FAM {fdapace}

Functional Additive Models

Description

Functional additive models with a single predictor process

Usage

FAM(
  Y,
  Lx,
  Lt,
  nEval = 51,
  newLx = NULL,
  newLt = NULL,
  bwMethod = 0,
  alpha = 0.7,
  supp = c(-2, 2),
  optns = NULL
)

Arguments

Y

An n-dimensional vector whose elements consist of scalar responses.

Lx

A list of n vectors containing the observed values for each individual. See FPCA for detail.

Lt

A list of n vectors containing the observation time points for each individual. Each vector should be sorted in ascending order. See FPCA for detail.

nEval

The number of evaluation grid points for kernel smoothing (default is 51. If it is specified as 0, then estimated FPC scores in the training set are used for evaluation grid instead of equal grid).

newLx

A list of the observed values for test set. See predict.FPCA for detail.

newLt

A list of the observed time points for test set. See predict.FPCA for detail.

bwMethod

The method of bandwidth selection for kernel smoothing, a positive value for designating K-fold cross-validtaion and zero for GCV (default is 50)

alpha

The shrinkage factor (positive number) for bandwidth selection. See Han et al. (2016) (default is 0.7).

supp

The lower and upper limits of kernel smoothing domain for studentized FPC scores, which FPC scores are divided by the square roots of eigenvalues (default is [-2,2]).

optns

A list of options control parameters specified by list(name=value). See FPCA.

Details

FAM fits functional additive models for a scalar response and single predictor process proposed by Müller and Yao (2007) that

E(Y | \mathbf{X}) = \sum_{k=1}^K g_{k}(\xi_{k}),

where \xi_{k} stand for the k-th FPC score of the the predictor process.

Value

A list containing the following fields:

mu

Mean estimator of EY

fam

A N by K matrix whose column vectors consist of the component function estimators at the given estimation points.

xi

An N by K matrix whose column vectors consist of N vectors of estimation points for each component function.

bw

A K-dimensional bandwidth vector.

lambda

A K-dimensional vector containing eigenvalues.

phi

An nWorkGrid by K matrix containing eigenfunctions, supported by WorkGrid. See FPCA.

workGrid

An nWorkGrid by K_j working grid, the internal regular grid on which the eigen analysis is carried on. See FPCA.

References

Müller, H.-G. and Yao, F. (2005), "Functional additive models", JASA, Vol.103, No.484, p.1534-1544.

Examples

set.seed(1000)

library(MASS)

f1 <- function(t) 0.5*t
f2 <- function(t) 2*cos(2*pi*t/4)
f3 <- function(t) 1.5*sin(2*pi*t/4)
f4 <- function(t) 2*atan(2*pi*t/4)

n <- 100
N <- 100

sig <- diag(c(4.0,2.0,1.5,1.2))

scoreX <- mvrnorm(n,mu=rep(0,4),Sigma=sig)
scoreXTest <- mvrnorm(N,mu=rep(0,4),Sigma=sig)

Y <- f1(scoreX[,1]) + f2(scoreX[,2]) + f3(scoreX[,3]) + f4(scoreX[,4]) + rnorm(n,0,0.1)
YTest <- f1(scoreXTest[,1]) + f2(scoreXTest[,2]) + 
  f3(scoreXTest[,3]) + f4(scoreXTest[,4]) + rnorm(N,0,0.1)

phi1 <- function(t) sqrt(2)*sin(2*pi*t)
phi2 <- function(t) sqrt(2)*sin(4*pi*t)
phi3 <- function(t) sqrt(2)*cos(2*pi*t)
phi4 <- function(t) sqrt(2)*cos(4*pi*t)

grid <- seq(0,1,length.out=21)
Lt <- Lx <- list()
for (i in 1:n) {
  Lt[[i]] <- grid
  Lx[[i]] <- scoreX[i,1]*phi1(grid) + scoreX[i,2]*phi2(grid) + 
    scoreX[i,3]*phi3(grid) + scoreX[i,4]*phi4(grid) + rnorm(1,0,0.01)
}

LtTest <- LxTest <- list()
for (i in 1:N) {
  LtTest[[i]] <- grid
  LxTest[[i]] <- scoreXTest[i,1]*phi1(grid) + scoreXTest[i,2]*phi2(grid) + 
    scoreXTest[i,3]*phi3(grid) + scoreXTest[i,4]*phi4(grid) + rnorm(1,0,0.01)
}


# estimation
fit <- FAM(Y=Y,Lx=Lx,Lt=Lt)

xi <- fit$xi

op <- par(mfrow=c(2,2))
j <- 1
g1 <- f1(sort(xi[,j]))
tmpSgn <- sign(sum(g1*fit$fam[,j]))
plot(sort(xi[,j]),g1,type='l',col=2,ylim=c(-2.5,2.5),xlab='xi1')
points(sort(xi[,j]),tmpSgn*fit$fam[order(xi[,j]),j],type='l')

j <- 2
g2 <- f2(sort(xi[,j]))
tmpSgn <- sign(sum(g2*fit$fam[,j]))
plot(sort(xi[,j]),g2,type='l',col=2,ylim=c(-2.5,2.5),xlab='xi2')
points(sort(xi[,j]),tmpSgn*fit$fam[order(xi[,j]),j],type='l')

j <- 3
g3 <- f3(sort(xi[,j]))
tmpSgn <- sign(sum(g3*fit$fam[,j]))
plot(sort(xi[,j]),g3,type='l',col=2,ylim=c(-2.5,2.5),xlab='xi3')
points(sort(xi[,j]),tmpSgn*fit$fam[order(xi[,j]),j],type='l')

j <- 4
g4 <- f4(sort(xi[,j]))
tmpSgn <- sign(sum(g4*fit$fam[,j]))
plot(sort(xi[,j]),g4,type='l',col=2,ylim=c(-2.5,2.5),xlab='xi4')
points(sort(xi[,j]),tmpSgn*fit$fam[order(xi[,j]),j],type='l')
par(op)

# fitting
fit <- FAM(Y=Y,Lx=Lx,Lt=Lt,nEval=0)
yHat <- fit$mu+apply(fit$fam,1,'sum')
plot(yHat,Y)
abline(coef=c(0,1),col=2)


# R^2
R2 <- 1-sum((Y-yHat)^2)/sum((Y-mean(Y))^2)
R2


# prediction
fit <- FAM(Y=Y,Lx=Lx,Lt=Lt,newLx=LxTest,newLt=LtTest)
yHat <- fit$mu+apply(fit$fam,1,'sum')
plot(yHat,YTest,xlim=c(-10,10))
abline(coef=c(0,1),col=2)