\section{Experiments}
	
	% introduction
	Evaluation took place within all floors (0 to 3) of the
	faculty building, each of which about \SI{77}{\meter} x \SI{55}{\meter} in size.
	%
	We conducted 4 distinct walks, for testing short distances, long distances, critical sections
	and ignoring the shortest-path suggested by the system.
	Due to an inhouse exhibition during that time, many places were crowded and \docWIFI{} signals
	are attenuated more than usual.
	Each acquired path is backed by ground truth information to enable error calculation.
	This ground truth is measured by recording a timestamp at a marked spot on the walking route.
	During the walk, the pedestrian had to click a button on the smartphone application
	when passing a marker. Between two consecutive points, a constant movement speed is assumed.
	Thus, the ground truth might not be \SI{100}{\percent} accurate, but fair enough to conduct
	error measurements. All walks were conducted using a Google Nexus 6 and a Samsung Galaxy S5.
	
	As the Samsung Galaxy S5's \docWIFI{} can not be limited to the \SI{2.4}{\giga\hertz} band only,
	its scans take much longer than those of the Google Nexus 6:
	\SI{3500}{\milli\second} vs. \SI{600}{\milli\second}.
	Also, the Nexus' barometer sensor provides readings both more frequent and far more accurate than
	the Galaxy does. This results in a much better localisation of the Nexus smartphone.
	
	Despite being fast enough to run in realtime on the smartphone itself, computation was done offline using
	the condensation algorithm with \SI{7500}{} particles as realization of the recursive density estimation \cite{todo}.
	The weighted arithmetic mean of the particles was used as state estimation.
	
	As mentioned earlier, the position of all \docAP{}s (about 5 per floor) is known beforhand. 
	Due to legal terms, we are not allowed to depict their positions and therefore omit this information within the figures.
	Additionally we used three \docIBeacon{}s for slight enhancements in some areas.
	The empirically chosen values for \docWIFI{} were $P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}$,
	and $\mPLE_{\text{ib}} = \SI{1.5}{}$ for the \docIBeacon{}s, respectively.
	%
	Due to omitting a time-consuming calibration process for those values we expect the localistation
	process to perform generally worse compared to fingerpring methods \todo{cite}. However,
	incorporating prior knowledge will often compensate for those poorly chosen system parameters.
	
	As uncertainties we used $\sigma_\text{wifi} = \sigma_\text{ib} = 8.0$, both growing with each measurement's age.
	While the pressure change was assumed to be \SI{0.105}{$\frac{\text{\hpa}}{\text{\meter}}$}, all other barometer-parameters
	are determined automatically (see \ref{sec:sensBaro}). The step size for the transition was configured to be \SI{70}{\centimeter}
	with an allowed derivation of \SI{10}{\percent}. The heading deviation in \refeq{eq:transSimple} was \SI{25}{\degree}.
	
	As we start with a uniformation distribution for $\mStateVec_0$ (random position and heading), the first few estimations
	are omitted from the error calculation to allow the system to somewhat settle its initial state. Even though, the error
	during the follwing few seconds is expected to be much higher than the error when starting with a well known initial
	position and heading.
	
	The follwing evaluations will depict the improvements prior path knowledge is able to provide
	even when other system parameters are badly chosen.
	Just adding importance-factors described in \ref{sec:wallAvoidance} and \ref{sec:doorDetection} 
	to the simple transition \refeq{eq:transSimple} addresses only minor local errors
	% like not sticking too close to walls. In most cases this lead only to slight improvements
	and is therefore not further evaluated. 
	%
	

	\commentByFrank{bergwerk\_path3\_galaxy}
	
	\begin{figure}
		\input{gfx/eval/paths}
		\caption{The four paths that were part of the evaluation.
		Starting positions are marked with black circles.
		For a better visualisation they were slightly shifted to avoid overlapping.}
		\label{fig:paths}
	\end{figure}

	\commentByFrank{verlassen vom shortest path fuehrt zu weniger verbesserung, aber es wird nach wie vor besser als ohne!}
	
	\commentByFrank{in den ersten paar sec ist die pfad-info teils hinderlich, da die genaue position noch sehr unklar ist und sich erst einstellen muss.
	deshalb geht der fehler hier oft leicht hoch}
		
	% error development over time while walking along a path
	\begin{figure}
		\input{gfx/eval/error_timed_nexus}
		\caption{Development of the error while walking along path 1 (upper) and path 4 (lower) using the Google Nexus 6.
		Path 4 shows increasing errors for our methods when leaving the shortest path (3) and when facing multimodalities between two
		staircases just before the destination (9).}
		\label{fig:errorTimedNexus}
	\end{figure}
	
