Particle Filter簡要介紹（Latex）

最近推導了一遍Particle Filter的原理

有了更詳細的理解，寫了一個小總結。主要是關於自己的理解。

因為是用Latex寫的，所以沒辦法黏貼過來了，

所以就把Latex的代碼黏貼在了下面。希望對同樣學習中的同學有所協助。

\documentclass[paper=a4, fontsize=11pt]{scrartcl} % A4 paper and 11pt font size\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs\usepackage{fourier} % Use the Adobe Utopia font for the document - comment this line to return to the LaTeX default\usepackage[english]{babel} % English language/hyphenation\usepackage{amsmath,amsfonts,amsthm} % Math packages\usepackage{lipsum} % Used for inserting dummy 'Lorem ipsum' text into the template\usepackage{sectsty} % Allows customizing section commands\allsectionsfont{\centering \normalfont\scshape} % Make all sections centered, the default font and small caps\usepackage{fancyhdr} % Custom headers and footers\pagestyle{fancyplain} % Makes all pages in the document conform to the custom headers and footers\fancyhead{} % No page header - if you want one, create it in the same way as the footers below\fancyfoot[L]{} % Empty left footer\fancyfoot[C]{} % Empty center footer\fancyfoot[R]{\thepage} % Page numbering for right footer\renewcommand{\headrulewidth}{0pt} % Remove header underlines\renewcommand{\footrulewidth}{0pt} % Remove footer underlines\setlength{\headheight}{13.6pt} % Customize the height of the header\numberwithin{equation}{section} % Number equations within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\numberwithin{figure}{section} % Number figures within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\setlength\parindent{0pt} % Removes all indentation from paragraphs - comment this line for an assignment with lots of text%----------------------------------------------------------------------------------------%TITLE SECTION%----------------------------------------------------------------------------------------\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height\title{\normalfont \normalsize\huge Particle Filter Study \\ % The assignment title}\author{Truman Nie} % Your name\date{\normalsize\today} % Today's date or a custom date\begin{document}\maketitle % Print the title%----------------------------------------------------------------------------------------%PROBLEM 1%----------------------------------------------------------------------------------------\section{What is Particle Filter?}Of course, Particle filter must be a best optimizing model. It can be used in may application such as tracking, locationrecommendation. I pay my attention in it because I do my research on tracking person and other target. I often use particlefilter to build some project. However, I find that I do not know it accurately. I decide some time to know about thisoptimizing model.\\\\Next section, I will explain it in my opinion.The goal of particle filter aims to estimate the sequence of hidden parameters, $x_k$ for $k=0,1,2,3.....$,based only on theobserved data $y_k$ for $k=0,1,2,3,....$, All Bayesian estimates of $x_k$ follow from the posterior distribution$p(x_k|y_0,y_1,.....,y_k)$. in contrast, the MCMC or improtance sampling approach would model the full posterior$p(x_0,x_1,....,x_k|y_0,y_1,.....y_k)$.\\\\I think all particle filter problem should convert to the follow type:\begin{align}\begin{split}x_k = g(x_{k-1})+w_k\\y_k = h(x_k)+v_k\end{split}\end{align}where both $w_k$ and $v_k$ are mutually independent and identically distributed sequences with known probability densityfunctions and $g(.)$ and $h(.)$ are known functions. These two equations can be view as state space equations and look similarto the state space equations for the Kalman filter. If the functions $g(.)$ and $h(.)$ are linear, and if both $w_k$ and $v_k$are Gaussian, the kalman fiter finds the exact Bayesian fitering distribution. If not, Kalman filter based methods are afirst-order approximation(EKF) or a second-order approximation(UKF in general, but if probability distribution is gaussian athird-order approximation is possible). Particle fiters are also an approximation, but with enough particles they can be muchmore accurate. \\\\In this filed, particle filter can get better result then Kalman filter. However, it also need enough particels, which willdirectly increase the calculated amount of system.