# American Institute of Mathematical Sciences

November  2014, 19(9): 2785-2808. doi: 10.3934/dcdsb.2014.19.2785

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

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

Received  December 2013 Revised  April 2014 Published  September 2014

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.
Citation: Simone Fiori. Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2785-2808. doi: 10.3934/dcdsb.2014.19.2785
##### References:
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Ray, An investigation of lag identification tools for vector nonlinear time series, Communications in Statistics - Theory and Methods, 29 (2000), 1677-1701. doi: 10.1080/03610920008832573.  Google Scholar [25] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778.  Google Scholar [26] T. Kaneko, S. Fiori and T. Tanaka, Empirical arithmetic averaging over the compact Stiefel manifold, IEEE Transactions on Signal Processing, 61 (2013), 883-894. doi: 10.1109/TSP.2012.2226167.  Google Scholar [27] R. Lenz, R. Mochizuki and J. Chao, Iwasawa decomposition and computational Riemannian geometry, in Proceedings of the $20^{th}$ International Conference on Pattern Recognition (ICPR), 2010, 4472-4475. doi: 10.1109/ICPR.2010.1086.  Google Scholar [28] N. L. Lim, Clustering and Sampling Algorithms on Shape Manifolds, Technical report of the Bio-Informatics Institute (Biopolis, Singapore), 2012. Google Scholar [29] G. Meyer, Regression on fixed-rank positive semidefinite matrices: A Riemannian approach, Journal of Machine Learning Research, 12 (2011), 593-625.  Google Scholar [30] F. Mezzadri, How to generate random matrices from the classical compact groups, Notices of the AMS, 54 (2007), 592-604.  Google Scholar [31] J. Navarro-Moreno, ARMA prediction of widely linear systems by using the innovations algorithm, IEEE Transactions on Signal Processing, 56 (2008), 3061-3068. doi: 10.1109/TSP.2008.919396.  Google Scholar [32] D. B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370. doi: 10.2307/2938260.  Google Scholar [33] P. Pajunen and M. Girolami, Implementing decisions in binary decision trees using independent component analysis, in Proceedings of the International Workshop on Independent Component Analysis and Blind Signal Separation (Helsinki, Finland, 2000), 2000, 477-481. Google Scholar [34] F. L. Ramsey, Characterization of the partial autocorrelation function, The Annals of Statistics, 2 (1974), 1296-1301. doi: 10.1214/aos/1176342881.  Google Scholar [35] H. Shimazaki, S.-i. Amari, E. N. Brown and S. Grün, State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data, PLoS Computional Biology, 8 (2012), e1002385. doi: 10.1371/journal.pcbi.1002385.  Google Scholar [36] M. Spivak, A Comprehensive Introduction to Differential Geometry, 1, $2^{nd}$ Edition, Berkeley, CA: Publish or Perish Press, 1979. Google Scholar [37] G. C. Tiao and R. S. Tsay, Model specification in multivariate time series, Journal of Royal Statistical Society, Series B (Methodological), 51 (1989), 157-213.  Google Scholar [38] H. Tidefelt and T. B. Schön, Robust point-mass filters on manifold, in Proceedings of the $15^{th}$ IFAC Symposium on System Identification (SYSID, Saint-Malo, France, 2009), 2009, 1-6. Google Scholar [39] N. T. Trendafilov and R. A. Lippert, The multimode Procrustes problem, Linear Algebra and iIts Applications, 349 (2002), 245-264. doi: 10.1016/S0024-3795(02)00253-7.  Google Scholar [40] P. Turaga, A. Veeraraghavan and R. Chellappa, Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008, Anchorage (AK), USA, 2008), 2008, 1-8. doi: 10.1109/CVPR.2008.4587733.  Google Scholar [41] P. A. Valdes-Sosa, Spatio-temporal autoregressive models defined over brain manifolds, Neuroinformatics, 2 (2004), 239-250. doi: 10.1385/NI:2:2:239.  