Regularization of error covariance with the Gaussian graphical model

In data assimilation, covariance matrices are introduced in order to prescribe the weights of the initial state, model dynamics, and observation, and suitable specification of the covariances is known to be essential for obtaining sensible state estimates. The covariance matrices are specified by sample covariances and are converted according to an assumed covariance structure. Modeling of the covariance structure consists of the regularization of a sample covariance and the constraint of a dynamic relationship. Regularization is required for converting the singular sample covariance into a non-singular sample covariance, removing spurious correlation between variables at distant points, and reducing the required number of parameters that specify the covariances. In previous studies, regularization of sample covariances has been carried out in physical (grid) space, spectral space, and wavelet space. We herein propose a method for covariance regularization in inverse space, in which we use the Gaussian graphical model. For each variable, we assume neighboring variables, i.e. a targeted variable is directly related to its neighbors and is conditionally independent of the non-neighboring variables. Conditional independence is expressed by specifying zero elements in the inverse covariance matrix. The non-zero elements are estimated numerically by the maximum likelihood using Newton's method. Appropriate neighbors can be selected with the AIC or BIC information criteria. We address some techniques for implementation when the covariance matrix has a large dimension. We present an illustrative example using a simple 3 × 3 matrix and an application to a sample covariance obtained from sea-surface height observations.