1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
|
\name{mkfun.tp}
\alias{mkfun.tp}
\alias{mkrk.tp}
\alias{mkrk.tp.p}
\alias{mkphi.tp}
\alias{mkphi.tp.p}
\alias{mkrk.sphere}
\title{
Crafting Building Blocks for Thin-Plate and Spherical Splines
}
\description{
Craft numerical functions to be used by \code{\link{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{
\item{dm}{Dimension of the variable \eqn{d}.}
\item{order}{Order of the differential operator \eqn{m}.}
\item{mesh}{Normalizing mesh.}
\item{weight}{Normalizing weights.}
}
\details{
\code{mkrk.tp}, \code{mkphi.tp}, \code{mkrk.tp.p}, and
\code{mkphi.tp.p} implement the construction in Gu (2002,
Sec. 4.4). Thin-plate splines are defined for \eqn{2m>d}.
\code{mkrk.tp.p} generates the pseudo kernel, and \code{mkphi.tp.p}
generates the \eqn{(m+d-1)!/d!/(m-1)!} lower order polynomials with
total order less than \eqn{m}.
\code{mkphi.tp} generates normalized lower order polynomials
orthonormal w.r.t. a norm specified by \code{mesh} and
\code{weight}, and \code{mkrk.tp} conditions the pseudo kernel to
generate the reproducing kernel orthogonal to the lower order
polynomials w.r.t. the norm.
\code{mkrk.sphere} implements the reproducing kernel construction of
Wahba (1981) for \eqn{m=2,3,4}.
}
\value{
A list of two components.
\item{fun}{Function definition.}
\item{env}{Portable local constants derived from the arguments.}
}
\note{
\code{mkrk.tp} and \code{mkrk.sphere} create a bivariate function
\code{fun(x,y,env,outer=FALSE)}, where \code{x}, \code{y} are real
arguments and local constants can be passed in through \code{env}.
\code{mkphi.tp} creates a collection of univariate functions
\code{fun(x,nu,env)}, where \code{x} is the argument and \code{nu}
is the index.
}
\author{Chong Gu, \email{chong@stat.purdue.edu}}
\references{
Gu, C. (2013), \emph{Smoothing Spline ANOVA Models (2nd Ed)}. New
York: Springer-Verlag.
Wahba, G. (1981), Spline interpolation and smoothing on the sphere.
\emph{SIAM Journal on Scientific and Statistical Computing},
\bold{2}, 5--16.
}
\seealso{
\code{\link{mkterm}}, \code{\link{mkfun.poly}}, and
\code{\link{mkrk.nominal}}.
}
\keyword{internal}
|
