## Improving the condition number of estimated covariance matrices. (arXiv:1810.10984v1 [math.OC])

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 wit查看全文