fgm {fgm}

Functional Gaussian Graphical Model

Description

Estimates a sparse adjacency matrix representing the conditional dependency structure between features of a multivariate Gaussian process

Usage

fgm(y, L, alpha, gamma, t = seq(0, 1, length.out = dim(y[[1]])[2]),
  thr.FVE = 95, include.Omega = FALSE)

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

L

the number of eigenfunctions used for dimension reduction using the partially separable Karhunen-Loeve (PSKL) expansion obtained using 'pfpca()'. This argument can take positive integer values greater or equal to 1.

alpha

penalty parameter for the common sparsity pattern taking values in [0,1].

gamma

penalty parameter for the overall sparsity pattern taking positive values.

t

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

thr.FVE

this parameter sets a threshold for the minimum percentage of functional variance explained (FVE) by the PSKL eigenfunctions (obtained using 'pfpca()'). This criterion is used only if a value for L is not provided or is greater than the maximum possible number of eigenfunctions estimated from y using pfpca().

include.Omega

logical variable indicating wheter to include the list of precision matrices in the output. Default value is FALSE.

Details

This function implements the functional graphical model in Zapata, Oh, and Petersen (2019). The arguments alpha and gamma are a reparameterization of the Group Graphical Lasso tuning parameters when using the JGL package. When using JGL::JGL, the tuning parameters are computed as lambda1 = alpha*gamma and lambda2 = (1-alpha)*gamma

Value

A list with letters and numbers.

A

Resulting adjacency matrix as the union of all the Omega matrices

L

number of PSKL expansion eigenfunctions considered for the estimation of the graphical model.

Omega

list of of precision matrices obtained using the multivariate functional principal component scores theta obtained using 'fpca()'

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
 fgm(y, alpha=0.5, gamma=0.8)