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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%   Presentation for:
+%
+%   Workshop on Adaptive Filters in Bucure\c{s}ti / Romania
+%   (March 2003)
+%
+%   (c) Matthias M\"{u}hlich, 03/2003
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+% this file must be processed with pdflatex!
+
+
+
+\documentclass[english,pdftex]{article}
+
+\usepackage{babel}          %% hyphenation patterns - takes global option english
+\usepackage{palatino}       %% Palatino fonts
+\usepackage{mathptm}        %% PostScript Type 1 math fonts
+\usepackage{textcomp}       %% symbols
+\usepackage[paneltoc,sectionbreak]{pdfwin}  %% my presentation style file
+                                            %% takes global options english + pdftex
+
+
+
+% character protruding; disabled for higher compatibility
+%\input{normprot.tex}
+
+% abbreviation for vectors
+\input{shortvec.tex}
+
+
+
+% SETUP OF PDFWIN PACKAGE
+%
+% define windows and margins
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+\SetWindow{text}{basex=0.2cm, basey=0.2cm, width=9.4cm,height=8.6cm,
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+\SetButtons{width=1.6cm, shadowdepth=0.3mm}
+\SetMargins{.5cm}{.5cm}{.5cm}{.5cm}
+%
+% define panel
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+\SetScreen{type=wallpaper, filename=marble.png}
+\SetPaneltext{\scriptsize\sffamily Filter-Workshop\\Bucure\c{s}ti
+2003}
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+
+
+% COLORS
+%
+% windows
+\definecolor{titlecolor}{rgb}{.7,.15,.1}
+\definecolor{TextBackgroundColor}{rgb}{.7,.8,1.0}
+\definecolor{TextBorderColor}{rgb}{0,0,.5}
+%
+% colors for presentation
+\definecolor{StateColor}{rgb}{0,0,.6}
+\definecolor{StateFuncColor}{rgb}{.4,0,.3}
+\definecolor{StateNoiseColor}{rgb}{.3,.3,.7}
+\definecolor{MeasColor}{rgb}{0,.5,0}
+\definecolor{MeasFuncColor}{rgb}{.4,.3,0}
+\definecolor{MeasNoiseColor}{rgb}{.3,.7,.3}
+\definecolor{InputColor}{rgb}{.7,0,0}
+%
+% colors for environments
+\definecolor{CodeTextColor}{rgb}{0,0.3,0}
+\definecolor{CodeMathColor}{rgb}{0.1,0.3,0.6}
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+%
+% misc
+\definecolor{dgreen}{rgb}{0,.6,0}
+
+
+% LAYOUT
+%
+% general layout
+\tolerance=2000
+\emergencystretch=5em
+\fboxsep=3mm
+\setlength{\parskip}{4pt plus 2pt minus 1pt}
+\setlength{\parindent}{0pt}
+\frenchspacing
+%
+% define page transition style
+\pdfpageattr{/Trans << /S /Dissolve /D 0.3 >>}
+%
+% replace default font (roman default) by computer modern sans serif;
+% Palatino only for panel
+% NOTE: brute force, no good style here...
+\renewcommand{\rmdefault}{cmss}
+%
+% redefine vector style (boldface instead of arrow on top)
+\renewcommand{\vec}[1]{\mathbf{#1}}
+%
+% redefine section numbering: only section (and no subsection or lower)
+% gets a number
+\setcounter{secnumdepth}{1}
+
+
+% NEW COMMANDS
+%
+\newcommand{\Realnumbers}{\mathrm{I\!R}}
+\newcommand{\Expect}[1]{\mbox{{\sf E}}\left[ #1 \right]}
+\newcommand{\Cov}[1]{\mbox{\sf Cov}\left[ #1 \right]}
+%
+\newcommand{\mybox}[1]{\begin{center}%
+  \fboxrule=.5mm%
+  \fcolorbox{MyBoxColor1}{MyBoxColor2}{%
+  \parbox[c]{.9\textwidth}{#1}}%
+\end{center}}
+%
+\newcommand{\myboxii}[1]{\begin{center}%
+  \fboxrule=.5mm%
+  \fcolorbox{MyBoxiiColor1}{MyBoxiiColor2}{%
+  \parbox[c]{.9\textwidth}{#1}}%
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+%
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+  \ttfamily\bfseries%
+  \color{CodeTextColor}%
+  \fcolorbox{MyCodeBoxColor1}{MyCodeBoxColor2}{%
+  \parbox[c]{.9\textwidth}{#1}}%
+\end{center}}
+%
+\newcommand{\codemath}[1]{\textcolor{CodeMathColor}{$#1$}}
+\newcommand{\codetext}[1]{\textsf{\mdseries\small #1}}
+
+
+
+%====================================================================
+
+
+
+\begin{document}
+
+
+
+% produce panel TOC entry for title page
+\AddPanelTocEntry{Title Page}%
+
+{\centering {\Huge\bfseries\color{titlecolor}
+  Particle Filters\rule[-.6ex]{0pt}{5ex}\\
+  \LARGE an overview\rule[-.6ex]{0pt}{3ex}\\
+} \vspace*{8mm}
+{\Large Matthias M\"{u}hlich}\\[3ex]
+  \large Institut f{\"u}r Angewandte Physik\\
+        J.W.Goethe-Universit{\"a}t Frankfurt\\
+  \href{mailto:muehlich@iap.uni-frankfurt.de}{\color{black}muehlich@iap.uni-frankfurt.de}\\
+}
+
+
+
+
+\newpage
+
+% replacement title page
+{\centering {\Huge\bfseries\color{titlecolor}
+  Particle Filters\rule[-.6ex]{0pt}{5ex}\\
+  \LARGE a tutorial\rule[-.6ex]{0pt}{3ex}\\
+} \vspace*{8mm}
+{\Large Matthias M\"{u}hlich}\\[3ex]
+  \large Institut f{\"u}r Angewandte Physik\\
+        J.W.Goethe-Universit{\"a}t Frankfurt\\
+  \href{mailto:muehlich@iap.uni-frankfurt.de}{\color{black}muehlich@iap.uni-frankfurt.de}\\
+}
+
+
+
+
+\section{Introduction}
+\label{Intro}
+
+% change text window style from normal (-> title page) to transparent
+% (-> rest of talk); must appear after page break, i.e. section command
+% ("sectionbreak" option for pdfwin produces page break at each \section
+% command)
