https://github.com/cran/fda
Tip revision: 84fa120851854f541d5c305af8a7e4067a0a7df0 authored by J. O. Ramsay on 04 July 2022, 12:20:06 UTC
version 6.0.5
version 6.0.5
Tip revision: 84fa120
Fperm.fd.Rd
\name{Fperm.fd}
\alias{Fperm.fd}
\title{
Permutation F-test for functional linear regression.
}
\description{
\code{Fperm.fd} creates a null distribution for a test of no effect in
functional linear regression. It makes generic use of \code{fRegress}
and permutes the \code{yfdPar} input.
}
\usage{
Fperm.fd(yfdPar, xfdlist, betalist, wt=NULL, nperm=200,
argvals=NULL, q=0.05, plotres=TRUE, ...)
}
\arguments{
\item{yfdPar}{
the dependent variable object. It may be an object of three
possible classes:
\itemize{
\item{vector}{ if the dependent variable is scalar.}
\item{fd}{
a functional data object if the dependent variable is
functional.
}
\item{fdPar}{
a functional parameter object if the dependent variable is
functional, and if it is necessary to smooth the prediction of
the dependent variable.
}
}
}
\item{xfdlist}{
a list of length equal to the number of independent
variables. Members of this list are the independent variables. They
be objects of either of these two classes:
\itemize{
\item a vector if the independent dependent variable is scalar.
\item {
a functional data object if the dependent variable is
functional.
}
}
In either case, the object must have the same number of replications
as the dependent variable object. That is, if it is a scalar, it
must be of the same length as the dependent variable, and if it is
functional, it must have the same number of replications as the
dependent variable.
}
\item{betalist}{
a list of length equal to the number of independent variables.
Members of this list define the regression functions to be
estimated. They are functional parameter objects. Note that even
if corresponding independent variable is scalar, its regression
coefficient will be functional if the dependent variable is
functional. Each of these functional parameter objects defines a
single functional data object, that is, with only one replication.
}
\item{wt}{
weights for weighted least squares, defaults to all 1.
}
\item{nperm}{
number of permutations to use in creating the null distribution.
}
\item{argvals}{
If \code{yfdPar} is a \code{fd} object, the points at which to
evaluate the point-wise F-statistic.
}
\item{q}{
Critical upper-tail quantile of the null distribution to compare to
the observed F-statistic.
}
\item{plotres}{
Argument to plot a visual display of the null distribution
displaying the \code{q}th quantile and observed F-statistic.
}
\item{...}{
Additional plotting arguments that can be used with \code{plot}.
}
}
\details{
An F-statistic is calculated as the ratio of residual variance to
predicted variance. The observed F-statistic is returned along with
the permutation distribution.
If \code{yfdPar} is a \code{fd} object, the maximal value of the
pointwise F-statistic is calculated. The pointwise F-statistics are
also returned.
The default of setting \code{q = 0.95} is, by now, fairly
standard. The default \code{nperm = 200} may be small, depending on
the amount of computing time available.
If \code{argvals} is not specified and \code{yfdPar} is a \code{fd}
object, it defaults to 101 equally-spaced points on the range of
\code{yfdPar}.
}
\value{
A list with the following components:
\item{pval}{the observed p-value of the permutation test.}
\item{qval}{the \code{q}th quantile of the null distribution.}
\item{Fobs}{the observed maximal F-statistic.}
\item{Fnull}{
a vector of length \code{nperm} giving the observed values of the
permutation distribution.
}
\item{Fvals}{the pointwise values of the observed F-statistic.}
\item{Fnullvals}{
the pointwise values of of the permutation observations.
}
\item{pvals.pts}{pointwise p-values of the F-statistic.}
\item{qvals.pts}{
pointwise \code{q}th quantiles of the null distribution
}
\item{fRegressList}{
the result of \code{fRegress} on the observed data
}
\item{argvals}{
argument values for evaluating the F-statistic if \code{yfdPar} is a
functional data object.
}
}
\source{
Ramsay, James O., and Silverman, Bernard W. (2006), \emph{Functional
Data Analysis, 2nd ed.}, Springer, New York.
}
\section{Side Effects}{
a plot of the functional observations
} \seealso{
\code{\link{fRegress}},
\code{\link{Fstat.fd}}
}
\examples{
##
## 1. yfdPar = vector
##
annualprec <- log10(apply(
CanadianWeather$dailyAv[,,"Precipitation.mm"], 2,sum))
# set up a smaller basis using only 40 Fourier basis functions
# to save some computation time
smallnbasis <- 40
smallbasis <- create.fourier.basis(c(0, 365), smallnbasis)
tempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"],
smallbasis)$fd
constantfd <- fd(matrix(1,1,35), create.constant.basis(c(0, 365)))
xfdlist <- vector("list",2)
xfdlist[[1]] <- constantfd
xfdlist[[2]] <- tempfd[1:35]
betalist <- vector("list",2)
# set up the first regression function as a constant
betabasis1 <- create.constant.basis(c(0, 365))
betafd1 <- fd(0, betabasis1)
betafdPar1 <- fdPar(betafd1)
betalist[[1]] <- betafdPar1
nbetabasis <- 35
betabasis2 <- create.fourier.basis(c(0, 365), nbetabasis)
betafd2 <- fd(matrix(0,nbetabasis,1), betabasis2)
lambda <- 10^12.5
harmaccelLfd365 <- vec2Lfd(c(0,(2*pi/365)^2,0), c(0, 365))
betafdPar2 <- fdPar(betafd2, harmaccelLfd365, lambda)
betalist[[2]] <- betafdPar2
# Should use the default nperm = 200
# but use 10 to save test time for illustration
F.res2 = Fperm.fd(annualprec, xfdlist, betalist, nperm=10)
##
## 2. yfdpar = Functional data object (class fd)
##
# The very simplest example is the equivalent of the permutation
# t-test on the growth data.
# First set up a basis system to hold the smooths
\dontrun{
Knots <- growth$age
norder <- 6
nbasis <- length(Knots) + norder - 2
hgtbasis <- create.bspline.basis(range(Knots), nbasis, norder, Knots)
# Now smooth with a fourth-derivative penalty and a very small smoothing
# parameter
Lfdobj <- 4
lambda <- 1e-2
growfdPar <- fdPar(hgtbasis, Lfdobj, lambda)
hgtfd <- smooth.basis(growth$age,
cbind(growth$hgtm,growth$hgtf),growfdPar)$fd
# Now set up factors for fRegress:
cbasis = create.constant.basis(range(Knots))
maleind = c(rep(1,ncol(growth$hgtm)),rep(0,ncol(growth$hgtf)))
constfd = fd( matrix(1,1,length(maleind)),cbasis)
maleindfd = fd( matrix(maleind,1,length(maleind)),cbasis)
xfdlist = list(constfd,maleindfd)
# The fdPar object for the coefficients and call Fperm.fd
betalist = list(fdPar(hgtbasis,2,1e-6),fdPar(hgtbasis,2,1e-6))
# Should use nperm = 200 or so,
# but use 10 to save test time
Fres = Fperm.fd(hgtfd,xfdlist,betalist,nperm=10)
}
}
\keyword{smooth}