spEigenCov {sparseEigen}

Covariance Matrix Estimation with Sparse Eigenvectors

Description

Estimates the covariance matrix with sparse (orthogonal) eigenvectors (in other words, it jointly estimates the sparse eigenvectors and the eigenvalues).

Usage

spEigenCov(S, q = 1, rho = 0.5, thres = 1e-09)

Arguments

S

m-by-m sample covariance matrix. It is required that S is full-rank. Both real and complex matrices are accepted.

q

number of sparse eigenvectors.

rho

sparsity weight factor. Any nonnegative number (suggested range [0,1]).

thres

threshold value. All the entries of the sparse eigenvectors less or equal to thres are set to 0. The default value is 1e-9.

Value

A list with the following components:

vectors

m-by-m matrix, columns corresponding to eigenvectors.

values

m-by-1 vector corresponding to eigenvalues.

Author(s)

Konstantinos Benidis and Daniel P. Palomar

References

K. Benidis, Y. Sun, P. Babu, D. P. Palomar, "Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation," IEEE Transactions on Signal Processing, vol. 64, no. 23, pp. 6211-6226, Dec. 2016.

Examples

## Not run: 
library(sparseEigen)
n <- 600  # samples
m <- 500  # dimension
q <- 3  # number of sparse eigenvectors to be estimated
sp_card <- 0.1*m  # sparsity of each eigenvector

# generate covariance matrix with sparse eigenvectors
V <- matrix(0, m, q)
V[cbind(seq(1, q*sp_card), rep(1:q, each = sp_card))] <- 1/sqrt(sp_card)
V <- cbind(V, matrix(rnorm(m*(m-q)), m, m-q))
V <- qr.Q(qr(V))  # orthogonalize eigenvectors
lmd <- c(100*seq(from = q, to = 1), rep(1, m-q))  # generate eigenvalues
R <- V %*% diag(lmd) %*% t(V)  # covariance matrix

# generate data
X <- MASS::mvrnorm(n, rep(0, m), R)  # random data with underlying sparse structure

# standard and sparse estimation
res_standard <- eigen(cov(X))
res_sparse <- spEigenCov(cov(X), q)

# show inner product between estimated eigenvectors and originals (the closer to 1 the better)
abs(diag(t(res_standard$vectors) %*% V[, 1:q]))  #for standard estimated eigenvectors
abs(diag(t(res_sparse$vectors) %*% V[, 1:q]))    #for sparse estimated eigenvectors

# show error between estimated and true covariance
norm(cov(X) - R, type = 'F') #for sample covariance matrix
norm(res_sparse$cov - R, type = 'F') #for covariance with sparse eigenvectors

## End(Not run)