Une approche geometrique a l'analyse en composantes independantes

This master's thesis is devoted to independent component analysis. We consider random signals S of the form S=i=1pai Vi, where the vectors V i ∈ RN are unknown and the ais are independent zero-mean random variables with unknown probability distributions. The problem that we consider is the estimation of the V is, called independent components, given a set of realizations of the signals S. This problem has applications in signal analysis and for the synthesis of random signals.;The signal S belongs to the p-dimensional space, (p ≤ N), E := span{V1,..., V p}. Our approach is to look for vectors X i ∈ E, i = 1..., p, such that XTiVj = 0 if i ≠ j, with XTiVj ≠ 0. If the Xis are known, then the p - 1 orthogonality conditions XTiVj = 0. i = 1.2, ..., j - 1, j + 1, ..., p allow the determination of Vj ∈ E because E has dimension p. To estimate the Xis, we take advantage of the independence of the factors XTiS , which results from the identity XTiS = ai XTiVj and the independence of the ais. We show that the independence of the factors XTiS and XTjS for i ≠ j has two consequences. Firstly, the orthogonality conditions XTiZnX j=0,i≠j,∀n∈N , where Zn(X) := E((XT S)nS), are satisfied. Secondly, these conditions imply the colinearity of the vector collection ZXi:= ZnXi ,n∈N.;For p = 2, we developed two estimation methods based on the colinearity of the vectors in Z(X i). For p = 3, we developed a method based on the orthogonality conditions. We minimise an objective function via a cyclic descent procedure. The convergence of this descent to a minimal value of the objective function is established.;For p = 2, we compared the precision of the estimated Vis obtained with our methods and with the method FastICA, using synthetic signals. Our methods are significantly more accurate than FastICA, with estimation errors which are at least twice smaller.