ReMFPCA (version 2.0.0)

remfpca: A Class for `ReMFPCA` objects

Description

The `remfpca` class represents regularized functional principal components components.

The `remfpca` class represents regularized functional principal components ('ReMFPCs') components.

Usage

Remfpca(
  mvmfd_obj,
  method = "power",
  ncomp,
  smooth_tuning = NULL,
  sparse_tuning = NULL,
  centerfns = TRUE,
  alpha_orth = FALSE,
  smoothing_type = "basispen",
  sparse_type = "soft",
  K_fold = 30,
  sparse_CV = TRUE,
  smooth_GCV = TRUE
)

Arguments

mvmfd_obj

An `mvmfd` object representing the multivariate functional data.

method

A character string specifying the approach to be used for MFPCA computation. Options are "power" (the default), which uses the power algorithm, or "eigen", which uses the eigen decomposition approach.

ncomp

The number of functional principal components to retain.

smooth_tuning

A list or vector specifying the smoothing regularization parameter(s) for each variable. If NULL, non-smoothing MFPCA is estimated.

sparse_tuning

A list or vector specifying the sparsity regularization parameter(s) for each variable. If NULL, non-sparse MFPCA is estimated.

centerfns

Logical indicating whether to center the functional data before analysis. Default is TRUE.

alpha_orth

Logical indicating whether to perform orthogonalization of the regularization parameters. If `method` is "power", setting `alpha_orth = FALSE` (default) uses the sequential power approach, while setting `alpha_orth = TRUE` uses the joint power approach.

smoothing_type

The type of smoothing penalty to be applied on the estimated functional PCs. The types "basispen" and "coefpen" is supported. Default is "basispen".

sparse_type

The type of sparse penalty to be applied on the estimated functional PCs. The types "soft-threshold", "hard-threshold" and "SCAD" is supported. Default is "soft-threshold".

K_fold

An integer specifying the number of folds in the sparse cross-validation process. Default is 30.

sparse_CV

@param sparse_CV Logical indicating whether cross-validation should be applied to select the optimal sparse tuning parameter in sequential power approach. If `sparse_CV = TRUE`, a series of tuning parameters should be provided as a vector with positive number with max equals to number of subjects. If `sparse_CV = FALSE`, specific tuning parameters are given directly to each principal components. Tuning parameters should be provided as a vector with length equal to `ncomp`. If the dimensions of input tuning parameters are incorrect, it will be converted to a list internally, and a warning will be issued.

smooth_GCV

@param smooth_GCV Logical indicating whether generalized cross-validation should be applied to select the optimal smooth tuning parameter. If `smooth_GCV = TRUE`, a series of tuning parameters should be provided as a list with length equal to the number of variables. If a list with incorrect dimensions is provided, it will be converted to a correct list internally, and a warning will be issued. If `smooth_GCV = FALSE`, specific tuning parameters are given directly. If `method` is "power" and `alpha_orth = FALSE` (sequential power), tuning parameters should be provided as a list with length equal to the number of variables, where each element is a vector of length `ncomp`. If `method` is "power" and `alpha_orth = TRUE` (joint power), tuning parameters should be provided as a vector with length equal to the number of variables. If the dimensions of input tuning parameters are incorrect, it will be converted to a list internally, and a warning will be issued.

Active bindings

pc_mfd

An object of class `mvmfd` where the first indices (fields) represents harmonics and second indices represents variables

lsv

= Left singular values vectors

values

= The set of eigenvalues

smooth_tuning

= The list of smoothing penalties parameters

sparse_tuning

= The list of sparse penalties parameters

GCVs

= Generalized cross validations scores of smoothing penalties parameters. If both smoothing and sparse tuning penalties are used in the ReMFPCA method, this represents the conditional generalized cross-validation scores, which means it is computed based on the optimal sparse tuning parameter selected via cross validation.

CVs

= Cross validations scores of sparse penalties parameters

mean_mfd

A multivariate functional data object giving the mean function

Methods

Public methods

• remfpca$new()

• remfpca$clone()

---

Method new()

Initialize the `remfpca` class.

Usage

remfpca$new(
  mvmfd_obj,
  method = "power",
  ncomp,
  smooth_tuning = NULL,
  sparse_tuning = NULL,
  centerfns = TRUE,
  alpha_orth = FALSE,
  smoothing_type = "coefpen",
  sparse_type = "soft",
  K_fold = 30,
  sparse_CV,
  smooth_GCV
)

Arguments

mvmfd_obj

An `mvmfd` object representing the multivariate functional data.

method

A character string specifying the approach to be used for MFPCA computation. Options are "power" (the default) or "eigen".

ncomp

The number of functional principal components to retain.

smooth_tuning

A list or vector specifying the smoothing regularization parameter(s).

sparse_tuning

A list or vector specifying the sparsity regularization parameter(s).

centerfns

Logical. Whether to center the functional data before analysis.

alpha_orth

Logical. Whether to perform orthogonalization of the regularization parameters.

smoothing_type

The type of smoothing penalty to be applied.

sparse_type

The type of sparse penalty to be applied.

K_fold

An integer specifying the number of folds in cross-validation.

sparse_CV

Logical. Whether cross-validation should be applied for sparse tuning.

smooth_GCV

Logical. Whether generalized cross-validation should be applied for smoothing tuning.

---

Method clone()

The objects of this class are cloneable with this method.

Usage

remfpca$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

See Also

mvmfd

Examples

require(fda)
# Brownian Bridge simulation on [0,1]
M <- 110 # number of components
N <- 20 # number of instances
n <- 100 # number of grides
t0 <- seq(0, 1, len = n)
j <- 1:M
alpha1 <- list(a1 = 2^seq(0, 1, length.out = 3), a2 = 2^seq(0, 1, length.out = 3))
sparse_tuning = as.integer(seq(1, N-1, length.out = 10))
psi_1 <- function(t, m) sin(m * pi * t) # eigenfunction of BB
psi_2 <- function(t, m) sin((2 * m - 1) * pi / 2 * t) # eigenfunction of BM
PC_1 <- outer(t0, j, FUN = psi_1) # n by M matrix
PC_2 <- outer(t0, j, FUN = psi_2) # n by M matrix
Z <- matrix(rnorm(N * M), nr = M)
lambda <- matrix(2 / (pi * (2 * j - 1)), nr = M, nc = N)
X_1t <- PC_1 %*% (lambda * Z)
X_2t <- PC_2 %*% (lambda * Z)
noise <- rnorm(n * N, 0, 0.1)
X_1 <- X_1t + noise
X_2 <- X_2t + noise
bs <- create.bspline.basis(c(0, 1), 51)
mdbs <- Basismfd(bs)
mfd1 <- Mfd(X = X_1, mdbs = mdbs)
mfd2 <- Mfd(X = X_2, mdbs = mdbs)
mvmfd_obj <- Mvmfd(mfd1, mfd2)
k <- 2
# Non Regularized MFPCA based on sequential power algorithm
Re0 <- Remfpca(mvmfd_obj, ncomp = k, smooth_GCV = FALSE, sparse_CV = FALSE)
fpc0 <- Re0$pc_mfd
scores0 <- inprod_mvmfd(mvmfd_obj, fpc0)
dim(scores0)
# Smooth MFPCA based on sequential power algorithm
Re1 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1)
# Smooth and sparse MFPCA based on sequential power algorithm
Re2 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, sparse_tuning = sparse_tuning)
# Smooth MFPCA based on joint power algorithm
Re3 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, alpha_orth = TRUE)
# Smooth MFPCA based on eigen decomposition algorithm
Re4 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, method = "eigen")