spTest (version 0.2.5)

GuanTestGrid: Nonparametric Test of Isotropy Using the Sample Semivariogram

Description

This function performs the nonparametric test of isotropy using the sample semivariogram from Guan et. al. (2004) for spatial data with sampling locations on a grid. See Guan et. al. (2004) for more details.

Usage

GuanTestGrid(spdata, delta = 1, lagmat = rbind(c(1, 0), c(0, 1), c(1, 1),
  c(-1, 1)), A = rbind(c(1, -1, 0, 0), c(0, 0, 1, -1)), df = 2,
  window.dims = c(2, 2), pt.est.edge = TRUE, sig.est.edge = TRUE,
  sig.est.finite = TRUE)

Arguments

spdata

An \(n \times 3\) matrix. The first two columns provide \((x,y)\) spatial coordinates. The third column provides data values at the coordinates. This argument can also be an object of class geodata from the package geoR, or of class SpatialGridDataFame or SpatialPixelsDataFame from the package sp. If the class is SpatialPixelsDataFame, then the parameter delta is extracted from the grid attributes.

delta

A scalar indicating the distance between grid locations. Defaults to 1 (integer grid) and assumes equal spacing between locations in the x and y directions.

lagmat

A \(k \times 2\) matrix of spatial lags. Each row corresponds to a lag of the form \((x.lag, y.lag)\) for which the semivariogram value will be estimated. The scale of the lags provided in 'lagmat' are in units of delta. See details for more information.

A

A \(d \times k\) contrast matrix. The contrasts correspond to contrasts of the estimated semivariogram at the lags given in lagmat.

df

A scalar indicating the row rank of the matrix A. This value gives the degrees of freedom for the asymptotic \(\chi^2\) distribution used to compute the p-value.

window.dims

A vector of length two corresponding to the width and height (in number of columns and rows, respectively) of the moving windows used to estimate the asymptotic variance-covariance matrix. If window width does not evenly divide the number of columns of spatial data, some data will be ommited during subsampling, i.e., function does not handle partial windows. Same applies to window height and number of rows of spatial data.

pt.est.edge

Logical. TRUE corrects for edge effects in the point estimate (see Guan et. al. (2004), Section 4.2.1 for details).

sig.est.edge

Logical. Defaults to TRUE, which corrects for edge effects when estimating the semivariogram in the moving windows (see Guan et. al. (2004), Section 4.2.1 for details).

sig.est.finite

Logical. Defaults to TRUE, which provides a finite sample correction in estimating \(\Sigma\), the asymptotic variance-covariance matrix. (see Guan et. al. (2004) Section 3.2.1, Equation 5). False provides the empirical variance-covariance matrix of sample semivariogram values computed via the moving windows.

Value

gamma.hat

A matrix of the spatial lags provided and the semivariogram point estimates,\(\hat{\gamma}()\), at the lags used to construct the test statistic.

sigma.hat

The estimate of asymptotic variance-covariance matrix, \(\widehat{\Sigma}\), used to construct the test statistic.

n.subblocks

The number of subblocks created by the moving window used to estimate \(\Sigma\).

test.stat

The calculated test statistic.

pvalue.finite

The approximate, finite-sample adjusted p-value computed by using the subblocks created by the moving windows (see Guan et. al. (2004), Section 3.3 for details).

pvalue.chisq

The p-value computed using the asymptotic \(\chi^2\) distribution.

Details

This function currently only supports square and rectangular sampling regions and does not currently support partial blocks. For example, suppose the sampling grid contains 20 columns and 30 rows of data. Then an ideal value of window.dims would be (2,3) since its entries evenly divide the number of columns (20) and rows (30), respectively, of data. To preserve the spatial dependence structure, the moving window should have the same shape (i.e., square or rectangle) and orientation as the entire sampling domain.

The parameter delta serves to scale the samplng locations to the integer grid. Thus the lags provided in lagmat are automatically scaled by delta by the function. For example, suppose spatial locations are observed on grid boxes of 0.5 degrees by 0.5 degrees and referenced by longitude and latitude coordinates in degrees. Then, delta should be 0.5 and a spatial lag of (0,1) corresponds to a change in coordinates of (0, 0.5), i.e., moving one sampling location north in the y-direction.

References

Guan, Y., Sherman, M., & Calvin, J. A. (2004). A nonparametric test for spatial isotropy using subsampling. Journal of the American Statistical Association, 99(467), 810-821.

See Also

GuanTestUnif MaityTest

Examples

# NOT RUN {
library(mvtnorm)
set.seed(1)
#number of rows and columns
nr <- 12
nc <- 18
n <- nr*nc
#Set up the coordinates
coords <- expand.grid(0:(nr-1), 0:(nc-1))
coords <- cbind(coords[,2], coords[,1])
#compute the distance between sampling locations
D <- as.matrix(dist(coords))
#Set parameter values for exponential covariance function
sigma.sq <- 1
tau.sq <- 0.0
phi <- 1/4
R <- sigma.sq * exp(-phi*D)
R <- R + diag(tau.sq, nrow = n, ncol = n)
#Simulate Gaussian spatial data
z <- rmvnorm(1,rep(0,n), R, method = c("chol"))
z <-  z-mean(z)
z <- t(z)
mydata <- cbind(coords, z)
mylags <-  rbind(c(1,0), c(0, 1), c(1, 1), c(-1,1))
myA <-  rbind(c(1, -1, 0 , 0), c(0, 0, 1, -1))
tr <- GuanTestGrid(mydata, delta = 1, mylags, myA, df = 2, window.dims = c(3,2), 
pt.est.edge = TRUE, sig.est.edge = TRUE, sig.est.finite = TRUE )
tr

library(geoR)
#Simulate data from anisotropic covariance function
aniso.angle <- pi/4
aniso.ratio <- 2
coordsA <- coords.aniso(coords, c(aniso.angle, aniso.ratio))
Da <- as.matrix(dist(coordsA))
R <- sigma.sq * exp(-phi*Da)
R <- R + diag(tau.sq, nrow = n, ncol = n)
z <- rmvnorm(1,rep(0,n), R, method = c("chol"))
z <-  z-mean(z)
z <- t(z)
mydata <- cbind(coords, z)
#Run the test on the data generated from an anisotropic covariance function
tr <- GuanTestGrid(mydata, delta = 1, mylags, myA, df = 2, window.dims = c(3,2), 
pt.est.edge = TRUE,sig.est.edge = TRUE, sig.est.finite = TRUE)
tr
# }