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

	Our smartphone-based indoor localization system estimates a pedestrian's current location and heading
	using recursive density estimation seen in \refeq{eq:recursiveDensity}.
	
		\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}
	
	A movement model, based on random walks on a graph, samples only those transitions,
	that are allowed by the buildings floorplan.
	%$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$
	The smartphone's accelerometer, gyroscope, magnetometer, GPS- and \docWIFI{}-module provide
	the observations for both, the transition and the following evaluation step to infer the hidden state,
	namely the pedestrian's location and heading
	\cite{Ebner2016OPN, Fetzer2016OMC}.

	
	This 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, \mPressure, \mObsGPS) \enspace.
	\end{equation}
	%
	$\mRssiVecWiFi$ contains the signal strength measurements of all \docAP{}s (\docAPshort{}s) currently visible to the smartphone,
	$\mObsSteps$ describes the number of steps detected since the last filter-step,
	$\mObsHeadingRel$ the (relative) angular change since the last filter-step,
	$\mObsHeadingAbs$ the vague absolute heading as provided by the magnetometer,
	$\mPressure$ the ambient pressure in hPa and
	$\mObsGPS = ( \mObsGPSlat, \mObsGPSlon, \mObsGPSaccuracy)$ the current location (if available) given by the GPS.
	
	
	Assuming statistical independence, the state-evaluation density can be written as 
	%
	\begin{equation}
		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}\enskip
		p(\vec{o}_t \mid \vec{q}_t)_\text{abshead}\enskip
		p(\vec{o}_t \mid \vec{q}_t)_\text{activity}\enskip
		\enspace.
		\label{eq:evalDensity}
	\end{equation}
	%

	Absolute location information is provided by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ and 
	$p(\vec{o}_t \mid \vec{q}_t)_\text{gps}$, if available.
	The vague absolute heading provided by
	the smartphone's magnetometer is included using a simple heuristic for 
	$p(\vec{o}_t \mid \vec{q}_t)_\text{abshead}$. Finally, the barometer is used
	to distinguish between normal walking and climbing stairs within
	$p(\vec{o}_t \mid \vec{q}_t)_\text{activity}$.
	%
	The remaining observations, derived from aforementioned smartphone sensors,
	namely: detected steps, and relative heading are 
	used within the transition model, where potential movements 
	$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ 
	are not only constrained by the buildings floorplan but also by
	those additional observations.

	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{Ebner2016OPN}.
	%
	Compared to this reference, absolute heading and GPS have been added as additional sensors
	to further enhance the localization. Their values are incorporated by simply
	comparing the sensor readings against a distribution that models the sensor's uncertainty.
	
	\todo{verteilung fuer gps und abs-heading}
	
	%\todo{neues resampling? je nach dem was sich noch in der eval zeigt}
	
	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 \docWIFI{}.
	Therefore its crucial for this component to supply location estimations 
	that are as accurate as possible, while ensuring fast setup and
	maintenance times.

	\todo{ueberleitung holprig?}