https://github.com/cran/fields
Tip revision: 6769ffc81115fbf0bf7d9c566cf7ac81be0049dc authored by Doug Nychka on 25 July 2005, 00:00:00 UTC
version 3.04
version 3.04
Tip revision: 6769ffc
Tps.Rd
\name{Tps}
\alias{Tps}
\title{
Thin plate spline regression
}
\description{
Fits a thin plate spline surface to irregularly spaced data. The
smoothing parameter is chosen by generalized cross-validation. The assumed
model is additive Y = f(X) +e where f(X) is a d dimensional surface.
This is a special case of the spatial process estimate.
}
\usage{
Tps(x, Y, m = NULL, p = NULL, scale.type = "range", ...)
}
\arguments{
To be helpful, a more complete list of arguments are described that are the
same as those for the Krig function.
\item{x}{
Matrix of independent variables. Each row is a location or a set of
independent covariates.
}
\item{Y}{
Vector of dependent variables.
}
\item{m}{
A polynomial function of degree (m-1) will be
included in the model as the drift (or spatial trend) component.
Default is the value such that 2m-d is greater than zero where d is the
dimension of x.
}
\item{p}{
Exponent for radial basis functions. Default is 2m-d.
}
\item{scale.type}{
The independent variables and knots are scaled to the specified
scale.type.
By default the scale type is "range", whereby
the locations are transformed
to the interval (0,1) by forming (x-min(x))/range(x) for each x.
Scale type of "user" allows specification of an x.center and x.scale by
the
user. The default for "user" is mean 0 and standard deviation 1. Scale
type of "unscaled" does not scale the data.
}
\item{\dots}{
Any argument that is valid for the Krig function. Some of the main ones
are listed below.
\describe{
\item{lambda}{
Smoothing parameter that is the ratio of the error variance (sigma**2)
to the scale parameter of the
covariance function. If omitted this is estimated by GCV.
}
\item{df}{
The effective number of parameters for the fitted surface. Conversely,
N- df, where N is the total number of observations is the degrees of
freedom associated with the residuals.
This is an alternative to specifying lambda and much more interpretable.
}
\item{cost}{
Cost value used in GCV criterion. Corresponds to a penalty for
increased number of parameters. The default is 1.0 and corresponds to the
usual GCV.
}
\item{weights}{
Weights are proportional to the reciprocal variance of the measurement
error. The default is no weighting i.e. vector of unit weights.
}
\item{nstep.cv}{
Number of grid points for minimum GCV search.
}
\item{x.center}{
Centering values are subtracted from each column of the x matrix.
Must
have scale.type="user".
}
\item{x.scale}{
Scale values that divided into each column after centering.
Must
have scale.type="user".
}
\item{rho}{
Scale factor for covariance.
}
\item{sigma2}{
Variance of errors or if weights are not equal to 1 the variance is
sigma**2/weight.
}
\item{method}{
Determines what "smoothing" parameter should be used. The default
is to estimate standard GCV
Other choices are: GCV.model, GCV.one, RMSE, pure error and REML. The
differences are explained below.
}
\item{verbose}{
If true will print out all kinds of intermediate stuff.
}
\item{mean.obj}{
Object to predict the mean of the spatial process.
}
\item{sd.obj}{
Object to predict the marginal standard deviation of the spatial process.
}
\item{null.function}{
An R function that creates the matrices for the null space model.
The default is fields.mkpoly, an R function that creates a polynomial
regression matrix with all terms up to degree m-1. (See Details)
}
\item{offset}{
The offset to be used in the GCV criterion. Default is 0. This would be
used when Krig/Tps is part of a backfitting algorithm and the offset has
to be included to reflect other model degrees of freedom.
}
}
}
}
\value{
A list of class Krig. This includes the predicted surface of
fitted.values and the residuals. The results of the grid
search minimizing the generalized cross validation function is
returned in gcv.grid.
Please see the documentation on Krig for details of the returned
arguments.
}
\details{
A thin plate spline is result of minimizing the residual sum of
squares subject to a constraint that the function have a certain
level of smoothness (or roughness penalty). Roughness is
quantified by the integral of squared m-th order derivatives. For one
dimension and m=2 the roughness penalty is the integrated square of
the second derivative of the function. For two dimensions the
roughness penalty is the integral of
(Dxx(f))**22 + 2(Dxy(f))**2 + (Dyy(f))**22
(where Duv denotes the second partial derivative with respect to u
and v.) Besides controlling the order of the derivatives, the value of
m also determines the base polynomial that is fit to the data.
The degree of this polynomial is (m-1).
The smoothing parameter controls the amount that the data is
smoothed. In the usual form this is denoted by lambda, the Lagrange
multiplier of the minimization problem. Although this is an awkward
scale, lambda =0 corresponds to no smoothness constraints and the data
is interpolated. lambda=infinity corresponds to just fitting the
polynomial base model by ordinary least squares.
This estimator is implemented by passing the right generalized covariance
function based on radial basis functions to the more general function
Krig. One advantage of this implementation is that once a Tps/Krig object
is created the estimator can be found rapidly for other data and smoothing
parameters provided the locations remain unchanged. This makes simulation
within R efficient (see example below). Tps does not currenty support the
knots argument where one can use a reduced set of basis functions. This is
mainly to simplify and a good alternative using knots would be to use a
valid covariance from the Matern family and a large range parameter.
}
\section{References}{
See "Nonparametric Regression and Generalized Linear Models"
by Green and Silverman.
See "Additive Models" by Hastie and Tibshirani.
}
\seealso{
Krig, summary.Krig, predict.Krig, predict.se.Krig, plot.Krig,
\code{\link{surface.Krig}},
\code{\link{sreg}}
}
\examples{
#2-d example
fit<- Tps(ozone$x, ozone$y) # fits a surface to ozone measurements.
set.panel(2,2)
plot(fit) # four diagnostic plots of fit and residuals.
set.panel()
summary(fit)
# predict onto a grid that matches the ranges of the data.
out.p<-predict.surface( fit)
image( out.p)
surface(out.p) # perspective and contour plots of GCV spline fit
# predict at different effective
# number of parameters
out.p<-predict.surface( fit,df=10)
#1-d example
out<-Tps( rat.diet$t, rat.diet$trt) # lambda found by GCV
plot( out$x, out$y)
lines( out$x, out$fitted.values)
#
# compare to the ( much faster) one spline algorithm
# sreg(rat.diet$t, rat.diet$trt)
#
#
# simulation reusing Tps/Krig object
#
fit<- Tps( rat.diet$t, rat.diet$trt)
true<- fit$fitted.values
N<- length( fit$y)
temp<- matrix( NA, ncol=50, nrow=N)
sigma<- fit$shat.GCV
for ( k in 1:50){
ysim<- true + sigma* rnorm(N)
temp[,k]<- predict(fit, y= ysim)
}
matplot( fit$x, temp, type="l")
#
#4-d example
fit<- Tps(BD[,1:4],BD$lnya,scale.type="range")
# plots fitted surface and contours
# default is to hold 3rd and 4th fixed at median values
surface(fit)
}
\keyword{smooth}
% docclass is function
% Converted by Sd2Rd version 1.21.