refund (version 0.1-1)

ffpc: Construct a PC-based function-on-function regression term

Description

Defines a term $\int X_i(s)\beta(t,s)ds$ for inclusion in an mgcv::gam-formula (or bam or gamm or gamm4:::gamm4) as constructed by pffr.

Usage

ffpc(X, yind = NULL, xind = seq(0, 1, length = ncol(X)),
  splinepars = list(bs = "ps", m = c(2, 1), k = 8), decomppars = list(pve =
  0.99, useSymm = TRUE), npc.max = 15)

Arguments

X

an n by ncol(xind) matrix of function evaluations $X_i(s_{i1}),\dots, X_i(s_{iS})$; $i=1,\dots,n$.

yind

DEPRECATED used to supply matrix (or vector) of indices of evaluations of $Y_i(t)$, no longer used.

xind

matrix (or vector) of indices of evaluations of $X_i(t)$, defaults to seq(0, 1, length=ncol(X)).

splinepars

optional arguments supplied to the basistype-term. Defaults to a cubic B-spline with first difference penalties and 8 basis functions for each $\tilde \beta_k(t)$.

decomppars

parameters for the FPCA performed with fpca.sc.

npc.max

maximal number $K$ of FPCs to use, regardless of decomppars; defaults to 15

Value

• A list containing the necessary information to construct a term to be included in a mgcv::gam-formula.

Details

In contrast to ff, ffpc does an FPCA decomposition $X(s) \approx \sum^K_{k=1} \xi_{ik} \Phi_k(s)$ using fpca.sc and represents $\beta(t,s)$ in the function space spanned by these $\Phi_k(s)$. That is, since $$\int X_i(s)\beta(t,s)ds = \sum^K_{k=1} \xi_{ik} \int \Phi_k(s) \beta(s,t) ds = \sum^K_{k=1} \xi_{ik} \tilde \beta_k(t),$$ the function-on-function term can be represented as a sum of $K$ univariate functions $\tilde \beta_k(t)$ in $t$ each multiplied by the FPC loadings $\xi_{ik}$. The truncation parameter $K$ is chosen as described in fpca.sc. Using this instead of ff() can be beneficial if the covariance operator of the $X_i(s)$ has low effective rank (i.e., if $K$ is small). If the covariance operator of the $X_i(s)$ is of (very) high rank, i.e., if $K$ is large, ffpc() will not be very efficient.

To reduce model complexity, the $\tilde \beta_k(t)$ all have a single joint smoothing parameter (in mgcv, they get the same id, see s). Please see pffr for details on model specification and implementation.

Examples

set.seed(1122)
n <- 55
S <- 60
T <- 50
s <- seq(0,1, l=S)
t <- seq(0,1, l=T)

#generate X from a polynomial FPC-basis:
rankX <- 5
Phi <- cbind(1/sqrt(S), poly(s, degree=rankX-1))
lambda <- rankX:1
Xi <- sapply(lambda, function(l)
            scale(rnorm(n, sd=sqrt(l)), scale=FALSE))
X <- Xi %*% t(Phi)

beta.st <- outer(s, t, function(s, t) cos(2 * pi * s * t))

y <- (1/S*X) %*% beta.st + 0.1 * matrix(rnorm(n * T), nrow=n, ncol=T)

data <- list(y=y, X=X)
# set number of FPCs to true rank of process for this example:
m.pc <- pffr(y ~ c(1) + 0 + ffpc(X, yind=t, decomppars=list(npc=rankX)),
        data=data, yind=t)
summary(m.pc)
m.ff <- pffr(y ~ c(1) + 0 + ff(X, yind=t), data=data, yind=t)
summary(m.ff)

# fits are very similar:
all.equal(fitted(m.pc), fitted(m.ff))

# plot implied coefficient surfaces:
layout(t(1:3))
persp(t, s, t(beta.st), theta=50, phi=40, main="Truth",
    ticktype="detailed")
plot(m.ff, select=1, pers=TRUE, zlim=range(beta.st), theta=50, phi=40,
    ticktype="detailed")
title(main="ff()")
ffpcplot(m.pc, type="surf", auto.layout=FALSE, theta = 50, phi = 40)
title(main="ffpc()")

# show default ffpcplot:
ffpcplot(m.pc)