Spatial statistics for covariance matrices

The statistical analysis of covariance matrices as data objects is becoming increasingly relevant. The aim of this work is to introduce models for spatial dependence among covariance matrices. Little attention has been paid to this problem, while in many applications data are spatially distributed. We introduce a semivariogram for a covariance matrix field and an estimator for the mean covariance matrix, considering both the non Euclidean nature of the data and their spatial correlation. Simulated data are used to evaluate the performance of the proposed estimator: taking into account spatial dependence leads to better estimates when observations are irregularly spaced in the region of interest. We apply the proposed methodology to the exploration of covariance matrices between temperature and precipitation in the province of Quebec, Canada. We obtain estimates that are in better agreement with previous analysis of Canadian climate than those obtained ignoring spatial dependence. Finally, we propose a kriging estimator for covariance matrix fields based on a tangent space approximation.