Performances analysis of a Givens parametrized adaptive eigenspace algorithm

Abstract

In this paper, we address an adaptive estimation method for eigenspaces of covariance matrices. We are interested in a gradient procedure based on coupled maximizations or minimizations of Rayleigh quotients where the constraints are replaced by a Givens parametrization. This enables us to provide a canonic orthonormal eigenbasis estimator. We study the convergence of this algorithm with the help of the associated ordinary differential equation (ODE), and propose a performance evaluation by computing the variances of the estimated eigenvectors and of the estimated projection matrices on eigenspaces for fixed gain factors. In particular, we show that these misadjustments depend on whether the successive analyzed vector signals are correlated or not, and thus greatly depend on the origin of the covariance matrices of interest (spatial, temporal, spatio-temporal). More precisely, we show that these misadjustments can be smaller in the case of correlated observations than in the case of independent observations. Finally, we show that performance can be improved when the symmetric-centrosymmetric property of some of those covariance matrices is exploited. 1998 Elsevier Science B.V. All rights reserved. ZusammenfassungIn diesem Artikel wird eine adaptive Methode zur Scha¨tzung von Eigenra¨umen von Kovarianzmatrizen behandelt. Wir interessieren uns fu¨r ein Gradientenverfahren, welches auf gekoppelten Maximierungen oder Minimierungen von Rayleigh-Quotienten beruht, wobei die Nebenbedingungen durch eine Givens-Parametrisierung ersetzt werden. Dies ermo¨glicht es, einen kanonischen Scha¨tzer fu¨r orthonormale Eigenbasen anzugeben. Wir studieren die Konvergenz dieses Algorithmus mit Hilfe der zugeho¨rigen gewo¨hnlichen Differentialgleichung. Zur Beurteilung der Leistungsfa¨higkeit schlagen wir vor, die Varianzen der gescha¨tzten Eigenvektoren und der gescha¨tzten Eigenraum-Projektionsmatrizen bei festen Versta¨rkungsfaktoren zu berechnen. Insbesondere zeigen wir, da{ diese Fehlanpassungen davon abha¨ngen, ob die nacheinander analysierten Vektorsignale korreliert sind, wodurch sich eine starke Abha¨ngigkeit von der Herkunft der Kovarianzmatrizen (ra¨umlich, zeitlich, ra¨umlich-zeitlich) ergibt. Wir zeigen konkret, da{ diese Fehlanpassungen im Fall korrelierter Beobachtungen kleiner sein ko¨nnen als im Fall unabha¨ngiger Beobachtungen. Schlie{lich zeigen wir, da{ die Leistungsfa¨higkeit erho¨ht werden kann, wenn die Symmetrie-Zentrosymmetrie-Eigenschaft einiger dieser Kovarianzmatrizen ausgenu¨tzt wird.1998 Elsevier Science B.V. All rights reserved. Re´sumeŃ ous conside´rons dans cet article une me´thode d'estimation de sous espaces propes de matrices de covariance. Nous nous inte´ressons a`une me´thode de gradient base´e sur des minimisations ou des maximisations de quotients de Rayleigh dans lesquelles les contraintes sont remplace´es par une parame´trisation de Givens. Cela permet de fournir de facon structurelle un estimateur orthonorme´de bases orthonorme´es. Nous e´tudions la convergence de cet algorithme graˆ...

Subspace Tracking for Signal Processing

“…, r. This rather intuitive computational process was confirmed by simulation results [60]. Later a formal analysis of the convergence and performance had been performed in [23] where it has been proved that the stationary points of the associated ODE are globally asymptotically stable (see Subsection VII-A) and that the stochastic algorithm (VI.1) converges almost surely to these points for stationary data x(k) when µ k is decreasing with lim k→∞ µ k = 0 and k µ k = ∞. We note that this algorithm yields exactly orthonormal r dominant or minor estimated eigenvectors by a simple change of sign in its step size, and requires O(nr) operations at each iteration but without accounting for the trigonometric functions.…”

“…T . This property has been exploited [23], [26] to reduce the computational cost of the previously introduced eigenvectors adaptive algorithms. Furthermore, the conditioning of these two independent EVD is improved with respect to the EVD of C x since the difference between two consecutive eigenvalues increases in general.…”

“…Expressions (VII.16) and (VII.19) extend provided that λ i λ j is replaced by λ i λ j + λ i,j where λ i,j is defined in [25]. Note that when x(k) = (x k , x k−1 , ..., x k−n+1 ) T with x k being an ARMA stationary process, the covariance of the field (VII.7) and thus λ i,j can be expressed in closed form with the help of a finite sum [23].…”

Optimal Controller Realisations with the Smallest Dynamic Range

“…In order to remove the constraint V T V = I in the optimisation problem (44), we can represent an orthogonal V by its independent parameters based on Givens rotation parameterisation [19]. Specifically, when m = 2, any orthogonal V can be written as…”

On eigenvalue decomposition estimators of centro-symmetric covariance matrices

“…In the real case, we use the property that an orthonormal eigenbasis of a symmetric centro-symmetric matrix can be obtained from orthonormal eigenbases of two half-sized symmetric real matrices [5]. This property has already been used in [8,9] and in [19] for, respectively, a parameterized adaptive eigenspace algorithm and an adaptive eigen"lter bank. But no asymptotic performance analysis has been done yet.…”