\section{Experiments}


The experiments were carried out on all floors (0 to 3) of the faculty building. 
Each floor is about \SI{77}{\meter} x \SI{55}{\meter} in size, with a ceiling height of \SI{3}{\meter}.
To resemble real-world conditions, the evaluation took place during an in-house exhibition. 
Thus, many places were crowded and Wi-Fi signals attenuated.
As can be seen in fig. \ref{fig:paths} we arranged 4 distinct walks, covering different distances, critical sections and uncertain decisions leading to multimodalities. 
The ground truth is measured by recording a timestamp at marked spots on the walking route.
When passing a marker, the pedestrian clicked a button on the smartphone application. 
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.
The approximation error is then calculated by comparing the interpolated ground truth position with the current estimation.
Especially in the context of smoothing, it is also very interesting to exclude the temporal delay from the error calculations and measure only the positional difference between estimated and ground truth path. 
This gives a statement about the extent to which the smoothed path superficially improves compared to the filtered one. 

All walks start with a uniform distribution (random position and heading) as prior for $\mStateVec_0$. 
To allow the system to stabilize its initial state, the first few estimations are omitted from error calculations.
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.

\commentByToni{Absatz drunter muss ich noch rumschreiben.}

The measurements were recorded 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}.
Additionally, the Galaxy's barometer sensor provides fare more inaccurate and less frequent readings than the Nexus does. 
This results in a better localisation using the Nexus smartphone.	
The computation for both filtering and smoothing was done offline using the aforementioned \mbox{CONDENSATION} algorithm. 
The state was then estimated using the weighted arithmetic mean of the particles. 
However, the filter itself would be fast enough to run on the smartphone itself ($ \approx \SI{100}{\milli\second} $ per transition, single-core Intel\textsuperscript{\textregistered} Atom{\texttrademark} C2750).
The computational times of the different smoothing algorithm will be discussed later. 

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 five \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.

\commentByToni{Hier eure noetigen Werte eintragen.}
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 $\mUsePath = 0.9$.


	%



% 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}


%Smoothing mit großen lag kann die zeitliche information schwer halten. das liegt hauptsächlich daran, das im smoothing nur die relativen positionsinfos genutzt werden. das wi-fi wird nicht beachtet und deswegen können absolute justierungen der position (sprünge) nur sehr schlecht abgefedert werden. 


%Evaluation:
%\begin{itemize}
%	\item Filter ist immer der gleiche mit MultiPathPrediction und Importance Factors
%	\item FBS Interval mit 500 und 2500 Partikeln auf 4 Pfaden mit SimpleSmoothingTrans
%	\item BS Interval mit 500 zu 100 und 2500 zu 500 Partikeln auf auf 4 Pfaden mit SimpleSmoothingTrans
%	\item FBS Lag = 5 mit 500 und 2500 Partikeln auf 4 Pfaden mit SimpleSmoothingTrans
%	\item BS Lag = 5 mit 500 zu 100 und 2500 zu 500 Partikeln auf auf 4 Pfaden mit SimpleSmoothingTrans
%	\item BS Lag zu Error Plot. Lag von 0 bis 100, wie verhält sich der Error. Am besten auf Pfad 4 mit SimpleSmoothingTrans.
%\end{itemize}



