6 research outputs found
Identification of Sparse Reciprocal Graphical Models
In this paper we propose an identification procedure of a sparse graphical
model associated to a Gaussian stationary stochastic process. The
identification paradigm exploits the approximation of autoregressive processes
through reciprocal processes in order to improve the robustness of the
identification algorithm, especially when the order of the autoregressive
process becomes large. We show that the proposed paradigm leads to a
regularized, circulant matrix completion problem whose solution only requires
computations of the eigenvalues of matrices of dimension equal to the dimension
of the process
A Maximum Entropy solution of the Covariance Extension Problem for Reciprocal Processes
Stationary reciprocal processes defined on a finite interval of the integer
line can be seen as a special class of Markov random fields restricted to one
dimension. Non stationary reciprocal processes have been extensively studied in
the past especially by Jamison, Krener, Levy and co-workers. The specialization
of the non-stationary theory to the stationary case, however, does not seem to
have been pursued in sufficient depth in the literature. Stationary reciprocal
processes (and reciprocal stochastic models) are potentially useful for
describing signals which naturally live in a finite region of the time (or
space) line. Estimation or identification of these models starting from
observed data seems still to be an open problem which can lead to many
interesting applications in signal and image processing. In this paper, we
discuss a class of reciprocal processes which is the acausal analog of
auto-regressive (AR) processes, familiar in control and signal processing. We
show that maximum likelihood identification of these processes leads to a
covariance extension problem for block-circulant covariance matrices. This
generalizes the famous covariance band extension problem for stationary
processes on the integer line. As in the usual stationary setting on the
integer line, the covariance extension problem turns out to be a basic
conceptual and practical step in solving the identification problem. We show
that the maximum entropy principle leads to a complete solution of the problem.Comment: 33 pages, to appear in the IEEE Trans. Aut. Cont
On the Geometry of Maximum Entropy Problems
We show that a simple geometric result suffices to derive the form of the
optimal solution in a large class of finite and infinite-dimensional maximum
entropy problems concerning probability distributions, spectral densities and
covariance matrices. These include Burg's spectral estimation method and
Dempster's covariance completion, as well as various recent generalizations of
the above. We then apply this orthogonality principle to the new problem of
completing a block-circulant covariance matrix when an a priori estimate is
available.Comment: 22 page