The Arov-Grossman Model and Burg’s Entropy

Stat., optim. inf. comput.

Morán

2019

Self Cite

In this paper we obtain a version of the Levinson algorithm for block Toeplitz matrices in an infinite dimensional setting from a geometrical approach. With this methodology we obtain a sequence of operators in the Levinson recurrences whose norms in geometric terms represent angles between subspaces. Additionally, under this geometric framework a block LU decomposition for a block Toeplitz matrix is obtained. 749 this, it is necessary to obtain the LU decomposition of T p . In [9], this decomposition is used to demonstrate that the maximum entropy density subject to the first p + 1 autocovariance matrices is the spectrum of a multivariate autoregressive process of p − th order. Problems involving Toeplitz linear equation arise in several applications, e.g. in the prediction of stationary processes, in inverse scattering problem, in buffer analysis for data communication system, among others (cf.[5], [18] and [27]). In [24] and [25] some previous results are obtained using orthogonal decomposition for the finite dimensional case.The main contribution of this work is to obtain a new version of the Levinson algorithm for block Toeplitz matrices in an infinite dimensional setting using orthogonal decomposition. The parameters obtained in this algorithm are a sequence of contractive operators whose norm can be expressed as angles between subspaces, similar to the finite dimensional case. Also, we obtain a block LU decomposition for block Toeplitz matrices where the triangular matrices can be obtained through Levinson recurrences.Additionally, our methodology could be useful to solve extension problems in statistics and for prediction problems in an infinite dimensional context (cf. [10], [11], [12], [15], [16], [19], [20], [28], [33] and [34]). The scheme to obtain the Levinson algorithm is as follows: the data is a sequence of bounded operators {R k } p k=0 , defined in a separable Hilbert space G. More specifically, based on the sequence of bounded operators {R k } p k=0 , we built a Hilbert space H p and a surjective isometry the Levinson recurrences are obtained as a consequence of the orthogonal decompositions where the subspaces in this decompositions are the defect spaces.The paper is organized as follows: in the second section; we introduce some notations and preliminary results; in the third section, we obtain a block LU decomposition for a block Toeplitz matrix; in fourth section, we state the main result of this work and in the last section, we present the conclusions and discussions.

Section: Conclusion and Discussionmentioning

confidence: 99%

A Geometrical Approach of the Levinson Algorithm for Block Toeplitz Matrices

Stat., optim. inf. comput.

Morán

2019

Self Cite

“…on T. First, we state the scalar covariance extension problem (see [10]): Given p ∈ N, and c 0 , c 1 , · · · , c p , complex numbers with c 0 > 0 and c −k = c k , k = 1, . .…”

Section: The Covariance Extension Problem and The Krein Entropymentioning

confidence: 99%

“…Since then many different authors have studied this problem, see for example [3], [8] and [10]. In 1993, Gabardo (cf.…”

Section: Introductionmentioning

confidence: 99%

On a generalization of the maximum entropy Theorem of Burg

Infante

Sánchez

2017

RMTA

In this article we introduce some matrix manipulations that allow us to obtain a version of the original Christoffel-Darboux formula, which is of interest in many applications of linear algebra. Using these developments matrix and Jensen's inequality, we obtain the main result of this proposal, which is the generalization of the maximum entropy theorem of Burg for multivariate processes.Keywords: multivariate processes; maximum entropy theorem; Christoffel-Darboux formula. ResumenEn este artículo se introducen algunas manipulaciones matriciales que nos permiten obtener una versión de la fórmula original de ChristoffelDarboux, la cual resulta de interés en muchas aplicaciones del álgebra lineal. Usando estos desarrollos matriciales y la desigualdad de Jensen, se obtiene el resultado principal de esta propuesta, que consiste en la generalización del teorema de entropía máxima de Burg para procesos multivariados.Palabras clave: procesos multivariados; teorema de entropía máxima; fórmula de Christoffel-Darboux.Mathematics Subject Classification: 94A17, 46E22.

“…. , q is solution of the scalar Covariance Extension Problem with data { R k E j , E j }, from [14] we obtain that H(z)H * (z) < I, z ∈D.…”

mentioning

confidence: 99%

The Arov?Grossman Model and the Burg Multivariate Entropy

Journal of Fourier Analysis and Applications

Morán

2003

Self Cite

In this article we deal with the Arov-Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy. This description is based on a one-to-one correspondence between the set of all solutions of the Covariance Extension Problem and the set of all contractive analytic functions H from the open unit disk with valueson the space of q × q matrices. With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H ≡ 0. Also, from the information that the Arov-Grossman functional model provides we obtain a version of the Levinson algorithm. The partial autocorrelation coefficient matrices are computed directly from Levinson's recursions.