\section{Experiments}

As explained at the very beginning of this work, we wanted to explore the limits of the here presented localization system.
By utilizing it to a 13th century historic building, we created a challenging scenario not only because of the various architectural factors, but also because of its function as a museum.
During all experiments, the museum was open to the public and had a varying number of \SI{10}{} to \SI{50}{} visitors while recording.

The \SI{2500}{\square\meter} building consists of \SI{6}{} different levels, which are grouped into 3 floors (see fig. \ref{fig:apfingerprint}). 
Thus, the ceiling height is not constant over one floor and varies between \SI{2.6}{\meter} to \SI{3.6}{\meter}.
In the middle of the building is an outdoor area, which is only accessible from one side. 
While most of the exterior and ground level walls are made of massive stones, the floors above are half-timbered constructions.
Due to different objects like exhibits, cabinets or signs not all positions within the building were walkable. 
For the sake of simplicity we did not incorporate such knowledge into the floorplan. 
Thus, the floorplan consists only of walls, ceilings, doors, windows and stairs.
It was created using our 3D map editor software based on architectural drawings from the 1980s. 

Sensor measurements are recorded using a simple mobile application that implements the standard Android sensor functionalities.
As smartphones we used either a Samsung Note 2, Google Pixel One or Motorola Nexus 6.
The computation of the state estimation as well as the \docWIFI{} optimization are done offline using an Intel Core i7-4702HQ CPU with a frequency of \SI{2.2}{GHz} running \SI{8}{cores} and \SI{16}{GB} main memory. 
However, similar to our previous, award-winning system, the setup is able to run completely on commercial smartphones as it written in high performant C++ code \cite{torres2017smartphone}. 
%Sensor measurements are recorded using a simple mobile application that implements the standard Android SensorManager. 

The experiments are separated into four sections:
At first, we discuss the performance of the novel transition model and compare it to a grid-based approach.  
In section \ref{sec:exp:opti} we have a look at \docWIFI{} optimization and how the real \docAPshort{} positions differ from it.
Following, we conducted several test walks throughout the building to examine the estimation accuracy (in \SI{}{\meter}) of the localisation system and discuss the here presented solutions for sample impoverishment.
Finally, the respective estimation methods are discussed in section \ref{sec:eval:est}.

\subsection{Transition}

\begin{figure}[t]
	\centering
	\begin{subfigure}{0.4\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/transEval/mesh_25_final.eps_tex}
        \caption{Mesh after 25 steps}
		 \label{fig:transitionEval:a}
    \end{subfigure}
	\hspace{2cm}
	\begin{subfigure}{0.4\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/transEval/grid_25_final.eps_tex}
        \caption{Graph after 25 steps}
 		\label{fig:transitionEval:b}
    \end{subfigure}
	\begin{subfigure}{0.4\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/transEval/mesh_180_final.eps_tex}
        \caption{Mesh after 180 steps}
		 \label{fig:transitionEval:c}
    \end{subfigure}
	\hspace{2cm}
	\begin{subfigure}{0.4\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/transEval/grid_180_final.eps_tex}
        \caption{Graph after 180 steps}
 		\label{fig:transitionEval:d}
    \end{subfigure}

    \caption{Simple staircase scenario to compare the graph-based model with the navigation mesh. The black line indicates the current position and the green line gives the estimated path until 25 or 180 steps, both using weighted average. The particles are coloured according to their height. A pedestrian walks up and down the stairs several times in a row. After 25 steps, both methods produce good results, although there are already some outliers (blue particles). After 180 steps, the outliers using the graph have multiplied, leading to a multimodal situation. In contrast, the mesh offers the possibility to remove particles that hit a wall and can thus prevent such a situation.}
    \label{fig:transitionEval}
\end{figure}

To compare our old graph-based model with our novel transition model presented within section \ref {sec:transition}, we chose a simple scenario, in which a tester walks up and down a staircase several times. 
We used 1000 particles and did not perform an evaluation and resampling step to maintain the pure performance of the transition (step and heading).
The filter starts at a fixed position and is updated after every newly recognized step. 
We set $\sigma_\text{step} = 0.1$ and $\sigma_\text{turn} = 0.1$ likewise. 
The cells of the gridded graph were \SI{20}{} x \SI{20}{\centi\meter} in size and the transition implemented as described in \cite{Ebner-16}. 
As described in section \ref {sec:transition}, the mesh demands for a strategy, how to handle unreachable destinations. 
We chose a simple, yet effective strategy: whenever a destination is unreachable to a particle, it is removed and the last correct transitioning particle is duplicated.
Of course, the graph does not require for such a rule, since particles are only allowed to move on nodes and search for neighbours. 

