Crossing-Free Probabilistic K-Line Forecasts Without Retraining

Probabilistic K-line forecasting describes uncertainty in four complementary prices, namely open--high--low--close (OHLC). However, it introduces two consistency problems: quantile crossing and K-line crossing. Quantile crossing occurs when a higher-quantile forecast falls below a lower-quantile forecast, while K-line crossing occurs when the forecast low exceeds the open or close, or the forecast high falls below the open or close. Existing solutions generally address only one problem through output reordering, specialized architectures, or penalized training objectives. We propose K-line--Quantile Sequential Projection (KQSP), a parameter-free and training-free reconciliation method applicable to forecasts produced by any model. Compared with other crossing solutions, KQSP preserves predictive accuracy while producing substantially smaller corrections to the original forecasts. To mitigate model bias, we evaluate KQSP using various models, including pretrained foundation models. KQSP reduces both quantile and K-line crossing rates to zero for all test data undertaken. These results show that probabilistic K-line consistency can be enforced independently of forecast generation and without retraining.

Hierarchical Forecast Reconciliation for Urban Rail Transit Demand Prediction under Operational Disruptions

Accurate and coherent passenger demand forecasting is essential for Urban Rail Transit (URT) operations. Passenger demand has a hierarchical structure in which origin-destination (OD) flows aggregate to station-level inflows and outflows through conservation constraints. In practice, station-level and OD-level forecasts are often generated independently, producing incoherent predictions that violate these constraints and introduce inconsistencies into operational decision-making. Such issues become more severe during disruptions, when forecasting reliability is most critical. This paper presents the first hierarchical forecast reconciliation framework for joint station-level and OD-level URT demand prediction. A neural Fully Connected Reconciler (FCR) learns a non-linear mapping from incoherent base forecasts to coherent hierarchical predictions while guaranteeing exact structural consistency by construction. The method is benchmarked against OLS, WLS, and Minimum Trace (MinT) variants using Rejsekort smart-card data from the Copenhagen S-train network under one-step, multi-step, and disruption forecasting scenarios. Results show that reconciliation consistently improves OD forecasting accuracy while ensuring hierarchical coherence. Under normal conditions, FCR performs competitively with MinT-based methods. An oracle analysis indicates that perfect station-level forecasts could reduce OD prediction error by up to 34 percent, highlighting the value of improved base forecasts. Under severe disruptions, FCR outperforms classical methods, reducing OD forecasting error by up to 17.45 percent in multi-step destination-side delay scenarios. These findings establish hierarchical reconciliation as an effective mechanism for improving forecast robustness, with the largest benefits occurring under the most challenging operating conditions.