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toni
2016-05-09 16:20:21 +02:00
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@@ -82,8 +82,6 @@
\usepackage{algorithm} \usepackage{algorithm}
\usepackage{algpseudocode} \usepackage{algpseudocode}
\usepackage{caption}
\usepackage{subcaption}
% replacement for the SI package % replacement for the SI package

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tex/chapters/experiments.tex
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@@ -32,9 +32,10 @@ As the Galaxy's \docWIFI{} can not be limited to the \SI{2.4}{\giga\hertz} band
Additionally, the Galaxy's barometer sensor provides fare more inaccurate and less frequent readings than the Nexus does. 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. 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 and multinomal (cumulative) resampling. The computation for both filtering and smoothing was done offline using the aforementioned \mbox{CONDENSATION} algorithm and multinomal (cumulative) resampling.
For each path we deployed 10 MC runs using \SI{2500}{} particles and $500$ sample realisations for BS.
%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). %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. %The computational times of the different smoothing algorithm will be discussed later.
Unless explicitly stated, the state was estimated using the weighted arithmetic mean of the particles. Unless explicitly stated, the state was estimated using the weighted arithmetic mean of the particle set.
As mentioned earlier, the position of all \docAP{}s (about 5 per floor) is known beforehand. 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. Due to legal terms, we are not allowed to depict their positions and therefore omit this information within the figures.
@@ -84,23 +85,15 @@ This focuses the computational resources of the FBS into regions with high proba
Since smoothing operates on known states, the danger of sample impoverishment is negligible. Since smoothing operates on known states, the danger of sample impoverishment is negligible.
We deployed 10 MC runs using \SI{2500}{} particles for approximation and a multinomial resampling step to every smoothing interval of the FBS. We deployed a multinomial resampling step to every smoothing interval of the FBS.
Now, the positional average error along all 4 paths using the Nexus and the Galaxy could be improved from \SI{2.08}{\meter} to \SI{1.37}{\meter}. Now, the positional average error along all 4 paths using the Nexus and the Galaxy could be improved from \SI{2.08}{\meter} to \SI{1.37}{\meter}.
Using the same number of particles and $500$ sample realisations the BS performs with an average error of \SI{2.21}{\meter} for filtering and \SI{1.51}{\meter} for smoothing. The BS performs with an average error of \SI{2.21}{\meter} for filtering and \SI{1.51}{\meter} for smoothing.
The difference between both filtering steps is of course based upon the randomized behaviour of the respective probabilistic models. The difference between both filtering steps is of course based upon the randomized behaviour of the respective probabilistic models.
It is interesting to note, that the positional error is very similar for both used smartphones, although the approximation error varies greatly. It is interesting to note, that the positional error is very similar for both used smartphones, although the approximation error varies greatly.
Using the FBS, the Galaxy donates an average approximation error of \SI{4.03}{\meter} by filtering with \SI{7.74}{\meter}. Using the FBS, the Galaxy donates an average approximation error of \SI{4.03}{\meter} by filtering with \SI{7.74}{\meter}.
In contrast the Nexus 6 filters at \SI{5.11}{\meter} and results in \SI{3.87}{\meter} for smoothing. In contrast the Nexus 6 filters at \SI{5.11}{\meter} and results in \SI{3.87}{\meter} for smoothing.
The BS has a similar improvement rate. The BS has a similar improvement rate.
A visual example of the smoothing outcome for path 3 is illustrated in fig. \ref{fig:int_path3_a}.
It can be clearly seen, how the smoothing compensates for the faulty detected floor change using future knowledge.
The estimation of BS looks way more realistic and adapts better to the ground truth path.
%However, in this particular example the FBS starts at an earlier position (cf. fig. \ref{fig:intcomp} seg. 2), better handling the initial uniform distribution.
%Another advantage of BS over FBS, is the ability to still improve the filtering results even while reducing the number of particles radical.
%For example \SI{50}{} particles and \SI{25}{} sample realisations are providing reliable estimations similar to above experiments, though the risk of losing track is higher.
\begin{figure} \begin{figure}
\begin{subfigure}{0.175\textwidth} \begin{subfigure}{0.175\textwidth}
@@ -113,23 +106,25 @@ The estimation of BS looks way more realistic and adapts better to the ground tr
\caption{} \caption{}
\label{fig:int_path2_b} \label{fig:int_path2_b}
\end{subfigure} \end{subfigure}
\caption{a) Exemplary results for path 2 where BS (blue) and filtering (green) using 2500 particles and 500 sample realisations. b) A situation where BS smoothing was not able to improve the filtering results. Two main factors are causing this: an initial position within a detached room and inaccurate pressure readings given by the Galaxy S5.} \caption{a) Exemplary results for path 2 where BS (blue) and filtering (green) using 2500 particles and 500 sample realisations. b) A situation where smoothing provides a worse error in regard to the ground truth, but obviously a more realistic path.}
\label{fig:int_path3_comp} \label{fig:int_path2}
\end{figure} \end{figure}
Despite the very good outcomes provided by both interval smoother, there are some rare situations in which smoothing does not improve the filtered estimation or even improves the visual path. %
For example fig. \ref{fig:int_path3_b} depicts such a situation for path 3 using BS and measurements provided by the Galaxy S5. Two visual examples of the smoothing outcome for path 2 are illustrated in fig. \ref{fig:int_path2}.