Fig. \ref{fig:transitionEval:a} and \ref{fig:transitionEval:b} illustrate the results after \SI{25}{steps} for each method. 
The particles are coloured according to their height and the walking path (green line) is estimated using weighted-average.
It can be seen that both methods provide similar results.
Due to the discrete grid structure, the purple particles on the graph scatter more strongly, while the mesh provides a truly continuous structure and thus a more compact representation.
It is important to note that outliers have already appeared in both scenarios (blue particles). 
Due to the included sensor noise, they covered a too short distance for several times and thus the upcoming left turn leads upwards instead of downwards.
Going straightforward to \SI{180} steps, this phenomenon has multiplied for the graph (cf. fig. \ref{fig:transitionEval:d}), but not for the mesh (cf. fig. \ref{fig:transitionEval:c}). 
This is due to the above-mentioned strategy for the mesh.
Compared to this approach, the graph is not able to remove any particles and thus they walk according to the recognized steps and heading changes, even if they theoretically hit a wall several times.
The resulting effects are obvious. 
After walking up and down twice, several particle groups have formed, which no longer allows an accurate position estimation.

Of course, a similar strategy could be developed for a graph. 
We have already shown how to identify the nodes nearest to walls in one of our previous works \cite{Ebner-16}. 
However, the limitation to walk in \SI{45}{\degree} angles as well as the discrete cell sizes lead to restrictions for small rooms, narrow hallways or bigger cells. 
For example walking through a door, would result in a strong reduction of differing particles.
If the state evaluation is then used to assigned weights to particles, the crucial problem of sample degeneracy often occurs. 
With a mesh, on the other hand, walkable destinations can also be located in a room behind a wall.
In combination with the continues movement, this allows for a high versatility of particles even in such situations. 
Another method to fix the problems shown in fig. \ref{fig:transitionEval:d}, is by adding an activity recognition (walking up, down straight) or to incorporate a barometer. 
Nevertheless, in most cases it is an advantage if a sensor model delivers good results on its own, without further dependencies. 
For example, if a sensor is currently unavailable or damaged, the system is still able to provide prober results. 

Besides the advantages the mesh offers, it also has a few disadvantages compared to the graph.
The computation time has increased due to the calculation of reachable destinations. 
With the graph, only the direct neighbours are of interest, which can be implemented very efficiently using a tree structure.
Further, the graph allows the easily store meta information on its nodes, for example Wi-Fi fingerprints or precalculations for shortest-path methods.
This is more difficult using the mesh and requires the handling of baricentric coordinates.


\subsection{\docWIFI{} Optimization}
\label{sec:exp:opti}

\commentByToni{Work in Progress... Irgendwie passt die Grafik nicht so wirklich. Im Gegensatz zum 2017 Paper würde ich gerne ein wenig über die geschätzten Positionen reden. Die Unterschiede zwischen Local und Global dabei. Warum machne Schätzungen gar so weit weg von der Realität sind und das es oft auch gar nicht so schlimm ist, falls das passiert. Tipps sind Willkommen}

%wie viele ap sind es insgesamt?
As described in section \ref{sec:wifi} we used \SI{42}{} WEMOS D1 mini to provide a \docWIFI{} infrastructure throughout the building.
Within all Wi-Fi observations, we only consider the beacons, which are identified by their well-known MAC address. 
Other transmitters like smart TVs or smartphone hotspots are ignored as they might cause estimation errors.
The position of every installed beacon was measured using a laser scanner. 
This allows a comparison with the optimized \docAPshort{} positions.
Fig. \ref{fig:apfingerprint} illustrates the results of the global and the per-floor method for all \docAPshort{}'s installed to ground level. 
%
\begin{figure}[t]
	\centering
	\begin{subfigure}{0.49\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/optimization/wifiOptTopView.eps_tex}
        \caption{Top view}
    \end{subfigure}
	\begin{subfigure}{0.49\textwidth}
		\def\svgwidth{\columnwidth}
		\input{gfx/optimization/side/sideplot2.eps_tex}
        \caption{Side view}
    	\label{fig:apfingerprint:b}
    \end{subfigure}
    \caption{Position of Ap's optimized with global and per floor and real.}
    \label{fig:apfingerprint}
\end{figure}
%
Here, the respective optimized positions are connected by a grey line with the corresponding ground truth.
In order not to be confused by overlaps, the side view in fig. \ref{fig:apfingerprint:b} only shows the eight \docAPshort{}'s installed on the right side of the building.


