Nadia Oudjane

Improving Regularised Particle Filters

The optimal filter computes the posterior probability distribution of the state in a dynamical system, given noisy measurements, by iterative application of prediction steps according to the dynamics of the state, and correction steps taking the measurements into account. A new class of approximate nonlinear filter has been recently proposed, the idea being to produce a sample of independent random variables, called a particle system, (approximately) distributed according to this posterior probability distribution. The method is very easy to implement, even in high-dimensional problems, since it is sufficient in principle to simulate independent sample paths of the hidden dynamical system.

A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals☆

We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a robust filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The robust filter is obtained here by truncation of the likelihood function, and the robustification result is proved under the assumption that the Markov transition kernel satisfies a pseudo-mixing condition (weaker than the usual mixing condition), and that the observations are sufficiently good. This robustification approach allows us to prove also the uniform convergence of several particle approximations to the optimal filter, in some cases of nonergodic signals.

A sequential particle algorithm that keeps the particle system alive

We consider the problem of approximating a nonlinear (unnormalized) Feynman-Kac flow, in the special case where the selection functions can take the zero value. We begin with a list of several important practical situations where this characteristics is present. We study next a sequential particle algorithm, proposed by Oudjane (2000), which guarantees that the particle system does not die. Among other results, we obtain a central limit theorem which relies on the result of Rényi (1957) for the sum of a random number of independent random variables.

L/sup 2/-density estimation with negative kernels

In this paper, we are interested in density estimation using kernels that can take negative values, also called negative kernels. On the one hand, using negative kernels allows reducing the bias of the approximation, but on the other hand it implies that the resulting approximation can take negative values. To obtain a new approximation which is a probability density, we propose to replace the approximation by its L/sup 2/-projection on the space of L/sup 2/-probability densities. A similar approach has been proposed in I.K. Glad et al. (2003) but, in this paper, we describe how to compute this projection and how to generate random variables from it. This approach can be useful for particle filtering, particularly for the regularization step in regularized particle filters (C. Musso and N. Oudjane, June 1998) or kernel filters (M. Hurzeler and H.R. Kunsch, June 1998).

Data reduction for particle filters

In this paper, we are interested in nonlinear filtering approximations. Approximate filters (such as the extended Kalman filter or particle filters) are known to converge to the optimal filter when the local error (committed at each step of time) vanishes. But this convergence is in general not uniform in time. Error bounds obtained in the general case suggest that the approximation error could grow exponentially with time. This divergent phenomena is actually observed in some simulations. To avoid that divergence of approximate filters with the number of observations, an idea is to reduce the number of observations without losing too much information. This paper proposes an optimal approach to reduce the number of observations for filtering. This new approach is applied to particle filtering and tested in the case of the bearing only tracking problem.