Package 'SLFPCA'

Generate binary functional data

Description

Generate binary functional data through latent process.

Usage

GenBinaryFD(n, interval, sparse, regular, meanfun, score, eigfd)

Arguments

n	An integer denoting the number of sample size.
interval	A vector of length two denoting the supporting interval.
sparse	A vector denoting the possible numbers of observation size. The elements are chosen with equal chance. The length of sparse must be one if regular = TRUE.
regular	Logical; If TRUE, the observation grids are equally-spaced.
meanfun	A function for the mean.
score	A n by npc matrix containing the FPC scores, where npc is the number of FPCs.
eigfd	A list containing functional objects for the eigenfunctions.

Value

A list containing the following components:

Lt	A list of n vectors, where n is the sample size. Each entry contains the observation time in ascending order for each subject.
Lx	A list of n vectors, where n is the sample size. Each entry contains vales of the latent process of each subject at the observation time correspond to Lt.
Ly	A list of n vectors, where n is the sample size. Each entry contains the binary measurements of each subject at the observation time correspond to Lt.

Examples

n <- 100
npc <- 2
interval <- c(0, 10)
gridequal <- seq(0, 10, length.out = 51)
basis <- fda::create.bspline.basis(c(0, 10), nbasis = 13, norder = 4,
         breaks = seq(0, 10, length.out = 11))
meanfun <- function(t){2 * sin(pi * t/5)/sqrt(5)}
lambda_1 <- 3^2 #the first eigenvalue
lambda_2 <- 2^2 #the second eigenvalue
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
eigfun <- list()
eigfun[[1]] <- function(t){cos(pi * t/5)/sqrt(5)}
eigfun[[2]] <- function(t){sin(pi * t/5)/sqrt(5)}
eigfd <- list()
for(i in 1:npc){
  eigfd[[i]] <- fda::smooth.basis(gridequal, eigfun[[i]](gridequal), basis)$fd
}
DataNew <- GenBinaryFD(n, interval, sparse = 8:12, regular = FALSE,
           meanfun = meanfun, score, eigfd)

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Sparse logistic functional principal component analysis

Description

Sparse logistic functional principal component analysis (SLFPCA) for binary data. The estimated eigenfunctions from SLFPCA can be strictly zero on some sub-intervals, which is helpful for interpretation.

Usage

SLFPCA(
  Ly,
  Lt,
  interval,
  npc,
  L_list,
  norder,
  kappa_theta,
  sparse_pen,
  nRegGrid = 51,
  bwmu_init = 0.5,
  bwcov_init = 1,
  kappa_mu,
  itermax = 100,
  tol = 10
)

Arguments

Ly	A list of n vectors, where n is the sample size. Each entry contains the binary measurements of each subject at the observation time correspond to Lt.
Lt	A list of n vectors, where n is the sample size. Each entry contains the observation time in ascending order for each subject.
interval	A vector of length two denoting the supporting interval.
npc	An integer denoting the number of FPCs.
L_list	A vector denoting the candidates for the number of B-spline basis functions.
norder	An integer denoting the order of the using B-spline basis, which is one higher than their degree.
kappa_theta	A vector denoting the smoothing parameters for eigenfunctions, the optimal tuning parameter is chosen from them.
sparse_pen	A vector denoting the sparseness parameters for eigenfunctions, the optimal tuning parameter is chosen from them.
nRegGrid	An integer denoting the number of equally spaced time points in the supporting interval. The eigenfunctions and mean function are estimated at these equally spaced time points first, before transforming into functional data object. (default: 51)
bwmu_init	A scalar denoting the bandwidth for mean function estimation in the setting of initial values. (default: 0.5)
bwcov_init	A scalar denoting the bandwidth for covariance function estimation in the setting of initial values. (default: 1)
kappa_mu	A vector denoting the smoothing parameters for mean function, the optimal tuning parameter is chosen from them.
itermax	An integer denoting the maximum number of iterations. (default: 100)
tol	A scalar. When difference of the loglikelihood functions between two consecutive iteration is less than tol, the convergence is supposed to be reached. (default: 10)

Value

A list containing the following components:

mufd	A functional data object for the mean function estimate.
eigfd_list	A list containing npc functional data objects, which are the eigenfunction estimates.
score	A n by npc matrix containing the estimates of the FPC scores, where n is the sample size.
kappa_mu	A scalar denoting the selected smoothing parameter for mean function.
kappa_theta	A scalar denoting the selected smoothing parameter for eigenfunctions.
sparse_pen	A scalar denoting the selected sparseness parameter for eigenfunctions.
L_select	A scalar denoting the selected number of B-spline basis functions.
EBICscore	A vector denoting the selected EBIC scores corresponding to different numbers of B-spline basis functions in L_list.

References

Rou Zhong, Shishi Liu, Haocheng Li, Jingxiao Zhang. Sparse logistic functional principal component analysis for binary data. Statistics and Computing, 33, 15 (2023). https://doi.org/10.1007/s11222-022-10190-3

Examples

#Generate data
n <- 100
npc <- 1
interval <- c(0, 10)
gridequal <- seq(0, 10, length.out = 51)
basis <- fda::create.bspline.basis(c(0, 10), nbasis = 13, norder = 4,
         breaks = seq(0, 10, length.out = 11))
meanfun <- function(t){2 * sin(pi * t/5)/sqrt(5)}
lambda_1 <- 3^2 #the first eigenvalue
score <- cbind(rnorm(n, 0, sqrt(lambda_1)))
eigfun <- list()
eigfun[[1]] <- function(t){cos(pi * t/5)/sqrt(5)}
eigfd <- list()
for(i in 1:npc){
  eigfd[[i]] <- fda::smooth.basis(gridequal, eigfun[[i]](gridequal), basis)$fd
}
DataNew <- GenBinaryFD(n, interval, sparse = 8:12, regular = FALSE,
           meanfun = meanfun, score, eigfd)
SLFPCA_list <- SLFPCA(DataNew$Ly, DataNew$Lt, interval, npc, L_list = 13,
               norder = 4, kappa_theta = 0.2, sparse_pen = 0, kappa_mu = 0)
plot(SLFPCA_list$eigfd_list[[1]])

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