Robust Resampling Confidence Intervals for Empirical Variograms
- PDF / 701,448 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 66 Downloads / 192 Views
Robust Resampling Confidence Intervals for Empirical Variograms Robert Graham Clark · Samuel Allingham
Received: 22 February 2010 / Accepted: 4 August 2010 / Published online: 3 December 2010 © International Association for Mathematical Geosciences 2010
Abstract The variogram function is an important measure of the spatial dependencies of a geostatistical or other spatial dataset. It plays a central role in kriging, designing spatial studies, and in understanding the spatial properties of geological and environmental phenomena. It is therefore important to understand the variability attached to estimates of the variogram. Existing methods for constructing confidence intervals around the empirical variogram either rely on strong assumptions, such as normality or known variogram function, or are based on resampling blocks and subject to edge effect biases. This paper proposes two new procedures for addressing these concerns: a quasi-block-bootstrap and a quasi-block-jackknife. The new methods are based on transforming the data to decorrelate it based on a fitted variogram model, resampling blocks from the decorrelated data, and then recorrelating. The coverage properties of the new confidence intervals are compared by simulation to a number of existing resampling-based intervals. The proposed quasi-block-jackknife confidence interval is found to have the best properties of all of the methods considered across a range of scenarios, including normally and lognormally distributed data and misspecification of the variogram function used to decorrelate the data. Keywords Spatial analysis · Variograms · Bootstrap · Jackknife · Block bootstrap · Block jackknife
R.G. Clark () Centre for Statistical and Survey Methodology, University of Wollongong, Wollongong, NSW 2522, Australia e-mail: [email protected] S. Allingham Centre for Health Services Development, University of Wollongong, Wollongong, NSW 2522, Australia e-mail: [email protected]
244
Math Geosci (2011) 43: 243–259
1 Introduction Spatial datasets consist of one or more variables measured at a number of locations defined by a coordinate system. One of the aims of many spatial analyses is to understand how the relationship between the variables measured at two points depends on their positions. Typically nearby points tend to be similar while distant points are less so, but the way in which the relationship depends on distance can be very different in different examples, and a good understanding of this is crucial in the design and analysis of spatial studies. The sample variogram is a widely used tool for this purpose (Cressie and Hawkins 1980). Variograms can be generalized to the cross-variogram for multivariate data (Chiles and Delfiner 1999), but we will consider situations where a single variable Z is measured on a set of points defined in two dimensions. We write Z(s) for the value of Z at a point s. First and second order stationarity is assumed; that is, the first two moments of Z(s) are assumed to not depend on s. It will also be assumed th
Data Loading...
