Testing for spatial isotropy under general designs

Abstract

Spatial modeling is typically composed of a specification of a mean function and a model for the correlation structure. A common assumption on the spatial correlation is that it is isotropic. This means that the correlation between any two observations depends only on the distance between those sites and not on their relative orientation. The assumption of isotropy is often made due to a simpler interpretation of correlation behavior and to an easier estimation problem under an assumed isotropy. The assumption of isotropy, however, can have serious deleterious effects when not appropriate. In this paper we formulate a test of isotropy for spatial observations located according to a general class of stochastic designs. Distribution theory of our test statistic is derived and we carry out extensive simulations which verify the efficacy of our approach. We apply our methodology to a data set on longleaf pine trees from an oldgrowth forest in the southern United States.

spTest: An R Package Implementing Nonparametric Tests of Isotropy

“…Lu (1994) and Lu and Zimmerman (2001) pioneered a popular approach to testing secondorder properties when they used the joint asymptotic normality of the sample semivariogram computed at different spatial lags to evaluate symmetry and isotropy properties. The subsequent works of Guan et al (2004); Guan, Sherman, and Calvin (2007) and Maity and Sherman (2012) built on these ideas and are the primary methods programmed in package spTest. Here we give an overview of the tests in Guan et al (2004) and Maity and Sherman (2012).…”

“…The subsequent works of Guan et al (2004); Guan, Sherman, and Calvin (2007) and Maity and Sherman (2012) built on these ideas and are the primary methods programmed in package spTest. Here we give an overview of the tests in Guan et al (2004) and Maity and Sherman (2012).…”

“…The tests in Guan et al (2004) and Maity and Sherman (2012) require estimating the semivariogram and covariance function values, respectively, at the set of chosen lags. We denote the vector of point estimates of the semivariogram/covariance function at the chosen lags as G n , the true values as G, and the asymptotic variance-covariance matrix of G n as Σ.…”

“…The bandwidth of the smoothing kernel used to estimate the semivariogram on the subblocks of data created by the moving window is indicated by subblock.h and a finite sample adjustment to the estimate of the asymptotic variance-covariance matrix is made by setting sig.est.finte = TRUE. For more information about the different arguments and guidelines on how to choose them, see Weller and Hoeting (2016), the spTest manual, and the original works (Guan et al 2004(Guan et al , 2007Maity and Sherman 2012).…”

“…A number of nonparametric tests of isotropy, which avoid the choice of a parametric covariance function, have been developed (see, e.g., Lu and Zimmerman 2001;Guan et al 2004;Lu and Zimmerman 2005;Maity and Sherman 2012). In Weller and Hoeting (2016), we provide a review of the different nonparametric methods available for testing isotropy and symmetry properties, including an extensive simulation study.…”