Covariance Tapering for Interpolation of Large Spatial Datasets

Abstract

Interpolation of a spatially correlated random process is used in many areas. The best unbiased linear predictor, often called kriging predictor in geostatistical science, requires the solution of a large linear system based on the covariance matrix of the observations. In this article, we show that tapering the correct covariance matrix with an appropriate compactly supported covariance function reduces the computational burden significantly and still has an asymptotic optimal mean squared error. The effect of tapering is to create a sparse approximate linear system that can then be solved using sparse matrix algorithms. Extensive Monte Carlo simulations support the theoretical results. An application to a large climatological precipitation dataset is presented as a concrete practical illustration.

Interpolation of spatial data – A stochastic or a deterministic problem?

“…When all sampling locations are on a (near-)regular lattice, spectral methods to approximate the likelihood can be used and allow to reduce the computational cost to an order of O(n log(n)) [31,13,28]. These techniques cannot be applied to scattered data, but other approaches to approximating likelihoods [79,78,5,46], covariance tapering [29], or simplified Gaussian models of low rank [3,12,20] have been proposed and have been shown to be quite effective in reducing the computational effort to an order that allows the application of REML in most practical situations.…”

Review on statistical methods for large spatial Gaussian data

“…Another approach to remark is to reduce dimension of covariance matrix by tapering method (see e.g., Furrer et al, 2006;Kaufman et al, 2008), which essentially set elements zero in a covariance matrix when a distance between two sites is located far in way to keep the resulting tapered matrix positive definite. Specifically, using tapering function K taper to the original covariance matrix Σ, we can obtain the tapered covariance matrix by Σ(θ) • K(γ), where the dot notation • refers to Schur product and K(γ) is a tapering matrix with parameter γ.…”

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

“…where, R k (x, y) is the remainder term in the Taylor series expansion which can be bounded as follows: (25) provided that diam(X s ) ≤ η dist(X t , X s ) and α small. Repeating the same argument interchanging the roles of x and y, a similar result can be obtained with diam(…”

“…One approach is to use covariance tapering [25,26], in which the correct covariance matrix is tapered using an appropriately chosen compactly supported radial basis function which results in a sparse approximation of the covariance matrix that can be solved using sparse matrix algorithms. Another approach is to choose classes of covariance functions for which kriging can be done exactly using a multiresolution spatial process [27][28][29].…”