refund (version 0.1-37)

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
)

Value

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

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

Author

Fabian Scheipl

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 scores \(\xi_{ik}\). 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

if (FALSE) {
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, 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)
}