Abstract: An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linearity or Gaussian noise: it may be applied to any state transition or measurement model. A simulation example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter. Bayesian methods provide a rigorous general framework for dynamic state estimation problems. The Bayesian approach is to construct the probability density function (PDF) of the state based on all the available information. For the linear-Gaussian estimation problem, the required PDF remains Gaussian at every iteration of the filter, and the Kalman filter relations propagate and update the mean and covariance of the distribution. For nonlinear or non-Gaussian problems there is no general analytic (closed form) expression for the required PDF. The extended Kalman filter (EKF) is the most popular approach to recursive nonlinear estimation 113. Here the estimation problem is linearised about the predicted state so that the Kalman filter can be applied. In this case the required PDF is still approximated by a Gaussian, which may be a gross distortion of the true underlying structure and may lead to filter divergence. In this paper we propose a new way of representing and recursively generating an approximation to the state PDF. The central idea is to represent the required PDF as a set of random samples, rather than as a function over state space [lo-121. As the number of samples becomes very large, they effectively provide an exact, equivalent, representation of the required PDF. Estimates of moments (such as mean and covariance) of percentiles of the state vector PDF can be obtained directly from the samples. If necessary, a functional estimate of the PDF could also be constructed from the samples [13] and from this estimates of highest posterior density (HPD) intervals or the mode can be obtained. In Reference 11, an adaptive importance sampling algorithm is presented. This results in large mixture distributions and the technique is computationally expensive. A recursive algorithm, which we call the bootstrap filter, for propagating and updating these samples for the discrete time problem (see Section 2) is presented and justified in Section 3. The filter is so called because the key update stage of the algorithm (Bayes rule) is implemented as a weighted bootstrap. This was motivated by a result from Smith and Gelfand 1141. A related algorithm is described in the paper by Muller 1121. These sampling techniques avoid the need to define a grid in state space; the samples being naturally concentrated in regions of high probability density. They also have the great advantage of being able to handle any functional nonlinearity, and system or measurement noise of any distribution. In Section 4, two simulation examples are presented. The second of these is the bearings-only tracking problem. The performance of the bootstrap filter is compared with that of the EKF. We are concerned with the discrete time estimation problem. The state vector, xk E R" is assumed to evolve according to the following system model where f k : [w" x R" + R" is the system transition function and wk E R" is a zero mean, white-noise sequence independent of past and current states. The PDF of wk is assumed to be known. At discrete times, measurements This work has been carried out with the support of DRA Farnborough.
As a result, in this case, the original PF REF will require too many particles for accurate tracking and hence becomes impractical to use.
Novel approach to nonlinear/non-Gaussian Bayesian state estimation
IEE Proceedings F Radar and Signal Processing
IEE Proceedings F Radar and Signal Processing
Mathematics