Negative ridge regression parameters for improving the covariance structure of multivariate linear downscaling models

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Abstract
A downscaling model for multivariate data, e.g. weather elements recorded at multiple sites, should not only be able to fit each of the observed series well, but it should also be able to reproduce observed relationships between the variables. In a linear sense, this means accurately simulating the observed covariance matrix. Multivariate ridge regression with negative ridge parameters is introduced as a means of accomplishing this goal. The procedure is conceptually similar to expanded downscaling: both force the covariance structure of the predictions to match that of observations. Unlike expanded downscaling, an explicit constraint on the covariance matrix is not added to the regression cost function. Instead, regression coefficients are estimated directly via a matrix equation, while ridge parameters, which are free to take positive or negative values, are adjusted iteratively such that the discrepancy between modelled and observed covariance matrices is minimized. Results from multi‐site temperature and precipitation data suggest that the proposed method is capable of constraining the predicted covariance matrix to closely match the observed. © Crown in the right of Canada. Published by John Wiley & Sons, Ltd

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