Looking at fig. \ref{fig:apfingerprint}, it can be seen that most optimized positions differ in the lower single-digit meter range.

gives a good idea on how optimizing a simplified signal-strength prediction model behaves. 



of course, the position alone does not provide sufficient information of the overall performance of the optimiziation since they give no information about the other optimized parameters (bla, blub and bliib). 
%Nevertheless, fig. \ref{} gives an idea on how optimizing a simplified signal-strength prediction model behaves. 
By only considering ceillings, the attenuation factore... and thus most z coordinates for the ap's are similiar. 
The main message from this is, that wherever the ap's are optimized they are optimized to perfectly fit the underlying signal-strength model. 
thus the optimized parameters provide far better localization results compared to just using the real ones. simply because modelling the realistic incidents is very time consuming. 

difference stockwerk, global
looking at the optimziation errors, this can be varified... etc pp

%wie fingerprints aufgenommen, wie viele ...

\todo{die werte "results from the (absolute) difference between model predictions and real-world values for each reference measurement", wie im other 2017 und dann darauf referenzieren.}

%kurze beschreibung was wir jetzt alles testen wollen.


%was kommt bei der optimierung raus. vergleichen mit ground truth. auch den fehler gegenüberstellen. 
%man sollte sehen das ohne optimierung gar nichts geht.

\subsection{Localization Error}


\begin{figure}[t]
	\centering
	\begin{subfigure}{0.32\textwidth}
		\def\svgwidth{\columnwidth}
		{\input{gfx/groundTruth/gt_unten_final.eps_tex}}
        \caption{Ground floor}
    \end{subfigure}
	\begin{subfigure}{0.32\textwidth}
		\def\svgwidth{\columnwidth}
		{\input{gfx/groundTruth/gt_mitte_final.eps_tex}}
        \caption{First floor}
    \end{subfigure}
	\begin{subfigure}{0.32\textwidth}
		\def\svgwidth{\columnwidth}
		{\input{gfx/groundTruth/gt_oben_final.eps_tex}}
        \caption{Second floor}
    \end{subfigure}
	\caption{All conducted walks within the building. The arrows indicate the running direction and a cross marks the end. For a better overview we have divided the building into 3 floors. However, each floor consists of different high levels. They are separated from each other by different shades of grey, dark is lower then light.}
	\label{fig:floorplan}
\end{figure}
%
The 4 chosen walking paths can be seen in fig. \ref{fig:floorplan}. 
Walk 0 is \SI{152}{\meter} long and took about \SI{2.30}{\minute} to walk.
Walk 2 has a length of \SI{223}{\meter} and Walk 3 a length of \SI{231}{\meter}, both required about \SI{6}{\minute} to walk. 
Finally, walk 3 is \SI{310}{\meter} long and takes \SI{10}{\minute} to walk.
All walks were carried out by 4 different male testers using either a Samsung Note 2, Google Pixel One or Motorola Nexus 6 for recording the measurements. 
All in all, we recorded \SI{28}{} distinct measurement series, \SI{7}{} for each walk.
The picked walks intentionally contain erroneous situations, in which many of the above treated problems occur. 
This allows us to discuss everything in detail.  
A walk is indicated by a set of numbered markers, fixed to the ground. 
Small icons on those markers give the direction of the next marker and in some cases provide instructions to pause walking for a certain time.
The intervals for pausing vary between \SI{10}{\second} to \SI{60}{\second}. 
The ground truth is then measured by recording a timestamp while passing a marker. 
For this, the tester clicks 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{Fetzer-16}.
An estimation on the wrong floor has a great impact on the location awareness of an pedestrian, but only provides a relatively small error.
Therefore, errors in $z$-direction are penalized by tripling the $z$-value.

