geostatsp (version 0.5.8)

matern: Evaluate the Matern correlation function

Description

Returns the Matern covariance for the distances supplied.

Usage

matern( x, y=NULL, param=c(range=1, variance=1, rough=1))
	## S3 method for class 'SpatialPoints':
matern(x,  y=NULL, param)
	## S3 method for class 'default':
matern( x, y=NULL, param)
	## S3 method for class 'dist':
matern( x, y=NULL, param)
	## S3 method for class 'Raster':
matern( x, y=NULL, param)
	## S3 method for class 'SpatialPointsDataFrame':
matern(x, y=NULL, param)

Arguments

x

A vector or matrix of distances, or Raster or SpatialPoints of locations, see Details below.

y

Covariance is calculated for the distance between locations in x and y. If y=NULL, covariance of x with itself is produced. However, if x is a matrix or vector it is assumed to b

param

A vector of named model parameters with, at a minimum names range and rough (see Details), and optionally variance (defaults to 1). For Geometric Anisotropy add aniso.ratio and either aniso.

Value

• When x is a vector or matrix or object of class dist, a vector or matrix of covariances is returned. With x being SpatialPoints, y must also be SpatialPoints and a matrix of correlations between x and y is returned. When x is a Raster, and y is a single location a Raster of covariances between each pixel centre of x and y is returned.

Details

The formula for the Matern correlation function is $$M(x) = \frac{variance}{\Gamma(rough)} 2^{rough-1} x (\sqrt{8 rough}/ range)^{rough} besselK(x \sqrt{8 rough}/ range, rough)$$

The range argument is sqrt(8*rough)*phi.geoR, sqrt(8*rough)*scale.whittle.RandomFields, and 2*scale.matern.RandomFields.

Geometric anisotropy is only avaliable when x is a Raster or SpatialPoints.

When x or y are rasters, cells are indexed row-wise starting at the top left.

Examples

# example with raster
myraster = raster(nrows=20,ncols=30,xmn=-3,xmx=3,ymn=-2,ymx=2)
param = c(range=2, rough=2,	aniso.ratio=0.6, aniso.angle.degrees=30)

# plot correlation of each cell with the origin
myMatern = matern(myraster, c(0,0), param)

plot(myMatern)


# correlation matrix for all cells with each other
myraster = raster(nrows=4,ncols=6,xmn=-3,xmx=3,ymn=-2,ymx=2)
myMatern = matern(myraster, param=c(range=2, rough=2))
dim(myMatern)

# plot the cell ID's
values(myraster) = seq(1, ncell(myraster))
mydf = as.data.frame(myraster, xy=TRUE)
plot(mydf$x, mydf$y, type='n')
text(mydf$x, mydf$y, mydf$layer)
# correlation between bottom-right cell and top right cell is
myMatern[6,24]

# example with points
mypoints = SpatialPointsDataFrame(cbind(runif(5), runif(5)),data=data.frame(id=1:5))
matern(mypoints, param=param)
#x=mypoints;y=NULL;param=param

# example with vector of distances
range=3
distVec = seq(0, 2*range, len=100)
roughSeq = c(0.5, 1, 2,20)
theCov = NULL
for(D in roughSeq) {
	theCov = cbind(theCov, matern(distVec, param=c(range=range, rough=D)))
}
matplot(distVec, theCov, type='l', lty=1, xlab='distance', ylab='correlation')
legend("topright", fill=1:length(roughSeq), legend=roughSeq,title='rough')
# exponential

distVec2 = seq(0, max(distVec), len=20)
points(distVec2, exp(-2*(distVec2/range)),cex=0.5)
# gaussian
points(distVec2, exp(-2*(distVec2/range)^2), col='blue',cex=0.5)
legend("right", pch=1, col=c('black','blue'), legend=c('exp','gau'))

# comparing to geoR and RandomFields

covGeoR = covRandomFields = NULL

for(D in roughSeq) {
	covGeoR = cbind(covGeoR, 
		geoR::matern(distVec, phi=param["range"]/sqrt(8*D), kappa=D))
	covRandomFields = cbind(covRandomFields,
		RandomFields::CovarianceFct(distVec, model="matern", 
			param = c(mean=0, variance=1, nugget=0, 
			scale=param["range"]/2, nu=D) ))
}

matplot(distVec, theCov, type='l', lty=1, xlab='distance', ylab='correlation')
matpoints(distVec, covGeoR, cex=0.5, pch=1)
matpoints(distVec, covGeoR, cex=0.5, pch=2)