pcre.Rd

% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/pffr-pcre.R
\name{pcre}
\alias{pcre}
\title{pffr-constructor for functional principal component-based functional random intercepts.}
\usage{
pcre(id, efunctions, evalues, yind, ...)
}
\arguments{
\item{id}{grouping variable a factor}

\item{efunctions}{matrix of eigenfunction evaluations on gridpoints \code{yind} (<length of \code{yind}> x <no. of used eigenfunctions>)}

\item{evalues}{eigenvalues associated with \code{efunctions}}

\item{yind}{vector of gridpoints on which \code{efunctions} are evaluated.}

\item{...}{not used}
}
\value{
a list used internally for constructing an appropriate call to \code{mgcv::gam}
}
\description{
pffr-constructor for functional principal component-based functional random intercepts.
}
\section{Details}{
 Fits functional random intercepts \eqn{B_i(t)} for a grouping variable \code{id}
using as a basis the functions \eqn{\phi_m(t)} in \code{efunctions} with variances \eqn{\lambda_m} in \code{evalues}:
\eqn{B_i(t) \approx \sum_m^M \phi_m(t)\delta_{im}} with
independent \eqn{\delta_{im} \sim N(0, \sigma^2\lambda_m)}, where \eqn{\sigma^2}
is (usually) estimated and controls the overall contribution of the \eqn{B_i(t)} while the relative importance
of the \eqn{M} basisfunctions is controlled by the supplied variances \code{lambda_m}.
Can be used to model smooth residuals if \code{id} is simply an index of observations.
Differing from scalar random effects in \code{mgcv}, these effects are estimated under a "sum-to-zero-for-each-t"-constraint --
specifically \eqn{\sum_i \hat b_i(t) = 0} (not \eqn{\sum_i n_i \hat b_i(t) = 0}) where $n_i$ is the number of observed curves for
subject i, so the intercept curve for models with unbalanced group sizes no longer corresponds to the global mean function.

\code{efunctions} and \code{evalues} are typically eigenfunctions and eigenvalues of an estimated
covariance operator for the functional process to be modeled, i.e., they are
a functional principal components basis.
}

\examples{
\dontrun{
residualfunction <- function(t){
#generate quintic polynomial error functions
    drop(poly(t, 5)\%*\%rnorm(5, sd=sqrt(2:6)))
}
# generate data Y(t) = mu(t) + E(t) + white noise
set.seed(1122)
n <- 50
T <- 30
t <- seq(0,1, l=T)
# E(t): smooth residual functions
E <- t(replicate(n, residualfunction(t)))
int <- matrix(scale(3*dnorm(t, m=.5, sd=.5) - dbeta(t, 5, 2)), byrow=T, n, T)
Y <- int + E + matrix(.2*rnorm(n*T), n, T)
data <- data.frame(Y=I(Y))
# fit model under independence assumption:
summary(m0 <- pffr(Y ~ 1, yind=t, data=data))
# get first 5 eigenfunctions of residual covariance
# (i.e. first 5 functional PCs of empirical residual process)
Ehat <- resid(m0)
fpcE <- fpca.sc(Ehat, npc=5)
efunctions <- fpcE$efunctions
evalues <- fpcE$evalues
data$id <- factor(1:nrow(data))
# refit model with fpc-based residuals
m1 <- pffr(Y ~ 1 + pcre(id=id, efunctions=efunctions, evalues=evalues, yind=t), yind=t, data=data)
t1 <- predict(m1, type="terms")
summary(m1)
#compare squared errors
mean((int-fitted(m0))^2)
mean((int-t1[[1]])^2)
mean((E-t1[[2]])^2)
# compare fitted & true smooth residuals and fitted intercept functions:
layout(t(matrix(1:4,2,2)))
matplot(t(E), lty=1, type="l", ylim=range(E, t1[[2]]))
matplot(t(t1[[2]]), lty=1, type="l", ylim=range(E, t1[[2]]))
plot(m1, select=1, main="m1", ylim=range(Y))
lines(t, int[1,], col=rgb(1,0,0,.5))
plot(m0, select=1, main="m0", ylim=range(Y))
lines(t, int[1,], col=rgb(1,0,0,.5))
}
}
\author{
Fabian Scheipl
}