Inference for variograms

Abstract

The empirical variogram is a standard tool in the investigation and modelling of spatial covariance. However, its properties can be difficult to identify and exploit in the context of exploring the characteristics of individual datasets. This is particularly true when seeking to move beyond description towards inferential statements about the structure of the spatial covariance which may be present. A robust form of empirical variogram based on a fourth-root transformation is used. This takes advantage of the normal approximation which gives an excellent description of the variation exhibited on this scale. Calculations of mean, variance and covariance of the binned empirical variogram then allow useful computations such as confidence intervals to be added to the underlying estimator. The comparison of variograms for different datasets provides an illustration of this. The suitability of simplifying assumptions such as isotropy and stationarity can then also be investigated through the construction of appropriate test statistics and the distributional calculations required in the associated p-values can be performed through quadratic form methods. Examples of the use of these methods in assessing the form of spatial covariance present in datasets are shown, both through hypothesis tests and in graphical form. A simulation study explores the properties of the tests while pollution data on mosses in Galicia (North-West Spain) are used to provide a real data illustration.

Analysis of Spatial Data with a Nested Correlation Structure

“…(b) shows that the dependence between malaria counts is stronger in the east–west direction than in the north–south direction, which is a suggestion of anisotropy. This directional dependence is confirmed by a statistically significant non‐parametric global isotropy test (Bowman and Crujeiras, ) on log‐transformed data with a p ‐value of less than 0.0001.…”

“…An elliptical contour suggests geometric anisotropy. Lastly, the non-parametric test of isotropy that was proposed by Bowman and Crujeiras (2013) can be used to confirm the directional dependence.…”

The effect of spatial variation for predicting aphid outbreaks

“…The n site-specific initial occurrence probabilities are assumed to arise from a Gaussian process model, where µ 0 is the mean initial occurrence probability, 2 0 is the variance, and ρ d i,j = ρ 0 Exp − d i,j α with d i,j being the distance between site i and site j, α is the scale of the spatial effect that is set to 10 km, and ρ 0 is a parameter that measures the spatial covariance (Haran, 2011; Ovaskainen, Roy, Fox, & Anderson, 2016). The covariance matrix by The semivariograms were calculated from the untransformed occurrence probability data where the distances were binned into intervals of 1,000 m (Bowman & Crujeiras, 2013).…”

Variogram calculations for random fields on regular lattices using quadrature methods

“…:/ of a spatial random field Z.x/ plays a central role in describing the structure of spatial variability of Z.x/. The early works of Matheron (1963Matheron ( , 1971) and Cressie (Cressie and Hawkins 1980;Cressie, 1985Cressie, , 1989Cressie, , 1990, the works of Diggle et al (1998Diggle et al ( , 2010, Huang et al (2011), Christensen (2011), Nordman and Caragea (2012), Bowman and Crujeiras (2013), Pigoli et al (2016), the geoR package (Ribeiro and Diggle, 2001), and the books by Journel and Huijbregts (1978), Isaaks and Srivastava (1989), Cressie (1993), Diggle and Ribeiro (2007), and Chilès and Delfiner (2009) provide an excellent account of variogram function and outline its importance in geostatistical inference. When it exists, the theoretical variogram is defined as half the variance of the difference between values Z.x/ and Z.x 0 / at two locations x and x 0 .…”