Geodesic
Efficient measurement of higher-order statistics of stochastic processes
Kybernetika, Tome 54 (2018) no. 5, pp. 865-887
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This paper is devoted to analysis of block multi-indexed higher-order covariance matrices, which can be used for the least-squares estimation problem. The formulation of linear and nonlinear least squares estimation problems is proposed, showing that their statements and solutions lead to generalized `normal equations', employing covariance matrices of the underlying processes. Then, we provide a class of efficient algorithms to estimate higher-order statistics (generalized multi-indexed covariance matrices), which are necessary taking in mind practical aspects of the nonlinear treatment of the least-squares estimation problem. The algorithms are examined for different higher-order and non-Gaussian processes (time-series) and an impact of signal properties on covariance matrices is analysed.
This paper is devoted to analysis of block multi-indexed higher-order covariance matrices, which can be used for the least-squares estimation problem. The formulation of linear and nonlinear least squares estimation problems is proposed, showing that their statements and solutions lead to generalized `normal equations', employing covariance matrices of the underlying processes. Then, we provide a class of efficient algorithms to estimate higher-order statistics (generalized multi-indexed covariance matrices), which are necessary taking in mind practical aspects of the nonlinear treatment of the least-squares estimation problem. The algorithms are examined for different higher-order and non-Gaussian processes (time-series) and an impact of signal properties on covariance matrices is analysed.
DOI : 10.14736/kyb-2018-5-0865
Classification : 15B05, 15B51, 60G10, 60G15, 93E24
Keywords: covariance matrix; higher-order statistics; adaptive; nonlinear 
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Magiera, Wladyslaw; Libal, Urszula; Wielgus, Agnieszka. Efficient measurement of higher-order statistics of stochastic processes. Kybernetika, Tome 54 (2018) no. 5, pp. 865-887. doi: 10.14736/kyb-2018-5-0865

[1] Dewilde, P.: Stochastic modelling with orthogonal filters. In: Outils et modeles mathematiques pour l'automatique, l'analyse de systemes et le traitement du signal, CNRS (ed.), Paris 1982, pp. 331-398. | MR

[2] Lee, D. T. L., Morf, M., Friedlander, B.: Recursive least-squares ladder estimation algorithms. IEEE Trans. Circuit Systems CAS 28 (1981), 467-481. | DOI | MR

[3] Jurečková, J.: Regression quantiles and trimmed least squares estimator under a general design. Kybernetika 20 (1984), 5, 345-357. | MR | Zbl

[4] Levinson, N.: The Wiener RMS error criterion in filter design and prediction. J. Math. Physics 25 (1947), 261-278. | DOI | MR

[5] Mandl, P., Duncan, T. E., Pasik-Duncan, B.: On the consistency of a least squares identification procedure. Kybernetika 24 (1988), 5, 340-346. | MR

[6] Mendel, J. M.: Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. Proc. IEEE 79 (1991), 3, 278-305. | DOI

[7] Pázman, A.: Probability distribution of the multivariate nonlinear least squares estimates. Kybernetika 20 (1984), 3, 209-230. | MR

[8] Pronzato, L., Pázman, A.: Second-order approximation of the entropy in nonlinear least-squares estimation. Kybernetika 30 (1994), 2, 187-198. | MR

[9] Schur, I.: Methods in 0perator Theory and Signal Processing. Operator Theory: Advances and Applications 18, Springer-Verlag 1086. | DOI

[10] Stellakis, H. M., Manolakos, E. M.: Adaptive computation of higher order moments and its systolic realization. Int. J. Adaptive Control Signal Process. 10 (1996), 283-302. | DOI

[11] Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, 1958. | MR

[12] Zarzycki, J.: Multidimensional nonlinear Schur parametrization of non-gaussian stochastic signals - Part one: Statement of the problem. MDSSP J. 15 (2004), 3, 217-241. | DOI | MR

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