ML_covariance_est.m
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ML_covariance_est.m
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			% Section 7.1.1: Covariance estimation for Gaussian variables
			% Boyd & Vandenberghe "Convex Optimization"
			% Joëlle 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);