\section{Indoor Positioning System}
\label{sec:system}

	Our smartphone-based indoor localization system estimates the current location and heading
	using recursive density estimation.
	A graph based movement model provides the transition,
	%$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$
	while the smartphone's accelerometer, gyroscope, magnetometer provide the observations  
	for the following evaluation step to infer the hidden state, namely the pedestrian's location and heading
	\cite{Ebner-16, Fetzer-16}.
	
	\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}} \enspace,
		\end{array}
		\label{eq:recursiveDensity}
	\end{equation}
	
	The hidden state $\mStateVec$ is given by
	\begin{equation}
		\mStateVec = (x, y, z, \mStateHeading),\enskip
		x, y, z, \mStateHeading \in \R \enspace,
	\end{equation}
	%
	where $x, y, z$ represent the pedestrian's position in 3D space
	and $\mStateHeading$ his current (absolute) heading.
	%
	The corresponding observation vector is defined as
	%
	\begin{equation}
		\mObsVec = (\mRssiVecWiFi{}, \mObsSteps, \mObsHeadingRel, \mObsHeadingAbs, \mObsGPS) \enspace.
	\end{equation}
	%
	$\mRssiVecWiFi$ contains the measurements of all nearby \docAP{}s (\docAPshort{}s),
	$\mObsSteps$ describes the number of steps detected since the last filter-step,
	$\mObsHeadingRel$ the (relative) angular change since the last filter-step,
	$\mObsHeadingAbs$ the current, vague absolute heading and
	$\mObsGPS = ( \mObsGPSlat, \mObsGPSlon )$ the current location (if available) given by the GPS.
	
	Assuming statistical independence, the state evaluation's density can be written as 
	%
	\begin{equation}
		%\begin{split}
			p(\vec{o}_t \mid \vec{q}_t) = 
			p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}\enskip
			p(\vec{o}_t \mid \vec{q}_t)_\text{gps}
			\enspace.
		\label{eq:evalDensity}
	\end{equation}
	%

	The remaining observations, 
	namely: detected steps, relative- and absolute heading are 
	used within the transition model, where potential movements 
	$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ are sampled 
	based on those sensor values.

	As this work focuses on \docWIFI{} optimization, not all parts of
	the localization system are discussed in detail.
	For missing explanations please refer to \cite{Ebner-16}.
	%
	Since then, absolute heading and GPS have been added as additional sensors
	to further enhance the localization by comparing the sensor values
	using some distribution.
	
	\todo{neues resampling?}
	
	\todo{ueberleitung}
	\todo{
		die absolute positionierung kommt aus dem wlant,
		dafür braucht man entweder viele fingerprints oder ein modell
	}
	As GPS will only work outdoors, e.g. when moving from one building into another,
	the system's absolute position indoors is solely provided by the \docWIFI{} component.
	Therefore its crucial for this component to provide location estimations 
	that are as accurate as possible, while ensuring fast setup and
	maintenance times.