Article Contents
Article Contents

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

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

 Citation:

•  [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. [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. [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. [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. [5] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327.doi: 10.1016/0304-4076(86)90063-1. [6] P. Brockwell and R. Davis, Time Series: Theory and Methods, $2^{nd}$ Edition, Springer, 2009.doi: 10.1007/978-1-4419-0320-4. [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. [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. [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. [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. [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. [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. [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. [14] L. Eldén and H. Park, A Procrustes problem on the Stiefel manifold, Numerical Mathematics, 82 (1999), 599-619.doi: 10.1007/s002110050432. [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. [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. [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. [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. [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. [20] G. Fonseca, On the Stability of Nonlinear ARMA Mmodels, Technical report 2005/03, Facoltà di Economia, Università dell'Insubria, 2005 [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. [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. [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. [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. [25] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008.doi: 10.1137/1.9780898717778. [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. [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. [28] N. L. Lim, Clustering and Sampling Algorithms on Shape Manifolds, Technical report of the Bio-Informatics Institute (Biopolis, Singapore), 2012. [29] G. Meyer, Regression on fixed-rank positive semidefinite matrices: A Riemannian approach, Journal of Machine Learning Research, 12 (2011), 593-625. [30] F. Mezzadri, How to generate random matrices from the classical compact groups, Notices of the AMS, 54 (2007), 592-604. [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. [32] D. B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370.doi: 10.2307/2938260. [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. [34] F. L. Ramsey, Characterization of the partial autocorrelation function, The Annals of Statistics, 2 (1974), 1296-1301.doi: 10.1214/aos/1176342881. [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. [36] M. Spivak, A Comprehensive Introduction to Differential Geometry, 1, $2^{nd}$ Edition, Berkeley, CA: Publish or Perish Press, 1979. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.