Cholesky-based multivariate Gaussian regression

TL;DR

This paper introduces a Cholesky-based approach for multivariate Gaussian regression that ensures positive definiteness of the covariance matrix in higher dimensions, enabling more flexible and reliable modeling of complex multivariate responses.

Contribution

The paper proposes a novel Cholesky decomposition parameterization for multivariate Gaussian regression, overcoming limitations of traditional methods in higher dimensions and allowing for efficient estimation and regularization.

Findings

Successfully applied to artificial data demonstrating model flexibility.

Achieved improved probabilistic weather forecasts by modeling temporal correlations.

Ensured positive definiteness of covariance matrices regardless of parameter values.

Abstract

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.