+\SetWindow{text}{type=transparent, borderthickness=0.5mm}
+
+
+An increasing number of researchers is using a family of
+techniques and algorithms called
+\begin{itemize}
+\itemsep=-1mm
+    \item \textit{condensation algorithms}
+    \item \textit{bootstrap filtering}
+    \item \textit{particle filters}
+    \item \textit{interacting particle approximations}
+    \item \textit{sequential Monte Carlo methods}
+    \item \textit{SIS, SIR, ASIR, RPF, \dots}
+\end{itemize}
+Time scale: last 10 years [e.g. Isard \& Blake 1996; Kitagawa
+1996; Gordon, Salmond \& Smith 1993]
+%
+\mybox{The question of this talk is: What is behind all that?}
+
+
+
+\newpage
+\subsection{General Classification of %On-line
+Filter Strategies}
+
+Gaussian models:
+\begin{itemize}
+   \item Kalman filter
+   \item extended Kalman filter
+   \item linear-update filter / linear regression filter /\\
+      statistical linearization filter
+   \begin{itemize}
+      \item unscented filter
+      \item central difference filter
+      \item divided difference filter
+   \end{itemize}
+   \item assumed density filter / moment matching
+\end{itemize}
+
+\newpage
+Mixture of Gaussian models:
+\begin{itemize}
+   \item assumed density filter / pseudo-Bayes
+   \item Gaussian-sum filter
+\end{itemize}
+
+\bigskip
+Nonparametric models:
+\begin{itemize}
+   \item \textcolor{red}{\bfseries particle filter class}
+   \item histogram filter
+\end{itemize}
+
+
+\newpage
+\subsection{Some Basic Remarks}
+
+\begin{itemize}
+  \item various applications: computer vision (i.e. tracking),
+  control theory, econometrics (stock markets, monetary flow, interest
+  rates), \dots
+  \item we deal with discrete time systems only
+  \item no out-of-sequence measurements
+  \item we are mainly interested in estimating the state at time
+  $k$ from measurements up to time $k'=k$ (opposite: smoothing
+  ($k'>k$) and prediction ($k'<k$); furthermore $k'$ need not be
+  fixed\dots)
+  \item no restrictions to linear processes or Gaussian noise!
+\end{itemize}
+
+
+\inithighlight{\subsection*{Overview of this Talk}
+\begin{center}
+\begin{itemize}
+\item
+    {\color{color1} The Dynamic System Model}
+\item
+    {\color{color2} Bayesian Filter Approach}
+\item
+    {\color{color3} Optimal and Suboptimal Solutions}
+\item
+    {\color{color4} The Particle Filter}
+\item
+    {\color{color5} Experiments and Summary}
+\end{itemize}
+\end{center}
+\vspace{4mm} }
+
+
+\highlightnext
+\pagenumbering{incremental}
+
+-- states of a system and state transition equation
+
+-- measurement equation
+
+
+\highlightnext
+
+-- estimation of the state
+
+-- probabilistic modelling
+
+-- Bayesian filter
+
+
+\highlightnext
+
+-- filtered pdf can be written down easily, but it is not always
+tractable ($\rightarrow$ ugly integrals \dots)
+
+-- conditions under which optimal solutions exist: Kalman filter
+and grid-based filter
+
+-- what can be done in other cases: suboptimal approaches
+
+
+\highlightnext
+
+-- standard particle filter
+
+-- various improved versions
+
+
+\highlightnext
+
+-- some experimental data and conclusion
+
+
+
+\newpage
+\pagenumbering{restore}
+
+
+\section{Dynamic System}
+
+A dynamic system can be modelled with two equations:
+
+\subsection{State Transition or Evolution Equation}
+ \[
+    {\color{StateColor}\vx_k} = {\color{StateFuncColor}f_k}({\color{StateColor}\vx_{k-1}},
+    {\color{InputColor}\vu_{k-1}},{\color{StateNoiseColor}\vv_{k-1}})
+ \]
+
+${\color{StateFuncColor}f}(\cdot,\cdot,\cdot)$:
+{\color{StateFuncColor}evolution
+function} (possible non-linear) \\
+${\color{StateColor}\vx_k}, {\color{StateColor}\vx_{k-1}} \in
+\Realnumbers^{n_x}$:
+current and previous {\color{StateColor}state} \\
+${\color{StateNoiseColor}\vv_{k-1}} \in \Realnumbers^{n_v}$:
+{\color{StateNoiseColor}state noise}
+(usually \emph{not} Gaussian)\\
+${\color{InputColor}\vu_{k-1}} \in
+\Realnumbers^{n_u}$: known {\color{InputColor}input}
+
+\bigskip
+Note: state only depends on previous state, i.e. first order
+Markov process
+
+
+
+
+\newpage
+\subsection{Measurement Equation}
+ \[
+    {\color{MeasColor}\vz_k} = {\color{MeasFuncColor}h_k}({\color{StateColor}\vx_k},
+    {\color{InputColor}\vu_k},{\color{MeasNoiseColor}\vn_k})
+ \]
+
+${\color{MeasFuncColor}h}(\cdot,\cdot,\cdot)$:
+{\color{MeasFuncColor}measurement
+function} (possible non-linear) \\
+${\color{MeasColor}\vz_k} \in
+\Realnumbers^{n_z}$: {\color{MeasColor}measurement} \\
+${\color{StateColor}\vx_k} \in
+\Realnumbers^{n_x}$: {\color{StateColor}state} \\
+${\color{MeasNoiseColor}\vn_k} \in \Realnumbers^{n_n}$:
+{\color{MeasNoiseColor}measurement noise}
+(usually \emph{not} Gaussian)\\
+${\color{InputColor}\vu_k} \in
+\Realnumbers^{n_u}$: known {\color{InputColor}input}
+
+\bigskip
+(dimensionality of {\color{StateColor}state},
+{\color{MeasColor}measurement}, {\color{InputColor}input},
+{\color{StateNoiseColor}state noise}, and
+{\color{MeasNoiseColor}measurement noise} can all be different!)
