\subsection{Real World}

To demonstrate the real time capabilities of the proposed method a real world scenario was chosen, namely indoor localization. 
The given problem is to localize a pedestrian walking inside a building. 
Ebner et al. proposed a method, which incorporate multiple sensors, e.g. Wi-Fi, barometer, step-detection and turn-detection \cite{Ebner-15}. 
At a given time $t$ the system estimates a state consisting of the three-dimensional position.
It is implemented using a particle filter with sample importance resampling and \SI{5000} particles.
The dynamics are modelled realistically, which constrains the movement according to walls, doors and stairs. 

We arranged a \SI{223}{\meter} long walk within the first floor of a \SI{2500}{m$^2$} museum, which was build in the 13th century and therefore offers non-optimal conditions for localization. 
%The measurements for the walks were recorded using a Motorola Nexus 6 at 2.4 GHz band only.
%
Since this work only focuses on processing a given sample set, further details of the localisation system and the described scenario can be looked up in \cite{Ebner17} and \cite{Fetzer17}.
The spacing $\delta$ of the grid was set to \SI{20}{\centimeter} and a state estimation was calculated whenever a step was recognized, about every \SI{500}{\milli  \second}.
%The intention of a real world experiment is to investigate the advantages and disadvantages of the here proposed method for finding a best estimate of the pedestrian's position in the wild, compared to conventional used methods like the weighted-average or choosing the maximum weighted particle.

\begin{figure}
	\input{gfx/walk.tex}
	\caption{Occurring bimodal distribution, caused by an unknown heading and bad Wi-Fi coverage. After \SI{20.8}{\second}, the distribution gets unimodal. The weigted-average estimation (blue) provides an high error compared to the ground truth (solid black), while the boxKDE approach (green) does not. }
	\label{fig:realWorldMulti}
\end{figure}
%
Fig. \ref{fig:realWorldMulti} illustrates a frequently occurring situation, where the particle set splits apart, due to uncertain measurements and multiple possible walking directions.
This results in a bimodal posterior distribution, which reaches its maximum distances between the modes at \SI{13.4}{\second} (black dotted line).
Thus estimating the most probable state using the weighted-average results in the blue line, describing the pedestrian's position to be somewhere outside the building (light green area). 
In contrast, the here proposed method (green line) is able to retrieve a good estimate compared the the ground truth path shown by the black solid line.
Due to a right turn, the distribution gets unimodal after \SI{20.8}{\second}. 
This happens since the lower red particles are walking against a wall and thus punished with a low weight.

This example highlights the main benefits using our approach. 
While being fast enough to be computed in real time the proposed method reduces the estimation error of the state in this situation, as it is possible to distinguish the two modes of the density.
It is clearly visible, that it enables the system to recover the real state if multimodalities arise



\begin{figure}
	\input{gfx/errorOverTime.tex}
	\caption{}
	\label{fig:realWorldTime}
\end{figure}
