A Fast, Optimal Spatial-Prediction Method for Massive Datasets

Abstract

This article considers a class of multiresolution tree-structured models that are spatially shifted versions of each other and proposes a new spatial-prediction method that averages over the optimal spatial predictors produced from members of this class of models. As a consequence, the resulting predicted surface is smooth, even when the predictors generated separately from individual multiresolution treestructured models are not. We call the new predictor the multiresolution spatial (MURS) predictor and develop a computationally efficient algorithm for it. The algorithm can handle massive datasets even when some observations are missing. Moreover, the MURS predictor can be shown to be the minimum mean squared error predictor for a large class of covariance functions. A simulation example for massive datasets shows that the MURS method consistently outperforms two commonly used filtering methods. Total column ozone data remotely sensed from a satellite are analyzed using the new methodology.

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

“…The numerical calculations involve the use of the Trench [24] algorithm for inverses of Toeplitz matrices mentioned above and truncation of the sums in (25). Moreover, since the big blocks estimator is the maximum likelihood estimator based on the block means, the expected value of * 2 p means () is the same as (25) and therefore, in this case, the information sandwich approximation to the asymptotic variance is just the reciprocal of (25).…”

“…Since the summands in (25) include elements of the gradient of the inverse covariance matrix for the block means (V means ), we have not attempted to evaluate (25) analytically but instead give numerical results. The numerical calculations involve the use of the Trench [24] algorithm for inverses of Toeplitz matrices mentioned above and truncation of the sums in (25).…”

“…it is resolution consistent) and uses a change of resolution Kalman filter. Building on these ideas, Johannesson and Cressie [16] and Tzeng et al [25] propose several extensions of these models to improve the "blocky'' structure of the initially proposed covariances and to extend it to spatio-temporal modeling. We have not taken this approach in the present work.…”

Estimation of missing data points from remotely sensed datasets

“…This step takes the finest resolution available and moves up to the coarsest defined resolution, following procedure introduced in [7] and outlined below. The process begins by first using the available data to obtain estimates at the finest resolution, factoring in whether the cell was informed or not, as well as the measurement error…”

“…After the estimation of the variance terms, a measure of total variance by resolution [7] can be calculated…”