+
+
+
+
+\newpage
+\pagenumbering{incremental}
+\includegraphics[width=.99\textwidth]{BucSystem1}
+
+
+\newpage
+\includegraphics[width=.99\textwidth]{BucSystem2}
+
+
+\newpage
+\includegraphics[width=.99\textwidth]{BucSystem3}
+
+
+\newpage
+\includegraphics[width=.99\textwidth]{BucSystem4}
+
+
+\newpage
+\includegraphics[width=.99\textwidth]{BucSystem5}
+
+
+\newpage
+\includegraphics[width=.99\textwidth]{BucSystem6}
+
+Assumptions:
+
+The observations are conditionally independent given the state:
+$p(\vz_k|\vx_k)$.
+
+Hidden Markov Model (HMM):\\
+$p(\vx_0)$ given and $p(\vx_k|\vx_{k-1})$ defines state transition probability for $k \ge 1$.
+
+
+\newpage
+\pagenumbering{restore}
+
+
+
+
+\section{Bayesian Filters}
+
+
+\subsection{Estimating the Posterior}
+
+Bayesian approach: We attempt to construct the posterior pdf of
+the state given all measurements.
+
+$\Rightarrow$ can be termed a complete solution to the estimation
+problem because all available information is used; from the pdf,
+an optimal estimate can theoretically be found for any criterion.
+
+in detail: We seek estimates of $\vx_k$ based on all available
+measurements up to time $k$ (abbreviated as $\vz_{1:k}$) by
+constructing the posterior $p(\vx_k|\vz_{1:k})$.
+
+Assumption: initial state pdf (prior) $p(\vx_0)$ is given
+
+
+
+\newpage
+\subsection{The Use of Knowing the Posterior}
+
+Let $f_k : \Realnumbers^{(k+1)\times n_x} \to \Realnumbers$ be any
+arbitrary (integrable) function that can depend
+\begin{itemize}
+  \item on all components of the state $\vx$
+  \item on the whole trajectory in state-space
+\end{itemize}
+
+Examples: This function can be an estimator for the current state
+or for future observations.
+
+Then we can compute its expectation using
+\[
+    \Expect{f_k(\vx_{0:k})} = \int f(\vx_{0:k})
+    p(\vx_{0:k}|\vz_{1:k}) d\vx_{0:k}
+\]
+
+MMSE estimate of state: $\hat{\vx} = \Expect{\vx_k}$. Other
+estimates that can be computed: median, modes, confidence
+intervals, kurtosis, \dots
+
+
+
+\newpage
+\subsection{Recursive Filters}
+
+recursive filters (i.e. sequential update of previous estimate)
+$\leftrightarrow$ batch processing (computation with all data in
+one step)
+
+not only faster: allow on-line processing of data (lower storage
+costs, rapid adaption to changing signals characteristics)
+
+essentially consist of two steps:
+ \begin{description}
+ \item[prediction step:] $p(\vx_{k-1}|\vz_{1:k-1}) \to
+ p(\vx_{k}|\vz_{1:k-1})$ \\
+ (usually deforms / translates / spreads state pdf due to noise)
+ \item[update step:] $p(\vx_{k}|\vz_{1:k-1}),\vz_k \to p(\vx_{k}|\vz_{1:k})$\\
+ (combines likelihood of current measurement with predicted state;
+ usually concentrates state pdf)
+ \end{description}
+
+
+\newpage
+\subsection{General Prediction-Update Framework}
+
+Assume that pdf $p(\vx_{k-1}|\vz_{1:k-1})$ is available at time
+$k-1$.
+
+Prediction step: (using Chapman-Kolmogoroff equation)
+\begin{equation}
+ \label{eq:predict}
+  p(\vx_k|\vz_{1:k-1}) = \int
+  p(\vx_k|\vx_{k-1}) p(\vx_{k-1}|\vz_{1:k-1}) d\vx_{k-1}
+\end{equation}
+This is the prior of the state $\vx_k$ at time $k$ \emph{without
+knowledge of the measurement $\vz_k$}, i.e. the probability
+\emph{given only previous measurements}.