%------------------------------------------------\section{Monte Carlo Approximation}Particle methods, like all sampling-based approaches(e.g.,MCMC), generate a set of samples that approximate the filteringdistribution $p(x_k|y_0,....,y_k)$. So, with $P$ samples, expectations with respect to the filtering distribution areapproximated by\begin{align}\int f(x_k)p(x_k|y_0,...,y_k)dx_k\approx\frac{1}{P}\sum_{L=1}^{P}f(x_{k}^{L})\end{align}and $f(.)$, in the usual way for monte Carlo, can give all the moments etc. of the distribution up to some degree ofapproximation.\\\\Monte Carlo only is a function or define. It allows us using some or little samples to estimate the whole distribution. We justneed to remember this formula.\\%------------------------------------------------\section{Sequential Importance Resampling (SIR)}Sequential importance resampling (SIR), the original particle filtering algorithm, is a very commonly used particle filteringalgorithm, which approximates the filtering distribution $p(x_k|y_0,...,y_k)$ by a weighted set of $P$ particles. such as:\begin{align*}\begin{Bmatrix}(w_k^L,x_k^L):L\in{1,...,P}\end{Bmatrix}\end{align*}The importance weights $w_k$ must satisfy\begin{align*}\sum_{L=1}^{P}w_k^L=1\end{align*}Resampling is used to avoid the problem of degeneracy of the algorithm, that is , avoiding the situation that all but one ofthe importance weights are close to zero. The performance of the algorithm can be also affected by proper choice of resamplingmethod. In the next section, I will explain about particle filter step.%----------------------------------------------------------------------------------------%PROBLEM 2%----------------------------------------------------------------------------------------\section{Particle Filter Step}In this section. I will give a single step of sequential of particle filter. \\1)We need to use sample to estimate the PDF, so we need to initialize the first set of sample. $G_0={x_0,...x_k,w_0,...,w_k}$\\2)we use function 1.1 to get the next state of $x_k$.\begin{align}x_k^L\sim f(x_k|x_{0:k-1}^L,y_{0:k})\end{align}3)for $L=1,....,P$ update the importance weights up to a normalizing constant:\begin{align}w_k^L=w_{k-1}^Lp(y_k|x_k^L)\end{align}4)then we need to compute the normalized importance weights:\begin{align}w_k^L=frac{\hat{w}_k^L}{sum_{J=1}^{P}\hat{w}_k^J}\end{align}5)compute an estimate of the effective number of particles as\begin{align}\hat{N}_eff=frac{1}{sum_{L=1}{P}(w_k^L)^2}\end{align}6)if the effective number of particles is less than a given threshold $hat{N}_eff<N_thr$, then perform resampling:\\\\a)draw $P$ particles from the current particle set with probabilities proportional to their weights. Replace the currentparticle set with this new one.\\b)for $L=1,...,P$ set $w_k^L=1/P$ \\\\This is a the original particle filter. The term sampling importance resampling is also sometimes used then referring to SIRfilters.\section{Some Question}I think some people must want to ask Monte Carlo do what? This is my question when I first see particle filter. \\\\In the above section. we could find function of updating weight. How to get this update function? Yes. we need Monte Carlo toget this update function. In the next step, I will explain it carefully.\\\\Why we need weight? Because, we need know which element is important and which elements can be deleted. In the other word, weneed to know the probability of element or particles. Now, we assume that we have the prior distribution $p(U)$ and the sample${u_i}$ and weight $w_i$. We need to get the posterior distribution $p(U|V=v_0)$ and the weight $w'_i$.\begin{align}\begin{split}p(U|V=v_0)&=\frac{p(V=v_0|U)p(U)}{\int p(V=v_0|U)p(U)dU}\\&=\frac{p(V=v_0|U)p(U)}{K}\end{split}\end{align}Then we should get a approximate explain:\begin{align}k=\int p(V=v_0|U)p(U)dU\simeq \frac{1}{N}\sum_{i=1}^{N}p(V=v_0|u_i)w_i\end{align}So we could get the expectation of posterior distribution:\begin{align}\begin{split}\int p(U|V=v_0)g(U)du&=\frac{1}{K}\int p(V=v_0|U)p(U)g(U)dU\\&\simeq \frac{1}{K}\frac{1}{N}\sum_{i-1}^N p(V=v_0|U_i)g(u_i)w_i\\&\simeq frac{\sum_{i-1}^{N}p(V=v_0|U)g(u_i)w_i}{\sum_{i-1}^N p(V=v_0|u_i)w_i}\end{split}\end{align}Now we could find that $p(V=v_0|u_i)w_i$ equal the weight of posterior distribution. In other word is:\begin{align}w'_i=p(V=v_0|u_i)w_i\end{align}Until now, we could get the update function of weight, which is used in the particle system. \\\\In my opinion, particle filter is not only a filter, but also a framework. If we could build a match model like the function1.1, we could use particle filter to get the best optimal solution in many different project.\end{document}