refund (version 0.1-13)

pcre: pffr-constructor for functional principal component-based functional random intercepts.

Description

pffr-constructor for functional principal component-based functional random intercepts.

Usage

pcre(id, efunctions, evalues, yind, ...)

Arguments

id

grouping variable a factor

efunctions

matrix of eigenfunction evaluations on gridpoints yind (yind> x )

evalues

eigenvalues associated with efunctions

yind

vector of gridpoints on which efunctions are evaluated.

...

not used

Value

• a list used internally for constructing an appropriate call to mgcv::gam

Details

Fits functional random intercepts $B_i(t)$ for a grouping variable id using as a basis the functions $\phi_m(t)$ in efunctions with variances $\lambda_m$ in evalues: $B_i(t) \approx \sum_m^M \phi_m(t)\delta_{im}$ with independent $\delta_{im} \sim N(0, \sigma^2\lambda_m)$, where $\sigma^2$ is (usually) estimated and controls the overall contribution of the $B_i(t)$ while the relative importance of the $M$ basisfunctions is controlled by the supplied variances lambda_m. Can be used to model smooth residuals if id is simply an index of observations. Differing from random effects in mgcv, these effects are estimated under a "sum-to-zero-for-each-t"-constraint. efunctions and 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

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))