+
+Update step: (compute posterior pdf from
+predicted prior pdf and new measurement)
+\begin{equation}
+ \label{eq:update}
+  p(\vx_k|\vz_{1:k}) = \frac{p(\vz_k|\vx_k)p(\vx_k|\vz_{1:k-1})}{p(\vz_{k}|\vz_{1:k-1})}
+\end{equation}
+
+
+
+
+
+\newpage
+\pagenumbering{incremental}
+
+\def\temp{Let us prove formula (\ref{eq:update}) (just in order to train calculations
+with joint and conditional probabilities\dots)}
+\temp
+\begin{eqnarray*}
+    && \hspace*{75mm} \\[-5mm]
+    && \textcolor{blue}{p(\vx_k|\vz_{1:k})} \\
+    &&= \frac{\textcolor{blue}{p(\vz_{1:k}|\vx_k)p(\vx_k)}}{\textcolor{blue}{p(\vz_{1:k})}}
+\end{eqnarray*}
+(Bayes rule)
+
+
+\newpage
+\temp
+\begin{eqnarray*}
+    && \hspace*{75mm} \\[-5mm]
+    && p(\vx_k|\vz_{1:k}) \\
+    &&= \frac{\textcolor{blue}{p(\vz_{1:k}|\vx_k)}p(\vx_k)}{\textcolor{dgreen}{p(\vz_{1:k})}} \\
+    &&= \frac{\textcolor{blue}{p(\vz_{k},\vz_{1:k-1}|\vx_k)}p(\vx_k)}{\textcolor{dgreen}{p(\vz_{k},\vz_{1:k-1})}}
+\end{eqnarray*}
+(separate $p(\vz_{1:k})$ into $p(\vz_k,\vz_{1:k-1})$)
+
+
+\newpage
+\temp
+\begin{eqnarray*}
+    && \hspace*{75mm} \\[-5mm]
+    && p(\vx_k|\vz_{1:k}) \\
+    &&= \frac{p(\vz_{1:k}|\vx_k)p(\vx_k)}{p(\vz_{1:k})} \\
+    &&= \frac{\textcolor{blue}{p(\vz_{k},\vz_{1:k-1}|\vx_k)}p(\vx_k)}{\textcolor{dgreen}{p(\vz_{k},\vz_{1:k-1})}} \\
+    &&= \frac{\textcolor{blue}{p(\vz_{k}|\vz_{1:k-1},\vx_k)p(\vz_{1:k-1}|\vx_k)}
+        p(\vx_k)}{\textcolor{dgreen}{p(\vz_{k}|\vz_{1:k-1})p(\vz_{1:k-1})}}
+\end{eqnarray*}
+(factorize joint probability: $p(a,b|c) = p(a|b,c)\cdot p(b|c)$
+and $p(a,b) = p(a|b)\cdot p(b)$)
+
+
+\newpage
+\temp
+\begin{eqnarray*}
+    && \hspace*{75mm} \\[-5mm]
+    && p(\vx_k|\vz_{1:k}) \\
+    &&= \frac{p(\vz_{1:k}|\vx_k)p(\vx_k)}{p(\vz_{1:k})} \\
+    &&= \frac{p(\vz_{k},\vz_{1:k-1}|\vx_k)p(\vx_k)}{p(\vz_{k},\vz_{1:k-1})} \\
+    &&= \frac{p(\vz_{k}|\vz_{1:k-1},\vx_k)\textcolor{blue}{p(\vz_{1:k-1}|\vx_k)}
+        p(\vx_k)}{p(\vz_{k}|\vz_{1:k-1})p(\vz_{1:(k-1}))} \\
+    &&= \frac{p(\vz_{k}|\vz_{1:k-1},\vx_k)\textcolor{blue}{p(\vx_k|\vz_{1:k-1})p(\vz_{1:k-1})}
+        p(\vx_k)}{p(\vz_{k}|\vz_{1:k-1})p(\vz_{1:k-1})\textcolor{blue}{p(\vx_k)}}
+\end{eqnarray*}
+(Bayes rule)
+
+
+\newpage
+\temp
+\begin{eqnarray*}
+    && \hspace*{75mm} \\[-5mm]
+    && p(\vx_k|\vz_{1:k}) \\
+    &&= \frac{p(\vz_{1:k}|\vx_k)p(\vx_k)}{p(\vz_{1:k})} \\
+    &&= \frac{p(\vz_{k},\vz_{1:k-1}|\vx_k)p(\vx_k)}{p(\vz_{k},\vz_{1:k-1})} \\
+    &&= \frac{p(\vz_{k}|\vz_{1:k-1},\vx_k)p(\vz_{1:k-1}|\vx_k)
+        p(\vx_k)}{p(\vz_{k}|\vz_{1:k-1})p(\vz_{1:(k-1}))} \\
+    &&= \frac{\textcolor{blue}{p(\vz_{k}|\vz_{1:k-1},\vx_k)}p(\vx_k|\vz_{1:k-1})\textcolor{dgreen}{p(\vz_{1:k-1})
+        p(\vx_k)}}{p(\vz_{k}|\vz_{1:k-1})\textcolor{dgreen}{p(\vz_{1:k-1})p(\vx_k)}} \\
+    &&= \frac{\textcolor{blue}{p(\vz_k|\vx_k)}p(\vx_k|\vz_{1:k-1})}{p(\vz_{k}|\vz_{1:k-1})}
+\end{eqnarray*}
+(independence of observations; cancelling out terms)
+
+
+
+\newpage
+\pagenumbering{restore}
+
+\subsection{The Structure of the Update Equation}
+
+\begin{eqnarray*}
+  p(\vx_k|\vz_{1:k}) &=& \frac{p(\vz_k|\vx_k)\cdot p(\vx_k|\vz_{1:k-1})}{p(\vz_{k}|\vz_{1:k-1})}
+  \\[1.5ex]
+  \mbox{posterior} &=& \frac{\mbox{likelihood}\cdot\mbox{prior}}{\mbox{evidence}}
+\end{eqnarray*}
+
+prior: given by prediction equation
+
+likelihood: given by observation model
+
+evidence: the normalizing constant in the denominator
+\[
+  p(\vz_{k}|\vz_{1:k-1}) = \int p(\vz_k|\vx_k)p(\vx_k|\vz_{1:k-1}) d\vx_k
+\]
+
+
+
+
+\newpage
+
+This theoretically allows an optimal Bayesian solution (in the
+sense of computing the posterior pdf).
