Generalised particle filters with Gaussian mixtures

© 2015 Elsevier B.V. All rights reserved.Stochastic filtering is defined as the estimation of a partially observed dynamical system. Approximating the solution of the filtering problem with Gaussian mixtures has been a very popular method since the 1970s. Despite nearly fifty years of development, the existing work is based on the success of the numerical implementation and is not theoretically justified. This paper fills this gap and contains a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. We deduce the L2-convergence rate for the approximating system and show some numerical examples to test the new algorithm

Data Assimilation by Conditioning on Future Observations

Conventional recursive filtering approaches, designed for quantifying the state of an evolving uncertain dynamical system with intermittent observations, use a sequence of (i) an uncertainty propagation step followed by (ii) a step where the associated data is assimilated using Bayes' rule. In this paper we switch the order of the steps to: (i) one step ahead data assimilation followed by (ii) uncertainty propagation. This route leads to a class of filtering algorithms named \emph{smoothing filters}. For a system driven by random noise, our proposed methods require the probability distribution of the driving noise after the assimilation to be biased by a nonzero mean. The system noise, conditioned on future observations, in turn pushes forward the filtering solution in time closer to the true state and indeed helps to find a more accurate approximate solution for the state estimation problem

A track-before-detect labelled multi-Bernoulli particle filter with label switching

This paper presents a multitarget tracking particle filter (PF) for general track-before-detect measurement models. The PF is presented in the random finite set framework and uses a labelled multi-Bernoulli approximation. We also present a label switching improvement algorithm based on Markov chain Monte Carlo that is expected to increase filter performance if targets get in close proximity for a sufficiently long time. The PF is tested in two challenging numerical examples.Comment: Accepted for publication in IEEE Transactions on Aerospace and Electronic System

Generalised particle filters

The ability to analyse, interpret and make inferences about evolving dynamical systems is of great importance in different areas of the world we live in today. Various examples include the control of engineering systems, data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. In general, the dynamical systems are not directly observable, quite often only partial information, which is deteriorated by the presence noise, is available. This naturally leads us to the area of stochastic filtering, which is defined as the estimation of dynamical systems whose trajectory is modelled by a stochastic process called the signal, given the information accumulated from its partial observation. A massive scientific and computational effort is dedicated to the development of various tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions (one to three). In higher dimensions (up to ten), a class of numerical methods called particle filters have proved the most successful methods to-date. These methods produce approximations of the posterior distribution of the current state of the signal by using the empirical distribution of a cloud of particles that explore the signal’s state space. In this thesis, we discuss a more general class of numerical methods which involve generalised particles, that is, particles that evolve through spaces larger than the signal’s state space. Such generalised particles include Gaussian mixtures, wavelets, orthonormal polynomials, and finite elements in addition to the classical particle methods. This thesis contains a rigorous analysis of the approximation of the solution of the filtering problem using Gaussian mixtures. In particular we deduce the L2-convergence rate and obtain the central limit theorem for the approximating system. Finally, the filtering model associated to the Navier-Stokes equation will be discussed as an example

Application of Sequential Quasi-Monte Carlo to Autonomous Positioning

Sequential Monte Carlo algorithms (also known as particle filters) are popular methods to approximate filtering (and related) distributions of state-space models. However, they converge at the slow $1/\sqrt{N}$ rate, which may be an issue in real-time data-intensive scenarios. We give a brief outline of SQMC (Sequential Quasi-Monte Carlo), a variant of SMC based on low-discrepancy point sets proposed by Gerber and Chopin (2015), which converges at a faster rate, and we illustrate the greater performance of SQMC on autonomous positioning problems.Comment: 5 pages, 4 figure

Convergence Analysis of Ensemble Kalman Inversion: The Linear, Noisy Case

We present an analysis of ensemble Kalman inversion, based on the continuous time limit of the algorithm. The analysis of the dynamical behaviour of the ensemble allows us to establish well-posedness and convergence results for a fixed ensemble size. We will build on the results presented in [26] and generalise them to the case of noisy observational data, in particular the influence of the noise on the convergence will be investigated, both theoretically and numerically. We focus on linear inverse problems where a very complete theoretical analysis is possible