R: Crafting Building Blocks for Thin-Plate and Spherical Splines

mkfun.tp {gss} | R Documentation

## Crafting Building Blocks for Thin-Plate and Spherical Splines

### Description

Craft numerical functions to be used by `mkterm` to assemble model terms.

### Usage

    mkrk.tp(dm, order, mesh, weight)
    mkphi.tp(dm, order, mesh, weight)
    mkrk.tp.p(dm, order)
    mkphi.tp.p(dm, order)

    mkrk.sphere(order)

### Arguments

`dm` | Dimension of the variable d.
`order` | Order of the differential operator m.
`mesh` | Normalizing mesh.
`weight` | Normalizing weights.

### Details

`mkrk.tp`, `mkphi.tp`, `mkrk.tp.p`, and `mkphi.tp.p` implement the construction in Gu (2002, Sec. 4.4). Thin-plate splines are defined for 2m>d.

`mkrk.tp.p` generates the pseudo kernel, and `mkphi.tp.p` generates the (m+d-1)!/d!/(m-1)! lower order polynomials with total order less than m.

`mkphi.tp` generates normalized lower order polynomials orthonormal w.r.t. a norm specified by `mesh` and `weight`, and `mkrk.tp` conditions the pseudo kernel to generate the reproducing kernel orthogonal to the lower order polynomials w.r.t. the norm.

`mkrk.sphere` implements the reproducing kernel construction of Wahba (1981) for m=2,3,4.

### Value

A list of two elements.

`fun` | Function definition.
`env` | Portable local constants derived from the arguments.

### Note

`mkrk.tp` and `mkrk.sphere` create a bivariate function `fun(x,y,env,outer=FALSE)`, where `x`, `y` are real arguments and local constants can be passed in through `env`.

`mkphi.tp` creates a collection of univariate functions `fun(x,nu,env)`, where `x` is the argument and `nu` is the index.

### References

Gu, C. (2013), Smoothing Spline ANOVA Models (2nd Ed). New York: Springer-Verlag.

Wahba, G. (1981), Spline interpolation and smoothing on the sphere. SIAM Journal on Scientific and Statistical Computing, 2, 5–16.

### See Also

`mkterm`, `mkfun.poly`, and `mkrk.nominal`.

[Package gss version 2.2-2 Index]