Continuous-time approximations for the nonlinear filtering problem

1981 ◽  
Vol 7 (1) ◽  
pp. 233-245 ◽  
Author(s):  
G. B. Di Masi ◽  
W. J. Runggaldier
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Filtering Problem

scholarly journals A Kusuoka–Lyons–Victoir particle filter

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130076 ◽  
Author(s):  
Dan Crisan ◽  
Salvador Ortiz-Latorre
Keyword(s):  
Particle Filter ◽  
Continuous Time ◽  
Nonlinear Filtering ◽  
Computational Effort ◽  
Observation Data ◽  
Minimal Variance ◽  
Discretization Grid ◽  
Filtering Problem ◽  
Grid Mesh ◽  
Number Of Particles

The aim of this paper is to introduce a new numerical algorithm for solving the continuous time nonlinear filtering problem. In particular, we present a particle filter that combines the Kusuoka–Lyons–Victoir (KLV) cubature method on Wiener space to approximate the law of the signal with a minimal variance ‘thinning’ method, called the tree-based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle filter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Beneš filter).

scholarly journals Information geometric nonlinear filtering

2015 ◽  
Vol 18 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Riemannian Metric ◽  
Point Of View ◽  
Posterior Distributions ◽  
Hilbert Manifold ◽  
Information Theoretic ◽  
Finite Dimensional ◽  
Geometric Representations ◽  
Filtering Problem

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

scholarly journals Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Amirhossein Taghvaei ◽  
Jana de Wiljes ◽  
Prashant G. Mehta ◽  
Sebastian Reich
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Nonlinear Problems ◽  
Improved Stability ◽  
Consistent Solution ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Filtering Problem

This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman–Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman–Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman–Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

Finite-dimensional models for hidden Markov chains

1995 ◽  
Vol 27 (01) ◽  
pp. 146-160
Author(s):  
Lakhdar Aggoun ◽  
Robert J. Elliott
Keyword(s):  
Markov Chains ◽  
Expectation Maximization ◽  
Continuous Time ◽  
Hidden Markov ◽  
Expectation Maximization Algorithm ◽  
Linear Filtering ◽  
Occupation Times ◽  
Finite Dimensional ◽  
Self Tuning ◽  
Filtering Problem

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.

Monte Carlo Methods for a Special Nonlinear Filtering Problem

2000 ◽  
Vol 33 (16) ◽  
pp. 347-352 ◽  
Author(s):  
O.A. Stepanov ◽  
V.M. Ivanov ◽  
M.L. Korenevski
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Nonlinear Filtering ◽  
Filtering Problem

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

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