A Geometrical Approach of the Levinson Algorithm for Block Toeplitz Matrices
AbstractIn this paper we obtain a version of the Levinson algorithm for block Toeplitz matrices in an infinity 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 LU decomposition for a block Toeplitz matrix is obtained.
. R. Arocena, Extensiones unitarias de isometrías, entropía máxima y parámetros de Schur de un proceso estacionario, Spanish, Actas III Congreso Latinoamericano: Prob. y Est. Mat., 1-8, 1990.
. M. Bakonyi and T. Constantinescu, Schur’s algorithm and several applications, Pitman Research Notes in Mathematics Series, 261,1992.
. D. Bosq, Linear Processes in Function Space, Lecture Notes in Statistics, Springer, 2000.
. D. Bosq, Computing the best linear predictor in a Hilbert space. Applications to general ARMAH processes, Journal of Multivariate Analysis, 124, 436-450, 2014.
. P. Brockwel and R. Davis, Time Series: Theory and Methods. Second Edition, Springer Series in Statistics, New York, 2006.
. J. Burg, Maximum Entropy Spectral Analysis, Thesis for the degree of Doctor of Philosophy, Department of Geophysics, Stanford University, California, 1975.
. G. Castro, Coeficientes de réflexion géneralisés. Extension de covariances multidimensionalles et autres applications, Thesis for the degree of Doctor of Philosophy, Universit d'Orsay, 1996.
. J. Claerbout, Fundamentals of geophysical data processing, McGraw-Hill, 1976.
. B. Choi, Multivariable maximum entropy spectrum, Journal of Multivariate Analysis, 46, 56-60, 1993.
S. Degerine, Canonical partial autocorrelation function of a multivariate time series, Annals of Statistics, 18, No. 2, 961-971, 1990.
P. Delsarte , Y. Genin and Y. Kamp , Orthogonal polynomial matrices on the unit circles, IEEE Trans. Circuits Syst., CAS-25,149-160, 1978.
P. Delsarte, Y. Genin y Y. Kamp, Schur parametrization of positive definite block-Toeplitz system, Siam J. Appl. Math, 36, 34-46,1979.
D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, Englewood Cliffs, New Jersey, 1984.
M. Dugast, G. Bouleux and E. Marcon, Representation and characterization of nonstationary processes by dilation operator and induced shape space manifolds, Entropy, 20, No. 9, 2018.
H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel, Pontryagin spaces, interpolation and extension, Operator Theory: Advances and Applications, 34, 1988.
C. Foias and A. Frazho, The commutant lifting approach to interpolation problem, Operator Theory: Advances and Applications, 44,1990.
Ya. Geronimus, Ortoghonal Polynomial, New York Consultans Bureau, 1961.
T. Kailath, A theorem of I. Schur and its impact on modern signal processing, Operator Theory: Advances and Applications, 18, 9-29, 1986.
H. Landau, Maximum entropy and the moment problem, Bull. Am. Math. Soc. 16, No. 1, 47-77, 1987.
N. Levinson, The Wiener RMS (root-mean-square) error criterion in filter design and prediction, J. Math. Phys., 25, 261-278, 1947.
M. Luz and M. Moklyachuck, Filtering problem for functionals of stationary sequences, Statistics, Optimization and Information Computing, 4, No. 1, 68-83, 2016.
P. Masani, Recent trends in multivariate prediction theory, Multivariate Anal., Academic Press, New York, 351-382, 1966.
J. Marcano, S. Infante and L. Sánchez, On a generalizaion of maximum entropy of Burg, Revista de Matemáticas: Teoría y Aplicaciones, 24, No, 1, 97-113, 2017.
J. Marcano and M. Morán, The Arov-Grossman model and Burg multivariate entropy, J. Fourier Anal. and Applications, Vol. 9, No.6, 623-647, 2003.
J. Marcano and M. Morán, The Arov-Grossman model and Burg’s entropy, in Recent Advances in Applied Probability, Ed Baeza-Yates R., Glaz J.,Gzyl H., Husler J. and Palacios J.L., 329-348, 2005.
S. Marple, Digital Spectral Analysis with applications, Prentice Hall, Englewood Cliffs, New Jersey, 1987.
M. Moklyachuck, Minimax-robust estimation problem for stationary stochastic sequences, Statistics, Optimization and Information Computing, 3, No. 4, 348-419, 2015.
M. Morf, A. Viera and T. Kailath, Covariance characterization by partial autocorrelation matrices, Annals of Statistics, 6, No. 3, 643-648, 1978.
M. Morf, A. Vieira, D. Lee and T. Kailath, Recursive multichannel maximum entropy method, IEEE Transaction on Geoscience Electronics, 16(2), 85-94, 1978.
B. Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Spaces, North-Holland, Amsterdan, 1970.
I. Schur, On power series which are bounded in the interior of the unit circle I, English translation in I. Schur Method in operator theory and signal processing, Op. theory: Adv. and Appl., Ed. I. Gohberg, 18, 31-59, 1986.
I. Schur, On power series which are bounded in the interior of the unit circle II, English translation in I. Schur Method in operator theory and signal processing, Op. theory: Adv. and Appl., Ed. I. Gohberg, 18, 61-88, 1986.
P. Whittle, On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix, Biometrika, 50, 129-134, 1963.
R. Wigging and E. Robinson, Recursive solution to the multichannel filtering problem, J. Geophys. Res., 70, 1885-1891, 1965.
H. Woerdeman, Matrix and operator extensions, Thesis for the degree of Doctor of Philosophy, Amsterdam, 1989.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).