原创 Kalman filter toolbox for Matlab

http://mirandali910.blog.sohu.com/34574160.html

该工具箱为线性系统提供滤波、平滑和参数估计（用EM），本章包括

• 下载该工具箱
• 什么是kalman滤波器
• kalman滤波和平滑追踪的例子
• 关于非线性系统和非高斯系统
• 其他kalman滤波器软件
• 推荐阅读
  

功能

• kalman滤波器
• kalman平滑器——用RTS等式实现
• 学习kalman——用EM寻找最大化可能的估计
• 采样_lds——产生随机采样
• AR 到SS——转换k的自回归模型为逐个状态空间形式
• SS 到AR
• 学习_AR——使用最小2乘法寻找最大化的可能状态的参数

什么是kalman滤波器？

一个线性动力学系统是一个可以用线性动力学和线性观测（它们都服从高斯噪声）的部分服从随机分布的过程。它可以被以下这样定义，这里X(t)是时间t的隐含状态， Y(t)是观测器。

x(t+1) = F*x(t) + w(t), w ~ N(0, Q), x(0) ~ N(X(0), V(0))
y(t) = H*x(t) + v(t), v ~ N(0, R)

在该模块中，kalman滤波器是用于滤波的算法，例如，计算P(X(t) | Y(1), ..., Y(t)).

Rauch-Tung-Striebel (RTS)算法执行混合时间间隔离线平滑，例如，计算computing P(X(t) | Y(1), ..., Y(T)), t <= T.

kalman滤波的例子 

这是一个简单的例子，考虑一个以恒定速率运动，在平面上移动的质点，它的轨迹服从随机分布。新的位置(x1, x2)是老位置加上速度(dx1, dx2)再加上噪声w。

[ x1(t) ] = [1 0 1 0] [ x1(t-1) ] + [ wx1 ]
[ x2(t) ] [0 1 0 1] [ x2(t-1) ] [ wx2 ]
[ dx1(t) ] [0 0 1 0] [ dx1(t-1) ] [ wdx1 ]
[ dx2(t) ] [0 0 0 1] [ dx2(t-1) ] [ wdx2 ]我们假设只能观测到质点的位置[ y1(t) ] = [1 0 0 0] [ x1(t) ] + [ vx1 ]
[ y2(t) ] [0 1 0 0] [ x2(t) ] [ vx2 ]
[ dx1(t) ]
[ dx2(t) ]

这表示滤波估计的均方差是4.9，而平滑估计的是3.2。不仅平滑估计效果比较好，而且我们知道通过更小的不定椭圆来表示效果会更好，这可以在例子中数据联合问题中得到解决。注意到平滑椭圆在最后会变大，因为这些点所含的数据比较少。并观察滤波椭圆到达他们稳定状态(Ricatti)值的速度。点击这里下载产生该图的代码，并图解如何使用该工具的。 

什么是非线性和非高斯系统？

对于非高斯噪声系统, 推荐Particle filtering (PF),它是一种流行的有序蒙特卡罗技术. 参阅discussion on pros/cons of particle filters. 和以下教程: M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, "A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking," IEEE Transactions on Signal Processing, Volume 50, Number 2, February 2002, pp 174-189 (pdf cached here

 EKF 可为作为一个推荐的使用分配给PF。该方法比其他方法好. The Unscented Particle Filter, by R van der Merwe, A Doucet, JFG de Freitas and E Wan, May 2000.UPF的 Matlab software 也可以获得。

Gatsby reading group on nonlinear dynamical systems

其他 Kalman 滤波工具包和状态空间模块

• The system identification toolbox Mathworks的工具，可以运行很多经典算法
• ARfit 是自回归模块的很好工具包
• Zoubin Ghahramani  拥有LDS EM的matlab代码，和本人写的类似，并用c语言编写
• KBF, 用矩阵实现Kalman滤波平滑器，一个matlab的更快版本
• Le Sage′s econometrics toolbox, 包括很多matlab的时间序列模块化函数
• Econometric Links Econometrics Journal. 在matlab中，大多数软件是在高斯分布下写的
• SSF pack  是为状态空间滤波用C语言写的一条指令
• Stamp  是一个为结构时间序列分析开发的商业工具包。Statistical Time Series Analysis Toolbox O matrix Statistical Time Series Analysis Toolbox

推荐阅读

• Welch & Bishop, Kalman filter web page, 初学者乐园.
• T. Minka, "From HMMs to LDSs", 学术报告.
• T. Minka, Bayesian inference in dynamic models -- an overview , 学术报告, 2002
• K. Murphy. "Filtering, Smoothing, and the Junction Tree Algorithm",. tech. 报告, 1998.
• Roweis, S. and Ghahramani, Z. (1999) A Unifying Review of Linear Gaussian Models Neural Computation 11(2):305--345.
• M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, "A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking," IEEE Transactions on Signal Processing, Volume 50, Number 2, February 2002, pp 174-189.
• A. Doucet, N. de Freitas and N.J. Gordon, "Sequential Monte Carlo Methods in Practice", Springer-Verlag, 2000

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英文版：

http://www.cs.ubc.ca/~murphyk/Software/Kalman/kalman.html

Kalman filter toolbox for Matlab

Written by Kevin Murphy, 1998.
Last updated: 7 June 2004.

This toolbox supports filtering, smoothing and parameter estimation (using EM) for Linear Dynamical Systems.

• Download toolbox
• What is a Kalman filter?
• Example of Kalman filtering and smoothing for tracking
• What about non-linear and non-Gaussian systems?
• Other software for Kalman filtering, etc.
• Recommended reading
  

Functions kalman_filter kalman_smoother - implements the RTS equations learn_kalman - finds maximum likelihood estimates of the parameters using EM sample_lds - generate random samples AR_to_SS - convert Auto Regressive model of order k to State Space form SS_to_AR learn_AR - finds maximum likelihood estimates of the parameters using least squares What is a Kalman filter?

For an excellent web site, see . For a brief intro, read on...

A Linear Dynamical System is a partially observed stochastic process with linear dynamics and linear observations, both subject to Gaussian noise. It can be defined as follows, where X(t) is the hidden state at time t, and Y(t) is the observation. x(t+1) = F*x(t) + w(t), w ~ N(0, Q), x(0) ~ N(X(0), V(0)) y(t) = H*x(t) + v(t), v ~ N(0, R)

The Kalman filter is an algorithm for performing filtering on this model, i.e., computing P(X(t) | Y(1), ..., Y(t)).
The Rauch-Tung-Striebel (RTS) algorithm performs fixed-interval offline smoothing, i.e., computing P(X(t) | Y(1), ..., Y(T)), for t <= T.

The mean squared error of the filtered estimate is 4.9; for the smoothed estimate it is 3.2. Not only is the smoothed estimate better, but we know that it is better, as illustrated by the smaller uncertainty ellipses; this can help in e.g., data association problems. Note how the smoothed ellipses are larger at the ends, because these points have seen less data. Also, note how rapidly the filtered ellipses reach their steady-state (Ricatti) values. (Click here to see the code used to generate this picture, which illustrates how easy it is to use the toolkit.) What about non-linear and non-Gaussian systems?

For non-linear systems, I highly recommend the Matlab package, which implements the extended Kalman filter, the unscented Kalman filter, etc. (See , S Julier and J Uhlmann, Proc. IEEE, 92(3), 401-422, 2004. Also, a small .)

For systems with non-Gaussian noise, I recommend (PF), which is a popular sequential Monte Carlo technique. See also this . and the following tutorial: M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, IEEE Transactions on Signal Processing, Volume 50, Number 2, February 2002, pp 174-189 (