Google Scholar [42] J. F. Vasconcelos, G. Elkaim, C. Silvestre, P. Oliveira and B. Cardeira, Geometric approach to strapdown magnetometer calibration in sensor frame, IEEE Transactions on Aerospace and Electronic Systems, 47 (2011), 1293-1306. doi: 10.1109/TAES.2011.5751259.  Google Scholar [43] A. Veeraraghavan, A. K. Roy-Chowdhury and R. Chellappa, Matching shape sequences in video with applications in human movement analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 1896-1909. doi: 10.1109/TPAMI.2005.246.  Google Scholar [44] N. Wessel, H. Malberg, R. Bauernschmitt, A. Schirdewan and J. Kurths, Nonlinear additive autoregressive model-based analysis of short-term heart rate variability, Medical and Biological Engineering and Computing, 44 (2006), 321-330. doi: 10.1007/s11517-006-0038-0.  Google Scholar [45] A. Yershova and S. M. LaValle, Deterministic sampling methods for spheres and $SO(3)$, in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA 2004, New Orleans (LA, USA), 2004), 4 (2004), 3974-3980. doi: 10.1007/s004539910020.  Google Scholar [46] F. Yger, M. Berar, G. Gasso and A. Rakotomamonjy, Oblique principal subspace tracking on manifold, in Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, Kyoto (Japan)), 2012, 2429-2432. doi: 10.1109/ICASSP.2012.6288406.  Google Scholar [47] J. Yoo and S. Choi, Orthogonal nonnegative matrix factorization: multiplicative updates on Stiefel manifolds, in Proceedings of the Intelligent Data Engineering and Automated Learning (IDEAL 2008), Springer Berlin/Heidelberg, 2008, 140-147. doi: 10.1007/978-3-540-88906-9_18.  Google Scholar [48] Y. Zhu, W. Mio and X. Liu, Optimal dimension reduction for image retrieval with correlation metrics, in Proceedings of the International Conference on Neural Networks (IJCNN 2009, Atlanta (GA, USA)), 2009, 3565-3570. doi: 10.1109/IJCNN.2009.5179020.  Google Scholar

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##### References:
 [1] S.-i. Amari, Differential geometry of a parametric family of invertible linear systems - Riemannian metric, dual affine connections, and divergence, Mathematical Systems Theory, 20 (1987), 53-82. doi: 10.1007/BF01692059.  Google Scholar [2] S.-i. Amari, Natural gradient learning for over- and under-complete bases in ICA, Neural Computation, 11 (1989), 1875-1883. doi: 10.1162/089976699300015990.  Google Scholar [3] T. W. Anderson, I. Olkin and L. G. Underhill, Generation of random orthogonal matrices, SIAM Journal on Scientific and Statistical Computing, 8 (1987), 625-629. doi: 10.1137/0908055.  Google Scholar [4] Y. K. Baik, J. Kwon, H. S. Lee and K. M. Lee, Geometric particle swarm optimization for robust visual ego-motion estimation via particle filtering, Image and Vision Computing, 31 (2013), 565-579. doi: 10.1016/j.imavis.2013.04.004.  Google Scholar [5] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327. doi: 10.1016/0304-4076(86)90063-1.  Google Scholar [6] P. Brockwell and R. Davis, Time Series: Theory and Methods, $2^{nd}$ Edition, Springer, 2009. doi: 10.1007/978-1-4419-0320-4.  Google Scholar [7] E. Celledoni and S. Fiori, Neural learning by geometric integration of reduced 'rigid-body' equations, Journal of Computational and Applied Mathematics, (JCAM), 172 (2004), 247-269. doi: 10.1016/j.cam.2004.02.007.  Google Scholar [8] Y. Chen and J. E. McInroy, Estimation of symmetric positive-definite matrices from imperfect measurements, IEEE Transactions on Automatic Control, 47 (2002), 1721-1725. doi: 10.1109/TAC.2002.803545.  Google Scholar [9] G. S. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods, Birkhäuser, 2009. doi: 10.1007/978-0-8176-4803-9.  Google Scholar [10] P. Chossat and O. Faugeras, Hyperbolic planforms in relation to visual edges and textures perception, PLoS Computational Biology, 5 (2009), e1000625. doi: 10.1371/journal.pcbi.1000625.  