Improving the condition number of estimated covariance matrices
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High dimensional error covariance matrices are used to weight the contribution of observation and background terms in data assimilation procedures. As error covariance matrices are often obtained by sampling methods, the resulting matrices are often degenerate or ill-conditioned, making them too expensive to use in practice. In order to combat these problems, reconditioning methods are used. In this paper we present new theory for two existing methods that can be used to reduce the condition number of (or 'recondition') any covariance matrix: ridge regression, and the minimum eigenvalue method. These methods are used in practice at numerical weather prediction centres, but their theoretical impact on the covariance matrix itself is not well understood. Here we address this by investigating the impact of reconditioning on variances and covariances of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users with respect to both method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, and that the ridge regression method results in a larger increase to the variances compared to the original matrix than the minimum eigenvalue method for any covariance matrix. We also prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. We apply the reconditioning methods to two examples: a simple general correlation function, and an error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than the ridge regression method, but in contrast can increase off-diagonal correlations.
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