%computation und monte carlo runs
For each walk we deployed 100 runs using \SI{5000}{particles} and set $N_{\text{eff}} = 0.85$ for resampling. 
Instead of an initial position and heading, all walks start with a uniform distribution (random position and heading) as prior.
The overall localisation results can be see in table \ref{table:overall}. 
Here, we differ between the respective anti-impoverishment techniques presented in chapter \ref{sec:impo}.
The simple anti-impoverishment method is added to the resampling step and thus uses the transition method presented in chapter \ref{sec:transition}.
In contrast, the $D_\text{KL}$-based method extends the transition and thus uses a standard cumulative resampling step. 
We set $l_\text{max} =$ \SI{-75}{dBm} and $l_\text{min} =$ \SI{-90}{dBm}.
For a better overview, we only used the KDE-based estimation, as the errors compared to the weighted-average estimation differ by only a few centimetres.

\newcommand{\STAB}[1]{\begin{tabular}{@{}c@{}}#1\end{tabular}}

\begin{table}[t]
	\centering
	\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
		\hline
		Method & \multicolumn{3}{c|}{none} & \multicolumn{3}{c|}{simple} & \multicolumn{3}{c|}{$D_\text{KL}$}\\
		\hline
		  & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ \\ 
		\hline \hline
     	walk 0 & \SI{1340}{\centi\meter} & \SI{1115}{\centi\meter} & \SI{2265}{\centi\meter} & \SI{715}{\centi\meter} & \SI{660}{\centi\meter} & \SI{939}{\centi\meter} & \SI{576}{\centi\meter} & \SI{494}{\centi\meter} & \SI{734}{\centi\meter} \\ \hline
		walk 1 & \SI{320}{\centi\meter} & \SI{242}{\centi\meter} & \SI{406}{\centi\meter} & \SI{322}{\centi\meter} & \SI{258}{\centi\meter} & \SI{404}{\centi\meter} & \SI{379}{\centi\meter} & \SI{317}{\centi\meter} & \SI{463}{\centi\meter} \\ \hline
		walk 2 & \SI{834}{\centi\meter} & \SI{412}{\centi\meter} & \SI{1092}{\centi\meter} & \SI{356}{\centi\meter} & \SI{232}{\centi\meter} & \SI{486}{\centi\meter} & \SI{362}{\centi\meter} & \SI{234}{\centi\meter} & \SI{484}{\centi\meter} \\ \hline
		walk 3 & \SI{704}{\centi\meter} & \SI{589}{\centi\meter} & \SI{1350}{\centi\meter} & \SI{538}{\centi\meter} & \SI{469}{\centi\meter} & \SI{782}{\centi\meter} & \SI{476}{\centi\meter} & \SI{431}{\centi\meter} & \SI{648}{\centi\meter} \\
		\hline
	\end{tabular}
		\caption{Overall localization results using the different impoverishment methods. The error is given by the \SI{75}{\percent}-quantil Used only kde for estimation, since kde and avg nehmen sich nicht viel. fehler kleiner als 10 cm im durchschnitt deshalb der übersichtshalber weggelassen. }
	\label{table:overall}
\end{table}

All walks, except for walk 1, suffer in some way from sample impoverishment. 
We discuss the single results of table \ref{table:overall} starting with walk 0. 
Here, the pedestrians started at the top most level, walking down to the lowest point of the building. 
The first critical situation occurs immediately after the start.
While walking down the small staircase, many particles are getting dragged into the room to the right due to erroneous Wi-Fi readings. 
At this point, the activity "walking down" is recognized, however only for a very short period. 
This is caused by the short length of the stairs.
After this period, only a small number of particles changed the floor correctly, while a majority is stuck within the right-hand room.
The activity based evaluation $p(\vec{o}_t \mid \vec{q}_t)_\text{act}$ prevents particles from further walking down the stairs, while the resampling step mainly draws particles in already populated areas.
In \SI{10}{\percent} of the runs using none of the anti-impoverishment methods, the system is unable to recover and thus unable to finish the walk somewhere near the correct position or even on the same floor. 
Yet, the other \SI{90}{\percent} of runs suffer from a very high error. 
Only by using one of the here presented methods to prevent impoverishment, the system is able to recover in \SI{100}{\percent} of cases.
Fig. \ref{fig:errorOverTimeWalk0} compares the error over time between the different methods for an exemplary run.
The above described situation, causing the system to stuck after \SI{10}{\second}, is clearly visible.  
Both, the simple and the $D_\text{KL}$ method are able to recover early and thus decrease the overall error dramatically.
Between \SI{65}{\second} and \SI{74}{\second} the simple method produces high errors due to some uncertain Wi-Fi measurements coming from an \docAP{} below, causing those particles who are randomly drawn near this \docAPshort{} to be rewarded with a very high weight. 
This leads to newly sampled particles in this area and therefore a jump of the estimation. 
The situation is resolved after entering another room, which is now shielded by stone walls instead of wooden ones. 
Walking down the stairs at \SI{80}{\second} does also recover the localization system using none of the methods.
%
\begin{figure}
	\centering
	\input{gfx/errorOverTimeWalk0/errorOverTime.tex}
	\caption{Error development over time of a single Monte Carlo run of walk 0. Between \SI{10}{\second} and \SI{24}{\second} the Wi-Fi signal was highly attenuated, causing the system to get stuck and producing high errors. Both, the simple and the $D_\text{KL}$ anti-impoverishment method are able to recover early. However, between \SI{65}{\second} and \SI{74}{\second} the simple method produces high errors due to the high random factor involved.} 
	\label{fig:errorOverTimeWalk0}
\end{figure}

