# Multivariate Probabilistic Time Series Forecasting with Correlated Errors

arXiv: https://arxiv.org/abs/2402.01000 · HTML: https://mlsanity.com/paper/2402.01000/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 53 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Temporal Pattern Attention for Multivariate Time Series Forecasting](https://mlsanity.com/paper/1809.04206/) (2018)
- [Deep Learning for Time Series Forecasting: Tutorial and Literature Survey](https://mlsanity.com/paper/2004.10240/) (2020)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [TACTiS: Transformer-Attentional Copulas for Time Series](https://mlsanity.com/paper/2202.03528/) (2022)
- [TACTiS-2: Better, Faster, Simpler Attentional Copulas for Multivariate Time Series](https://mlsanity.com/paper/2310.01327/) (2023)
- [Adjusting for Autocorrelated Errors in Neural Networks for Time Series](https://mlsanity.com/paper/2101.12578/) (2021)
- [Neural networks for geospatial data](https://mlsanity.com/paper/2304.09157/) (2023)
- [Random Noise vs State-of-the-Art Probabilistic Forecasting Methods : A Case Study on CRPS-Sum Discrimination Ability](https://mlsanity.com/paper/2201.08671/) (2022)
- [Regions of Reliability in the Evaluation of Multivariate Probabilistic Forecasts](https://mlsanity.com/paper/2304.09836/) (2023)
- [Nonstationary Multivariate Gaussian Processes for Electronic Health Records](https://mlsanity.com/paper/1910.05851/) (2019)
- [Better Batch for Deep Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2305.17028/) (2023)
- [Scalable Dynamic Mixture Model with Full Covariance for Probabilistic Traffic Forecasting](https://mlsanity.com/paper/2212.06653/) (2022)

## Built on by (most cited first, 6 of 6)

- [MVG-CRPS: A Robust Loss Function for Multivariate Probabilistic Forecasting](https://mlsanity.com/paper/2410.09133/) (2024)
- [Univariate to Multivariate: LLMs as Zero-Shot Predictors for Time-Series Forecasting](https://mlsanity.com/paper/2506.02389/) (2025)
- [Deep Time-Series Forecasting in 10 Years: A Survey](https://mlsanity.com/paper/2603.19899/) (2026)
- [Teger: Spatiotemporal Covariance for Probabilistic Traffic Forecasting](https://mlsanity.com/paper/2605.18068/) (2026)
- [Parametric Prior Mapping Framework for Non-stationary Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2605.23402/) (2026)
- [TSSM: Triaxial State Space Model for Global Station Weather Forecasting with Temporal-Variable-Historical Modeling](https://mlsanity.com/paper/2607.13101/) (2026)

## Abstract (arXiv)

Accurately modeling the correlation structure of errors is critical for reliable uncertainty quantification in probabilistic time series forecasting. While recent deep learning models for multivariate time series have developed efficient parameterizations for time-varying contemporaneous covariance, but they often assume temporal independence of errors for simplicity. However, real-world data often exhibit significant error autocorrelation and cross-lag correlation due to factors such as missing covariates. In this paper, we introduce a plug-and-play method that learns the covariance structure of errors over multiple steps for autoregressive models with Gaussian-distributed errors. To ensure scalable inference and computational efficiency, we model the contemporaneous covariance using a low-rank-plus-diagonal parameterization and capture cross-covariance through a group of independent latent temporal processes. The learned covariance matrix is then used to calibrate predictions based on observed residuals. We evaluate our method on probabilistic models built on RNNs and Transformer architectures, and the results confirm the effectiveness of our approach in improving predictive accuracy and uncertainty quantification without significantly increasing the parameter size.