exp.cov {fields} | R Documentation |
Given two sets of locations computes the cross covariance matrix for covariances among all pairings.
exp.cov(x1, x2, theta = rep(1, ncol(x1)), p = 1, C = NA) exp.cov.simple(x1, x2, theta =1)
x1 |
Matrix of first set of locations where each row gives the coordinates of a particular point. |
x2 |
Matrix of second set of locations where each row gives the coordinates of a particular point. If this is missing x1 is used. |
theta |
Range (or scale) parameter. This can be a scalar or a vector that is the same length as the dimension of the locations. Default is theta=1. |
p |
Exponent in the exponential form. p=1 gives an exponential and p=2 gives a Gaussian. Default is the exponential form. |
C |
A vector with the same length as the number of rows of x2. If specified the covariance matrix will be multiplied by this vector. |
Functional Form: If x1 and x2 are matrices where nrow(x1)=m and nrow( x2)=n then this function should return a mXn matrix where the (i,j) element is the covariance between the locations x1[i,] and x2[j,]. The covariance is found as exp( -(D.ij **p)) where D.ij is the Euclidean distance between x1[i,] and x2[j,] but having first been scaled by theta. Specifically
D.ij = sqrt( sum.k (( x1[i,k] - x2[j,k]) /theta[k])**2 ).
Note that if theta is a scalar then this defines an isotropic covariance function.
Implementation: The function r.dist is a useful FIELDS function that finds the cross distance matrix ( D defined above) for two sets of locations. Thus in compact S code we have
u <- t(t(x1)/theta)
v <- t(t(x2)/theta)
exp(-rdist(u, v)**p)
FORTRAN: The actual function calls FORTRAN to make the evaluation more
efficient this is especially important when the C argument is supplied.
So unfortunately the actual code is not as crisp as the few lines given
above. For purposes of illustration, the function
exp.cov.simple
is provided as
a simple example and implements the R code above. It can also serve as a
template for creating new covariance functions for the Krig
function.
If the argument C is NULL the cross covariance matrix. Moreover if x1 is
equal to x2 then this is the covariance matrix for this set of locations.
In general if nrow(x1)=m and nrow(
x2)=n then the returned matrix, Sigma will be mXn.
If C is a vector of length n,
then returned value is the multiplication of the cross covariance matrix
with this vector: Sigma%*%C
Krig, matern.cov, rdist, rdist.earth, gauss.cov, exp.image.cov
# exponential covariance matrix ( marginal variance =1) for the ozone #locations out<- exp.cov( ozone$x, theta=100) # out is a 20X20 matrix out2<- exp.cov( ozone$x[6:20,],ozone$x[1:2,], theta=100) # out2 is 15X2 matrix # Kriging fit where the nugget variance is found by GCV fit<- Krig( ozone$x, ozone$y, exp.cov, theta=100)