A similar behaviour as the above can be seen in walk 3. 
Without a method to recover from impoverishment, the system lost track in \SI{100}{\percent} of the runs due to a not detected floor change in the last third of the walk. 
By using the simple method, the overall error can be reduced and the impoverishment resolved. Nevertheless, unpredictable jumps of the estimation are causing the system to be highly uncertain in some situations, even if those jumps do not last to long.
Only the use of the $D_\text{KL}$ method is able to produce reasonable results.

As described in chapter \ref{sec:wifi}, we use a Wi-Fi model optimized for each floor instead of a single global one.
A good example why we do this, can be seen in fig. \ref{fig:walk3:wifiopt}, considering a small section of walk 3.
Here, the system using the global Wi-Fi model makes a big jump into the right-hand corridor and requires \SI{5}{\second} to recover.
This happens through a combination of environmental occurrences, like the many different materials and thus attenuation factors, as well as the limitation of the here used Wi-Fi model, only considering ceilings and ignoring walls.
Following, \docAPshort{}'s on the same floor level, which are highly attenuated by \SI{2}{\meter} thick stone walls, are neglected and \docAPshort{}'s from the floor above, which are only separated by a thin wooden ceiling, have a greater influence within the state evaluation process.
Of course, we optimize the attenuation per floor, but at the end this is just an average value summing up the \docAPshort{}'s surrounding materials. 
Therefore, the calculated signal strength predictions do not fit the measurements received from the above in a optimal way.
In contrast, the model optimized for each floor only considers the respective \docAPshort{}'s on that floor, allowing to calculate better fitting parameters. 
A major disadvantage of the method is the reduced number of visible \docAPshort{}'s and thus measurements within an area. 
This could lead to an underrepresentation of \docAPshort{}'s for triangulation. 
Such a scenario can be seen in fig. \ref{fig:walk3:time} between \SI{200}{\second} and \SI{220}{\second}, where the pedestrian enters an isolated room. 
Only two \docAPshort{}'s provide a solid signal within this area, leading to a higher error, while the global scheme still receives RSSI readings from above.

\begin{figure}[t!]
	\centering
  	\begin{subfigure}[t]{0.45\textwidth}
		\def\svgwidth{\columnwidth}
	    {\input{gfx/wifiOptGlobalFloor/wifiOptGlobalFloor.eps_tex}}
        \caption{}
        \label{fig:walk3:wifiopt}
    \end{subfigure}
    \hfil
	\begin{subfigure}[t]{0.45\textwidth}
		\resizebox{1\textwidth}{!}{\input{gfx/errorOverTimeWalk3/errorOverTime.tex}}
        \caption{}
        \label{fig:walk3:time}
    \end{subfigure}
	\caption{(a) A small section of walk 3. Optimizing the system with a global Wi-Fi optimization scheme (blue) causes a big jump and thus high errors. This happens due to highly attenuated Wi-Fi signals and inappropriate Wi-Fi parameters. We compare this to a system optimized for each floor individually (orange), resolving the situation a producing reasonable results. (b) Error development over time for this section. The high error can be seen at \SI{190}{\second}. }
	\label{fig:walk3}
\end{figure}


%walk 1 
Looking at the results of table \ref{table:overall} again, it can be seen that the $D_\text{KL}$ method is able to improve the results in three of the four walks. 
Those walks have in common, that they suffer in some way from sample impoverishment or other problems causing the system to stuck. 
The only exception is walk 1. 
It was set up to provide a challenging scenario, leading to as many multimodalities as possible.
We intentionally searched for situations in which there was a great chance that the particle set would separate, e.g. by providing multiple possible whereabouts through crossings or by blocking and thus separating a straight path with objects like movable walls. 
Similar to the other walks, we added different pausing intervals of \SI{10}{\second} to \SI{60}{\second}. 
This helps to analyse how the particles behave in such situations, especially in this multimodal setting.

