pfpca {fgm}

Partially Separable Karhunen-Loeve Expansion

Description

Estimates the Karhunen-Loeve expansion for a partially separable multivariate Gaussian process.

Usage

pfpca(y, t = seq(0, 1, length.out = dim(y[[1]])[2]))

Arguments

y

list of length p containing densely observed multivariate (p-dimensional) functional data . y[[j]] is an nxm matrix of functional data for n subjects observed on a grid of length m

t

(optional) grid on which functional data is observed, defaults to seq(0, 1, m) where m = dim(data[[1]])[2]

Details

This function implements the functional graphical model in Zapata, Oh, and Petersen (2019). This code uses functions from the testing version of fdapace available at: https://github.com/functionaldata/tPACE.

Value

A list with three variables:

phi

Lxm matrix where each row denotes the value of a basis function evaluated at a grid of length m

theta

list of length L of functional principal component scores. theta[[l]] is an nxp matrix of vector scores corresponding to the basis function phi[l,]

FVE

fraction of functional variance explained (FVE) by the first L components

Author(s)

Javier Zapata, Sang-Yun Oh and Alexander Petersen

References

Zapata J., Oh S. and Petersen A. (2019) - Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes. Available at https://arxiv.org/abs/1910.03134.

Examples

## Variables
# Omega - list of precision matrices, one per eigenfunction
# Sigma - list of covariance matrices, one per eigenfunction
# theta - list of functional  principal component scores
# phi - list of eigenfunctions densely observed on a time grid
# y - list containing densely observed multivariate (p-dimensional) functional data 

library(mvtnorm)
library(fda)

## Generate data y
 source(system.file("exec", "getOmegaSigma.R", package = "fgm"))
 theta = lapply(1:nbasis, function(b) t(rmvnorm(n = 100, sigma = Sigma[[b]])))
 theta.reshaped = lapply( 1:p, function(j){
     t(sapply(1:nbasis, function(i) theta[[i]][j,]))
 })
 phi.basis=create.fourier.basis(rangeval=c(0,1), nbasis=21, period=1)
 t = seq(0, 1, length.out = time.grid.length)
 chosen.basis = c(2, 3, 6, 7, 10, 11, 16, 17, 20, 21)
 phi = t(predict(phi.basis, t))[chosen.basis,]
 y = lapply(theta.reshaped, function(th) t(th)%*%phi)
 
## Solve  
 pfpca(y)