Two dimensional radial basis and tensor functions based on a Wendland function and using sparse matrix format to reduce the storage.
1 2 3 4 5 6 7 8 9 10 11 12  Radial.basis(x1, centers, basis.delta, max.points = NULL,
mean.neighbor = 50,
BasisFunction = "WendlandFunction",
distance.type = "Euclidean",
verbose = FALSE)
Tensor.basis(x1, centers, basis.delta, max.points = NULL, mean.neighbor = 50,
BasisFunction = "WendlandFunction", distance.type = "Euclidean")
WendlandFunction(d)
triWeight(d)

x1 
A matrix of locations to evaluate the basis
functions. Each row of 
centers 
A matrix specifying the basis function centers. 
d 
A vector of distances. 
basis.delta 
A vector of scale parameters for the basis functions. 
max.points 
Maximum number of nonzero entries expected for the returned matrix. 
distance.type 
The distance metric. See

mean.neighbor 
Average number of centers that are within delta of each x1 location. For centers on a regular grid this is often easy to estimate. 
BasisFunction 
A function that will take a nonnegative argument and be zero outside [0,1]. This is applied to distance(s) to generate the basis functions. For tensor basis functions, the function is applied to the distance components for each dimension. 
verbose 
Print out debugging information if TRUE. 
This function finds the pairwise distances between the points x1 and
centers and evaluates the function RadialBasisFunction at these
distances scaled by delta. In most applications delta is constant, but
a variable delta could be useful for lon/lat regular grids. The
Wendland function is for 2 dimensions and smoothness order 2. See
WendlandFunction
for the polynomial form. This code has a very
similar function to the fields function wendland.cov
.
In pseudo R code for delta a scalar Radial.basis
evaluates as
1 2  BigD< rdist( x1,centers)
WendlandFunction(BigD/basis.delta)

The actual code uses a FORTRAN subroutine to search over distances less than delta and also returns the matrix in sparse format.
The function Tensor.basis
has similar function as the radial
option. The main difference is that a slightly different distance function is
used to return the component distances for each dimension. In pseudo R code
for delta a scalar and for just two dimensions Tensor.basis evaluates as
1 2 3  BigD1< rdist( x1[,1],centers[,1])
BigD2< rdist( x1[,2],centers[,2])
WendlandFunction(BigD1/basis.delta) *WendlandFunction(BigD1/basis.delta)

The function LKrig.cyl
transforms coordinates on a cylinder,
e.g. lon/lat when taken as a Mercator projection, and returns the 3d
coordinates. It is these 3d coordinates that are used to find distances
to define the radial basis functions. For points that are close this
"chordal" type distance will be close to the geodesic distance on a
cylinder but not identical.
For Wendland.basis
a matrix in sparse format with number of
rows equal to nrow(x1) and columns equal to nrow(center).
Doug Nychka
LKrig.basis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  set.seed(12)
x< cbind( runif(100), runif(100))
center< expand.grid( seq( 0,1,,5), seq(0,1,,5))
# coerce to matrix
center< as.matrix(center)
PHI1< Radial.basis(x, center, basis.delta = .5)
PHI2< Tensor.basis( x, center, basis.delta = .5 )
# similarity of radial and tensor product forms
plot( c(0,1.1), c(0,1.1), type="p")
for( k in 1:25){
points( PHI1[,k], PHI2[,k])
}
# LKrig with a different radial basis function.
#
data(ozone2)
x<ozone2$lon.lat
y< ozone2$y[16,]
# Find location that are not 'NA'.
# (LKrig is not set up to handle missing observations.)
good < !is.na( y)
x< x[good,]
y< y[good]
obj< LKrig(x,y,NC=30,nlevel=1, alpha=1, lambda=.01, a.wght=5)
obj1< LKrig(x,y,NC=30,nlevel=1, alpha=1,
lambda=.01, a.wght=5, BasisFunction="triWeight", overlap=1.8)

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