pffr-constructor for functional principal component-based functional random intercepts.
pffr-constructor for functional principal component-based functional random intercepts.
pcre(id, efunctions, evalues, yind, ...)
id |
grouping variable a factor |
efunctions |
matrix of eigenfunction evaluations on gridpoints |
evalues |
eigenvalues associated with |
yind |
vector of gridpoints on which |
... |
not used |
a list used internally for constructing an appropriate call to mgcv::gam
Fits functional random intercepts B_i(t) for a grouping variable id
using as a basis the functions φ_m(t) in efunctions with variances λ_m in evalues:
B_i(t) \approx ∑_m^M φ_m(t)δ_{im} with
independent δ_{im} \sim N(0, σ^2λ_m), where σ^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.
Fabian Scheipl
## Not run:
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))
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