Probabilistic Traffic Forecasting with Dynamic Regression

TL;DR

The paper addresses the mismatch between common isotropic, time-independent error assumptions in deep spatiotemporal traffic forecasting and real-world error structure. It introduces a dynamic regression (DR) framework that models the base model's error as a matrix-valued first-order autoregression with Kronecker-structured, non-isotropic covariance, enabling probabilistic forecasts while preserving base model outputs. The loss combines a non-isotropic negative log-likelihood with an L1 penalty for sparsity, and the AR coefficients together with the covariance parameters are learned jointly with the base model. Across PEMSD7 (M) and PEMS08 datasets and multiple strong baselines, DR improves probabilistic metrics such as CRPS and quantile risks and provides interpretable error dynamics, though with higher training cost and scalability considerations for very large spatial dimensions. This modular, model-agnostic approach enhances uncertainty quantification in multivariate Seq2Seq traffic forecasting and can be extended to other spatiotemporal tasks.

Abstract

This paper proposes a dynamic regression (DR) framework that enhances existing deep spatiotemporal models by incorporating structured learning for the error process in traffic forecasting. The framework relaxes the assumption of time independence by modeling the error series of the base model (i.e., a well-established traffic forecasting model) using a matrix-variate autoregressive (AR) model. The AR model is integrated into training by redesigning the loss function. The newly designed loss function is based on the likelihood of a non-isotropic error term, enabling the model to generate probabilistic forecasts while preserving the original outputs of the base model. Importantly, the additional parameters introduced by the DR framework can be jointly optimized alongside the base model. Evaluation on state-of-the-art (SOTA) traffic forecasting models using speed and flow datasets demonstrates improved performance, with interpretable AR coefficients and spatiotemporal covariance matrices enhancing the understanding of the model.

Figures (17)

• Figure 1: Correlation matrices of residuals obtained from the original ASTGCN model on the PEMS08 dataset, where $\boldsymbol{\eta}_t=\operatorname{vec}(\boldsymbol{R}_t) \in \mathbb{R}^{NQ}$ is the vectorized residual ($N=170$ and $Q=12$). (a) Contemporaneous correlations. (b) Cross-correlations at lag 2016 (i.e., one week).
• Figure 2: Overview of the proposed DR framework, highlighting its structure and comparison with conventional methods.
• Figure 3: Correlation matrices of residuals obtained from the original Graph WaveNet model on the PEMS08 dataset. From (a) to (d): Contemporaneous correlations, Cross-correlations at lag 12 (i.e., most recent), lag 288 (i.e., 1 day), lag 2016 (i.e., 1 week).
• Figure 4: Correlation matrices of residuals obtained from the Graph WaveNet model on the PEMS08 dataset after applying our method. (a) Contemporaneous correlations; (b) Cross-correlations at lag 2016 (i.e., one week).
• Figure 5: Learned AR coefficient matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ for the Graph WaveNet model on the PEMS08 dataset.