-model without introducing target leakage, the framework employs grouped cross-validation structured around unique train-run trajectory identifiers. The resulting out-of-fold estimates ensure that the higher-level meta-learner trains exclusively on base predictions produced by models that have never encountered those specific trips during training, preserving the integrity of real-time decision-support simulations.

The study validated the methodology using real-world operational records spanning from 2016 through 2022 from a regional rail corridor in central Germany. The study environment comprises a two-leg passenger journey running from Fulda to Göttingen with a scheduled intra-modal transfer at Bebra station. The feeder segment corresponds to regional line RB5 connecting Fulda to Bebra, while the connecting downstream segment corresponds to regional line RB87 running from Bebra to Göttingen. Both routes operate as regional passenger lines making intermediate stops, but they present markedly different operational dynamics. To account for exogenous weather shocks that disrupt regular timetable execution, the operational logs were matched with daily meteorological observations from the open-access mCloud platform, including wind speed, precipitation volume, precipitation form, snow depth, and surface temperatures.

Prior to corridor integration, the authors performed candidate model screening using a forward-looking, time-aware holdout protocol to evaluate Ridge regression, Random Forest, Gradient Boosting, and XGBoost. The training sets covered earlier operational years while test sets examined future operational periods—specifically June 2021 to July 2022 for RB5 and April 2021 to July 2022 for RB87. This forward-looking evaluation confirmed that XGBoost provided the most balanced performance across both segments, offering regularized shrinkage, robust tail-error mitigation, and computational efficiency. Hyperparameter configurations were subsequently optimized using randomized search across five-fold grouped cross-validation over train trajectory identifiers, minimizing mean absolute error across the candidate spaces.

The empirical results from the leg-level models revealed a striking asymmetry in delay predictability between the upstream and downstream segments. The upstream RB5 model attained high predictive accuracy, yielding an out-of-fold mean absolute error of approximately 0.50 minutes, a median absolute error below 0.30 minutes, and a coefficient of determination of 0.84, with 95 percent of predictions falling within 1.36 minutes of actual arrivals. In contrast, the downstream RB87 segment proved substantially more volatile. Although the downstream model retained a low median absolute error of 0.42 minutes and a 95th percentile error of 1.74 minutes, its squared-error metrics deteriorated sharply: the root mean squared error expanded significantly while the coefficient of determination dropped to 0.07.

Analysis of corridor-wide operational logs explained the physical mechanisms underlying this statistical divergence. Upstream delay standard deviations along RB5 remained stable at approximately 3 to 4 minutes, governed primarily by local timetable adherence and short-term persistence. At Bebra, delay variability dropped slightly due to timetable buffer allocations. Downstream along RB87, however, delay standard deviations exceeded 12 minutes, approaching 30 minutes at Eichenberg station. Downstream operations absorb upstream disturbances, transfer-point interactions, and stochastic recovery constraints. Residual diagnostics on RB87 predictions confirmed significant heteroskedasticity, with a positive Spearman rank correlation of 0.142 between predicted delay magnitudes and absolute residuals. Small disturbances propagated predictably, whereas large disruption events produced heavy-tailed distributions that distorted squared-error metrics without undermining typical minute-level accuracy.

To synthesize these divergent operational signals, the secondary XGBoost meta-model integrates out-of-fold feeder predictions at Bebra with near-term propagation estimates from the initial downstream stop, scheduled transfer slack, calendar context, and meteorological covariates. Evaluated across held-out corridor journeys, the stacked meta-model achieved a mean absolute error of 1.53 minutes and a median absolute error under 1.00 minute. Furthermore, 90 percent of journey-level arrival delay predictions fell within 3.3 minutes of realized arrivals, and 95 percent remained within 5.0 minutes. By explicitly reflecting corridor topology rather than pooling disparate legs into an undifferentiated predictor, the hierarchical framework preserves operational interpretability while maintaining robust performance despite extreme downstream variance.

Beyond continuous delay estimation, the framework embeds a deterministic transfer-feasibility mapping layer to support dispatching and passenger information systems. Based on predicted upstream arrival delays and a required ten-minute operational buffer, the system calculates whether passengers will reach the connecting platform prior to the scheduled departure of the planned connecting service. If the feeder delay renders the connection infeasible, the model searches for the closest scheduled departure within a six-hour operational horizon. Across 8,868 mapped corridor instances, the model classified 79.3 percent of planned connections as feasible and 20.7 percent as broken. For missed transfers with an available service within the horizon, the mean predicted waiting time was 166.5 minutes, with median and 90th percentile waiting times reaching 175.4 and 235.9 minutes, respectively.

Visualizing predicted waiting times revealed distinct multi-modal distributions that directly reflect timetable geometry and train headways. When a connection is broken, passenger delay escalates discontinuously to the fixed interval of the subsequent train run, occasionally compounded by late-night service gaps where no alternative departure operates within the six-hour threshold, an outcome observed in 505 instances. These discrete jumps illustrate why traditional regression metrics alone cannot capture the operational stakes of multi-leg transit management. Slight forecast errors near the feasibility margin carry asymmetric real-world consequences, turning what appears to be a small minute-level variance into multi-hour travel disruptions.

The authors note that the architecture is designed for direct integration into real-time transit management, as all required input variables are fully observable at the interchange node prior to downstream departures. While the present implementation models a two-leg corridor without dynamic dispatch optimization, it provides the structural foundation for advanced disruption management. Future research directions outlined by the team include extending the hierarchical formulation to multi-transfer transit networks, incorporating real-time passenger flow counts, modeling infrastructure constraints, and developing multi-criteria optimization routines that balance passenger waiting penalties against system-wide delay propagation.

Subject of Research: Hierarchical machine learning framework for multi-leg passenger rail arrival delay prediction

Article Title: A hierarchical machine learning framework for aggregated multi-leg arrival time prediction and transfer-feasibility integration in regional rail corridors

Article References: Ahmadi, E., Ludwig, A., & Leopold, H. (2026). A hierarchical machine learning framework for aggregated multi-leg arrival time prediction and transfer-feasibility integration in regional rail corridors. European Transport Research Review, 18(1), Article 55. https://doi.org/10.1186/s12544-026-00814-4

Image Credits: AI Generated

DOI: 10.1186/s12544-026-00814-4

Keywords: passenger rail, delay prediction, machine learning, XGBoost, hierarchical modeling, stacked generalization, transfer feasibility, regional rail, interchange stations, out-of-fold predictions, delay propagation, railway operations, timetable slack, transit reliability, gradient boosting, decision support, multi-leg journeys