Article information

2017 , Volume 22, ¹ 3, p.45-60

Kulikova M.V., Tsyganova Y.V.

Numerically stable Kalman filter implementations for estimating linear pairwise Markov models in the presence of Gaussian noise

This paper studies numerical methods of Kalman filtering for vector state estimation of linear Gaussian pairwise models. The pairwise Markov model generalizes the hidden Markov model and it attracted an increasing attention in recent years. For instance, the use of the pairwise Markov models instead of hidden Markov models in segmentation problems allows for dividing the error ratio by two. This paper explores such effective state estimation methods as the square-root pairwise Kalman filtering algorithms, including their array implementations. These filtering methods and their performance are studied in detail. The new UD factorization-based pairwise Kalman filtering approach has been developed. Other filtering algorithms applicable to the linear pairwise Markov model are discussed and numerically compared to the newly developed method on two examples.

[full text]
Keywords: pairwise Markov model, linear stochastic system, optimal estimation of the state vector, pairwise Kalman filter

Author(s):
Kulikova Maria Vyacheslavovna
Office: CEMAT Instituto Superior Tecnico Universidade de Lisboa
Address: Portugal, Lissabone, 1049-001 Lisboa

Tsyganova Yulia Vladimirovna
Dr. , Associate Professor
Position: Professor
Office: Ulyanovsk State University
Address: 432017, Russia, Ulyanovsk, Leo Tolstoy str., 42
E-mail: tsyganovajv@gmail.com

