Beyond MSE: Ordinal Cross-Entropy for Probabilistic Time Series Forecasting
This addresses uncertainty estimation and outlier robustness in time series forecasting for applications like finance or weather prediction, though it appears incremental as it modifies loss functions rather than introducing a new paradigm.
The paper tackles the limitations of MSE loss in time series forecasting by proposing OCE-TS, which uses ordinal cross-entropy loss for probabilistic forecasting, and shows it consistently outperforms five baseline models on seven datasets using MSE and MAE metrics.
Time series forecasting is an important task that involves analyzing temporal dependencies and underlying patterns (such as trends, cyclicality, and seasonality) in historical data to predict future values or trends. Current deep learning-based forecasting models primarily employ Mean Squared Error (MSE) loss functions for regression modeling. Despite enabling direct value prediction, this method offers no uncertainty estimation and exhibits poor outlier robustness. To address these limitations, we propose OCE-TS, a novel ordinal classification approach for time series forecasting that replaces MSE with Ordinal Cross-Entropy (OCE) loss, preserving prediction order while quantifying uncertainty through probability output. Specifically, OCE-TS begins by discretizing observed values into ordered intervals and deriving their probabilities via a parametric distribution as supervision signals. Using a simple linear model, we then predict probability distributions for each timestep. The OCE loss is computed between the cumulative distributions of predicted and ground-truth probabilities, explicitly preserving ordinal relationships among forecasted values. Through theoretical analysis using influence functions, we establish that cross-entropy (CE) loss exhibits superior stability and outlier robustness compared to MSE loss. Empirically, we compared OCE-TS with five baseline models-Autoformer, DLinear, iTransformer, TimeXer, and TimeBridge-on seven public time series datasets. Using MSE and Mean Absolute Error (MAE) as evaluation metrics, the results demonstrate that OCE-TS consistently outperforms benchmark models. The code will be published.