Two-stage Odd Residual Flows for Mean-Preserving Probabilistic Time Series Forecasting

2026-08-11

AI Summary

Existing probabilistic time series forecasting methods struggle to simultaneously achieve distributional flexibility and accurate mean prediction. This work proposes a Two-stage Odd Residual Flow (TORF) framework that first leverages a pretrained deterministic model to generate high-accuracy point forecasts, then models the residual uncertainty using a constrained normalizing flow composed exclusively of odd functions centered at the predicted mean. TORF is the first approach to decouple mean estimation from uncertainty modeling while guaranteeing exact mean preservation without requiring sampling, thanks to the odd-function constraint. By jointly optimizing negative log-likelihood and Continuous Ranked Probability Score (CRPS), TORF achieves state-of-the-art performance in both deterministic accuracy (measured by NMAE) and probabilistic forecasting quality (measured by CRPS) across short- and long-horizon time series prediction tasks.

Abstract

Probabilistic forecasting plays an essential role in risk-sensitive decision-making, particularly in long-horizon settings. However, existing approaches often face a fundamental trade-off between distributional flexibility and accurate mean prediction. Traditional parametric methods, such as Mean Variance Estimation (MVE), can suffer from degraded point accuracy when trained under joint Negative Log-Likelihood (NLL) objectives, while modern-flexible generative models, including Normalizing Flows and Diffusion Models, typically rely on costly Monte Carlo sampling and may yield suboptimal mean estimates. To address this limitation, we propose Two-stage Odd Residual Flows (TORF), a framework that decouples mean forecasting from uncertainty estimation. In the first stage, a pre-trained deterministic model is used to produce an accurate mean prediction. In the second stage, a Restricted Normalizing Flow, with strictly odd functions learns flexible residual distributions around the point forecast, guaranteeing mean preservation from the first stage without sampling. Experiments show that TORF achieves state-of-the-art deterministic accuracy (NMAE) while providing strong density estimation performance (CRPS) on short and long-horizon forecasting.

Problem

• probabilistic forecasting
• mean prediction
• distributional flexibility
• time series forecasting
• uncertainty estimation

Innovation

• Two-stage Odd Residual Flows
• mean-preserving
• probabilistic forecasting
• restricted normalizing flows
• decoupled uncertainty estimation