Summary The need for accurate monitoring and analysis of sequential data arises in many scientific, in- dustrial and financial problems. Although the Kalman filter is effective in the linear-Gaussian case, new methods of dealing with sequential data are required with non-standard models. Recently, there has been renewed interest in simulation-based techniques. The basic idea be- hind these techniques is that the current state of knowledge is encapsulated in a representative sample from the appropriate posterior distribution. As time goes on, the sample evolves and adapts recursively in accordance with newly acquired data. We give a critical review of re- cent developments, by reference to oil well monitoring, ion channel monitoring and tracking problems, and propose some alternative algorithms that avoid the weaknesses of the current methods.

The Kalman filter provides an effective solution to the linear Gaussian filtering problem. However where there is nonlinearity, either in the model specification or the observation process, other methods are required. Methods known generically as `particle filters' are considered. These include the condensation algorithm and the Bayesian bootstrap or sampling importance resampling (SIR) filter. These filters represent the posterior distribution of the state variables by a system of particles which evolves and adapts recursively as new information becomes available. In practice, large numbers of particles may be required to provide adequate approximations and for certain applications, after a sequence of updates, the particle system will often collapse to a single point. A method of monitoring the efficiency of these filters is introduced which provides a simple quantitative assessment of sample impoverishment and the authors show how to construct improved particle filters that are both structurally efficient in terms of preventing the collapse of the particle system and computationally efficient in their implementation. This is illustrated with the classic bearings-only tracking problem

We illustrate the potential pitfalls in the choice of sampling strategies for simulating the posterior distribution in a linear system with a non-linear observation process. We focus on the example of bearings only tracking, and use simulation to compare the convergence rate of various Metropolis-Hastings strategies, in the context of a simple model. The results indicate that the inclusion of a global 'scale move' in the Metropolis-Hastings sampler dramatically increases the convergence rate. Using such a Metropolis-Hastings sampler as a benchmark, we are able to explore the efficacy of other sampling strategies, such as the Gibbs sampler, and the effect of the prior distribution on the posterior distribution. We are further able to evaluate various recursive filtering algorithms, such as the Kalman filter and the SIR filter. Finally, we discuss ways in which the 'global' Metropolis-Hastings approach could be modified to take account of the computational advantages of recursive filtering techniques. In the context of bearings only tracking, these advantages accrue from assuming that all our prior knowledge about the target position at time t is accurately encapsulated in our current estimate of the target distribution at time t, and can therefore be discarded.