Search of weights in the problem of finite-rank signal estimation in presence of noise
DOI:
https://doi.org/10.21638/spbu01.2021.304Abstract
The problem of weighted finite-rank time-series approximation is considered for signal estimation in “signal plus noise” model, where the inverse covariance matrix of noise is (2p+1)-diagonal. Finding of weights, which improve the estimation accuracy, is examined. An effective method for the numerical search of the weights is constructed and proved. Numerical simulations are performed to study the improvement of the estimation accuracy for several noise models.Keywords:
time series, Cadzow’s iterative method, time series of finite rank, weighted lowrank approximation, Kullback-Leibler divergence, optimization methods
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References
Литература
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2. Golyandina N., Zhigljavsky A. Singular spectrum analysis for time series. 2nd ed. Berlin, Heidelberg, Springer (2020).
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6. Adachi K. Matrix-based introduction to multivariate data analysis. Springer (2016).
7. Allen G. I., Grosenick L., Taylor J. A generalized least-square matrix decomposition. Journal of the American Statistical Association 109 (505), 145–159 (2014).
8. Golyandina N., Zhigljavsky A. Blind deconvolution of covariance matrix inverses for autoregressive processes. Linear Algebra and its Applications 593, 188–211 (2020).
9. Zvonarev N., Golyandina N. Iterative algorithms for weighted and unweighted finite-rank timeseries approximations. Statistics and Its Interface 10 (1), 5–18 (2017).
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11. Wiesel A., Globerson A. Covariance estimation in time varying ARMA processes. IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM) / IEEE, 357–360 (2012).
12. Verbyla A. A note on the inverse covariance matrix of the autoregressive process. Australian & New Zealand Journal of Statistics 27 (2), 221–224 (1985).
13. Cadzow J.A. Signal enhancement: a composite property mapping algorithm. IEEE Trans. Acoust. 36 (1), 49–62 (1988).
14. Byrd R.H., Lu P., Nocedal J., Zhu C. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16 (5), 1190–1208 (1995).
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References
1. Golyandina N., Nekrutkin V., Zhigljavsky A. Analysis of Time Series Structure: SSA and Related Techniques. Chapman & Hall/CRC (2001).
2. Golyandina N., Zhigljavsky A. Singular spectrum analysis for time series. 2nd ed. Berlin, Heidelberg, Springer (2020).
3. Van Der Veen A.-J., Deprettere E.F., Swindlehurst A.L. Subspace-based signal analysis using singular value decomposition. Proceedings of the IEEE 81 (9), 1277–1308 (1993).
4. Gonen E., Mendel J.M. Subspace-based direction finding methods. In: Madisetti V.K., Williams D.B. (eds.) The Digital Signal Processing Handbook, vol. 62. Boca Raton, CRC Press LLC (1999).
5. Heinig G., Rost K. Algebraic Methods for Toeplitz-like Matrices and Operators. In: Operator Theory: Advances and Applications. Birkh¨auser, Verlag (1985).
6. Adachi K. Matrix-based introduction to multivariate data analysis. Springer (2016).
7. Allen G. I., Grosenick L., Taylor J. A generalized least-square matrix decomposition. Journal of the American Statistical Association 109 (505), 145–159 (2014).
8. Golyandina N., Zhigljavsky A. Blind deconvolution of covariance matrix inverses for autoregressive processes. Linear Algebra and its Applications 593, 188–211 (2020).
9. Zvonarev N., Golyandina N. Iterative algorithms for weighted and unweighted finite-rank timeseries approximations. Statistics and Its Interface 10 (1), 5–18 (2017).
10. Zvonarev N.K. Search of weights in problem of weighted finite-rank time-series approximation. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy 3 (61), iss. 4, 570–581 (2016). https://doi.org/10.21638/11701/spbu01.2016.406 (In Russian)
11. Wiesel A., Globerson A. Covariance estimation in time varying ARMA processes. IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM) / IEEE, 357–360 (2012).
12. Verbyla A. A note on the inverse covariance matrix of the autoregressive process. Australian & New Zealand Journal of Statistics 27 (2), 221–224 (1985).
13. Cadzow J.A. Signal enhancement: a composite property mapping algorithm. IEEE Trans. Acoust. 36 (1), 49–62 (1988).
14. Byrd R.H., Lu P., Nocedal J., Zhu C. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16 (5), 1190–1208 (1995).
15. Kullback S., Leibler R.A. On information and sufficiency. The annals of mathematical statistics 22 (1), 79–86 (1951).
16. Duchi J. Derivations for linear algebra and optimization (2007). Available at: https://web.stanford.edu/jduchi/projects/general_notes.pdf (accessed: April 1, 2021).
17. Golub G.H., Van Loan C.F. Matrix computations. 3rd ed. Baltimore, MD, USA, Johns Hopkins University Press (1996).
18. Gillard J. Cadzow’s basic algorithm, alternating projections and singular spectrum analysis. Stat. Interface 3 (3), 335–343 (2010).
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Published
2021-09-26
How to Cite
Zvonarev, N. K. (2021). Search of weights in the problem of finite-rank signal estimation in presence of noise. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 417–429. https://doi.org/10.21638/spbu01.2021.304
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
