Cholesky-based multivariate Gaussian regression

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