Non-stationary Covariance Function

Analyzing datasets, such as sea-level records, pose a challenging statistical problem for reasons including non-stationarity, non-uniformly smooth spatial boundaries, and sparsity in the data. In this paper, we propose a framework to estimate the non-stationary covariance function by employing intrinsic statistics on the local covariates. These local covariates represent the underlying local correlation in the measurements, and they are assumed to lie on a Riemannian manifold of positive definite matrices. Additionally, we provide a technique for data-assimilation of correlated natural processes in order to improve the regression estimates arising from spatially sparse datasets. Experiments on a synthetic and real dataset of relative sealevel changes across the world demonstrate improvements in the error metrics for the regression estimates using our newly proposed approach.