Spatiotemporal Covariance Estimation by Shifted Partial Tracing
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We consider the problem of covariance estimation for replicated space-time processes from the functional data analysis perspective. Due to the challenges to computational and statistical efficiency posed by the dimensionality of the problem, common paradigms in the space-time processes literature typically adopt parametric models, invariances, and/or separability. Replicated outcomes may allow one to do away with parametric specifications, but considerations of statistical and computational efficiency often compel the use of separability, even though the assumption may fail in practice. In this paper, we consider the problem of non-parametric covariance estimation, under "local" departures from separability. Specifically, we consider a setting where the underlying random field's second order structure is nearly separable, in that it may fail to be separable only locally (either due to noise contamination or due to the presence of a non-separable short-range dependent signal component). That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on the novel concept of shifted partial tracing, which is capable of estimating the model computationally efficiently under dense observation. Due to the denoising properties of shifted partial tracing, our methods are shown to yield consistent estimators of the separable part of the covariance even under noisy discrete observation, without the need for smoothing. Further to deriving the convergence rates and limit theorems, we also show that the implementation of our estimators, including for the purpose of prediction, comes at no computational overhead relative to a separable model. Finally, we demonstrate empirical performance and computational feasibility of our methods in an extensive simulation study and on a real data set.
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