% Section 7.1.1: Covariance estimation for Gaussian variables
% Boyd & Vandenberghe "Convex Optimization"
% JoĆ«lle Skaf - 04/24/08
%
% Suppose y \in\reals^n is a Gaussian random variable with zero mean and
% covariance matrix R = \Expect(yy^T). We want to estimate the covariance
% matrix R based on N independent samples y1,...,yN drawn from the
% distribution, and using prior knowledge about R (lower and upper bounds
% on R)
% L <= R <= U
% Let S be R^{-1}. The maximum likelihood (ML) estimate of S is found
% by solving the problem
% maximize logdet(S) - tr(SY)
% subject to U^{-1} <= S <= L^{-1}
% where Y is the sample covariance of y1,...,yN.
% Input data
randn('state',0);
n = 10;
N = 1000;
tmp = randn(n);
L = tmp*tmp';
tmp = randn(n);
U = L + tmp*tmp';
R = (L+U)/2;
y_sample = sqrtm(R)*randn(n,N);
Y = cov(y_sample');
Ui = inv(U); Ui = 0.5*(Ui+Ui');
Li = inv(L); Li = 0.5*(Li+Li');
% Maximum likelihood estimate of R^{-1}
cvx_begin sdp
variable S(n,n) symmetric
maximize( log_det(S) - trace(S*Y) );
S >= Ui;
S <= Li;
cvx_end
R_hat = inv(S);