Advertisement

Biases and accuracy of, and an alternative to discrete nonlinear filters

  • Peiliang Xu
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 121)

Abstract

Dynamical systems encountered in reality are essentially nonlinear. A number of approaches were propped for nonlinear filter with different accuracy. They are mainly investigated from the Bayesian point of view, and may be classified into three kinds of methods: linearization, statistical approximation and Monte-Carlo simulation (Jazwinski 1970; Tanizaki 1993; Gelb 1994). Very often, linearization of the nonlinear system is done using either a precomputed nominal trajectory or the estimate of the state vector. This second linearization approach is better known as the extended Kalman filter (Jazwinski 1970). Since the solution based on one-step linearization may be poor for a highly nonlinear system, iteration in this case is expected in order to obtain a more accurate estimate. Higher order approaches should probably be taken into account. It should be noted, however, that if the system presents a significant nonlineaaity, even the mean and covariance matrix of the nonlinear filter can be misleading, since the means of the estimated state variables may deviate appreciably from their true parameter values. The basic idea of statistical approximation is to replace a nonlinear function of random variables by a series expansion (Gelb 1994) or to approximate thea posteriori conditional probability density function of the state vector (Sorenson & Stubberud 1968; Kramer & Sorenson 1988). The Monte Carlo simulation technique may be used to determine the mean and covariance matrix of the nonlinear filter (if properly designed), which requires a large sample to obtain statistically meaningful results (see e.g. Brown & Mariano 1989; Carlin et al. 1992; Gelb 1994; Tanizaki 1993)

Under the Bayesian. framework, first and second order nonlinear filters have been derived formally by computing the relevant (Bayesian) conditional means and covariances and substituting them into the standard linear Kalman filtering algorithm (Tanizaki 1993; Gelb 1994). Higher order nonlinear filters were proposed, with hope in mind that they might be less biased and more efficient. The Bayesian derivation of the second order nonlinear filter SONF has been critically challenged in this paper by examining the biases and accuracy of the SONF and the corresponding residuals.

We have taken a frequentist standpoint to deal with the problem of nonlinear filtering and derived the biases of the EKF and the SONF. For a nonlinear dynamical and nonlinear measurement system, the frequentist approach and Bayesian method have been shown in this paper to be fundamentally different. Assume that the noises of a nonlinear dynamical and nonlinear measurement system are normally distributed. The interface between Bayesians and frequentists on nonlinear filters has resulted in several new findings in the present paper, which are summarized in the following: (i) the Gaussian or truncated second order nonlinear filter not only cannot guarantee the improvement of the biases of the estimated state vector, but may also exaggerate them; (ii) the variance-covariance measure of the second order nonlinear filter currently given in the literature is correct, only if some extra terms obtained in this paper are included; and (iii) an alternative, almost unbiased second order nonlinear filter is proposed, if the randomness of some coefficient matrices is neglected. Practically, the new SONF filter in this paper is expected to significantly improve te EKF and the SONF given in the literature, if dynamical and measurement systems are highly nonlinear and if the ratio of signal to noise is not very large. Numerical confirmation requires a large scale of simulations with well designed experiments, which will be left for a future contribution. For other new findings and possible practical pitfalls due to a direct interface between Bayesians and frequentists, the reader is referred to Berger & Robert (1990), Xu (1992) and Xu & Rummel (1994).

Keywords

Nonlinear dynamical and nonlinear measurement system nonlinear filters bias and accuracy analysis almost unbiased second-order nonlinear filter 

