mkfun.tp: Crafting Building Blocks for Thin-Plate and Spherical Splines

In gss: General Smoothing Splines

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.

gss documentation built on April 14, 2025, 1:09 a.m.