References:
[1] Stratonovich, R.L. Uslovnye markovskie protsessy i ikh primenenie k teorii optimal'nogo upravleniya [MGU Conditional Markov processes and their application to the theory of optimal control]. Moscow: Izdatel'stvo MGU; 1966: 319. (In Russ.)
[2] Kalman, R.E. A new approach to linear filtering and prediction problems. Journal of Fluids Engineering. 1960; (82):35–45.
[3] Kalman, R.E., Bucy, R.S. New results in linear filtering and prediction theory. Journal of Basic Engineering. 1961; 83(1):95–108.
[4] Lipster, R.S., Shiryaev, A.N. Statistics of condi- tionally gaussian random sequences. Proceedings of the Sixth BerkeleySymposium on Mathematical Statistics and Probability. California: University of California Press; 1972; (2):389–422.
[5] Liptser, R.Sh., Shiryaev, A.N. Statistika sluchaynykh protsessov (nelineynaya fil'tratsiya i smezhnye voprosy) [Statistics of random processes (Nonlinear filtering and related problems)]. Moscow: Nauka; 1974: 696 . (In Russ.)
[6] Liptser, R.Sh. Equations of near-optimal Kalman filter with a singular matrix of noise covariations in observations. Automation and Remote Control. 1974; 35(1):29-35.
[7] Ershov, A.A., Liptser, R.Sh. A robust Kalman filter in discrete time. Automation and Remote Control. 1978; 39(3):359-367.
[8] Mil’shtein, G.N., P’yanzin, S.A. Digital simulation of Kalman-Bucy filter and an optimal filter with discrete input data, Automation and Remote Control. 1985, Vol. 46, P. 50-58.
[9] Derrode, S., Pieczynski, W. Signal and image segmentation using pairwise Markov chains. IEEE Transactions on Signal Processing. 2004; 52(9):2477–2489.
[10] Ait-el-Fquih, B., Desbouvries, F. Unsupervised signal restoration in partially observed Markov chains. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. France: Toulouse. 2006; (3):13–16.
[11] Derrode, S., Pieczynski, W. Exact fast computation of optimal filter in Gaussian switching linear systems. IEEE Signal Processing Letters. 2013; 20(7):701–704.
[12] Lanchantin, P., Lapuyade-Lahorgue, J., Pieczynski, W. Unsupervised segmentation of randomly switching data hidden with non-Gaussian correlated noise. Signal Processing. 2011; 91(2):163–175.
[13] Boudaren, M.E.Y., An, L., Pieczynski, W. Dempster–Shafer fusion of evidential pairwise Markov fields. International Journal of Approximate Reasoning. 2016; (74):13–29.
[14] Boudaren, M.E.Y., Pieczynski, W. Dempster-Shafer fusion of evidential pairwise Markov Chains. IEEE Transactions on Fuzzy Systems. 2016; 24(6):1598—1610.
[15] Pieczynski, W., Desbouvries, F. Kalman filtering using pairwise Gaussian models. Proceedings – ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing, Hong-Kong. 2003; (V. VI): 57–60.
[16] Ait El Fquih, B., Desbouvries, F. Unsupervised signal restoration in partially observed Markov chains. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Toulouse, France. 2006; (3). P. III. DOI: 10.1109/ICASSP.2006.1660578
[17] Verhaegen, M., Van Dooren, P. Numerical aspects of different Kalman filter Implementations. IEEE Transactions on Automatic Control. 1986; AC-31(10):907–917.
[18] N´emesin, V., Derrode, S. Robust blind pairwise Kalman algorithms using QR Decompositions. IEEE Transactions on Signal Processing.
2013; 61(1):5–9.
[19] N´emesin, V., Derrode, S. Robust partial-learning in linear Gaussian systems. IEEE Transactions on Automatic Control. 2015; 60(9):2518–2523.
[20] Dempster, A., Laird, N., Rubin, D. Maximum likelihood from incomplete data via the ÅÌ Algorithm. Journal of the Royal Statistical Society. Series B (Methodological). 1977; 39(1):1–38.
[21] Kulikova, M.V. Gradient-based parameter estimation in pairwise linear Gaussian system. IEEE Transactions on Automatic Control (in press). DOI: 10.1109/TAC.2016.2579745.
[22] Kaminski, P. G., Bryson, A. E., Schmidt, S.F. Discrete square-root filtering: a survey of current techniques. IEEE Transactions on Automatic Control. 1971; AC-16(6):727–735.
[23] N´emesin, V., Derrode, S. Inferring segmental pairwise Kalman filter with application to pupil tracking. Traitement et Analyse de l’Information Methodes et Applications, Hammamet (Tunisia), May 13-18, 2013.
[24] Bierman, G.J. Factorization methods for discrete sequential estimation. New York: Academic Press; 1977: 241.
[25] Potter, J. E., Stern, R. G. Statistical filtering of space navigation measurements. Proc. 1963 AIAA Guidance and Control Conf. New York: AIAA; 1963: 13.
[26] Dyer, P., McReynolds, S. Extension of square-root filtering to include process noise. Journal of Optimization Theory and Applications. 1969; (3):444-459.
[27] Golub, G.H., Van Loan, C.F. Matrix Computations. Baltimore: Johns Hopkins University Press; 1996: 694.
[28] Kailath, T., Sayed, A. H., Hassibi, B. Linear estimation. New Jersey: Prentice Hall; 2000: 854.
[29] Grewal, M.S., Andrews, A.P. Kalman filtering: theory and practice. New Jersey: PrenticeHall; 2001: 401.
[30] Park, P., Kailath, T. New square-root algorithms for Kalman filtering. IEEE Transactions on Automatic Control. 1995; 40(5):895–899.
[31] Bakhvalov, N.S., Zhidkov, N.P., Kobel’kov, G.M. Chislennye metody [Numerical methods]. Moscow: Nauka; 1987: 600. (In Russ.)
[32] Samarskiy, A.A., Gulin, A.V. Chislennye metody [Numerical methods]. Moscow: Nauka; 1989: 432. (In Russ.)
[33] Bjorck, A. Solving least squares problems by orthogonalization. BIT. 1967; (7):1–21.
[34] Jover, J.M., Kailath, T. A parallel architecture for kalman filter measurement update and parameter estimation. Automatica. 1986; 22(1):43–57.

Bibliography link:
Kulikova M.V., Tsyganova Y.V. Numerically stable Kalman filter implementations for estimating linear pairwise Markov models in the presence of Gaussian noise // Computational technologies. 2017. V. 22. ¹ 3. P. 45-60
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