refund (version 0.1-37)

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, ...)

Value

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

Arguments

id

grouping variable a factor

efunctions

matrix of eigenfunction evaluations on gridpoints yind (<length of yind> x <no. of used eigenfunctions>)

evalues

eigenvalues associated with efunctions

yind

vector of gridpoints on which efunctions are evaluated.

...

not used

Author

Fabian Scheipl

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 scalar random effects in mgcv, these effects are estimated under a "sum-to-zero-for-each-t"-constraint -- specifically \(\sum_i \hat b_i(t) = 0\) (not \(\sum_i n_i \hat b_i(t) = 0\)) 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.

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

if (FALSE) {
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))
}