The Nonlinear Filtering Problem

1990 ◽  
pp. 115-150
Author(s):  
Harold J. Kushner
Keyword(s):  
Nonlinear Filtering ◽  
Filtering Problem

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.

Estimation of anthracnose dynamics by nonlinear filtering

2018 ◽  
Vol 11 (01) ◽  
pp. 1850005
Author(s):  
David Jaurès Fotsa-Mbogne
Keyword(s):  
Nonlinear Filtering ◽  
Space Variable ◽  
Initial Conditions ◽  
Confidence Region ◽  
Inhibition Rate ◽  
Deterministic Models ◽  
Incomplete Observations ◽  
Partially Observed ◽  
Uniform Law ◽  
Filtering Problem

In this work, we apply the nonlinear filtering theory to the estimation of the partially observed dynamics of anthracnose which is a phytopathology. The signal here is the inhibition rate and the observations are the fruit volume and the rotted volume. We propose stochastic models based on deterministic models studied previously in the literature, in order to represent the noise introduced by uncontrolled variations on parameters and errors on the measurements. Under the assumption of Brownian noises, we prove the well-posedness of the models in either they take into account the space variable or not. The filtering problem is solved for the nonspatial model giving Zakai and Kushner–Stratonovich equations satisfied respectively by the unnormalized and the normalized conditional distribution of the signal with respect to the observations. A prevision problem and a discrete filtering problem are also studied for the realistic cases of discrete and possibly incomplete observations. We illustrate the filter behavior through figures displaying the average estimation relative error and a 95% confidence region obtained after a hundred of numerical simulations with initial conditions taken randomly with respect to uniform law.

A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

1998 ◽  
Vol 35 (04) ◽  
pp. 873-884 ◽  
Author(s):  
P. Del Moral
Keyword(s):  
Monte Carlo ◽  
Uniform Convergence ◽  
Convergence Theorem ◽  
Nonlinear Filtering ◽  
Partial Information ◽  
Particle System ◽  
Convergence Result ◽  
Recursive Solution ◽  
Ergodic Condition ◽  
Filtering Problem

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

scholarly journals On the convergence of the wavelet-Galerkin method for nonlinear filtering

2010 ◽  
Vol 20 (1) ◽  
pp. 93-108 ◽  
Author(s):  
Łukasz Nowak ◽  
Monika Pasławska-Południak ◽  
Krystyna Twardowska
Keyword(s):  
Galerkin Method ◽  
Compact Support ◽  
Nonlinear Filtering ◽  
Euler Scheme ◽  
Zakai Equation ◽  
Implicit Euler Scheme ◽  
Galerkin Scheme ◽  
Wavelet Galerkin Method ◽  
Biorthogonal Basis ◽  
Filtering Problem

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

Sign in / Sign up

Export Citation Format

Share Document