Path sampling for particle filters with application to multi-target tracking

In recent work (arXiv:1006.3100v1), we have presented a novel approach for improving particle filters for multi-target tracking. The suggested approach was based on drift homotopy for stochastic differential equations. Drift homotopy was used to design a Markov Chain Monte Carlo step which is appended to the particle filter and aims to bring the particle filter samples closer to the observations. In the current work, we present an alternative way to append a Markov Chain Monte Carlo step to a particle filter to bring the particle filter samples closer to the observations. Both current and previous approaches stem from the general formulation of the filtering problem. We have used the currently proposed approach on the problem of multi-target tracking for both linear and nonlinear observation models. The numerical results show that the suggested approach can improve significantly the performance of a particle filter.Comment: Minor corrections, 23 pages, 8 figures. This is a companion paper to arXiv:1006.3100v

Sequential Empirical Bayes method for filtering dynamic spatiotemporal processes

We consider online prediction of a latent dynamic spatiotemporal process and estimation of the associated model parameters based on noisy data. The problem is motivated by the analysis of spatial data arriving in real-time and the current parameter estimates and predictions are updated using the new data at a fixed computational cost. Estimation and prediction is performed within an empirical Bayes framework with the aid of Markov chain Monte Carlo samples. Samples for the latent spatial field are generated using a sampling importance resampling algorithm with a skewed-normal proposal and for the temporal parameters using Gibbs sampling with their full conditionals written in terms of sufficient quantities which are updated online. The spatial range parameter is estimated by a novel online implementation of an empirical Bayes method, called herein sequential empirical Bayes method. A simulation study shows that our method gives similar results as an offline Bayesian method. We also find that the skewed-normal proposal improves over the traditional Gaussian proposal. The application of our method is demonstrated for online monitoring of radiation after the Fukushima nuclear accident

Large deviations for stochastic flows of diffeomorphisms

A large deviation principle is established for a general class of stochastic flows in the small noise limit. This result is then applied to a Bayesian formulation of an image matching problem, and an approximate maximum likelihood property is shown for the solution of an optimization problem involving the large deviations rate function.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ203 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Small noise large deviations for infinite dimensional stochastic dynamical systems

Large deviations theory concerns with the study of precise asymptotics governing the decay rate of probabilities of rare events. A classical area of large deviations is the Freidlin–Wentzell (FW) theory that deals with path probability asymptotics for small noise Stochastic Dynamical Systems (SDS). For finite dimensional SDS, FW theory has been very well studied. The goal of the present work is to develop a systematic framework for the study of FW asymptotics for infinite dimensional SDS. Our first result is a general LDP for a broad family of functionals of an infinite dimensional small noise Brownian motion (BM). Depending on the application, the driving infinite dimensional BM may be given as a space–time white noise, a Hilbert space valued BM or a cylindrical BM. We provide sufficient conditions for LDP to hold for all such different model settings. As a first application of these results we study FW LDP for a class of stochastic reaction diffusion equations. The model that we consider has been widely studied by several authors. Two main assumptions imposed in all previous studies are the boundedness of the diffusion coefficient and a certain geometric condition on the underlying domain. These restrictive conditions are needed in proofs of certain exponential probability estimates that form the basis of classical proofs of LDPs. Our proofs instead rely on some basic qualitative properties, eg. existence, uniqueness, tightness, of certain controlled analogues of the original systems. As a result, we are able to relax the two restrictive requirements described above. As a second application, we study large deviation properties of certain stochastic diffeomorphic flows driven by an infinite sequence of i.i.d. standard real BMs. LDP for small noise finite dimensional flows has been studied by several authors. Typical space– time stochastic models with a realistic correlation structure in the spatial parameter naturally leads to infinite dimensional flows. We establish a LDP for such flows in the small noise limit. We also apply our result to a Bayesian formulation of an image analysis problem. An approximate maximum likelihood property is shown for the solution of an optimal image matching problem that involves the large deviation rate function