References

  1. Baheti, R., O’Hallaron, D. & Itzkowitz, H. (1990). Mapping extended Kalman filters onto linear arrays, IEEE Trans. Auto. Contr., AC-35, 1310–1319CrossRefGoogle Scholar
  2. Bates, D. & Watts, D.G. (1980). Relative curvature measures of nonlinearity (with discussions), J. R. statist. Soc., B42, 1–25Google Scholar
  3. Bates, D. & Watts, D.G. (1988). Applied nonlinear regression, John Wiley & Sons, New YorkCrossRefGoogle Scholar
  4. Beale, E.M.L. (1960). Confidence regions in nonlinear estimation (with discussions), J. R. statist. Soc., B22, 41–88Google Scholar
  5. Berger, J.O. & Robert, C. (1990). Subjective hierarchical Bayes estimation of a multivariate normal mean: on the frequentist interface, Ann. Statist., 18, 617– 651CrossRefGoogle Scholar
  6. Box, M.J. (1971). Bias in nonlinear estimation (with discussions), J. R. statist. Soc., B33, 71–201Google Scholar
  7. Brown, B. & Mariano, R. (1989). Measures of deterministic prediction bias in nonlinear models, Int. Economic Rev., 30, 667–684CrossRefGoogle Scholar
  8. Carlin, B., Poison, N. & Stoffer, D. (1992). A Monte Carlo approach to nonnormal and nonlinear state space modelling, J. Amer. Stat. Assoc., 87, 493–500CrossRefGoogle Scholar
  9. Clark, G.P.Y. (1980). Moments of the least squares estimators in a nonlinear regression model, J. R. statist, Soc., B42, 227–237Google Scholar
  10. Denham, W. & Pines, S. (1966). Sequential estimation when measurement function nonlinearity is comparable to measurement error, AIAA J., 4, 1071–1078CrossRefGoogle Scholar
  11. Friedland, B. & Bernstein, I. (1966). Estimation of the state of a nonlinear process in the presence of nongaussian noise and disturbances, J. Franklin Inst., 281, 455–480CrossRefGoogle Scholar
  12. Gelb, A. (ed) (1994). Applied optimal estimation, The MIT Press, CambridgeGoogle Scholar
  13. Jazwinski, A. (1970). Stochastic processes and filtering theory, Academic Press, New YorkGoogle Scholar
  14. Kramer, S. & Sorenson, H. (1988). Recursive Bayesian estimation using piece-wise constant approximations, Automatica, 24, 789–801CrossRefGoogle Scholar
  15. Kushner, H . (1967a). Dynamical equations for optimal nonlinear filtering, J. diff. Eq., 3, 179–190CrossRefGoogle Scholar
  16. Kushner, H . (1967b). Approximations to optimal nonlinear filters, IEEE Trans. Auto. Contr., AC12, 546–556CrossRefGoogle Scholar
  17. Mahalanabis, A. & Farooq, M. (1971). A second – order method for state estimation of nonlinear dynamical systems, Int. J. Control, 14, 631–639CrossRefGoogle Scholar
  18. Schaffrin, B. (1991). Generalized robustified Kalman filters for the integration of GPS and INS, Geodetic Institute, Stuttgart University, Tech. Report No.15, StuttgartGoogle Scholar
  19. Searle, S.R. (1971). Linear models, John Wiley & Sons, New YorkGoogle Scholar
  20. Seber, G. & Wild, C. (1989). Nonlinear Regression, John Wiley & Sons, New YorkCrossRefGoogle Scholar
  21. Sorenson, H. & Stubberud, A. (1968). Non-linear filtering by approximation of thea posterioridensity, Int. J. Control, 8, 33–51CrossRefGoogle Scholar
  22. Tanizaki, H . (1993). Nonlinear filters — Estimation and applications, Springer Verlag, BerlinGoogle Scholar
  23. Wishner, R., Tabaczynski, J. & Athans, M. (1989). A comparison of three non-linear filters, Automatica, 5, 487–496CrossRefGoogle Scholar
  24. Xu, P.L. (1991). Least squares collocation with incorrect prior information, ZfV, 116, 266–273Google Scholar
  25. Xu, P.L. (1992). The value of minimum norm estimation of geopotential fields, Geophys. J. Int., 111,170–178CrossRefGoogle Scholar
  26. Xu, P.L. & Rwnmel, R. (1994). A simulation study of smoothness methods in recovery of regional gravity fields, Geophys. J. Int., 127, 472–486CrossRefGoogle Scholar

Copyright information

© SPringer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Peiliang Xu
    • 1
  1. 1.Disaster Prevention Research InstituteKyoto University at UjiKyotoJapan

Personalised recommendations