Multivariate functional principal component analysis for variable domain data

Performs a multivariate functional principal component analysis (MFPCA) for functional data observed on subject-specific (variable) domains. Each functional variable is first decomposed through a variable domain FPCA, and the resulting univariate scores are combined into a domain-varying multivariate decomposition.

Usage

mfpca_vd(
  Data,
  Times = NULL,
  M_grid = NULL,
  Hz = 1,
  m_npcs = NULL,
  u_npcs = 5,
  k_m = 15,
  model_type = "gam"
)

Arguments

Data: A list (one entry per functional variable) of matrices with subjects in rows and observation points in columns, with NA on the unobserved part of each domain. All variables must have the same number of subjects. Currently two variables are supported.
Times: A list of matrices, the same shape as Data, giving the actual observation time of each measurement. Mutually exclusive with M_grid.
M_grid: A numeric vector of length \(N\) with the domain length of each subject, used when the observations are equidistant. Mutually exclusive with Times.
Hz: Sampling rate used to build the equidistant grid when M_grid is supplied (default 1).
m_npcs: Number of multivariate principal components to retain. If NULL, the number explaining more than 90 percent of the variance is used.
u_npcs: Number of univariate principal components retained per variable (default 5).
k_m: Dimension of the basis used to model the score covariances along the domain (default 15).
model_type: Either "gam" (the default) or "sop", the method used to smooth the score covariances along the domain.

Value

A list with the following items:

scores_m: Matrix of multivariate scores.
efunctions_m: Multivariate eigenfunctions, as a list over domains, each a list over variables.
efunctions_u: Univariate eigenfunctions, as a list over variables.
scores_u: Matrix of univariate scores.
evalues_u: Univariate eigenvalues, as a list over variables.
evalues_m: Multivariate eigenvalues, as a list over domains.
var_u: Cumulative variance explained by the univariate components.
mean_model: The fitted univariate mean models (one per variable).
M_grid: The domain grid used.
argvals_u: The observation times of each subject, as a list over variables, each a list over subjects.

Details

The observation times can be supplied in two mutually exclusive ways. With Times, each variable carries a matrix (subjects in rows, observation points in columns) holding the actual observation time of every measurement, and the domain of subject \(i\) is taken as the maximum observed time. With M_grid, the observations are assumed equidistant and M_grid is a vector of length \(N\) giving the domain length of each subject, so subject \(i\) is evaluated on seq(0, M_grid[i], by = 1 / Hz). Exactly one of Times or M_grid must be provided.