Here, the estimation was not able to change floors correctly due to faulty pressure readings. Additionally, the initial position was located within a detached room. It can be clearly seen, how the smoothing compensates for the faulty detected floor changes using future knowledge.
This shows that the smoothing results are of course highly depend upon the filtering performance. Additionally, the initial error is reduced extremely, approximating the pedestrian's starting position down to a few centimetres.
In the context of reducing the error as far as possible, fig. \ref{fig:int_path2b} is a very interesting example.
Here, the filter offers a lower approximation and positional error in regard to the ground truth.
However it is obvious that smoothing causes the estimation to behave more natural instead of walking the supposed path.
This phenomena could be observed for both smoothers respectively.
At next, we discuss the advantages and disadvantages of utilizing FBS and BS as fixed-lag smoother. At next, we discuss the advantages and disadvantages of utilizing FBS and BS as fixed-lag smoother.
Compared to fixed-interval smoothing, timely errors are now of higher importance due to an interest on real-time localization. Compared to fixed-interval smoothing, timely errors are now of higher importance due to an interest on real-time localization.
Especially interesting in this context are small lags $\tau < 10$ considering filter updates near \SI{500}{\milli\second}. Especially interesting in this context are small lags $\tau < 10$ considering filter updates near \SI{500}{\milli\second}.
Fig. \ref{} illustrates the different estimations for path 4 using a fixed-lag $\tau = 5$.
The associated approximation errors alongside the path can additionally be seen in fig. \ref{fig:lag_error_path4}.
For better distinction, the path was divided into $10$ individual segments.
%Path 4 grafik mit fixed-lags
%
%wie gut ist fixed-lag mit einem lag = 5. was fällt so auf?
Fig. \ref{fig:lag_error_path4} illustrates the different approximation errors alongside path 4 using $500$ particles, \SI{100}{sample realisations} for BS and a fixed-lag $\tau = 5$.
% %
\begin{figure} \begin{figure}
\input{gfx/eval/lag_path4_error/error_timed_costum} \input{gfx/eval/lag_path4_error/error_timed_costum}
@@ -143,9 +138,9 @@ Besides the positional quality, also the timely error could be reduced by both a
Once more, the BS outperforms the FBS by providing an overall approximation error of $\SI{4.86}{\meter}$ for the Galaxy and $\SI{2.97}{\meter}$ for the Nexus by filtering with $\SI{5.68}{\meter}$ and $\SI{3.15}{\meter}$ respectively. Once more, the BS outperforms the FBS by providing an overall approximation error of $\SI{4.86}{\meter}$ for the Galaxy and $\SI{2.97}{\meter}$ for the Nexus by filtering with $\SI{5.68}{\meter}$ and $\SI{3.15}{\meter}$ respectively.
Whereas FBS is not able to reduce the filtering error, obtaining $\SI{5.59}{\meter}$ using the Nexus and $\SI{3.12}{\meter}$ the Galaxy. Whereas FBS is not able to reduce the filtering error, obtaining $\SI{5.59}{\meter}$ using the Nexus and $\SI{3.12}{\meter}$ the Galaxy.
However, this does not mean, that the FBS is unpracticable for problems of fixed-lag smoothing. However, this does not mean, that the FBS is unpracticable for problems of fixed-lag smoothing.
By looking at the data in detail, high errors similar as seen in fig. \ref{fig:lag_error_path4} occur on path 3, which cause the estimation to behave more natural instead of walking the supposed path. %By looking at the data in detail, high errors similar as seen in fig. \ref{fig:lag_error_path4} occur on path 3, which cause the estimation to behave more natural instead of walking the supposed path.
This phenomena could be observed for both smoothers respectively. %This phenomena could be observed for both smoothers respectively.
Also note the difference between this error, including timely information, and the positional error used before. %Also note the difference between this error, including timely information, and the positional error used before.
%The median errors for all conducted walks are listed in table \ref{}. %The median errors for all conducted walks are listed in table \ref{}.
Similar to fixed-interval smoothing, decreasing the number of particles does not necessarily worsen the estimation. Similar to fixed-interval smoothing, decreasing the number of particles does not necessarily worsen the estimation.
In most cases smoothing compensates for this reduction and maintains the good results. In most cases smoothing compensates for this reduction and maintains the good results.