Besides uncertain measurements, one of the main sources for multimodalities are restrictive transition models, e.g. no walking through walls. 
As shown in section \ref{sec:impo}, the $D_\text{KL}$ method compares the current posterior $p(\mStateVec_{t} \mid \mObsVec_{1:t})$ with the probability grid $\probGrid_{t, \text{wifi}}$ using the Kullback-Leibler divergence and a Wi-Fi quality factor. 
Environmental restriction like walls are not considered while creating $\probGrid_{t, \text{wifi}}$, that is why the grid is not effected by a transition-based multimodal setting. 
Given accurate Wi-Fi measurements, it is therefore very likely that $\probGrid_{t, \text{wifi}}$ represents a unimodal distribution, even if the particles got separated by an obstacle or wall. 
This leads to a situation, in which posterior and grid differ.
As a result, the radius $r_\text{sub}$ increases and thus the diversity of particles. 
We are able to confirm the above by examining the different scenarios integrated into walk 1.
For this, we compared the error development with the corresponding radius $r_\text{sub}$ over time. 
In situations where the errors given by the $D_\text{KL}$ method and the simple method differ the most, $r_\text{sub}$ also increases the most. 
Here, the radius grows to a maximum of $r_\text{sub} = $ \SI{8.4}{\meter}, using the same measurement series as in fig. \ref{fig:walk1:kdeovertime}.
In contrast, a real sample impoverishment scenario, as seen in walk 0 (cf. fig. \ref{fig:errorOverTimeWalk0}), has a maximum radius of \SI{19.6}{\meter}.
Nevertheless, such an slightly increased diversity is enough to influence the estimation error of the $D_\text{KL}$ in a negative way (cf. walk 1 in table \ref{table:overall}). 
Ironically, this is again some type of sample impoverishment, caused by the aforementioned environmental restrictions not allowing particles inside walls or other out of reach areas. 

%%estimation
\subsection{Estimation}
\label{sec:eval:est}

\todo{boxkde 0.2 point2(1,1);}

As mentioned before, the single estimation methods (cf. chapter \ref{sec:estimation}) only vary by a few centimetres in the overall localization error. 
That means, they differ mainly in the representation of the estimated locations. 
More easily spoken, in which way the estimated path is drawn and thus presented to the user.
Regarding the underlying particle set, different shapes of probability distributions need to be considered, especially those with multimodalities. 
%
\begin{figure}[t]
	\centering
	\begin{subfigure}{0.45\textwidth}
		\resizebox{1\textwidth}{!}{\input{gfx/walk.tex}}
        \caption{}
        \label{fig:walk1:kde}
    \end{subfigure}
    \hfil
	\begin{subfigure}{0.45\textwidth}
		\resizebox{1\textwidth}{!}{\input{gfx/errorOverTimeWalk1/errorOverTime.tex}}
        \caption{}
        \label{fig:walk1:kdeovertime}
    \end{subfigure}
	\caption{(a) Occurring bimodal distribution caused by uncertain measurements in the first \SI{13.4}{\second} of walk 1. After \SI{20.8}{\second}, the distribution gets unimodal. The weigted-average estimation (blue) provides a high error compared to the ground truth (solid black), while the KDE approach (orange) does not. (b) Error development over time for the complete walk. From \SI{230}{\second} to \SI{290}{\second} to pedestrian was not moving. }
	\label{fig:walk1}
\end{figure}
%
The main advantage of a KDE-based estimation is that it provides the "correct" mode of a density, even under a multimodal setting (cf. section \ref{sec:estimation}). 
That is why we again have a look at walk 1.
A situation in which the system highly benefits from this is illustrated in fig. \ref{fig:walk1:kde}.
Here, a set of particles splits apart, due to uncertain measurements and multiple possible walking directions.
Indicated by the black dotted line, the resulting bimodal posterior reaches its maximum distance between the modes at \SI{13.4}{\second}.
Thus, a weighted-average estimation (blue line) results in a position of the pedestrian somewhere outside the building (light green area). 
The ground truth is given by the black solid line.
The KDE-based estimation (orange line) is able to provide reasonable results by choosing the "correct" mode of the density.
After \SI{20.8}{\second} the setting returns to be unimodal again. 
Due to a right turn the lower red particles are walking against a wall and thus punished with a low weight.

