Estimation of the mean for spatially dependent data belonging to a Riemannian manifold

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The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications. The aim of this work is to introduce models for spatial dependence among Riemannian data, with a special focus on the case of positive definite symmetric matrices. First, the Riemannian semivariogram of a field of positive definite symmetric matrices is defined. Then, we propose an estimator for the mean which considers 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. Finally, we address a meteorological problem, namely, the estimation of the covariance matrix between temperature and precipitation for the province of Quebec in Canada.