This paper proposes to apply Gaussian graphical models to estimate the large-scale normal distribution in the context of data assimilation from a relatively small number of data from the satellite. Data assimilation is a field which fits simulation models to observation data developed mainly in meteorology and oceanography. The optimization problem tends to be huge as the nature of the simulation models and the observation systems. We consider the Gaussian graphical model which represents the variance-covariance matrix of the sea-surface heights over the equatorial Pacific and the global ocean. We consider several Gaussian graphical models which model the width of the neighborhood interaction and the best model is selected with the information criteria such as AIC and BIC. The maximum likelihood estimation of the Gaussian graphical model reduces to minimizing the minus log determinant function with an additional linear term. We succeeded in optimizing the variance-covariance matrix of up to the size 34,300*34,300 with 101,310 degrees of freedom, and the matrix of 8,585*8,585 with 150,381 degrees of freedom. This is accomplished by the Newton method (or ``dual interior-point algorithm") utilizing a supercomputer.
Manuscript, The Institute of Statistical Mathematics, April 2009 (To appear in The Quarterly Journal of the Royal Meteorological Society)