Biases and accuracy of, and an alternative to discrete nonlinear filters
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).
KeywordsNonlinear dynamical and nonlinear measurement system nonlinear filters bias and accuracy analysis almost unbiased second-order nonlinear filter
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