mkfun.tp: 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)

Value

A list of two elements.

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

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 \(m=2,3,4\).

References

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