+
+\mybox{Problem: only a conceptual solution; integrals are not
+tractable.}
+
+But: in some restricted cases, an optimal solution is possible.
+Two optimal solutions (under restrictive assumptions):
+\begin{itemize}
+  \item (standard) Kalman filter
+  \item grid-based filter
+\end{itemize}
+
+
+
+\section{Kalman Filter}
+
+\subsection{Introduction}
+
+Assumptions:
+\begin{itemize}
+  \item posterior at time $k-1$, i.e. $p(\vx_{k-1}|\vz_{k-1})$, is Gaussian
+  \item dynamic system characterized by
+  \begin{eqnarray*}
+    &{\color{StateColor}\vx_k} = {\color{StateFuncColor}\MF_k}{\color{StateColor}\vx_{k-1}}
+    +
+    {\color{StateFuncColor}\MG_k}{\color{StateNoiseColor}\vv_{k-1}}&
+    \\
+    &{\color{MeasColor}\vz_k} = {\color{MeasFuncColor}\MH_k}{\color{StateColor}\vx_k}
+    +
+    {\color{MeasFuncColor}\MJ_k}{\color{MeasNoiseColor}\vn_k}&
+  \end{eqnarray*}
+  \item both noise vectors Gaussian (covariance matrices are $\MQ_{k-1}$ and $\MR_k$)
+\end{itemize}
+Then new posterior $p(\vx_{k}|\vz_{k})$ is Gaussian, too, and can
+be computed using simple linear equations.
+
+optimal solution, but \emph{highly restrictive} assumptions must hold
+
+
+%\newpage
+\subsection{Prediction Equation}
+
+At time $k-1$: $p(\vx_{k-1}|\vz_{1:k-1}) = {\cal N}(\vm_{k-1|k-1}, \MP_{k-1|k-1})$
+
+Inserting into (\ref{eq:predict}) yields
+\[
+  p(\vx_k|\vz_{1:k-1}) = {\cal N}(\vm_{k|k-1}, \MP_{k|k-1})
+\]
+with
+\[
+  \vm_{k|k-1} = \MF_k\vm_{k-1|k-1}
+\]
+and
+\[
+  \MP_{k|k-1} = \MG_{k}\MQ_{k-1}\MG_{k}^T + \MF_k\MP_{k-1|k-1}\MF_k^T
+\]
+
+
+
+\newpage
+\subsection{Update Equation}
+
+Inserting into (\ref{eq:update}) yields
+\[
+  p(\vx_k|\vz_{1:k}) = {\cal N}(\vm_{k|k}, \MP_{k|k})
+\]
+with
+\[
+  \vm_{k|k} = \vm_{k|k-1} + \MK_k (\vz_k - \underbrace{\MH_k\vm_{k|k-1}}_{\mbox{\small estimated }\hat{\vz}_k})
+\]
+and
+\[
+  \MP_{k|k} = \MP_{k|k-1} - \MK_k\MH_k\MP_{k|k-1}
+\]
+Kalman Gain:
+\[
+  \MK_k = \MP_{k|k-1}\MH_k^T(\underbrace{\MH_k\MP_{k|k-1}\MH_k^T + \MJ_k\MR_k\MJ_k^T}_{\mbox{$\Cov{\hat{\vz}_k}$}})^{-1}
+\]
+
+
+
+
+
+
+\section{Grid-Based Filter}
+
+
+\subsection{Introduction}
+
+Assumptions:
+\begin{itemize}
+  \item state space is discrete
+  \item number of different states ($N_s$) is limited \\
+  {\footnotesize (Note: implicitly includes discreteness)}
+\end{itemize}
+
+Suppose at time $k-1$ we have states $\vx^i$ with
+$i=1,\ldots,N_s$. Conditional probability of these states:
+\[
+  Pr(\vx_{k-1} = \vx^i|\vz_{1:k-1}) = w_{k-1|k-1}^i
+\]
+Then the (old) posterior at time $k-1$ is given by:
+\[
+  p(\vx_{k-1}|\vz_{1:k-1}) = \sum_{i=1}^{N_s} w_{k-1|k-1}^i \; \delta(\vx_{k-1}-\vx^i)
+\]
+
+\newpage
+
+\subsection{Results (Summary)}
+
+Both the (new) prior and the (new) posterior have the same
+structure: a sum of weighted Dirac peaks:
+
+\[
+  p(\vx_{k}|\vz_{1:k-1}) = \sum_{i=1}^{N_s} w_{k|k-1}^i \; \delta(\vx_{k-1}-\vx^i)
+\]
+\[
+  p(\vx_{k}|\vz_{1:k}) = \sum_{i=1}^{N_s} w_{k|k}^i \; \delta(\vx_{k-1}-\vx^i)
+\]
+
+Note: extension to different sets of states for each time step
+\[
+  \{\vx^i\}: i=1,\ldots,N_s \quad\longrightarrow\quad \{\vx^i_k\}: i=1,\ldots,N_{s,k}
+\]
+with time-varying index $k$ is easily possible; the `allowed'
+states need not be constant.
+
+
+
+\newpage
+\subsection{Prediction Equation (in Detail)}
+
+Inserting into (\ref{eq:predict}) yields
+\begin{eqnarray*}
+  p(\vx_k|\vz_{1:k-1})
+  &=& \sum_{i=1}^{N_s} \sum_{j=1}^{N_s} w_{k-1|k-1}^j
+      p(\vx^i|\vx^j) \delta(\vx_{k-1}-\vx^i)\\
+  &=& \sum_{i=1}^{N_s} w_{k|k-1}^i \delta(\vx_{k-1}-\vx^i)
+\end{eqnarray*}
+where $w_{k|k-1}^i = \sum_{j=1}^{N_s} w_{k-1|k-1}^j
+p(\vx^i|\vx^j)$
+
+\bigskip
+\mybox{(new) prior weights = old posterior weights,\\
+reweighted using state transition probabilities}
+
+
+
+
+\newpage
+\subsection{Update Equation (in Detail)}
+
+Inserting into (\ref{eq:update}) yields
+\[
+  p(\vx_k|\vz_{1:k}) = \sum_i^{N_s} w_{k|k}^i \delta(\vx_{k-1}-\vx^i)
+\]
+where $w_{k|k}^i = \frac{w_{k|k-1}^i
+p(\vz_{k}|\vx^i)}{\sum_j^{N_s} w_{k|k-1}^j
+p(\vz_{k}|\vx^j)}$.
