# Two dimensional radial and tensor basis functions based on a Wendland function. Two dimensional radial basis and tensor functions based on a Wendland function and using sparse matrix format to reduce the storage. ```r 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) ``` ## Arguments - `x1`: A matrix of locations to evaluate the basis functions. Each row of `x1` is a location. For spherical distances these locations are assumed to be in degrees. - `centers`: A matrix specifying the basis function centers. For spherical distances these locations are assumed to be in degrees. - `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`: A text string indicate type distance to use between spatial locations when evaluating the basis functions. Default is "Euclidean". Other choices when locations are in degrees ( e.g. longitude and latitude) are "Chordal" and "GreatCircle" with the default units being miles. See Details below how to change the radius that is used. - `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 non-negative 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` with the Wendland "shape" evaluates as ``` BigD<- rdist( x1,centers) WendlandFunction(BigD/basis.delta) ``` and is comparable to `Radial.basis(x1, centers,basis.delta)` The actual code uses a FORTRAN subroutine to search over distances less than delta and also returns the matrix in sparse format. Part of the flexibility in these function is to use different distance functions (metrics). See help on `LKDistance` for the S4 method to find distance and LKDist for the lower level functions that actually do the work. To change the radius used for the spherical distances one add a `Radius` attribute to the text string. For example to use kilometers for great circle distance ( R approximately 6371) ``` dtype <- "GreatCircle" attr( dtype, which="Radius") <- 6371 ``` `dtype` is still a text string but has the extra bit of information attached to it. Now use `distance.type = dtype` for the distance type argument. 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 ``` BigD1<- rdist( x1[,1],centers[,1]) BigD2<- rdist( x1[,2],centers[,2]) WendlandFunction(BigD1/basis.delta) *WendlandFunction(BigD1/basis.delta) ``` The user is also encourage to consider using the higher level function LKrig.basis paired with an LKinfo object to define the centers. But besure to use `normalize = FALSE` to avoid strange interactions of the SAR model, irrelevant in this case, with the basis function form. ## Returns For `Radial.basis` and `Tensor.basis` a matrix in sparse format with number of rows equal to nrow(x1) and columns equal to nrow(center) and value are the basis functions evaluated at these combinations For `WendlandFunction` and `triweight` a vector with the values for this function. ## Author(s) Doug Nychka ## See Also LKrig.basis , LKDist ## Examples ```r 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 locations 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) # It is usually better to create the LKinfo object first and then call LKrig LKinfo <- LKrigSetup( x,NC=30,nlevel=1, alpha=1, lambda=.01, a.wght=5, BasisFunction="triWeight", overlap=1.8 ) obj1B<- LKrig(x,y, LKinfo=LKinfo) ####################################################### ### a radial basis on the sphere ####################################################### # icosohedral grid at the second generation allLevels<- IcosahedronGrid(2) # extract second level centers3d<- allLevels[[2]] centers<- toSphere(centers3d) # take a look plot( centers, xlim=c(-180,180), ylim=c( -90, 90), pch=16) lonlat<- make.surface.grid( list( x= seq( -180,180,,80), y= seq( -90,90,,40))) delta<- 5000 # default distance is in miles # evaluate all basis functions on the grid bigX<- Radial.basis( lonlat, centers, basis.delta= delta, distance.type="GreatCircle") # look at the 12th basis function b12<- bigX[,12] image( as.surface( lonlat, b12), col=c( "white",tim.colors(256)), add=TRUE, xlim="longitude", ylim="latitude") points( centers, pch=16, col="magenta", cex=2) ```

Radial.Basis function