A Resampling-Based Stochastic Approximation Method for Analysis of Large Geostatistical Data

Abstract

The Gaussian geostatistical model has been widely used in modeling of spatial data. However, it is challenging to computationally implement this method because it requires the inversion of a large covariance matrix, particularly when there is a large number of observations. This article proposes a resampling-based stochastic approximation method to address this challenge. At each iteration of the proposed method, a small subsample is drawn from the full dataset, and then the current estimate of the parameters is updated accordingly under the framework of stochastic approximation. Since the proposed method makes use of only a small proportion of the data at each iteration, it avoids inverting large covariance matrices and thus is scalable to large datasets. The proposed method also leads to a general parameter estimation approach, maximum mean log-likelihood estimation, which includes the popular maximum (log)-likelihood estimation (MLE) approach as a special case and is expected to play an important role in analyzing large datasets. Under mild conditions, it is shown that the estimator resulting from the proposed method converges in probability to a set of parameter values of equivalent Gaussian probability measures, and that the estimator is asymptotically normally distributed. To the best of the authors’ knowledge, the present study is the first one on asymptotic normality under infill asymptotics for general covariance functions. The proposed method is illustrated with large datasets, both simulated and real. Supplementary materials for this article are available online.

Keywords:

• Asymptotic normality
• Infill asymptotics
• Large spatial data
• U-statistics

SUPPLEMENTARY MATERIALS

Section 1: A real data example, goldmine samples.

Section 2: The proof of Lemma 3.2.

Section 3: The proofs of Lemma 4.1, Theorem 4.1, and Theorem 4.2.

Liang’s research was partially supported by grants from the National Science Foundation (DMS-1007457 and DMS-1106494) and the award (KUS-C1-016-04) made by King Abdullah University of Science and Technology (KAUST). Yang’s research was partially supported by the endowment funds related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University. The authors thank the editor, associate editor, and the referees for their constructive comments that have led to significant improvement of this article.