Resampling Methods for Spatial Data

Abstract

In this chapter, we describe bootstrap methods for spatial processes observed at finitely many locations in a sampling region in ℝ d . Depending on the spatial sampling mechanism that generates the locations of these data-sites, one gets quite different behaviors of estimators and test statistics. As a result, formulation of resampling methods and their properties depend on the underlying spatial sampling mechanism. In Section 12.2, we describe some common frameworks that are often used for studying asymptotic properties of estimators based on spatial data. In Section 12.3, we consider the case where the sampling sites (also referred to as data-sites in this book) lie on the integer grid and describe a block bootstrap method that may be thought of as a direct extension of the MBB method to spatial data. Here, some care is needed to handle sampling regions that are not rectangular. We establish consistency of the bootstrap method and give some numerical examples to illustrate the use of the method. Section 12.4 gives a special application of the block resampling methods. Here, we make use of the resampling methods to formulate an asymptotically efficient least squares method of estimating spatial covariance parameters, and discuss its advantages over the existing estimation methods. In Section 12.5, we consider irregularly spaced spatial data, generated by a stochastic sampling design. Here, we present a block bootstrap method and show that it provides a valid approximation under nonuniform concentration of sampling sites even in presence of infill sampling. It may be noted that infill sam-pling leads to conditions of long-range dependence in the data, and thus, the block bootstrap method presented here provides a valid approximation under this form of long-range dependence. Resampling methods for spatial prediction are presented in Section 12.6.