Google Scholar [11] P. I. Davies and N. J. Higham, Numerically stable generation of correlation matrices and their factors, BIT, 40 (2000), 640-651. doi: 10.1023/A:1022384216930.  Google Scholar [12] P. Diaconis and M. Shahshahamni, The subgroup algorithm for generating uniform random variables, Probability in Engineering and Informational Sciences, 1 (1987), 15-32. doi: 10.1017/S0269964800000255.  Google Scholar [13] A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM Journal on Matrix Analysis Applications, 20 (1998), 303-353. doi: 10.1137/S0895479895290954.  Google Scholar [14] L. Eldén and H. Park, A Procrustes problem on the Stiefel manifold, Numerical Mathematics, 82 (1999), 599-619. doi: 10.1007/s002110050432.  Google Scholar [15] S. Fiori, A fast fixed-point neural blind deconvolution algorithm, IEEE Transactions on Neural Networks, 15 (2004), 455-459. doi: 10.1109/TNN.2004.824258.  Google Scholar [16] S. Fiori, Blind adaptation of stable discrete-time IIR filters in state-space form, IEEE Transactions on Signal Processing, 54 (2006), 2596-2605. doi: 10.1109/TSP.2006.874807.  Google Scholar [17] S. Fiori, Geodesic-based and projection-based neural blind deconvolution algorithms, Signal Processing, 88, (2008), 521-538. doi: 10.1016/j.sigpro.2007.08.014.  Google Scholar [18] S. Fiori, Learning the Fréchet mean over the manifold of symmetric positive-definite matrices, Cognitive Computation, 1 (2009), 279-291. doi: 10.1007/s12559-009-9026-7.  Google Scholar [19] S. Fiori, Blind deconvolution by a Newton method on the non-unitary hypersphere, International Journal of Adaptive Control and Signal Processing, 27 (2013), 488-518. doi: 10.1002/acs.2324.  Google Scholar [20] G. Fonseca, On the Stability of Nonlinear ARMA Mmodels, Technical report 2005/03, Facoltà di Economia, Università dell'Insubria, 2005 Google Scholar [21] A. Genz, Methods for Generating Random Orthogonal Matrices, in Monte Carlo and Quasi-Monte Carlo Methods 1998, (Eds. H. Niederreiter and J. Spanier), Springer-Verlag, Berlin, 1999, 199-213.  Google Scholar [22] W. Gonzalez-Manteiga, G. Henry and D. Rodriguez, Partly linear models on Riemannian manifolds, Journal of Applied Statistics, 39 (2012), 1797-1809. doi: 10.1080/02664763.2012.683169.  Google Scholar [23] R. Harman and V. Lacko, On decompositional algorithms for uniform sampling from $n$-spheres and $n$-balls, Journal of Multivariate Analysis, 101 (2010), 2297-2304. doi: 10.1016/j.jmva.2010.06.002.  Google Scholar [24] J. L. Harvill and B. K. Ray, An investigation of lag identification tools for vector nonlinear time series, Communications in Statistics - Theory and Methods, 29 (2000), 1677-1701. doi: 10.1080/03610920008832573.  Google Scholar [25] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778.  Google Scholar [26] T. Kaneko, S. Fiori and T. Tanaka, Empirical arithmetic averaging over the compact Stiefel manifold, IEEE Transactions on Signal Processing, 61 (2013), 883-894. doi: 10.1109/TSP.2012.2226167.  Google Scholar [27] R. Lenz, R. Mochizuki and J. Chao, Iwasawa decomposition and computational Riemannian geometry, in Proceedings of the $20^{th}$ International Conference on Pattern Recognition (ICPR), 2010, 4472-4475. doi: 10.1109/ICPR.2010.1086.  Google Scholar [28] N. L. Lim, Clustering and Sampling Algorithms on Shape Manifolds, Technical report of the Bio-Informatics Institute (Biopolis, Singapore), 2012. Google Scholar [29] G. Meyer, Regression on fixed-rank positive semidefinite matrices: A Riemannian approach, Journal of Machine Learning Research, 12 (2011), 593-625.  Google Scholar [30] F. Mezzadri, How to generate random matrices from the classical compact groups, Notices of the AMS, 54 (2007), 592-604.  Google Scholar [31] J. Navarro-Moreno, ARMA prediction of widely linear systems by using the innovations algorithm, IEEE Transactions on Signal Processing, 56 (2008), 3061-3068. doi: 10.1109/TSP.2008.919396.  