spTest (version 0.2.5)

GuanTestUnif: 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 uniformly distributed sampling locations. See Guan et. al. (2004) for more details.

Usage

GuanTestUnif(spdata, lagmat, A, df, h = 0.7, kernel = "norm",
  truncation = 1.5, xlims, ylims, grid.spacing = c(1, 1),
  window.dims = c(2, 2), subblock.h = h, 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 SpatialPointsDataFame from the package sp.

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.

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.

h

A scalar giving the bandwidth for the kernel smoother. The same bandwidth is used for lags in both the x and y directions.

kernel

A string taking one of the following values: norm, ep, cos, or unif, for the normal, Epanechnikov, cosine, or uniform kernel functions. Defaults to norm.

truncation

A scalar providing the truncation value for the normal density if kernel = "norm".

xlims

A vector of length two providing the lower and upper x-limits of the sampling region. To ensure all sampling locations are included in the subsampling procedure, the x-limits should be slightly wider than than the minimum and maximum observed x-coordinates of sampling locations.

ylims

A vector of length two providing the lower and upper y-limits of the sampling region. To ensure all sampling locations are included in the subsampling procedure, the y-limits should be slightly wider than than the minimum and maximum observed y-coordinates of sampling locations.

grid.spacing

A vector of length two providing the x (width) and y (height) spacing, respectively, of the underlying grid laid on the sampling region to create moving windows. If the grid spacing width does not evenly divide the width of the sampling region, some data will be ommited during subsampling, i.e., the function does not handle partial windows. Same applies to grid spacing height and height of sampling region. See details for an example.

window.dims

A vector of length two corresponding to the width and height of the moving windows used to estimate the asymptotic variance-covariance matrix. The width and height are given in terms of the spacing of the grid laid on the sampling region. See details for an example.

subblock.h

A scalar giving the bandwidth used for the kernel smoother when estimating the semivariogram on the moving windows (sub-blocks of data). Defaults to the same bandwidth used for the entire domain.

sig.est.finite

Logical. Defaults to TRUE, which provides a finite sample correction in estimating \(\Sigma\) (see Guan et. al. (2004) Section 4.2.2). 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 those 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 support partial blocks. For example, suppose the sampling region runs from 0 to 20 in the x-direction and from 0 to 30 in the y-direction and an underlying grid of 1 by 1 is laid over the sampling region. Then an ideal value of window.dims would be (2,3) since its entries evenly divide the width (20) and height (30), respectively, of the sampling region. Using window.dims (3, 4.5) would imply that some data will not be used in the moving windows since these values would create partial blocks in the sampling region.

The value window.dims provides the width and height of the moving window in terms of the underlying grid laid on the sampling region. For example, if a grid with dimensions of grid.spacing = c(0.1, 0.2) is laid on the sampling region and window.dims = c(2,3) then the dimensions of the subblocks created by the moving windows are (0.2, 0.6). Thus, the user must take care to ensure that the values of grid.spacing and window.dims are compatible with the dimensions of the sampling region. The easiest way to meet this constrain is to make the grid.spacing values a function of the xlims and ylims values. For example, to put down a \(10 \times 10\) grid on the domain, use grid.spacing = (xlims[2]-xlims[1], ylim[2]-ylims[1])/10. Then, setting window.dims = c(2,2) ensures that no data will be omitted during the subsampling.

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.

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

MaityTest GuanTestGrid

Examples

# NOT RUN {
library(mvtnorm)
set.seed(1)
#Sample Size
N <- 300
#Set parameter values for exponential covariance function
sigma.sq <- 1
tau.sq <- 0.0
phi <- 1/4
#Generate sampling locations
coords <-  cbind(runif(N,0,16), runif(N,0,16))
D <-  as.matrix(dist(coords))
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 = "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))
my.grid = c(1,1)
my.windims = c(4,4)
myh = 0.7
myh.sb = 0.8
my.xlims = c(0, 16)
my.ylims = c(0, 16)
tr <- GuanTestUnif(mydata, mylags, myA, df = 2, myh, "norm", 1.5,
 my.xlims, my.ylims, my.grid,my.windims, myh.sb)
tr


# }
# NOT RUN {
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 <- GuanTestUnif(mydata, mylags, myA, df = 2, myh, "norm", 1.5,
 my.xlims, my.ylims, my.grid,my.windims, myh.sb)
tr
# }