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Multivariate spectrum approximation in the Hellinger distance

Due to the work of the Byrnes-Georgiou-Lindquist (BGL) school in recent years, generalized moment problems are gaining popularity as tools to derive spectra of processes given second-order statistics estimates. Consider the following inverse problem: Given a discrete-time stationary process y(t), we feed it to a bank of filters represented by the input-to-state transfer function G(z) = (zI-A)-1 B. Suppose that we have estimated the asymptotic state covariance S of this system. We seek a spectral density consistent with S. As Georgiou pointed out, the structure of S poses notable limitations on the existence of such spectra. Nevertheless, when solutions exist, they are usually infinitely many (thus, in general, this problem is not well posed). In the BGL approach, this redundancy is dealt with resorting to convex optimization. A feasible spectrum is sought that minimizes some (pseudo-) distance from an "a priori spectrum" (the identity in the case of no prior information). This procedure has the great advantage that it allows to get rational solutions whose McMillan degree is a priori bounded in terms of the McMillan degrees of G and of the prior spectrum. In this spirit, after reviewing the conditions for existence of solutions, we show how constrained minimization la Lagrange allows to find the spectrum which best approximates a particular "a priori" spectrum. We discuss both minimization in the Kullback-Leibler divergence and in the Hellinger distance. We then introduce a multivariable extension of the Hellinger distance. This is apparently the only metric in which the general multivariate approximation problem is so far known to admit an explicit solution. In all of these cases,existence for the finite-dimensional dual optimization problem can be established. This result, however, is highly nontrivial since the optimization takes place on an unbounded, open set. The numerical solution of the dual problem is carried trough a suitable matricial, Newton-like method. We finally outline an application to multivariate spectral estimation that appears to outperform standard multivariable identification techniques such as MATLAB's PEM and MATLAB's N4SID in the case of a short observation record. -- References -- * A. Ferrante, M. Pavon and F. Ramponi, Constrained approximation in the Hellinger distance, Proc. of European Control Conference 2007 (ECC'07), pp. 322-327, Kos, Greece, July 2007. * A. Ferrante, M. Pavon and F. Ramponi, Further results on the Byrnes-Georgiou-Lindquist generalized moment problem, A. Chiuso, A. Ferrante and S. Pinzoni (eds), in Modeling, Estimation and Control: Festschrift in honor of Giorgio Picci on the occasion of his sixty-fifth Birthday, Springer-Verlag, pp. 73-83, 2007. * A. Ferrante, M. Pavon and F. Ramponi, Hellinger Versus Kullback-Leibler Multivariable Spectrum Approximation, IEEE Trans. Aut. Control, 53, Issue 4, May 2008, 954 - 967. * F. Ramponi, A. Ferrante and M. Pavon, Multivariate spectral approximation in the Hellinger distance, Proc. MTNS 2008, Blacksburg, Virginia, U.S.A. * F. Ramponi, A. Ferrante and M. Pavon, A globally convergent matricial algorithm for multivariate spectral estimation, Feb. 2008, to appear in the IEEE Trans. Aut. Control, arXiv:0809.5024v1 [math.OC] .
Type of Seminar:
IfA Seminar
Frederico Ramponi
Department of Information Engineering, University of Padova
Dec 19, 2008   14:00 /

ETH Zentrum, Building ETZ, Room J 91
Contact Person:

Prof. John Lygeros
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Biographical Sketch:
Federico Ramponi was born in Verona (Italy) in 1978. In 2004 he graduated in Computer Engineering at the University of Padova. In 2005 he got a master in Enterprise Application Integration. From 2006 he is a student in the control engineering Ph.D. program at the Department of Information Engineering of the University of Padova, under the supervision of Prof. Augusto Ferrante and Prof. Michele Pavon. His current interests reside in multivariate spectrum approximation and spectrum estimation by means of convex optimization methods.