第四章-(1)-概率基础及滤波器原理

Catalogue

1. 基于滤波的融合方法
2. 概率相关基础
3. 滤波器基本原理

基于滤波的融合方法

概率相关基础

贝叶斯公式和贝叶斯推断——核心

高斯概率密度函数——核心

上式说明，如果有(x,y)的联合概率密度函数，同时又有y的分布，那么就可以求出x关于y的条件概率密度，假设y是传感器观测的分布，而x是待求的状态量的话，那么相当于可以求出后验概率

上图专门用来求后面观测量的均值和方差的

滤波器基本原理

滤波器的目的是求最大后验概率

贝叶斯滤波器

详细推导

待估计的后验概率密度：

\[ \begin{aligned} p(x_k|x_0,v_{1:k},y_{0:k}) \end{aligned} \]

意思为，给定

初始状态\(x_0\)

1-k时刻的输入\(v_{1:k}\)

以及0-k时刻的观测\(y_{0:k}\)

发生状态为\(x_k\)的概率，我们后面的操作，就是为了求解\(x_k\)，使得这个概率最大化

\[ \begin{aligned} p(x_k|x_0,v_{1:k},y_{0:k})&= p(x_k|x_0,v_{1:k},y_{0:k-1},y_k) \\ &= \frac{p(x_k,x_0,v_{1:k},y_{0:-1},y_k)}{p(x_0,v_{1:k},y_{0:k-1},y_k)} \\ &=\frac{p(y_k|x_k,x_0,v_{1:k},y_{0:k-1}) p(x_k|x_0,v_{1:k},y_{0:k-1}) p(x_0,v_{1:k},y_{0:k-1})}{p(y_k|x_0,v_{1:k},y_{0:k-1}) p(x_0,v_{1:k},y_{0:k-1})} \\ &= \frac{p(y_k|x_k,x_0,v_{1:k},y_{0:k-1}) p(x_k|x_0,v_{1:k},y_{0:k-1})}{p(y_k|x_0,v_{1:k},y_{0:k-1})} \\ &= \eta p(y_k|x_k,x_0,v_{1:k},y_{0:k-1}) p(x_k|x_0,v_{1:k},y_{0:k-1}) \end{aligned} \]

又因为观测方程中，\(y_k\)仅和\(x_k\)有关，所以进一步简化：

\[ p(x_k|x_0,v_{1:k},y_{0:k})=\eta p(y_k|x_k) p(x_k|x_0,v_{1:k},y_{0:k-1}) \]

接下来进一步化简上式右侧一项\(p(x_k|x_0,v_{1:k},y_{0:k-1})\)

根据\(p(x)=\int p(x|y) p(y) dy\)可得:

\[ \begin{aligned} p(x_k|x_0,v_{1:k},y_{0:k-1}) &= \frac{p(x_k,x_0,v_{1:k},y_{0:k-1})}{p(x_0,v_{1:k},y_{0:k-1})} \\ &= \int \frac{p(x_k,x_0,v_{1:k},y_{0:k-1}|x_{k-1}) p(x_{k-1})}{p(x_0,v_{1:k},y_{0:k-1})} d x_{k-1} \\ &= \int \frac{p(x_k,x_{k-1},x_0,v_{1:k},y_{0:k-1})}{p(x_{k-1},x_0,v_{1:k},y_{0:k-1})} \frac{p(x_{k-1},x_0,v_{1:k},y_{0:k-1})}{p(x_0,v_{1:k},y_{0:k-1})} d x_{k-1} \\ &= \int p(x_k|x_{k-1},x_0,v_{1:k},y_{0:k-1}) p(x_{k-1}|x_0,v_{1:k},y_{0:k-1}) d x_{k-1} \\ &= \int p(x_k| x_{k-1},v_k) p(x_{k-1}|x_0,v_{1:k},y_{0:k-1}) d x_{k-1} \end{aligned} \]