Optimum nonlinear filtering

IEEE Transactions on Signal Processing

Volumn 45, Issue 11, 1997, Pages 2774-2786

  Haykin, Simon ^a   Yee, Paul ^a   Derbez, Eric ^a  

^a MCMASTER UNIVERSITY   (Canada)

Author keywords

Kaiman filter; Nonlinear filtering; Optimum filtering; Radial basis function neural networks; Stochastic calculus

Indexed keywords

DIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; NEURAL NETWORKS; RANDOM PROCESSES;

RADIAL BASIS FUNCTION (RBF) NETWORKS; STOCHASTIC CALCULUS;

KALMAN FILTERING;

EID: 0031268933     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/78.650104     Document Type: Article

References (55)

1. D. Allinger and S. K. Mitter, "New results in innovations problem for nonlinear filtering," Stochastics, vol. 5, 1981; "Differential equations I. Convergence rate of the distribution function," Probab. Theory Related Fields, vol. 104, pp. 43-60, 1996; "The Zakai equation using multiple Wiener and Stratonovich integral expansions," Stochastics Rep., vol. 56, pp. 271-315, 1996.
2. A. Bensoussan, "On some approximation techniques in nonlinear filtering," Stochastic Differential Systems, Stochastic Control Theory and Applications. New York-Berlin: Springer, 1988, vol. 10, pp. 17-31.
3. F. Campillo, F. Cefou, and F. Le Gland, "Particle and cell approximations for nonlinear filtering," Int. Pub. INRIA, Tech. Rep. 2567.
4. Y. Chen, "Orthogonal least-squares selection for radial basis functions, IEEE Trans. Neural Networks, 1991.
5. M. Fujisaki, G. Kallianpur, and H. Kunita, "Stochastic differential equations for the nonlinear filtering problem," Osaka J. Math., vol. 9, pp. 19-40, 1972.
6. S. Geman, E. Bienenstock, and R. Doursat, "Neural networks and the bias/variance dilemma," Neural Comput., vol. 4, pp. 1-58, 1992.
7. I. V. Girsanov, "On the transformation of a certain class of stochastic processes by absolutely continuous substitution of measures," Theory Prob., vol. 5, pp. 285-301, 1960.
8. U. G. Haussmann and E. Pardoux, "A conditionally almost linear filtering problem with non-Gaussian initial condition," Stochastics, vol. 23, no. 2, pp. 241-275, 1988.
9. S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
10. S. Haykin, Neural Networks: A Comprehensive Foundation. New York: Macmillan, 1994.
11. S. Haykin and L. Li, "Nonlinear adaptive prediction of nonstationary signals," IEEE Trans. Signal Processing, vol. 45, pp. 526-535, 1995.
12. S. Haykin and P. Yee, "Nonlinear state estimation using RBF networks," IEEE GLOBECOM, McLean, VA, 1996.
13. T. Kailath, "Innovations approach to least squares estimation, Part I: Filtering in additive white noise," IEEE Trans. Automat. Contr., vol. AC-13, pp. 646-654, 1968.
14. T. Kailath and P. Frost, "Innovations approach to least squares estimation, Part III: Nonlinear estimation in white Gaussian noise," IEEE Trans. Automat. Contr., vol. AC-16, pp. 217-226, 1971.
15. R. E. Kaiman, "A new approach to linear filtering and prediction problems," Trans. Amer. Soc. Mech. Eng., J. Basic Eng., vol. 82, pp. 355, 1960.
16. A. Kirsch, "An introduction to the mathematical theory of inverse problems," Appl. Math. Sei, vol. 120, 1996.
17. A. N. Kolmogorov, "Sur l'interpolation et extrapolation des suite stationaries," C. R. Acad. Sei, vol. 208, pp. 2043-2045, 1939.
18. V. Krishnan, Nonlinear Filtering and Smoothing: An Introduction to Martingales, Stochastic Integrals and Estimation. New York: Wiley, 1984.
19. H. J. Kushner, "Dynamical equations for optimal nonlinear filtering," J. Differential Equations, vol. 3, pp. 179-190, 1967.
20. _, "Probability methods for approximations in stochastic control and for elliptic equations." New York: Academic, 1977.
21. H. J. Kushner and P. Dupuis, "Numerical methods for stochastic control problems in continuous time," Applications of Mathematics, vol. 24. New York: Springer, 1992.
22. J. T. Lo, "Optimal estimation for the satellite attitude using star tracker measurements," Automatica, vol. 22, pp. 477-482, 1986.
23. J. T. Lo, "Neural network approach to optimal filtering," Tech. Rep. RLTR-94-197, Rome Lab., Air Force Material Command, Griffiss AFB, NY, Sept. 1995.
24. P. S. Maybeck, Stochastic Models, Estimation, and Control, vol. 2. New York: Academic, 1982.