+
+Note: denominator only needed for normalization
+
+\bigskip
+\mybox{posterior weights = prior weights, reweighted using
+likelihoods}
+
+
+
+
+
+\section{Particle Filter}
+
+
+\subsection{Suboptimal Approximations}
+
+If we want to preserve Kalman filter principle\dots
+\begin{itemize}
+  \item Extended Kalman Filter (EKF)
+  \item Unscented Kalman Filter (UKF)
+\end{itemize}
+...we get better results, \mybox{BUT: \emph{we cannot get rid off
+Gaussian approximations}}
+
+\newpage
+
+EKF / UKF: \mybox{All these approaches fail if we have
+\begin{itemize}
+  \item bimodal / multimodal pdfs
+  \item heavily skewed pdfs
+\end{itemize}
+}
+
+We need a more general scheme to tackle these problems.
+
+
+\newpage
+\subsection{Particle Filter -- General Concept}
+
+Many different names (do you remember the
+\href{#section.1}{introduction}?) but the general concept is
+rather simple:
+
+\mybox{PARTICLE FILTER:
+
+If we cannot solve the integrals required for a Bayesian recursive
+filter analytically \dots we represent the posterior probabilities
+by a set of randomly chosen weighted samples.}
+
+Note: ``randomly chosen'' $\equiv$ ``Monte Carlo'' \\
+(we are playing roulette / throwing the dice)
+
+Increasing number of samples $\Rightarrow$ (almost sure)
+convergence to true pdf
+
+
+
+\newpage
+\subsection{Sequential Importance Sampling (SIS)}
+
+SIS is the basic framework for most particle filter algorithms.
+Let
+\begin{eqnarray*}
+    \{\vx_{0:k}^i\} &:& \mbox{set of support points
+    (samples, particles)} \\
+    && i = 1,\ldots,N_s \\
+    && \mbox{(whole trajectory for each particle!)} \\
+    w_{k}^i &:& \mbox{associated weights, normalized to $\sum_i w_k^i = 1$}
+\end{eqnarray*}
+Then:
+\[
+    p(\vx_k|\vz_{1:k}) \approx \sum_{i=1}^{N_s} w_k^i \delta(\vx_{0:k} - \vx^i_{0:k})
+\]
+(discrete weighted approximation to the true posterior)
+
+
+\newpage
+\subsection*{SIS (continued)}
+
+Usually we cannot draw samples $\vx_k^i$ from $p(\cdot)$ directly.
+Assume we sample directly from a (different) \emph{importance
+function} $q(\cdot)$. Our approximation is still correct (up to
+normalization) if
+\[
+    w_k^i \propto \frac{p(\vx^i_{0:k}|\vz_{1:k})}{q(\vx^i_{0:k}|\vz_{1:k})}
+\]
+\mybox{The trick: we can choose $q(\cdot)$ freely!}
+
+If the importance function is chosen to factorize such that
+\[
+    q(\vx_{0:k}|\vz_{1:k}) = q(\vx_{k}|\vx_{0:k-1},\vz_{1:k})\;q(\vx_{0:k-1}|\vz_{1:k-1})
+\]
+then one can augment old particles $\vx^i_{0:k-1}$ by $\vx_k \sim
+q(\vx_{k}|\vx_{0:k-1},\vz_{1:k})$ to get new particles
+$\vx^i_{0:k}$.
+
+
+\newpage
+\subsection*{SIS (continued)}
+
+Weight update (after some lengthy computations\dots):
+\begin{equation}
+ \label{eq:WeightUpdate}
+    w_k^i = w^i_{k-1}
+    \frac{p(\vz_k|\vx^i_k)\,p(\vx^i_{k}|\vx^i_{k-1})}{q(\vx^i_{k}|\vx^i_{0:k-1},\vz_{1:k})}
+\end{equation}
+Furthermore, if $q(\vx_{k}|\vx_{0:k-1},\vz_{1:k}) =
+q(\vx_{k}|\vx_{k-1},\vz_{1:k})$\\
+(only dependent on \emph{last} state and observations):
+\[
+    p(\vx|\vz_{1:k}) \approx \sum_{i=1}^{N_s} w_k^i \,\delta(\vx_k -
+    \vx_k^i)
+\]
+(and we need not preserve trajectories $\vx^i_{0:k-1}$ and history
+of observations $\vz_{1:k-1}$)
+
+
+\newpage
+\subsection{SIS Algorithm -- Pseudo Code}
+
+\mycodebox{
+ [\codemath{\{\vx^i_k,w^i_k\}_{i=1}^{N_s}}] = SIS(\codemath{\{\vx^i_{k-1},w^i_{k-1}\}_{i=1}^{N_s}},
+ \codemath{\vz_k}) \\
+ \hspace*{7mm} FOR \codemath{i} = \codemath{1} : \codemath{N_s} \\
+ \hspace*{14mm}   \codetext{draw} \codemath{\vx_k^i \sim q(\vx_{k}|\vx^i_{k-1},\vz_{k})} \\
+ \hspace*{14mm}   \codetext{update weights according to (\ref{eq:WeightUpdate})} \\
+ \hspace*{7mm} END FOR \\
+ \hspace*{7mm} \codetext{normalize weights to} \codemath{\sum_{i=1}^{N_s} w_k^i = 1}
+}
+
+
+
+\newpage
+\subsection{PROBLEM: Degeneracy Problem}
+
+Problem with SIS approach: after a few iterations, most particles
+have negligible weight (the weight is concentrated on a few
+particles only)
+
+Counter measures:
+\begin{itemize}
+  \item brute force: many, many samples $N_s$
+  \item good choice of importance density
+  \item resampling
+\end{itemize}
+
+Note: amount of degeneracy can be estimated based on variance of
+weights [Liu 1996].
