fields (version 6.8)

Exponential, Matern, Radial Basis: Covariance functions

Description

Functional form of covariance function assuming the argument is a distance between locations.

Usage

Exponential(d, range = 1, alpha = 1/range, phi = 1)
Matern (d , scale = 1, range = 1,alpha=1/range,
     smoothness = 0.5, nu= smoothness, phi=scale) 
Matern.cor.to.range(d, nu, cor.target=.5, guess=NULL,...)
RadialBasis(d,M,dimension, derivative = 0)

Value

For the covariance functions: a vector of covariances.

For Matern.cor.to.range: the value of the range parameter.

Details

Exponential:

phi* exp( -d/range)

Matern:

phi*con*(d^nu) * besselK(d , nu )

Matern covariance function transcribed from Stein's book page 31 nu==smoothness, alpha == 1/range

GeoR parameters map to kappa==smoothness and phi == range check for negative distances

con is a constant that normalizes the expression to be 1.0 when phi=1.0 and d=0.

Matern.cor.to.range: This function is useful to find Matern covariance parameters that are comparable for different smoothness parameters. Given a distance d, smoothness nu, target correlation cor.target and range theta, this function determines numerically the value of theta so that

Matern( d, range=theta, nu=nu) == cor.target

See the example for how this might be used.

Radial basis functions: C.m,d r**(2m-d) d- odd

C.m,d r**(2m-d)ln(r) d-even where C.m.d is a constant based on spline theory and r is the radial distance between points. See radbas.constant for the computation of the constant. NOTE: Earlier versions of fields used ln(r^2) instead of ln(r) and so differ by a factor of 2.

References

Stein, M.L. (1999) Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.

See Also

stationary.cov, stationary.image.cov, Wendland,stationary.taper.cov rad.cov

Examples

# a Matern correlation function 
 d<- seq( 0,10,,200)
 y<- Matern( d, range=1.5, smoothness=1.0)
 plot( d,y, type="l")

# Several Materns of different smoothness with a similar correlation 
# range

# find ranges for nu = .5, 1.0 and 2.0 
# where the correlation drops to .1 at a distance of 10 units.

 r1<- Matern.cor.to.range( 10, nu=.5, cor.target=.1)
 r2<- Matern.cor.to.range( 10, nu=1.0, cor.target=.1)
 r3<- Matern.cor.to.range( 10, nu=2.0, cor.target=.1)

# note that these equivalent ranges
# with respect to this correlation length are quite different
# due the different smoothness parameters. 

 d<- seq( 0, 15,,200)
 y<- cbind(  Matern( d, range=r1, nu=.5),
             Matern( d, range=r2, nu=1.0),
             Matern( d, range=r3, nu=2.0))

 matplot( d, y, type="l", lty=1, lwd=2)
 xline( 10)
 yline( .1)