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Unscented Grid Filtering and Smoothing for Nonlinear Time Series Analysis

Authors: Nikolay Nikolaev, Evgueni Smirnov


This paper develops an unscented grid-based filter and a smoother for accurate nonlinear modeling and analysis of time series. The filter uses unscented deterministic sampling during both the time and measurement updating phases, to approximate directly the distributions of the latent state variable. A complementary grid smoother is also made to enable computing of the likelihood. This helps us to formulate an expectation maximisation algorithm for maximum likelihood estimation of the state noise and the observation noise. Empirical investigations show that the proposed unscented grid filter/smoother compares favourably to other similar filters on nonlinear estimation tasks.


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[1] I. Arasaratnam, S. Haykin and R.J. Elliott, "Discrete-Time Nonlinear Filtering Algorithms using Gauss-Hermite Quadrature", Proc. of the IEEE, vol.95, pp.953-977, 2007.
[2] T. Briegel and V. Tresp, "Robust Neural Network Regression for Offline and Online Learning", in Advances in NIPS 12, Solla,S. et. al, Eds., Cambridge, MA: The MIT Press, 2000, pp.407-413.
[3] J.F.G. de Freitas, M. Niranjan and A.G. Gee, "Hierarchical Bayesian Models for Regularization in Sequential Learning", Neural Computation, vol.12, pp.933-953, 2000.
[4] A. Doucet, N. de Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, New York: Springer-Verlag, 2001.
[5] J.C. Hull, Options, Futures and Other Derivatives, New Jersey: Prentice Hall, 2000.
[6] A. Jazwinsky, Stochastic Processes and Filtering Theory, New York: Academic Press, 1970.
[7] S.J. Julier and J.K. Uhlmann, "A New Extension of the Kalman Filter to Nonlinear Systems", Proc. SPIE Int. Soc. Opt. Eng., vol.3068, pp.182- 193, 1997.
[8] G. Kitagawa, "Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State-space Models", Journal of Computational and Graphical Statistics, vol.5, pp.1-25, 1996.
[9] H.J. Kushner and A.S. Budhiraja, "A Nonlinear Filtering Algorithm Based on an Approximation of the Conditional Distribution", IEEE Trans. on Automatic Control, vol.45, pp.580-585, 2000.
[10] N. Nikolaev and E. Smirnov, "A One-Step Unscented Particle Filter for Nonlinear Dynamical Systems", in Proc. Int. Conf. on Artificial Neural Networks, ICANN-2007, LNCS 4668, Springer-Verlag, 2007, pp.747- 756.
[11] M. Niranjan, "Sequential Tracking in Pricing Financial Options using Model Based and Neural Network Approaches", in Advances in NIPS-8, M.C.Mozer et al., Eds., Cambridge, MA: The MIT Press, 2000, pp.960- 966.
[12] M. Norgaard, N.K. Poulsen and O. Ravn, "New Developments in State Estimation for Nonlinear Systems", Automatica, vol.36, pp.1627-1638, 2000.
[13] M.L. Psiaki and M. Wada, "Derivation and Simulation Testing of a Sigma-points Smoother", Journal of Guidance, Control and Dynamics, vol.30, pp.78-86, 2007.
[14] B. Ristic, M.S. Arulampalam and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications, Artech House, 2004.
[15] S. S¨arkk¨a, "Recursive Bayesian Inference on Stochastic Differential Equations", PhD Thesis, Dept. El. and Comm. Eng., Helsinki Univ. of Technology, 2006.
[16] R.H. Shumway and D.S. Stoffer, "An Approach to Time Series Smoothing and Forecasting using the EM Algorithm", Journal of Time Series Analysis, vol.3, pp.253-264, 1982.
[17] H. Tanizaki, Nonlinear Filters: Estimation and Applications, 2nd Ed., Springer, 1996.
[18] E.A. Wan and R. van der Merwe, "The Unscented Kalman Filter", in Kalman Filtering and Neural Networks, S.Haykin Ed., New York: John Wiley and Sons, 2001, pp.221-282.
[19] R. van der Merwe, "Sigma-point Kalman Filters and Probabilistic Inference in Dynamic State-Space Models", PhD Thesis, OGI School of Science and Engineering, Oregon Health and Science University, 2004.
[20] O. Zoeter, A. Ypma and T. Heskes, "Improved Unscented Kalman Smoothing for Stock Volatility Estimation", in Machine Learning for Signal Processing, Proc. of the 14th IEEE Signal Proc. Society Workshop, A.Barros et al. Eds., New Jersey: IEEE Press, 2004, pp.143-152.