+
+
+
+\newpage
+\subsection{Optimal Impotance Density:}
+
+It can be shown that the optimal importance density is given by
+\[
+    q(\vx_k|\vx_{k-1},\vz_k)_{opt} = p(\vx_k|\vx_{k-1},\vz_k)
+\]
+Then
+\[
+    w^i_k = w^i_{k-1} \int p(\vz_k|\vx'_k) p(\vx'_k|\vx^i_{k-1})
+    d\vx'_k
+\]
+Two major drawbacks: usually neither sampling from $q_opt$ nor
+solving the integral in $w^i_k$ is possible\dots (but in some
+special cases, it works)
+
+Other alternative which is often convenient: \\
+$q(\cdot) = p(\vx_k|\vx_{k-1})$ (prior). Easy to implement, but does not take
+measurements into account.
+
+
+
+\newpage
+\subsection{Resampling Approaches}
+
+Basic idea of resampling:
+ \mybox{Whenever degeneracy rises above threshold: replace old set of samples
+ (+ weights) with new set of samples (+ weights), such that sample density
+ better reflects posterior pdf.}
+
+This eliminates particles with low weight and chooses more
+particles in more probable regions.
+
+Complexity: possible in $O(N_s)$ operations
+
+
+\newpage
+The resampling principle:
+
+\begin{center}
+\includegraphics[width=0.8\linewidth]{BucResampling}
+\end{center}
+
+(graphics taken from Van der Merwe et al.)
+
+
+
+\newpage
+\subsection{General Particle Filter -- Pseudo Code}
+
+\mycodebox{
+ [\codemath{\{\vx^i_k,w^i_k\}_{i=1}^{N_s}}] = PF(\codemath{\{\vx^i_{k-1},w^i_{k-1}\}_{i=1}^{N_s}},
+ \codemath{\vz_k}) \\
+ \hspace*{7mm} FOR \codemath{i} = \codemath{1} : \codemath{N_s} \\
+ \hspace*{14mm}   \codetext{draw} \codemath{\vx_k^i \sim q(\vx_{k}|\vx^i_{k-1},\vz_{k})} \\
+ \hspace*{14mm}   \codetext{update weights according to (\ref{eq:WeightUpdate})} \\
+ \hspace*{7mm} END FOR \\
+ \hspace*{7mm} \codetext{normalize weights to} \codemath{\sum_{i=1}^{N_s} w_k^i = 1}\\
+ \hspace*{7mm} IF \codetext{degeneracy too high} \\
+ \hspace*{14mm}   \codetext{resample} \codemath{\{\vx^i_k,w^i_k\}_{i=1}^{N_s}} \\
+ \hspace*{7mm} END IF
+}
+
+
+
+
+\newpage
+\subsection{PROBLEM: Loss of Diversity}
+
+No degeneracy problem but new problem arises:
+%
+\mybox{Particles with high weight are selected more and more
+often, others die out slowly
+
+$\Rightarrow$ \emph{loss of diversity} or \emph{sample
+impoverishment}}
+
+For small process noise, all particles can collapse into a single
+point within a few iterations.
+
+Other problem: resampling limits the ability to parallelize
+algorithm.
+
+
+
+\newpage
+\subsection{Other Particle Filter Variants}
+
+
+Methods to counteract loss of diversity and degeneracy problem:
+\begin{itemize}
+  \item resample-move algorithm
+  \item regularization
+  \item Rao-Blackwellisation
+  \item multiple Monte-Carlo
+\end{itemize}
+
+Other particle filter variants found in the literature:
+\begin{itemize}
+  \item sampling importance resampling (SIR)
+  \item auxiliary sampling importance resampling (ASIR)
+  \item regularized particle filter (RPF)
+  \item \dots
+\end{itemize}
+
+
+
+
+\section{Experiments}
+
+see videos...
+
+
+
+\section{Summary}
+
+First of all: what I did \emph{not} talk about\dots
+\begin{itemize}
+  \item speed of convergence
+  \item number of samples needed
+  \item complexity issues / tricks for speed-up of algorithms
+  \item advanced particle filter variants in detail
+\end{itemize}
+
+
+$\Rightarrow$ refer to the literature if you want to know more
+
+
+
+\newpage
+Advantages of particle filters (PFs):
+\begin{itemize}
+  \item can deal with non-linearities
+  \item can deal with non-Gaussian noise
+  \item can be implemented in $O(N_s)$
+  \item mostly parallelizable
+  \item easy to implement
+  \item in contrast to HMM filters (state-space discretized to $N$ fixed
+  states) : PFs focus adaptively on probable regions of state-space
+\end{itemize}
+
+
+
+\newpage
+{\bfseries\Large Thesis:}
+
+\myboxii{If you want to solve a filtering problem, then particle
+filters are the best filters you can use, much better than e.g.
+Kalman filters.}
+
+\textcolor{dgreen}{\bfseries Right} or \textcolor{red}{\bfseries wrong}?
+
+
+\newpage
+
+\textcolor{red}{\bfseries WRONG}!
+
+Particle filters include a random element; they only convergence
+to the true posterior pdf (almost surely) if $N_s \to \infty$.
