\section{Experiments}

% allgemeine infos über pfade und gebäude. wo 
% bild: mit pfaden drauf und eventl. wifi qualität in jeweiligen bereichen? (kann frank das)
All upcoming experiments were carried out on four floors of a \SI{77}{m} x \SI{55}{m} sized faculty building. 
It includes several staircases and elevators and has a ceiling height of about \SI{3}{m}. 
Nevertheless, the grid was generated for the complete campus and thus outdoor areas like the courtyard are also walkable.
As Wi-Fi is attenuated by obstacles and walls, it does not provide a consistent quality over the complete building. 
In fig. \ref{} we illustrate the quality obtained by the wall attenuation factor model presented earlier.
Here, green indicates a high coverage and thus a good quality for localisation, while red does the opposite.
To obtain this information we measured Wi-Fi at $666$ different points and interpolated the results as described in \cite{}.
As mentioned before, we omit any time-consuming calibration processes and use the same values for all access-points. That would be $P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}, \mWAF_{\text{wifi}} = \SI{8}{\dB}$.
The position of the access-points (about five 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.

% gewählte parameter (auch mal die optimieren wifi parameter testen)
We arranged three distinct walks (see also fig. \ref{}).
The measurements for the walks were recorded using a Motorola Nexus 6 at 2.4 GHz band only. 
The computation was done offline as described in algorithm \ref{}. 
For each walk we deployed $xx$ MC runs using 5000 Particles for each mode.
Instead of an initial position and heading, all walks start with a uniform distribution (random position and heading) as prior.
For the  filtering we used $\sigma_\text{wifi} = 8.0$ as uncertainties, 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. 
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 was set to \SI{25}{\degree}. 

% wie für die kld gezogen? begründen warum wir nun keine parzenschätzung machen (weil ähnliche ergebnisse)
To calculate \eqref{equ:KLD} and thus the Kullback-Leibler divergence, we need to sample densities from both modes likewise. 
The grid is suitable for this purpose. 
However, sampling at any vertex $\mVertexA$ of the grid, given just a set of random variables (particles), is not the easiest task. 
We need to estimate the posterior distribution given by the respective particle sets.
A common way is to deploy a kernel density estimation using a Gaussian distribution as kernel. 
The density of a specific point $\hat\mStateVec_{t} = \fPos{\mVertexA}$ is then given by 
	%
	\begin{equation}
	p(\hat\mStateVec_{t} \mid m_t, \mObsVec_{1:t}) = \sum_{i=1}^{N_{m_t}} \mathcal{N}(d^i_{\text{KL}} \mid 0, \sigma_{\text{KL}})
	\enspace ,
	\end{equation}
	%
while $d^i_{\text{KL}}$ is the euclidean distance between the considered point's $\hat\mStateVec_{t}$ and all particles $\fPos{\vec{X}_t^{i,m_t}}$ of the mode. The variance $\sigma_{\text{KL}}$ is set to \SI{1}{m}.
It is well known, that the computation of the kernel density estimation is rather slow, thus we also used a much simpler estimation by assuming a multivariate Gaussian distribution for both modes.  
Here, the mean is given by weighted arithmetic mean of the particles and the variance is defined by the sample covariance matrix. 

% ground truth
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 \cite{Fetzer2016OMC}.

% maß für die streuung der verteilung (diversity von partikeln)


error at the beginning always very high. about 44 meters. therefore the median is better value oder 75 quantil.

% zeigen das es stucken verhindert (eventl. hier eine andere aufnahme die mitten drinnen stecken bleibt)
% bild: stucken im raum + nicht mehr stucken im raum + kld mit anzeigen




% zeigen das schlechtes wi-fi (zu hohe diversity) behoben wird. 
% bild: lauf auf der rechten seite des gebäudes zeige mit und ohne wifi faktor (schlechtes wifi einzeichnen)

% zeigen das immpf nicht viel schlechter als normaler pf (ohne stucken) ist. 
% bild: er schafft es nicht die treppe rauf + er schafft es immpf + er schafft es normal filter

% gegenüberstellung aller pfade und werte in tabelle
