Non-stationary Multivariate Spatial Covariance Estimation via Low-Rank Regularization

Abstract

Optimal spatial prediction is challenging when the number of observations is large. A large data set also comes with a non-stationary structure more often than not. Fixed rank kriging (FRK) is a popular method that handles both massive observations and non-stationarity. A family of non-stationary covariance functions is constructed based on a set of basis functions, and it allows for quite a few computational simplifications in spatial prediction. Our work extends FRK in several aspects. ??? ?First, a multivariate version of FRK is proposed based on spatial random effect models. It includes the model of FRK as a univariate special case. The family of the multivariate models is flexible enough to incorporate not only spatial non-stationarity but also asymmetry in spatial cross-covariances. ???? Second, a regularization approach is introduced for improvement of spatial covariance estimation. By regularizing the eigenvalues of a spatial covariance matrix, our method can effectively control estimation variability even when the number of parameters is large. Then FRK is identical to that with a zero regularization parameter. ???? Third, a simple method is developed for choosing among various sets of basis functions. FRK, as its name suggests, requires that the form of basis functions and the number of bases are fixed and known. Our selection method relaxes such a restriction, and brings about a more practical framework. ??? Moreover, a fast estimation and prediction method is provided to avoid taking a high-dimensional matrix inversion. Some numerical examples are given to demonstrate the effectiveness of the proposed method. ??