	% overall error-distribution for nexus and galaxy
	\begin{figure}
		\input{gfx/eval/error_dist_nexus}
		\caption{Error distribution for all walks conducted with the Google Nexus 6. Our proposed methods
		clearly provide an enhancement for the overall localization process.}
		\label{fig:errorDistNexus}
	\end{figure}
	\begin{figure}
		\input{gfx/eval/error_dist_galaxy}
		\caption{Nicht so markant beim galaxy, denke aber der platz reicht eh nicht, also einfach kurz erwaehnen}
	\end{figure}

	\begin{table}
		\centering
		\begin{tabular}{|l|c|c|c|c|}
			\hline
														& Path1 				& Path2					& Path3 				& Path4 \\\hline
			Simple (\refeq{eq:transSimple})				&	\SI{6.68}{\meter}	&	\SI{5.25}{\meter}	&	\SI{4.32}{\meter}	&	\SI{3.84}{\meter}	\\\hline
			Shortest (\refeq{eq:transShortestPath})		&	\SI{2.72}{\meter}	&	\SI{2.98}{\meter}	&	\SI{2.48}{\meter}	&	\SI{3.06}{\meter}	\\\hline
			Multipath (\refeq{eq:transMultiPath})		&	\SI{2.62}{\meter}	&	\SI{2.14}{\meter}	&	\SI{2.46}{\meter}	&	\SI{2.75}{\meter}	\\\hline
		\end{tabular}
		\caption{Median error for walks conducted with the Nexus 6.}
	\end{table}
	
	\begin{table}
		\centering
		\begin{tabular}{|l|c|c|c|c|}
			\hline
														& Path1					& Path2					& Path3					& Path4 \\\hline
			Simple (\refeq{eq:transSimple})				&	\SI{10.03}{\meter}	&	\SI{7.65}{\meter}	&	\SI{6.03}{\meter}	&	\SI{7.54}{\meter}	\\\hline
			Shortest (\refeq{eq:transShortestPath})		&	\SI{ 5.86}{\meter}	&	\SI{4.14}{\meter}	&	\SI{5.14}{\meter}	&	\SI{5.20}{\meter}	\\\hline
			Multipath (\refeq{eq:transMultiPath})		&	\SI{ 6.35}{\meter}	&	\SI{4.21}{\meter}	&	\SI{5.03}{\meter}	&	\SI{6.79}{\meter}	\\\hline
		\end{tabular}
		\caption{Median error for walks conducted with the Galaxy S5.}
	\end{table}
	
	\begin{figure}
		\includegraphics{gfx/eval/bergwerk_path1_galaxy}
		\caption{Path 1 recorded with the Galaxy S5. Using prior knowledge improves the staircase (left) and the target area (right) where
		both the barometer and \docWIFI{} provided bad readings.}
		\label{fig:bergwerkPath1Galaxy}
	\end{figure}
	
	\begin{figure}
		\includegraphics{gfx/eval/bergwerk_path3_galaxy}
		\caption{Path 3 recorded with the Galaxy S5. Even though both paths look similar, the version with prior knowledge ended
		much closer to the real destination due to reduced delays.}
		\label{fig:bergwerkPath3Galaxy}
	\end{figure}

\begin{itemize}
  \item Nochmal kurz auf die Probleme des letzten Systems eingehen (schon teil der introduction)
	\item Da letztes mal nur 1 Pfad, machen wir dieses mal mehrere! 
	\item Stelle normale Lokalisation der Pfad Lokalisation gegenüber und überlege wo Probleme auftreten
	\item nutze den "natürlichen Pfad" und einen normalen dijkstra
	\item Analysiere Probleme ggf. mit schönen Grafiken.
	\item Vergleich zum Schluss das neue System mit dem Alten um eine schöne Conclusion der Verbesserungen einzuleiten. 
\end{itemize}

\commentByFrank{we start with a uniform distribution $\mStateVec_0$}

\commentByFrank{hinweis auf die verschiedenen geraete (smartphones) und unterschiede, wlan/baro}
\commentByFrank{
	PATH4 HAELT SICH NICHT AN DEN SHORTEST PATH.
	GUTES BEISPIEL.
	der pfad wechselt sogar 2x! (3. stock)
	der shortest wird am ende etwas ungenau bei der treppe
}

\commentByFrank{sensorausfall simulieren, z.b. in der mitte, oder auf einer treppe}
\commentByFrank{zwischendrin mal stehenbleiben und schauen ob auch das klappt}
\commentByFrank{pfad verlassen und ganz wo anders hingehen}

\commentByFrank{die reine importance selbst auf dem graphen hilft, aber nur minimal weiter}

\commentByFrank{pfad4 nexus. pfadlos laeuft mit ach und krach richtig (treppenhaus, wlan schlecht)
mit pfad laeuft es falsch, weil die andere treppe kuerzer zum ziel ist und das wlan dort besser passt}

\commentByFrank{zu grosser einfluss vom pfad ist also kein allheilmittel.. kann, wie beim treppenhaus, auch nach hinten los gehen}


\commentByFrank{path1: bad start due to nearby AP and bad parameters (path-loss too high): high starting errors: median better}
