\newcommand{\mPosVec}{\vec{\mPos}}						% position vector
\newcommand{\mRssiVec}{\vec{s}}							% client signal strength measurements

\newcommand{\mState}{q}									% state variable
\newcommand{\mStateVec}{\vec{q}}						% state vector variable
\newcommand{\mObs}{o}									% observation variable
\newcommand{\mObsVec}{\vec{o}}							% observation vector variable
\newcommand{\mObsWifi}{\vec{o}_{\text{wifi}}}			% wifi observation

\newcommand{\mPressure}{\rho}
\newcommand{\mObsPressure}{\mPressure_\text{rel}}				% symbol for observation pressure
\newcommand{\mStatePressure}{\hat{\mPressure}_\text{rel}}		% symbol for state pressure

\newcommand{\mHeading}{\theta}
\newcommand{\mObsHeading}{\Delta\mHeading}						% symbol used for the observation heading
\newcommand{\mStateHeading}{\mHeading}							% symbol used for the state heading

\newcommand{\mSteps}{n_\text{steps}}
\newcommand{\mObsSteps}{\mSteps}

\newcommand{\mActivity}{\Omega}
\newcommand{\mObsActivity}{\mActivity}

\newcommand{\R}{\mathbb{R}}
%\newcommand{\N}{\mathbb{N}}

\begin{frame}[fragile]
	\frametitle{System}
	
	%\includegraphics[width=4cm]{gfx/map1}
	
		\small
		
		\begin{minipage}{0.49\textwidth}
		Unbekannter Zustand
			\begin{equation*}
				\mStateVec = (\underbrace{x, y, z}_{\text{Position}}, \underbrace{\mStateHeading}_{\text{Richtung}}),\enskip
				x, y, z, \mStateHeading \in \R
			\end{equation*}
		\vspace{1mm}
		\end{minipage}
		%
		\begin{minipage}{0.49\textwidth}
			Smartphone Sensordaten
			\begin{equation*}
				\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{beacon}, \mObsSteps, \mObsHeading, \mStateHeading, \mObsActivity, ) \enspace .
			\end{equation*}
		\vspace{1mm}
		\end{minipage}
		
		Sensorfusion über rekursive Dichteschätzung
		\begin{equation*}
			\arraycolsep=1.2pt
			%\begin{array}{ll}
				p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto
					\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}}
					\int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})}_{\text{transition}}
					\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}}
			%\end{array}
			%\label{equ:bayesInt}
		\end{equation*}
		%\vspace{1mm}
		%
		\begin{equation*}
			\begin{aligned}
				p(\vec{o}_t \mid \vec{q}_t) &= 
				\,p(\mRssiVec_\text{wifi} \mid \vec{q}_t)%_\text{wifi}
				\,p(\mRssiVec_\text{beacon} \mid \vec{q}_t)%_\text{beacon}
				\,p(\mStateHeading \mid \vec{q}_t)%_\text{kompass}
				\,p(\mObsActivity \mid \vec{q}_t)%_\text{activity}
				\footnote{\tiny Annahme: Sensoren sind statistisch unabhängig}
				\\
				p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1}) &= \text{Bewegungsmodell mit Zusatzwissen}
			\end{aligned}
		\end{equation*}
		
		
	
\end{frame}
