Cholesky-based multivariate Gaussian regression

Distributional regression is extended to Gaussian response vectors of dimension greater than two by parameterizing the covariance matrix Σ of the response distribution using the entries of its Cholesky decomposition. The more common variance-correlation parameterization limits such regressions to bivariate responses because higher dimensions require complicated constraints among the correlations to ensure positive definite Σ and a well-defined probability density function. In contrast, Cholesky-based parameterizations ensure positive definiteness for all distributional dimensions regardless of the 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, model complexity can be further reduced by setting the corresponding Cholesky parameters 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 time points, leveraging temporal correlations over the prediction period to obtain more precise and meteorologically consistent probabilistic forecasts.