| Exponential, Matern, Radial Basis {fields} | R Documentation |
Functional form of covariance function assuming the argument is a distance between locations.
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)
d |
Vector of distances or for Matern.cor.to.range just a single distance. |
range |
Range parameter default is one. Note that the scale can also be specified through the "theta" scaling argument used in fields covariance functions) |
alpha |
1/range |
scale |
Same as phi |
phi |
Marginal variance. |
smoothness |
Smoothness parameter in Matern. Controls the number of derivatives in the process. Default is 1/2 corresponding to an exponential covariance. |
nu |
Same as smoothness |
M |
Interpreted as a spline M is the order of the derivatives in the penalty. |
dimension |
Dimension of function |
cor.target |
Correlation used to match the range parameter. Default is .5. |
guess |
An optional starting guess for solution. This should not be needed. |
... |
Additional arguments to pass to the bisection search function. |
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 covariance functions: a vector of covariances.
For Matern.cor.to.range: the value of the range parameter.
Doug Nychka
Stein, M.L. (1999) Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
stationary.cov, stationary.image.cov, Wendland,stationary.taper.cov rad.cov
# 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)