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covariance matrix; higher-order statistics; adaptive; nonlinear
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.
[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 0782526
[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 10.1109/tcs.1981.1085020 | MR 0629997
[3] Jurečková, J.: Regression quantiles and trimmed least squares estimator under a general design. Kybernetika 20 (1984), 5, 345-357. MR 0776325 | Zbl 0561.62027
[4] Levinson, N.: The Wiener RMS error criterion in filter design and prediction. J. Math. Physics 25 (1947), 261-278. DOI 10.1002/sapm1946251261 | MR 0019257
[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 0970211
[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 10.1109/5.75086
[7] Pázman, A.: Probability distribution of the multivariate nonlinear least squares estimates. Kybernetika 20 (1984), 3, 209-230. MR 0763647
[8] Pronzato, L., Pázman, A.: Second-order approximation of the entropy in nonlinear least-squares estimation. Kybernetika 30 (1994), 2, 187-198. MR 1283494
[9] Schur, I.: Methods in 0perator Theory and Signal Processing. Operator Theory: Advances and Applications 18, Springer-Verlag 1086. DOI 10.1007/978-3-0348-5483-2
[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 10.1002/(sici)1099-1115(199603)10:2/3<283::aid-acs351>;2-2
[11] Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, 1958. MR 0100912
[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 10.1023/b:mult.0000028007.05748.48 | MR 2075150
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