# Cholesky-based multivariate Gaussian regression

arXiv: https://arxiv.org/abs/2102.13518 · HTML: https://mlsanity.com/paper/2102.13518/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2021.

## Built on by (most cited first, 7 of 7)

- [Multi-Target XGBoostLSS Regression](https://mlsanity.com/paper/2210.06831/) (2022)
- [Online Multivariate Regularized Distributional Regression for High-dimensional Probabilistic Electricity Price Forecasting](https://mlsanity.com/paper/2504.02518/) (2025)
- [Improved probabilistic regression using diffusion models](https://mlsanity.com/paper/2510.04583/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Additive Covariance Matrix Models: Modelling Regional Electricity Net-Demand in Great Britain](https://mlsanity.com/paper/2211.07451/) (2022)
- [Stagewise Boosting Distributional Regression](https://mlsanity.com/paper/2405.18288/) (2024)
- [Scoring rule nets: beyond mean target prediction in multivariate regression](https://mlsanity.com/paper/2409.14456/) (2024)

## Abstract (arXiv)

Distributional regression is extended to Gaussian response vectors of dimension greater than two by parameterizing the covariance matrix Sigma of the response distribution using the entries of its Cholesky decomposition. The more common variance-correlation parameterization limits such regressions to bivariate responses -- higher dimensions require complicated constraints among the correlations to ensure positive definite Sigma and a well-defined probability density function. In contrast, Cholesky-based parameterizations ensure positive definiteness for all distributional dimensions no matter what values the parameters take, enabling estimation and regularization as for other distributional regression models. In cases where components of the response vector are assumed to be conditionally independent beyond a certain lag r, model complexity can be further reduced by setting Cholesky parameters beyond this lag to zero a priori. Cholesky-based multivariate Gaussian regression is first illustrated and assessed on artificial data and subsequently applied to a real-world 10-dimensional weather forecasting problem. There the regression is used to obtain reliable joint probabilities of temperature across ten future times, leveraging temporal correlations over the prediction period to obtain more precise and meteorologically consistent probabilistic forecasts.