\section{Experiments}
	
	\newcommand{\refSeg}[1]{seg. $#1$}
	
	% 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, to test short and long distances, critical sections
	and ignoring the shortest-path suggested by the system.
	Due to an in-house exhibition during that time, many places were crowded and \docWIFI{} signals
	are attenuated.
%	To enable error calculation, each acquired path is backed by ground truth information.
	The ground truth is measured by recording a timestamp at marked spots on the walking route.
	While walking, the pedestrian clicked 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 for error measurements. 
	
	All walks were performed using a Motorola Nexus 6 and a Samsung Galaxy S5.
	%
	As the Galaxy's \docWIFI{} can not be limited to the \SI{2.4}{\giga\hertz} band only,
	its scans take much longer than those of the Nexus:
	\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 better localisation using the Nexus smartphone.	
	%
	Despite being fast enough to run on the smartphone itself
	($ \approx \SI{100}{\milli\second} $ per transition, single-core Intel\textsuperscript{\textregistered} Atom{\texttrademark} C2750),
	computation was done offline using
	the \mbox{CONDENSATION} algorithm with \SI{7500}{} particles as realization.
	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 beforehand. 
	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
	$P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}, \mWAF_{\text{wifi}} = \SI{8}{\dB}$,
	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 localisation process to
	perform generally worse compared to standard fingerprinting methods \cite{Ville09}. 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 $\mStepSize$ 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}, \refeq{eq:transShortestPath} and \refeq{eq:transMultiPath} was \SI{25}{\degree}.
	Edges departing from the pedestrian's destination are downvoted using $\mUsePath = 0.9$.
	%\commentByFrank{$\mUsePath$ erklaert}
	
	
	As we start with a uniform distribution (random position and heading) for $\mStateVec_0$, the first few estimations
	are omitted from error calculations to allow the system to somewhat settle its initial state. Even though, the error
	during the following few seconds is expected to be much higher than the error when starting with a well known initial
	position and heading.
	%
	
	
	The following evaluations will depict the improvements that the 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.
	To examine the contribution our approach is able to provide, we will have a closer look at a long walk with many stairs, intentionally leaving the shortest path several times, named path 4 (see fig. \ref{fig:paths}). 
	%
	% all paths we evaluated
	\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.}
		%\commentByFrank{font war korrekt, aber die groesse war zu gross im vgl. zu den anderen}
		\label{fig:paths}
	\end{figure}
	% error development over time while walking along a path
	\begin{figure}
		\input{gfx/eval/error_timed_nexus}
		\caption{Error development while walking along Path 4 using the Nexus 6.
		When leaving the suggested route in \refSeg{3}, the error of \textbf{shortest} path \refeq{eq:transShortestPath}
		and \textbf{multi}path \refeq{eq:transMultiPath} increases.
		The same issues arise when facing multimodalities between two staircases in \refSeg{9}, just before the destination.}
		\label{fig:errorTimedNexus}
	\end{figure}
	% detailed analysis of path 4
	\begin{figure}
		\input{gfx/eval/path_nexus_detail}
		\caption{Detailed path analysis depicting the individual segments of path 4 using \refeq{eq:transMultiPath}.
		For a better visualisation, the segments are divided using an outline of alternating grey levels.
		The corresponding segment-error can be seen in fig. \ref{fig:errorTimedNexus}. Even though the shortest path
		suggested by the system is ignored multiple times (\refSeg{3'} and \refSeg{3''}) our approach is still able to improve
		the overall localisation error.}
		\label{fig:nexusPathDetails}
	\end{figure}
	%
	
