R: Kernel functional partial least squares method

KFPLS {KFPLS}

R Documentation

Kernel functional partial least squares method

Description

Kernel functional partial least squares (KFPLS) method for functional nonlinear models with scalar response and functional predictors. The Gaussian kernel is used.

Usage

KFPLS(X, Y, obser_time, nfold, n_comp, sigm_list, basis)

Arguments

`X`	An `array` with three indices. The (i, j, k)-th element of it corresponds to the measurment of the i-th subject for the k-th functional predictor at j-th observation grid.
`Y`	A `vector` with length n, where n is the sample size. The i-th element of it corresponds to the measurement of the scalar response for the i-th subject.
`obser_time`	A `vector` denoting the observation times of the functional predictors.
`nfold`	An `integer` denoting the number of folds for the selection of the tuning parameters by cross-validation.
`n_comp`	A `vector` denoting the candidates of the number of components.
`sigm_list`	A `vector` denoting the candidates of the tuning parameter for the Gaussian kernel.
`basis`	A basis object denoting the basis that used for the smoothing of the functional predictors. It is created by functions in `fda` package, such as `create.bspline.basis`.

Value

A list containing the following components:

`n`	A `scalar` denoting the sample size.
`p`	A `scalar` denoting the number of functional predictors.
`nk`	A `scalar` denoting the selected number of components.
`T`	A `matrix` denoting the value of T at convergence.
`U`	A `matrix` denoting the value of U at convergence.
`K`	A `matrix` denoting the Gram matrix.
`K_c`	A `matrix` denoting the centralized Gram matrix.
`Xfd_list`	A `list` of length `p`. The k-th entry corresponds to the functional data object of the k-th functional predictor.
`XX_list`	A `list` of length `p`. The k-th entry corresponds to the matrix that denotes the inner product of the k-th functional predictor for all subjects.
`Y_c`	A `vector` denoting the centralized scalar response.
`meanY`	A `scalar` denoting the sample mean of the scalar response.
`Y_hat`	A `vector` denoting the prediction of the scalar response.
`obser_time`	A `vector` denoting the observation times of the functional predictors.
`basis`	A basis object denoting the basis that used for the smoothing of the functional predictors.
`sigm`	A `scalar` denoting the selected tuning parameter for the Gaussian kernel.
`CVscore`	A `matrix` denoting the CV scores.
`time`	A `scalar` denoting the computation time.

Examples

# Generate data
n <- 200
t_range <- c(0, 1)
obser_time <- seq(0, 1, length.out = 51)
beta_fun <- function(t){2 * sin(2 * pi * t)}
basis <- fda::create.bspline.basis(t_range, nbasis = 13, norder = 4,
breaks = seq(0, 1, length.out = 11))
beta_fd <- fda::smooth.basis(obser_time, beta_fun(obser_time), basis)$fd
X_basis <- fda::create.bspline.basis(t_range, nbasis = 23, norder = 4,
breaks = seq(0, 1, length.out = 21))
Bbeta <- fda::inprod(X_basis, beta_fd)
Xi_B <- splines::bs(obser_time, knots = seq(0, 1, length.out = 21)[-c(1, 21)],
degree = 3, intercept = TRUE)
a <- array(0, dim = c(n, 23, 1))
X <- array(0, dim = c(n, 51, 1))
Y <- NULL
for(i in 1:n){
a[i, , 1] <- stats::rnorm(23)
X[i, , 1] <- Xi_B %*% a[i, , 1]
aBbeta <- as.numeric(t(a[i, , 1]) %*% Bbeta)
Y[i] <- aBbeta + stats::rnorm(1, mean = 0, sd = 0.05)
}
# KFPLS
KFPLS_list <- KFPLS(X, Y, obser_time, nfold = 5, n_comp = 5, sigm_list = 0.005, basis)
plot(KFPLS_list$Y_hat, Y)
lines(Y, Y)

[Package KFPLS version 1.0 Index]