Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds

Simone Fiori

doi:10.3934/dcdsb.2014.19.2785

Article Contents

2014, Volume 19, Issue 9: 2785-2808. Doi: 10.3934/dcdsb.2014.19.2785

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Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds

Simone Fiori^1,

1.
Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, Via Brecce Bianche, I-60131, Ancona

Received: December 2013

Revised: April 2014

Published: November 2014

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Abstract

The present research paper proposes an extension of the classical scalar Auto-Regressive Moving-Average (ARMA) model to real-valued Riemannian matrix manifolds. The resulting ARMA model on matrix manifolds is expressed as a non-linear discrete-time dynamical system in state-space form whose state evolves on the tangent bundle associated with the underlying manifold. A number of examples are discussed within the present contribution that aim at illustrating the numerical behavior of the proposed ARMA model. In order to measure the degree of temporal dependency between the state-values of the ARMA model, an extension of the classical autocorrelation function for scalar sequences is suggested on the basis of the geometrical features of the underlying real-valued matrix manifold.

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Mathematics Subject Classification: Primary: 37M10, 53C21; Secondary: 65P99.

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