A matrix of locations to evaluate the basis
functions. Each row of x1 is a location.

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
LKrigDistance for details.

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.

Details

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

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

The function LKrig.cyl transforms coordinates on a cylinder,
e.g. lon/lat when taken as a Mercator projection, and returns the 3-d
coordinates. It is these 3-d 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.

Value

For Wendland.basis a matrix in sparse format with number of
rows equal to nrow(x1) and columns equal to nrow(center).

Author(s)

Doug Nychka

See Also

LKrig.basis

Examples

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)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> library(LatticeKrig)
Loading required package: spam
Loading required package: grid
Spam version 1.3-0 (2015-10-24) is loaded.
Type 'help( Spam)' or 'demo( spam)' for a short introduction
and overview of this package.
Help for individual functions is also obtained by adding the
suffix '.spam' to the function name, e.g. 'help( chol.spam)'.
Attaching package: 'spam'
The following objects are masked from 'package:base':
backsolve, forwardsolve
Loading required package: fields
Loading required package: maps
# maps v3.1: updated 'world': all lakes moved to separate new #
# 'lakes' database. Type '?world' or 'news(package="maps")'. #
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/LatticeKrig/Radial.Basis.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Radial.basis
> ### Title: Two dimensional radial and tensor basis functions based on a
> ### Wendland function.
> ### Aliases: Radial.basis LKrig.cyl WendlandFunction Tensor.basis triWeight
> ### Keywords: spatial
>
> ### ** Examples
>
> 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)
>
>
>
>
>
> dev.off()
null device
1
>