+
+Therefore: \emph{If the assumptions for Kalman filters or
+grid-based filters are valid, no PF can outperform them!}
+
+Additionally: depending on the dynamic model, Gaussian sum
+filters, unscented Kalman filters or extended Kalman filters may
+produce satisfactory results at lower computational cost.
+
+(But you should at least try a PF; it is usually better than other
+suboptimal methods!)
+
+
+
+
+\newpage
+PF approaches proved their usefulness in a variety of
+applications.
+
+But:
+\begin{itemize}
+  \item choice of importance function $q(\cdot)$ is crucial in PF
+  design
+  \item large sample number $N_s$ increases computational effort
+  \item potential problems: \emph{degeneracy} and \emph{loss of diversity}
+\end{itemize}
+
+\mybox{If these points are taken into account, then particle
+filters are an extremely powerful tool for filtering /
+estimation.}
+
+(``black box usage'' vs ``know what you're doing!'')
+
+
+
+
+
+\newpage
+\vspace*{2cm}
+
+
+{\Huge Thank you!}
+
+\vspace{1.3cm}
+
+\begin{raggedleft}
+  \footnotesize
+  This presentation was made with \LaTeX.\\
+  (try to write Bucure\c{s}ti in Powerpoint\dots)\\
+\end{raggedleft}
+
+
+
+\end{document}
diff --git a/Master/texmf-dist/doc/latex/pdfwin/JWGU-Logo.png b/Master/texmf-dist/doc/latex/pdfwin/JWGU-Logo.png
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diff --git a/Master/texmf-dist/doc/latex/pdfwin/normprot.tex b/Master/texmf-dist/doc/latex/pdfwin/normprot.tex
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index 00000000000..6d4da0d7d45
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/pdfwin/normprot.tex
@@ -0,0 +1,12 @@
+\pdfprotrudechars=2
+
+\lpcode`\(=50 \lpcode`\A=50 \lpcode`\J=50 \lpcode`\T=50
+\lpcode`\V=50 \lpcode`\W=50 \lpcode`\X=50 \lpcode`\Y=50
+\lpcode`\v=50 \lpcode`\w=50 \lpcode`\x=50 \lpcode`\y=50
+\rpcode`\-=700 \rpcode`\.=700 \rpcode`\,=700 \rpcode`\´=700
+\rpcode`\;=500 \rpcode`\:=500 \rpcode`\!=200 \rpcode`\?=200
+\rpcode`\)=50 \rpcode`\A=50 \rpcode`\F=50 \rpcode`\K=50
+\rpcode`\L=50 \rpcode`\T=50 \rpcode`\V=50 \rpcode`\W=50
+\rpcode`\X=50 \rpcode`\Y=50 \rpcode`\k=50 \rpcode`\r=50
+\rpcode`\t=50 \rpcode`\v=50 \rpcode`\w=50 \rpcode`\x=50
+\rpcode`\y=50
diff --git a/Master/texmf-dist/doc/latex/pdfwin/shortvec.tex b/Master/texmf-dist/doc/latex/pdfwin/shortvec.tex
new file mode 100644
index 00000000000..17b00cc760a
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/pdfwin/shortvec.tex
@@ -0,0 +1,26 @@
+% shortvec.tex -- abbreviation for vectors
+
+% matrices
+\newcommand{\mat}[1]{\mathbf{#1}}
+\newcommand{\MA}{\mat{A}} \newcommand{\MB}{\mat{B}} \newcommand{\MC}{\mat{C}}
+\newcommand{\MD}{\mat{D}} \newcommand{\ME}{\mat{E}} \newcommand{\MF}{\mat{F}}
+\newcommand{\MG}{\mat{G}} \newcommand{\MH}{\mat{H}} \newcommand{\MI}{\mat{I}}
+\newcommand{\MJ}{\mat{J}} \newcommand{\MK}{\mat{K}} \newcommand{\ML}{\mat{L}}
+\newcommand{\MM}{\mat{M}} \newcommand{\MN}{\mat{N}} \newcommand{\MO}{\mat{O}}
+\newcommand{\MP}{\mat{P}} \newcommand{\MQ}{\mat{Q}} \newcommand{\MR}{\mat{R}}
+\newcommand{\MS}{\mat{S}} \newcommand{\MT}{\mat{T}} \newcommand{\MU}{\mat{U}}
+\newcommand{\MV}{\mat{V}} \newcommand{\MW}{\mat{W}} \newcommand{\MX}{\mat{X}}
+\newcommand{\MY}{\mat{Y}} \newcommand{\MZ}{\mat{Z}} \newcommand{\Mnull}{\mat{0}}
+
+% vectors
+\newcommand{\va}{\vec a} \newcommand{\vb}{\vec b} \newcommand{\vc}{\vec c}
+\newcommand{\vd}{\vec d} \newcommand{\ve}{\vec e} \newcommand{\vf}{\vec f}
+\newcommand{\vg}{\vec g} \newcommand{\vh}{\vec h} \newcommand{\vi}{\vec i}
+\newcommand{\vj}{\vec j} \newcommand{\vk}{\vec k} \newcommand{\vl}{\vec l}
+\newcommand{\vm}{\vec m} \newcommand{\vn}{\vec n} \newcommand{\vo}{\vec o}
+\newcommand{\vp}{\vec p} \newcommand{\vq}{\vec q} \newcommand{\vr}{\vec r}
+\newcommand{\vs}{\vec s} \newcommand{\vt}{\vec t} \newcommand{\vu}{\vec u}
+\newcommand{\vv}{\vec v} \newcommand{\vw}{\vec w} \newcommand{\vx}{\vec x}
+\newcommand{\vy}{\vec y} \newcommand{\vz}{\vec z}
+
+% EOF