	
	Fig. \ref{fig:errorTimedNexus} depicts the error for path 4 recorded with the Motorola Nexus 6.
	For a better understanding of the following discussion, the path was divided into $10$ individual segments.
	Remember that we start with a uniform distribution instead of a well known pedestrian location. 
	Therefore, the first few estimations
	reside somewhere near the centre of the building and result in a very high error contribution
	as illustrated in fig. \ref{fig:nexusPathDetails} \refSeg{1}.
	%
	Even when removing those initial estimations from the error calculation, the next few seconds are still erroneous
	due to (intentionally) bad system parameters introduced in section \ref{sec:sensors}.
	Furthermore, as the pedestrian is not yet walking, our proposed method is also not yet able to address those errors.
	This can be seen at the red area in the upper left corner of fig. \ref{fig:nexusPathDetails} \refSeg{1} and within
	 \refSeg{1} of fig. \ref{fig:errorTimedNexus}.
	%
	Starting instead with both, known position and heading, reduced the error by about \SI{15}{\percent} when using prior knowledge and
	by \SI{25}{\percent} when omitting prior knowledge. As prior knowledge directs the density towards the known target,
	it is able to compensate initially unknown headings which explains the \SI{10}{\percent} difference.
	%
	As soon as the pedestrian starts moving down the hallway \refSeg{2} the error is reduced dramatically.
	Adding prior knowledge centres the density in the middle of the floor, ensures that the heading is directed towards
	the shortest path and thus produces even better localisation results.
	%
	Directly hereafter, we ignore the shortest path \refSeg{3'} determined by the system and walk along \refSeg{3}
	instead. Of course, this leads to a temporarily increasing error, as the system needs to detect this path change
	and takes some time to recover (see fig. \ref{fig:errorTimedNexus} \refSeg{3}). The new path to the desired destination
	is \refSeg{3''} which is also ignored. Instead, we took a much longer route down the stairwell \refSeg{4}.
	After this change is detected by the system, prior knowledge is again able to reduce the error for \refSeg{5}.
	%
	Immediately hereafter follows a long, straight walk down the hallway. While the \docWIFI{} component pulls
	the pedestrian into the rooms on the right side, the actual walking route was located on the left side
	of the floor (see ground truth in fig. \ref{fig:nexusPathDetails} \refSeg{6}). While prior knowledge prevents
	the density from being dragged into the office-rooms, the estimated path is still located on the wrong side
	of the hallway. As both sides of the floor result in a route with almost the same length,
	just knowing the pedestrian's destination is not able to provide further improvements.
	Thus, a constant error of approximately the floor's width remains
as seen in fig. \ref{fig:nexusPathDetails} \refSeg{6}.
	%
	Due to the excellent barometer installed within the Nexus 6, changing the floor provides only small estimation
	errors in \refSeg{7}. 
	It follows a critical area with high errors and multimodalities. 
	Due to an in-house exhibition during the time of recording, we had to leave the ground truth by a few meters.
	Furthermore, the overcrowded areas lead to attenuated \docWIFI{} signals. This moves the
	density into another stairwell (see fig. \ref{fig:nexusPathDetails}, red lines in the lower right).
	The resulting multimodality (two staircases possible) leads to a rising error in
	\refSeg{8}, \refSeg{9}. At the end of the walk \refSeg{10} the system is able to recover, again.
	
	
	% overall error-distribution for nexus and galaxy
	\begin{figure}
		\input{gfx/eval/error_dist_nexus}
		\caption{Cumulative error distribution of all walks conducted with the \mbox{Nexus 6} for distinct percentile values.
		Both, \textbf{shortest} path \refeq{eq:transShortestPath} and \textbf{multi}path
		\refeq{eq:transMultiPath} clearly provide an enhancement compared to the reference \refeq{eq:transSimple}.}
		%\commentByFrank{percentile erwaehnt}
		\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}

	 The median errors for all conducted walks are listed in table \ref{tbl:errNexus}. Furthermore, fig. \ref{fig:errorDistNexus} 
	depicts the error development for several percentile values. As can be seen, adding prior
	knowledge improves the localisation for all examined situations, even when
	leaving the suggested path or when facing bad/slow sensor readings.
\newpage
	%\commentByFrank{fig. \ref{fig:errorDistNexus} erwaehnt}

	% error values
	%\begin{table}
	%	\caption{Median error for walks conducted with the Nexus 6.}
	%	\label{tbl:errNexus}
	%	\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}
	%\end{table}
	%
	%\begin{table}
	%	\caption{Median error for walks conducted with the Galaxy S5.}
	%	\label{tbl:errGalaxy}
	%	\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}
	%\end{table}
	
	% error values
	\definecolor{header}{rgb}{.8, .8, .8}
	\begin{table}
		\caption{Median error for all conducted walks.}
		\label{tbl:errNexus}
		\centering
		\begin{tabular}{|l|c|c|c|c|}
			\hline
			\textbf{Motorola Nexus 6}							& 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
			\hline
			\textbf{Samsung Galaxy S5}					& 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}
	\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}

%\commentByFrank{sensorausfall simulieren, z.b. in der mitte, oder auf einer treppe}
%\commentByFrank{zwischendrin mal stehenbleiben und schauen ob auch das klappt}
%\commentByFrank{zu grosser einfluss vom pfad ist also kein allheilmittel.. kann, wie beim treppenhaus, auch nach hinten los gehen}