Although, situations as displayed in fig. \ref{fig:walk1:kde} frequently occur, the KDE-estimation is not able to improve the overall estimation results.
This can be seen in the corresponding error development over time plot given by fig. \ref{fig:walk1:kdeovertime}. 
Here, the KDE-estimation performs slightly better then the weighted-average, however after deploying \SI{100}{} Monte Carlo runs, the difference becomes insignificant.
It is obvious, that the above mentioned "correct" mode, not always provides the lowest error. 
In some situations the weighted-average estimation is often closer to the ground truth. 
Within our experiments this happened especially when entering or leaving thick-walled rooms, causing slow and attenuated Wi-Fi signals. 
While the system’s dynamics are moving the particles outside, the faulty Wi-Fi readings are holding back a majority by assigning corresponding weights.
Only with new measurements coming from the hallway or other parts of the building, the distribution and thus the KDE-estimation are able to recover. 

This leads to the conclusion, that a weighted-average approach provides a more smooth representation of the estimated locations and thus a higher robustness.
A comparison between both methods is illustrated in fig. \ref{fig:estimationcomp} using a measuring sequence of walk 2.
We have highlighted some interesting areas with coloured squares.
The greatest difference between the respective estimation methods can be seen inside the green square, the gallery wing of the museum.
While the weighted-average (blue) produces a very straight estimated path, the KDE-based method (red) is much more volatile. 
This can be explained by the many small rooms that pedestrians pass through. 
The doors act like bottlenecks, which is why many particles run against walls and thus are either drawn on a new position within a reachable area (cf. section \ref{sec:estimation}) or walk along the wall towards the door.
This causes a higher uncertainty and diversity of the posterior, what is more likely to be reflected by the KDE method than by the weighted-average.
Additionally, the pedestrian was forced seven times to look at paintings (stop walking) between \SI{10}{\second} and \SI{20}{\second}, just in this small area. 
Nevertheless, even if both estimated paths look very different, they produce similar errors. 

The purple square displays a situation in which a sample impoverishment was successfully resolved. 
Due to a poorly working \docAPshort{}, in the lower corner of the big room the pedestrians passes before walking down the stairs, the majority of particles is dragged into the upper right corner of  that room and unable to walk down. 
By allowing some particles to walk through the wall and thus down the stairs, the impoverishment could be dissolved. 
The KDE-based estimation illustrates this behaviour very accurate.
Another situation in which the estimated paths do not provide sufficient results can be seen inside the teal square. 
The room is very isolated from the rest of the building, which is reflected by the fact that only 3 \docAPshort{}'s are detected.
The pedestrians have been asked to cross the room at a quick pace, leading to a higher step rate and therefore update rate of the filter.
The results within this area lead to the assumption, that even if Wi-Fi has a bad coverage, it influences the estimation results the most. 
The PDR based transition alone is able to walk alongside the ground truth in an accurate manner. 
However, this is of course only true if we consider this area individually, without the rest of the walk due to the accumulating bias of the relative sensors involved.
In the end, it is a question of optimal harmony between transition and evaluation. 
We hope to further improve such situations in future work by enabling the transition step to provide a weight to particles that walk very likely, especially in situation where Wi-Fi provides bad readings.

\begin{figure}[t]
	\centering
		\def\svgwidth{0.8\columnwidth}
		{\input{gfx/estimationPath2/est.eps_tex}}
	\caption{Estimation results of walk 2 using the KDE method (blue) and the weighted-average (orange). While the latter provides a more smooth representation of the estimated locations, the former provides a better idea of the quality of the underlying processes. In order to keep a better overview, the top level of the last floor was hidden. The coloured squares are used as references within the text.}
	\label{fig:estimationcomp}
\end{figure}

To summarize, the KDE-based approach for estimation is able to resolve multimodalities. 
It does not provide a smooth estimated path, since it depends more on an accurate sensor model than a weighted-average approach, but is suitable as a good indicator about the real performance of a sensor fusion system. 
At the end, in the here shown examples we only searched for a global maxima, even though the KDE approach opens a wide range of other possibilities for finding a best estimate.






