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added first draf real world experiments

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toni
2018-02-20 23:18:25 +01:00
parent 455a5cc121
commit 5bd226e1e8
3 changed files with 41 additions and 7 deletions
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15
tex/chapters/experiments.tex
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@@ -26,7 +26,7 @@ The MISE between $f$ and the estimates as a function of $h$ are evaluated, and t
\begin{figure} [t]
\label{fig:evalBandwidth}
\includegraphics[width=\columnwidth]{gfx/Eval1Bandwidth_abs.png}
\caption{Bla.} \label{fig:eval1GroundTruth}
\caption{Hier kommt Performance Plot 2 spaltig} \label{fig:eval1GroundTruth}
\end{figure}
Other test cases of theoretical relevance are error as a function of the grid size $G$ and the sample size $N$.
@@ -42,4 +42,17 @@ All tests are performed on a Intel Core \mbox{i5-7600K} CPU with a frequency of
We compare our C++ implementation of the box filter based KDE to the KernSmooth R package and the \qq{FastKDE} implementation \cite{oBrien2016fast}.
The KernSmooth packages provides a FFT-based BKDE implementation based on optimized C functions at its core.
\begin{figure} [t]
\label{fig:evalBandwidth}
\includegraphics[width=\columnwidth]{gfx/Eval1Bandwidth_abs.png}
\caption{Hier kommt Performance Plot 2 spaltig} \label{fig:eval1GroundTruth}
\end{figure}
\begin{figure} [b]
\label{fig:evalBandwidth}
\includegraphics[width=\columnwidth]{gfx/Eval1Bandwidth_abs.png}
\caption{Hier kommt bandwith error plot single} \label{fig:eval1GroundTruth}
\end{figure}
\input{chapters/realworld}

2
tex/chapters/introduction.tex
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@@ -36,7 +36,7 @@ Especially in time critical and time sequential sensor fusion scenarios, the her
In addition, it requires only a few elementary operations and is highly parallelizable.
\todo{Mehrdimensionen mit aufnehmen. das das abgedeckt ist! }
%\todo{Mehrdimensionen mit aufnehmen. das das abgedeckt ist! }
%linear complexity and easy parall
%ist immer gleich schnell.

31
tex/chapters/realworld.tex
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@@ -16,7 +16,7 @@ The spacing $\delta$ of the grid was set to \SI{20}{\centimeter} and a state est
\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. }
\caption{Occurring bimodal distribution, caused by uncertain measurements. 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}
%
@@ -28,13 +28,34 @@ 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
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.
However, in situations with highly uncertain measurements, the estimation error could further increase since the real estimate is not equal to the best estimate, i.e. the real position of the pedestrian.
The error over time for different estimation methods of the complete walk can be seen in fig. \ref{fig:realWorldTime}.
It is given by calculating the distance between estimation and ground truth at a specific time $t$.
Estimates provided by simply choosing the maximum particle stand out the most.
As one could have expected beforehand, this method provides many strong peaks through continues jumping between single particles.
Additionally, in most real world scenarios many particles share the same weight and thus multiple highest-weighted particles exist.
\begin{figure}
\input{gfx/errorOverTime.tex}
\caption{}
\caption{Error development over time calculated between estimation and ground truth. Between \SI{230}{\second} and \SI{290}{\second} to pedestrian was not moving.}
\label{fig:realWorldTime}
\end{figure}
Further investigating fig. \ref{fig:realWorldTime}, the boxKDE performs slightly better then the weighted-average, however after deploying \SI{100} MC runs, the difference becomes insignificant.
The main reason for this are again multimodalities caused by faulty or delayed measurements, especially when entering or leaving rooms.
Within our experiments the problem occurred due to slow and attenuated Wi-Fi signals inside thick-walled rooms.
While the system's dynamics are moving the particles outside, the faulty Wi-Fi readings are holding back a majority by assigning corresponding weights.
Therefore, the average between the modes of the distribution is often closer to the ground truth as the real estimate, which is located on the \qq{wrong} mode.
With new measurements coming from the hallway or other parts of the building, the distribution and thus the estimation are able to recover.
Nevertheless, it could be seen that our approach is able to resolve multimodalities even under real world conditions.
It does not always provide the lowest error, since it depends more on an accurate sensor model then a weighted-average approach, but is very 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 boxKDE approach opens a wide range of other possibilities for finding a best estimate.
%springt nicht so viel wie maximum
%sehr ähnlich zu weighted-average. in 1000 mc runs ist sind average und std sehr ähnlich.
%das lässt den schluss, dass boxKDE den Fehler nicht reduziert, aber in bestmmten situationen einfach einen realistischeren pfad liefert.