5,883 research outputs found
Adaptive estimation of spectral densities via wavelet thresholding and information projection
In this paper, we study the problem of adaptive estimation of the spectral
density of a stationary Gaussian process. For this purpose, we consider a
wavelet-based method which combines the ideas of wavelet approximation and
estimation by information projection in order to warrants that the solution is
a nonnegative function. The spectral density of the process is estimated by
projecting the wavelet thresholding expansion of the periodogram onto a family
of exponential functions. This ensures that the spectral density estimator is a
strictly positive function. Then, by Bochner's theorem, the corresponding
estimator of the covariance function is semidefinite positive. The theoretical
behavior of the estimator is established in terms of rate of convergence of the
Kullback-Leibler discrepancy over Besov classes. We also show the excellent
practical performance of the estimator in some numerical experiments
A new family of high-resolution multivariate spectral estimators
In this paper, we extend the Beta divergence family to multivariate power
spectral densities. Similarly to the scalar case, we show that it smoothly
connects the multivariate Kullback-Leibler divergence with the multivariate
Itakura-Saito distance. We successively study a spectrum approximation problem,
based on the Beta divergence family, which is related to a multivariate
extension of the THREE spectral estimation technique. It is then possible to
characterize a family of solutions to the problem. An upper bound on the
complexity of these solutions will also be provided. Simulations suggest that
the most suitable solution of this family depends on the specific features
required from the estimation problem
Rational approximations of spectral densities based on the Alpha divergence
We approximate a given rational spectral density by one that is consistent
with prescribed second-order statistics. Such an approximation is obtained by
minimizing a suitable distance from the given spectrum and under the
constraints corresponding to imposing the given second-order statistics. Here,
we consider the Alpha divergence family as a distance measure. We show that the
corresponding approximation problem leads to a family of rational solutions.
Secondly, such a family contains the solution which generalizes the
Kullback-Leibler solution proposed by Georgiou and Lindquist in 2003. Finally,
numerical simulations suggest that this family contains solutions close to the
non-rational solution given by the principle of minimum discrimination
information.Comment: to appear in the Mathematics of Control, Signals, and System
On the well-posedness of multivariate spectrum approximation and convergence of high-resolution spectral estimators
In this paper, we establish the well-posedness of the generalized moment
problems recently studied by Byrnes-Georgiou-Lindquist and coworkers, and by
Ferrante-Pavon-Ramponi. We then apply these continuity results to prove almost
sure convergence of a sequence of high-resolution spectral estimators indexed
by the sample size
Multivariate Spectral Estimation based on the concept of Optimal Prediction
In this technical note, we deal with a spectrum approximation problem arising
in THREE-like multivariate spectral estimation approaches. The solution to the
problem minimizes a suitable divergence index with respect to an a priori
spectral density. We derive a new divergence family between multivariate
spectral densities which takes root in the prediction theory. Under mild
assumptions on the a priori spectral density, the approximation problem, based
on this new divergence family, admits a family of solutions. Moreover, an upper
bound on the complexity degree of these solutions is provided
On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist
A parametric spectral estimation problem in the style of Byrnes, Georgiou,
and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was
only proved in a special case. Based on their results, we show that a solution
indeed exists given an arbitrary matrix-valued prior density. The main tool in
our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA
An Interpretation of the Dual Problem of the THREE-like Approaches
Spectral estimation can be preformed using the so called THREE-like approach.
Such method leads to a convex optimization problem whose solution is
characterized through its dual problem. In this paper, we show that the dual
problem can be seen as a new parametric spectral estimation problem. This
interpretation implies that the THREE-like solution is optimal in terms of
closeness to the correlogram over a certain parametric class of spectral
densities, enriching in this way its meaningfulness
A globally convergent matricial algorithm for multivariate spectral estimation
In this paper, we first describe a matricial Newton-type algorithm designed
to solve the multivariable spectrum approximation problem. We then prove its
global convergence. Finally, we apply this approximation procedure to
multivariate spectral estimation, and test its effectiveness through
simulation. Simulation shows that, in the case of short observation records,
this method may provide a valid alternative to standard multivariable
identification techniques such as MATLAB's PEM and MATLAB's N4SID
- …