25. E. Mayer-Wolf, M. Zakai, and O. Zeltôuni, "On the memory length of the optimal nonlinear filter," in Proc. Stochastic Differential Syst. Stochastic Contr. Theory Applicat, Minneapolis, Minn, 1986, pp. 311-322.
26. G. M. Milshtein, "Numerical integration of stochastic differential equations," Mathematics and Its Applications, vol. 313. Dordrecht: The Netherlands: Kluwer, 1995.
27. K. S. Narandra and K. Parthasarthy, "Identification and control of dynamical systems using neural networks," IEEE Trans. Neural Networks, vol. 1, pp. 4-27, 1990.
28. H. Kunita, "Stochastic partial differential equations connected with nonlinear filtering," in Nonlinear Filtering Stochastic Contr., Lecture Notes Math., S. Mitter and A. Moro, Eds., vol. 972, pp. 163-165, 1982.
29. D. Ocone and E. Pardoux, "A lie algebra criterion for the nonexistence of finite dimensionally computable filters," Lecture Notes Math., vol. 1390, pp. 197-204, 1988.
30. B. Oksendal, Stochastic Differential Equations, 4th ed. New York: Springer, 1995.
31. M. J. L. Orr, "Regularization in the forward selection of radial basis function centers," Neural Comput., vol. 7, pp. 606-623, 1995.
32. T. Parisini and R. Zoppoli, "Neural networks for nonlinear state estimation," Int. J. Robust Nonlinear Contr., vol. 4, pp. 231-248, 1994.
33. B. A. Pearlmutter, "Learning state trajectories in recurrent neural networks," Neural Computat., vol. 1, pp. 263-269, 1989.
34. T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE, vol. 78, pp. 1484-1487, 1990.
35. G. V. Puskorius, L. A. Feldkamp, and L. I. Davis, "Dynamic neural network methods applied to on-vehicle idle speed control," Proc. IEEE, vol. 84, p. 1407, 1996.
36. P. A. Raviart, "An analysis of particle methods," in Numerical Methods in Fluid Dynamics, Lecture Notes Math., F. Brezzi, Ed., vol. 1127, 1985.
37. H. Risken, The Fokker-Planck Equation. New York: Springer-Verlag, 1989.
38. A. H. Sayed and T. Kailath, "A state-space approach to adaptive RLS filtering," IEEE Signal Processing Mag., vol. 11, pp. 18-60.
39. H. W. Sorenson, Ed., Kaiman Filtering: Theory and Application. New York: IEEE, 1985.
40. M. Sun and R. Glowinski, "Pathwise approximation and simulation for the Zakai filtering equation through operator splitting," Calcolo., vol. 30, pp. 219-239, 1993.
41. G. Wahba, Spline Models for Observational Data. Philadelphia, PA: SIAM, 1990.
42. H. White, "Learning in artificial neural networks: A statistical perspective," Neural Comput., vol. 1, pp. 425-464, 1989.
43. H. White, "Connectionist nonparametric regression: Multilayer feedforward networks can learn arbitrary mappings," Neural Networks, vol. 3, pp. 533-550, 1990.
44. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications. Cambridge, MA: MIT Press, 1949.
45. Y. Yavin, "Numerical studies in nonlinear filtering," Lecture Notes in Control and Information Sciences, vol. 65, pp. 14-15, 1985.
46. P. Yee and S. Haykin, "A dynamic regularized Gaussian radial basis function network for nonlinear, nonstationary time series prediction," in Proc. ICASSP, Detroit, MI, May 8-11, 1995.
47. Z. Cai, F. Le Gland, and H. Zhang, "An adaptive local grid refinement method for nonlinear filtering," Internal Pub. INRIA, Tech. Rep. 954.
48. M. Zakai, "On the optimal filtering of diffusion processes," Wahrscheinlichkeitstheorie verw Gebiete, vol. 11, pp. 230-243, 1969.
49. L. Xu, A. Krzyzak, and A. Yuille, "On radial basis function nets andc kernel regression: Statistical consistency, convergence rates, and receptive field size," Neural Networks, vol. 7, pp. 609-628, 1994.
50. L. Györfi, W. Hardie, P. Sarda, and P. Vieu, "Nonparametric curve estimation from time series," Lecture Notes Statist., vol. 60, p. 47, 1989.
51. E. A. Nadaraya, "On estimating regression," Theory Probab. Appl., vol. 9, pp. 141-142, 1964.
52. E. A. Nadaraya, "On nonparametric estimation of density functions and regression curves," Theory Probab. Appl., vol. 10, pp. 186-190, 1965.
53. G. S. Watson, "Smooth regression analysis," Sankhyä, vol. 26, pp. 359-372, 1964.
54. P. Yee, "Regularized radial basis functions: Theory and applications to probability estimation, classification, and time series prediction," Ph.D. dissertation, Dept. Elect. Comput Eng., McMasterUniv., Hamilton, Ont., Canada, 1997.
55. D. Bosq, "Nonparametric statistics for stochastic processes," Lecture Notes Statist., vol. 110, p. 74, 1996.