Google Scholar [32] D. B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370. doi: 10.2307/2938260.  Google Scholar [33] P. Pajunen and M. Girolami, Implementing decisions in binary decision trees using independent component analysis, in Proceedings of the International Workshop on Independent Component Analysis and Blind Signal Separation (Helsinki, Finland, 2000), 2000, 477-481. Google Scholar [34] F. L. Ramsey, Characterization of the partial autocorrelation function, The Annals of Statistics, 2 (1974), 1296-1301. doi: 10.1214/aos/1176342881.  Google Scholar [35] H. Shimazaki, S.-i. Amari, E. N. Brown and S. Grün, State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data, PLoS Computional Biology, 8 (2012), e1002385. doi: 10.1371/journal.pcbi.1002385.  Google Scholar [36] M. Spivak, A Comprehensive Introduction to Differential Geometry, 1, $2^{nd}$ Edition, Berkeley, CA: Publish or Perish Press, 1979. Google Scholar [37] G. C. Tiao and R. S. Tsay, Model specification in multivariate time series, Journal of Royal Statistical Society, Series B (Methodological), 51 (1989), 157-213.  Google Scholar [38] H. Tidefelt and T. B. Schön, Robust point-mass filters on manifold, in Proceedings of the $15^{th}$ IFAC Symposium on System Identification (SYSID, Saint-Malo, France, 2009), 2009, 1-6. Google Scholar [39] N. T. Trendafilov and R. A. Lippert, The multimode Procrustes problem, Linear Algebra and iIts Applications, 349 (2002), 245-264. doi: 10.1016/S0024-3795(02)00253-7.  Google Scholar [40] P. Turaga, A. Veeraraghavan and R. Chellappa, Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008, Anchorage (AK), USA, 2008), 2008, 1-8. doi: 10.1109/CVPR.2008.4587733.  Google Scholar [41] P. A. Valdes-Sosa, Spatio-temporal autoregressive models defined over brain manifolds, Neuroinformatics, 2 (2004), 239-250. doi: 10.1385/NI:2:2:239.  Google Scholar [42] J. F. Vasconcelos, G. Elkaim, C. Silvestre, P. Oliveira and B. Cardeira, Geometric approach to strapdown magnetometer calibration in sensor frame, IEEE Transactions on Aerospace and Electronic Systems, 47 (2011), 1293-1306. doi: 10.1109/TAES.2011.5751259.  Google Scholar [43] A. Veeraraghavan, A. K. Roy-Chowdhury and R. Chellappa, Matching shape sequences in video with applications in human movement analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 1896-1909. doi: 10.1109/TPAMI.2005.246.  Google Scholar [44] N. Wessel, H. Malberg, R. Bauernschmitt, A. Schirdewan and J. Kurths, Nonlinear additive autoregressive model-based analysis of short-term heart rate variability, Medical and Biological Engineering and Computing, 44 (2006), 321-330. doi: 10.1007/s11517-006-0038-0.  Google Scholar [45] A. Yershova and S. M. LaValle, Deterministic sampling methods for spheres and $SO(3)$, in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA 2004, New Orleans (LA, USA), 2004), 4 (2004), 3974-3980. doi: 10.1007/s004539910020.  Google Scholar [46] F. Yger, M. Berar, G. Gasso and A. Rakotomamonjy, Oblique principal subspace tracking on manifold, in Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, Kyoto (Japan)), 2012, 2429-2432. doi: 10.1109/ICASSP.2012.6288406.  Google Scholar [47] J. Yoo and S. Choi, Orthogonal nonnegative matrix factorization: multiplicative updates on Stiefel manifolds, in Proceedings of the Intelligent Data Engineering and Automated Learning (IDEAL 2008), Springer Berlin/Heidelberg, 2008, 140-147. doi: 10.1007/978-3-540-88906-9_18.  Google Scholar [48] Y. Zhu, W. Mio and X. Liu, Optimal dimension reduction for image retrieval with correlation metrics, in Proceedings of the International Conference on Neural Networks (IJCNN 2009, Atlanta (GA, USA)), 2009, 3565-3570. doi: 10.1109/IJCNN.2009.5179020.  Google Scholar
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