Exponential, Matern, Radial Basis | R Documentation |
Functional form of covariance function assuming the argument is a
distance between locations. As they are defined here, they are in
fact correlation functions. To set the marginal variance (sill)
parameter, use the sigma
argument in mKrig
or Krig
.
To set the nugget variance, use the tau2
argument in
mKrig
or Krig
.
Exponential(d, aRange = 1, phi = 1, theta = NULL, range = NULL)
Matern(d , range = 1,alpha=1/range, smoothness = 0.5,
nu= smoothness, phi=1.0)
Matern.cor.to.range(d, nu, cor.target=.5, guess=NULL,...)
RadialBasis(d,M,dimension, derivative = 0)
aRange |
The usual range parameter for a covariance function. We use this names to be distinct from the "range"" function and the generic parameter name "theta"." |
d |
Vector of distances or for |
range |
Range parameter default is one. Note that the scale can also be specified through the "aRange" scaling argument used in fields covariance functions) |
alpha |
1/range |
theta |
Same as alpha |
phi |
This parameter option is added to be compatible with older
versions of fields and refers to the marginal variance of the process.
e.g. |
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. |
derivative |
If greater than zero finds the first derivative of this function. |
... |
Additional arguments to pass to the bisection search function. |
Exponential:
exp( -d/aRange)
Matern:
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 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 aRange
, this function determines numerically the value of
aRange so that
Matern( d, range=aRange, 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.
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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.