% Sparse covariance estimation for Gaussian variables
% JoĆ«lle Skaf - 04/24/08
% (a figure is generated)
%
% Suppose y \in\reals^n is a Gaussian random variable with zero mean and
% covariance matrix R = \Expect(yy^T), with sparse inverse S = R^{-1}
% (S_ij = 0 means that y_i and y_j are conditionally independent).
% We want to estimate the covariance matrix R based on N independent
% samples y1,...,yN drawn from the distribution, and using prior knowledge
% that S is sparse
% A good heuristic for estimating R is to solve the problem
% maximize logdet(S) - tr(SY)
% subject to sum(sum(abs(S))) <= alpha
% S >= 0
% where Y is the sample covariance of y1,...,yN, and alpha is a sparsity
% parameter to be chosen or tuned.
% Input data
rand('state',0);
randn('state',0);
n = 10;
N = 100;
Strue = sprandsym(n,0.5,0.01,1);
R = inv(full(Strue));
y_sample = sqrtm(R)*randn(n,N);
Y = cov(y_sample');
alpha = 50;
% Computing sparse estimate of R^{-1}
cvx_begin sdp
variable S(n,n) symmetric
maximize log_det(S) - trace(S*Y)
sum(sum(abs(S))) <= alpha
S >= 0
cvx_end
R_hat = inv(S);
S(find(S<1e-4)) = 0;
figure;
subplot(121);
spy(Strue);
title('Inverse of true covariance matrix')
subplot(122);
spy(S)
title('Inverse of estimated covariance matrix')