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Auto-Regressive Moving Average Models on Complex-Valued Matrix Lie Groups

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Abstract

The present contribution aims at extending the classical scalar autoregressive moving average (ARMA) model to generate random (as well as deterministic) paths on complex-valued matrix Lie groups. The numerical properties of the developed ARMA model are studied by recurring to a tailored version of the Z-transform on Lie groups and to statistical indicators tailored to Lie groups, such as correlation functions on tangent bundles. The numerical behavior of the devised ARMA model is also illustrated by numerical simulations.

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Notes

  1. The fact that the variable \(z\) be scalar is inherently related to the nature of the ‘temporal variable’ \(k\): Since the time is a scalar and since the operator \(z^{-1}\) may be interpreted as a unit delay or as a time marker, it is sufficient to take \(z\) as a scalar entity.

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Acknowledgments

The author wishes to gratefully thank the anonymous reviewers and the associate editor who coordinated the review of the present paper, Prof. Robert Newcomb, for the detailed, thoughtful and constructive comments to the manuscript that contributed substantially to improve the presentation of its scientific content. The author also wishes to thank Dr. Yacine Chitour (Supélec, Gif-sur-Yvette, France) for fruitful discussions about the extension of the classical notion of system transfer function to the Lie algebra setting.

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

    Simone Fiori

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Fiori, S. Auto-Regressive Moving Average Models on Complex-Valued Matrix Lie Groups. Circuits Syst Signal Process 33, 2449–2473 (2014). https://doi.org/10.1007/s00034-014-9745-1

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