added new figure comparing good vs bad int results

tex/chapters/experiments.tex

@@ -42,7 +42,7 @@ Additionally, we used five \docIBeacon{}s for slight enhancements in some areas.
 The empirically chosen values for \docWIFI{} were $P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}, \mWAF_{\text{wifi}} = \SI{8}{\dB}$, and $\mPLE_{\text{ib}} = \SI{1.5}{}$ for the \docIBeacon{}s, respectively.
 The system was tested by omitting any time-consuming calibration processes for those values.
 We therefore expect the localisation process to perform generally worse compared to standard fingerprinting methods \cite{Ville09}. 
-However, incorporating prior knowledge and smoothing will often compensate for those poorly chosen system parameters.
+However, smoothing will often compensate for those poorly chosen system parameters.
 
 For the  filtering we used $\sigma_\text{wifi} = \sigma_\text{ib} = 8.0$ 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 (see \ref{sec:eval}). 
@@ -63,7 +63,7 @@ By adding the activity recognition to the system of \cite{Ebner-16}, we are able
 The approximation error decreases by an average of \SI{XX}{\centimeter} for all 4 paths on 10 MC runs.
 Due to this additional knowledge, the state transition samples mostly depending upon the current activity and therefore limits the possibility of false floor changes.
 Fig. \ref{fig:activityRecognition} shows the recognized activities for path 4 using the Nexus 6. 
-Despite a short misdetection caused by faulty pressure readings, the recognition can be considered to be very robust and accurate.
+Despite a short misdetection in seg. 2, caused by faulty pressure readings, the recognition can be considered to be very robust and accurate.
 
 %Fixed Interval Smoothing
 At first, both FBS and BS are compared in context of fixed-interval smoothing.
@@ -75,35 +75,43 @@ Thus, we calculate only the positional error between estimation and ground truth
 	\caption{particles. green = avg50, black = avg. gnuplot zickt bei der legende}
 	\label{fig:particles}
 \end{figure}
-In contrast to BS, the FBS is not able to improve the results using the weighted arithmetic mean for estimating the current position. 
-As illustrated in fig. \ref{fig:particles}, the estimated position (black dot) for filtering and FBS is identical although the weights are highly different. 
-To address this problem, we are calculating the average state using the \SI{50}{\percent} best weighted particles (green dot).
-For evaluating the FBS this estimation method was applied to filtering and smoothing likewise.
-We deployed 10 MC runs using \SI{2500}{} particles for approximation. 
-%The resulting positional error between estimated and ground truth along path 4 can be seen in fig. \ref{}. 
-Now, the positional average error along all 4 paths could be improved from \SI{2.49}{\meter} to \SI{2.2}{\meter} for the Galaxy and from \SI{1.76}{\meter} to \SI{1.54}{\meter} for the Nexus.
-%Similar outcomes can be observed by adding a resampling step at the end of every smoothing iteration. 
+In contrast to BS, the FBS is not able to improve the results using the weighted arithmetic mean for estimating the current position.
+Fig. \ref{fig:particles} illustrates the filtered and smoothed particle set at a certain time step on path 4. 
+It can be seen that the estimated position (black dot) for filtering and FBS is identical although the particle weights are highly different. 
+To address this problem, we are extending the FBS by adding a resampling step. 
+Here, particles with large weights are likely to be drawn multiple times.
+This focuses the computational resources of the FBS into regions with high probability mass and finally improves the estimation.
+Since smoothing operates on known states, the danger of sample impoverishment is negligible.
 
-However, BS still outperforms the FBS with an average error of \SI{1.74}{\meter} for the Galaxy and \SI{1.41}{\meter} for the Nexus on all 4 paths using the same number of particles and $500$ sample realisations. 
-A visual example comparing both smoothing methods on path 4 is illustrated in fig. \ref{fig:intcomp}.
+
+We deployed 10 MC runs using \SI{2500}{} particles for approximation and 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}. 
+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 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 as we will see later. 
+
+A visual example of the smoothing outcome for path 3 is illustrated in fig. \ref{fig:int_path3_a}.
 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}
-	\input{gfx/eval/interval_path4_comp/path4_interval}
-	\caption{Comparison between FBS (red) and BS (blue) on path 4 (black). Both were approximated using $2500$ particles and \SI{500}{} sample realisations for BS. The measurements were recorded using the Nexus 6. }
-	\label{fig:intcomp}
-\todo{Filter mit rein}
-\end{figure}
+%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}
-	\input{gfx/eval/interval_path3_bad/path3_interval}
-	\caption{A situation where BS smoothing (blue) was not able to improve the filtering results (green). Two main factors are causing this: an initial position within a detached room and inaccurate pressure readings given by the Galaxy S5.}
-	\label{fig:intervalbad}
+	\begin{subfigure}{0.175\textwidth}
+		\input{gfx/eval/interval_path3_good/path3_interval_good}
+		\caption{}
+		\label{fig:int_path3_a}
+	\end{subfigure}
+	\begin{subfigure}{0.175\textwidth}
+		\input{gfx/eval/interval_path3_bad/path3_interval}
+		\caption{}
+		\label{fig:int_path3_b}
+	\end{subfigure}
+	\caption{a) Exemplary comparison between 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.}
+\label{fig:int_path3_comp}
 \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:intervalbad} depicts such a situation for path 3 using BS and measurements provided by the Galaxy S5. 
+For example fig. \ref{fig:int_path3_b} depicts such a situation for path 3 using BS and measurements provided by the Galaxy S5. 
 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.
 This shows that the smoothing results are of course highly depend upon the filtering performance.
 
@@ -142,7 +150,7 @@ It is obvious that a lag of \SI{30}{} time steps has access to much more future
 Considering an update interval of \SI{500}{\milli\second}, a lag of \SI{30}{} would however mean that the smoother is \SI{15}{\second} behind the filter.
 Nevertheless, there are practical applications like accurately verifying hit checkpoints or continuously optimizing a recurring segment of the path.
 
-
+%verlgeich noch zwischen bs und fbs. bs weniger partikel und somit schneller. ändern der estimations kann aber auch ein möglichkeit sein.