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Multivariate Probabilistic Time Series Forecasting with Correlated Errors

NeurIPS 2024 · Time series › Causal and generative time series · approach: Probabilistic and conformal forecasting · 2 citations (OpenAlex)

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Proposes a plug-and-play method that models cross-covariance of Gaussian errors across multiple steps in multivariate autoregressive probabilistic forecasters, using a low-rank-plus-diagonal covariance and latent temporal kernels, and applies it to GPVar (RNN) and a Transformer.

Problem

Probabilistic multivariate time series forecasters usually assume temporally independent errors, which ignores error autocorrelation and cross-lag correlation (e.g., from omitted covariates), hurting uncertainty quantification and accuracy.

Approach

Extend GLS-style likelihood training to multivariate models by letting independent latent temporal processes carry a dynamic correlation matrix C_t (weighted sum of SE kernels plus identity), giving a low-rank-plus-diagonal covariance over a batch of D steps whose inverse and determinant are computed via Woodbury and the matrix determinant lemma; residuals are then used to calibrate multistep pred

Key results

• Averaged over nine GluonTS datasets, the method improves CRPS_sum by 13.79% on GPVar (e.g., traffic 0.1095 to 0.0649 and m4_hourly 0.0613 to 0.0358).
• On the Transformer base model, the method yields an average 6.91% CRPS_sum improvement (e.g., wikipedia 0.0841 to 0.0500).
• The method is worse on exchange_rate for GPVar (0.0068 without vs 0.0117 with), so gains are not universal.

Measured effects reported in the paper

Surprising findings

• The method hurts GPVar on exchange_rate (CRPS_sum 0.0068 without vs 0.0117 with), and the Transformer on electricity and traffic.
• The learned weight on the identity component dominates, indicating mild error correlation for most of the data.

Limitations

• Gaussian error assumption may be misspecified; t-distribution baseline closes the gap only partially.
• Kronecker structure C_t kron I_R forces all latent processes to share the same autocorrelation, which may be too restrictive.
• Gains vary across datasets and base models and can be negative on some datasets; training costs are higher.

Experimental setup

Datasets: electricity m4_hourly exchange_rate m1_quarterly pems03 solar traffic uber_hourly wikipedia

Benchmarks: electricity m4_hourly exchange_rate m1_quarterly pems03 solar traffic uber_hourly wikipedia

Metrics: CRPS_sum CRPS energy score quantile loss RRMSE

Baselines: GPVar w/o correlated errors Transformer w/o correlated errors VAR GARCH (DCC-GARCH)

Builds on: GPVar DeepAR Gaussian copula linear model of coregionalization

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