\section{Indoor Positioning System}

	nur grob beschreiben wie unser system funktioniert, 
	dass die absolute positionierung aus dem wlan kommt,
	dass man dafür entweder viele fingerprints oder ein modell braucht
	dann kommts zu dem modell
	
	\subsection{Sensor Fusion}
	
		Gesamtsystem
		dann einzel-komponenten

	\subsection{Signal Strength Prediction}
		
		\begin{equation}
			x = \mTXP{} + 10 \mPLE{} + \log_{10} \frac{d}{d_0} + \mGaussNoise{}
			\label{eq:logDistModel}
		\end{equation}

		The log distance model \todo{cite} in \refeq{eq:logDistModel} is a commonly used signal strength prediction model that
		is intended for line-of-sight predictions. However, depending on the surroundings, the model is versatile enough
		to also serve for indoor purposes.
		%
		This model predicts an \docAP{}'s signal strength 
		for an arbitrary location given the distance between both and two environmental parameters:
		The \docAPshort{}'s signal strength \mTXP{} measurable at a known distance $d_0$ (usually \SI{1}{\meter}) and
		the signal's depletion over distance \mPLE{}, which depends on the \docAPshort{}'s surroundings like walls
		and other obstacles.
		\mGaussNoise{} is a zero-mean Gaussian noise and models the uncertainty.
			
		The log normal shadowing model is a slight modification, to adapt the log distance model to indoor use cases.
		It introduces an additional parameter, that models obstalces between (line-of-sight) the \docAPshort{} and the
		location in question by attenuating the signal with a constant value.
		%
		Depending on the use case, this value describes the number and type of walls, ceilings, floors etc. between both locations.
		For obstacles, this requires an intersection-test of each obstacle with the line-of-sight, which is costly
		for larger buildings. For realtime use on a smartphone, a (discretized) model pre-computation might thus be necessary
		\todo{cite competition}.
		
		\begin{equation}
			x = \mTXP{} + 10 \mPLE{} + \log_{10} \frac{d}{d_0} + \numFloors{} \mWAF{} + \mGaussNoise{}
			\label{eq:logNormShadowModel}
		\end{equation}
		
		Throughout this work, walls are ignored and only floors/ceilings are considered for the model.
		In \refeq{eq:logNormShadowModel}, floors/ceilings
		are included using a constant attenuation factor \mWAF{} multiplied by the number of floors/ceilings \numFloors{}
		between sender and the location in question. Assuming \todo{passendes wort?} buildings, this number can be determined
		without  costly intersection checks and thus allows for realtime use cases.
		The attenuation \mWAF{} depends on the building's architecture and for common, steel enforced concrete floors
		$\approx 8.0$ might be a viable choice \todo{cite}.
		
		
		
	\subsection {Model Setup}

		As previously mentioned, for the prediction model to work, one needs to know the locations of all 
		permanently installed \docAP{}s within the building plus their environmental parameters.
		%
		While it is possible to use empiric values for \mTXP, \mPLE and \mWAF  \cite{Ebner-15}, the positions are mandtatory.
		
		For many installations, there should be floorplans that include the locations of all installed transmitters. 
		If so, a model setup takes only several minutes to (vaguely) position the \docAPshort{}s within a virtual
		map and assigning them some fixed, empirically choosen parameters for \mTXP, \mPLE and \mWAF.
		Depending on the building's architecture this might already provide enough accuracy for some use-cases
		where a vague location information is sufficient.
		
		
	\subsection{Parameter Optimization}
		
		
		
		
		
		
		

		\begin{figure}
			\input{gfx/wifiop_show_optfunc_params}
			\label{fig:wifiOptFuncParams}
			\caption{The average error (in \SI{}{\decibel}) between reference measurements and model predictions for one \docAPshort{} dependent on \docTXP{} and \docEXP{} [fixed position and \mWAF{}] denotes a convex function.}
		\end{figure}
		
		\begin{figure}
			\input{gfx/wifiop_show_optfunc_pos_yz}
			\label{fig:wifiOptFuncPosYZ}
			\caption{The average error (in \SI{}{\decibel}) between reference measurements and model predictions for one \docAPshort{} dependent on $y$- and $z$-position [fixed $x$, \mTXP{}, \mPLE{} and \mWAF{}] usually denotes a non-convex function with multiple [here: two] local minima.}
		\end{figure}
		
		while optimizing txp and exp usually means optimizing a concave function,
		optimzing the positiong usually isn't. Especially when the z-coordinate
		[influencing the WAF] is involved. While this can be mitigated by introducing
		a continuous function for the WAF like a sigmoid, this will still not work
		for all situations



wie wird optimiert
a) bekannte pos + empirische params
b) bekannte pos + opt params (fur alle APs gleich)		[simplex]
c) bekannte pos + opt params (eigene je AP)				[simplex]
d) alles opt: pos und params (je ap)					[range-random]

wenn man nur die fingerprints des floors nimmt in dem gelaufen wird, ist alles gut
sobald man andere floors drueber/drunter dazu nimmt, ist es nicht mehr gnaz so gut, oder wird schlechter
das spricht dafuer dass das modell nicht gut passt
koennte man zeigen indem man den durchschnittlichen fehler je fingerprint plottet???

the optimization-result also depends on the optimzation target:
a) the (average) error between measurment and model prediction
b) the (average) probability for the model's prediction given the fingerprint, ...
c) ...





	\subsection {VAP grouping}
		VAP grouping erklaeren


probleme bei der optimierung beschreiben. convex usw..
wo geht simplex gut, wo eher nicht

harte WAF übergänge scheinen beim optimieren als auch beim matchen nicht so gut
gleitende übergänge mittels sigmoid wirken besser
war eine wichtige erkenntnis

die vom AP bekannte position wird NICHT als input fuer die alles-OPT funktion benutzt
die ist wirklich 'irgendwo'
 
range-random algo
domain bekannt [map groesse, txp/exp/waf in etwa]
genetic refinement mit cooling [= erst grob, dann fein]

optimierung ist tricky. auch wegen dem WAF der ja sprunghaft dazu kommt, sobald messung und AP in zwei unterschiedlichen
stockwerken liegen.. und das selbst wenn hier vlt sichtkontakt möglich wäre, da der test 2D ist und nicht 3D

aps sind (statistisch) unaebhaengig. d.h., jeder AP kann fuer sich optimiert werden.
optimierung des gesamtsystems ist nicht notwendig.

pro AP also 6 params. pos x/y/z, txp, exp, waf