# Gaussian filters for nonlinear filtering problems

@article{Ito2000GaussianFF, title={Gaussian filters for nonlinear filtering problems}, author={Kazufumi Ito and Kaiqi Xiong}, journal={IEEE Trans. Autom. Control.}, year={2000}, volume={45}, pages={910-927} }

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests…

## 1,362 Citations

### Gaussian filter for nonlinear filtering problems

- Computer ScienceProceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
- 2000

This work develops and analyzes real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions and develops efficient and accurate numerical integration of the proposed filter.

### Non-Gaussian Filters for Nonlinear Continuous-Discrete Models

- Computer Science
- 2017

This paper discretizes the Ito-type stochastic differential system model and applies the EnKF and PFs originally developed for nonlinear discrete-time models to the discretized system models, yielding the non-Gaussian filtering algorithms.

### A quasi-Gaussian Kalman filter

- Computer Science, Mathematics2006 American Control Conference
- 2006

A Gaussian approximation to the nonlinear filtering problem, namely the quasi-Gaussian Kalman filter is presented and two methods are proposed, one based on stochastic linearization and the other based on a direct evaluation of the innovations terms, to perform the measurement update in the Kalman recursion.

### Adaptive Gaussian Sum Filter for Nonlinear Bayesian Estimation

- Computer ScienceIEEE Transactions on Automatic Control
- 2011

The numerical simulation results show that updating the weights of different mixture components during propagation mode of the filter not only provides us with better state estimates but also with a more accurate state probability density function.

### Non-linear noise adaptive Kalman filtering via variational Bayes

- Computer Science, Mathematics2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)
- 2013

This work proposes a variational Bayes and Gaussian non-linear filtering based algorithm for efficient computation of the approximate filtering posterior distributions that allows the use of efficient Gaussian integration methods such as unscented transform, cubature integration and Gauss-Hermite integration along with the classical Taylor series approximations.

### Analysis of Kalman Filter Approximations for Nonlinear Measurements

- Computer ScienceIEEE Transactions on Signal Processing
- 2013

A theoretical analysis is presented of the correction step of the Kalman filter (KF) and its various approximations for the case of a nonlinear measurement equation with additive Gaussian noise. The…

### A Tutorial on Bayesian Estimation and Tracking Techniques Applicable to Nonlinear and Non-Gaussian Processes

- Computer Science
- 2005

An overview of techniques for nonlinear filtering for a wide variety of conditions on the nonlinearities and on the noise is presented and a general Bayesian approach to filtering is developed which is applicable to all linear or nonlinear stochastic systems.

### Variational Bayesian Adaptation of Noise Covariances in Non-Linear Kalman Filtering

- Computer Science, Mathematics
- 2013

A variational Bayes and Gaussian filtering based algorithm for e cient computation of the approximate filtering posterior distributions for non-linear stochastic state space models.

### Nonlinear Gaussian Filtering: Theory, Algorithms and Applications

- Computer Science
- 2015

By restricting to Gaussian distributions, the optimal Bayesian filtering problem can be transformed into an algebraically simple form, which allows for computationally efficient algorithms. Three…

### Kalman Filtering for a Generalized Class of Nonlinear Systems and a New Gaussian Quadrature Technique

- MathematicsIEEE Transactions on Automatic Control
- 2012

The class of nonlinear systems treated in this technical note consists of the discrete time non linear systems that are formed by the interconnection of linear systems through static nonlinearities with few inputs, and a new quadrature scheme suitable for nonlinear Kalman filtering is introduced.

## References

SHOWING 1-10 OF 37 REFERENCES

### Approximations to optimal nonlinear filters

- MathematicsIEEE Transactions on Automatic Control
- 1967

Let the signal and noise processes be given as solutions to nonlinear stochastic differential equations. The optimal filter for the problem, derived elsewhere, is usually infinite dimensional.…

### A new approach for filtering nonlinear systems

- Engineering, MathematicsProceedings of 1995 American Control Conference - ACC'95
- 1995

A new recursive linear estimator for filtering systems with nonlinear process and observation models which can be transformed directly by the system equations to give predictions of the transformed mean and covariance is described.

### Nonlinear Bayesian estimation using Gaussian sum approximations

- Computer Science
- 1972

In this paper an approximation that permits the explicit calculation of the a posteriori density from the Bayesian recursion relations is discussed and applied to the solution of the nonlinear filtering problem.

### A General Method for Approximating Nonlinear Transformations of Probability Distributions

- Mathematics, Computer Science
- 1996

A new approach for generalised nonlinear ltering is described, which is more accurate, more stable, and far easier to implement than an extended Kalman lter.

### Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements

- Mathematics
- 1968

This paper presents the derivation of the dynamical equations of a second-order filter which estimates the states of a non-linear plant on the basis of discrete noisy measurements. The filter…

### Stochastic differential equations for the non linear filtering problem

- Mathematics
- 1972

The general nonlinear filtering or estimation problem may be described as follows. xty (0<t<T)y called the signal or system process is a stochastic process direct observation is not possible. The…

### Approximation of the Zakai Equation for Nonlinear Filtering

- Mathematics, Computer Science
- 1996

Time discretization based on the implicit Milshtein and Euler methods and Galerkin approximation in the spatial coordinates and Convergence and rate of convergence of approximation methods are established.

### On approximate approximations using Gaussian kernels

- Computer Science
- 1996

Estimates are obtained showing a high-order approximation up to some saturation error negligible in numerical applications in quasi-interpolation and interpolation with Gaussians.

### Convergence of implicit discretization schemes for linear differential equations with application to filtering

- Computer Science
- 1987

The motivation for the present work arises from a well-known problem in nonlinear filtering, to design algorithms that discretize but still retain the representation, merely changing the process involved.