Multilevel ensemble transform particle filtering

This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles

Maximum Likelihood Estimator for Hidden Markov Models in continuous time

The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain.Comment: Warning: due to a flaw in the publishing process, some of the references in the published version of the article are confuse

On Robustness of Discrete Time Optimal Filters

A new result on stability of an optimal nonlinear filter for a Markov chain with respect to small perturbations on every step is established. An exponential recurrence of the signal is assumed

Likelihood Based Statistics For Partially Observed Diffusion Processes

The purpose of this paper is to study some statistical problems: parameter estimation, binary detection, change detection (disorder problem), etc. for partially observed diffusion processes, using the likelihood approach. It is shown that the stochastic PDE related to the state estimation problem, provides also a way to compute the likelihood function/ratio. A recent result on consistency of the MLE, in the small noise asymptotics, is also presented

Small Noise Asymptotics of the Bayesian Estimator in Nonidentifiable Nonlinear Regressions

: We study the asymptotic behaviour of the Bayesian estimator for a deterministic signal in additive Gaussian white noise, in the case where the set of minima of the Kullback--Leibler information is a submanifold of the parameter space. This problem includes as a special case the study of the asymptotic behaviour of the nonlinear filter, when the state equation is noise--free, and when the limiting deterministic system is non--observable. We present a practical example where this situation occurs. We give an explicit expression of the limit, as the noise intensity goes to zero, of the posterior probability distribution of the parameter, and we study the rate of convergence. 1 INTRODUCTION This paper is concerned with the small noise asymptotics of the Bayesian estimator, based on the continuous time nonlinear regression observation X t ; 0 t T with differential dX t = m t (`) + " dW ` t " ? 0 ; where ` 2 \Theta ae R p and W ` t is a standard Wiener process. The process X t c..

Variational particle smoothers and their localization

Given the success of 4D-variational methods (4D-Var) in numerical weather prediction, and recent efforts to merge ensemble Kalman filters with 4D-Var, we revisit how one can use importance sampling and particle filtering ideas within a 4D-Var framework. This leads us to variational particle smoothers (varPS) and we study how weight-localization can prevent the collapse of varPS in high-dimensional problems. We also discuss the relevance of (localized) weights in near-Gaussian problems. We test our ideas on the Lorenz'96 model of dimensions n = 40, n = 400, and n = 2, 000. In our numerical experiments the localized varPS does not collapse and yields results comparable to ensemble formulations of 4D-Var, while tuned EnKFs and the local particle filter lead to larger estimation errors. Additional numerical experiments suggest that using localized weights may not yield significant advantages over unweighted or linearized solutions in near-Gaussian problems.Office of Naval Research [N00173-17-2-C003, PE-0601153N]; National Science Foundation [DMS-1619630]; Alfred P. Sloan Foundation; National Research Council Research Associateship Program fellowship12 month embargo; published online: 10 February 2018This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]