final version paper

tex/chapters/experiments.tex

@@ -7,8 +7,8 @@ All upcoming experiments were carried out on four floors (0 to 3) of a \SI{77}{m
 It includes several staircases and elevators and has a ceiling height of about \SI{3}{m}. 
 The gridded graph was generated for the complete campus and thus outdoor areas like the courtyard are also walkable.
 As \docWIFI{} is attenuated by obstacles and walls, it does not provide a consistent quality over the complete building. 
-To get an idea about the \docWIFI{} quality, we interpolated the \docWIFI{} quality factor $q(\mObsVec_t^{\mRssiVec_\text{wifi}})$, with $l_{\text{max}} = \SI{-75}{\dBm}$ and $l_{\text{min}} = \SI{-90}{\dBm}$, between some measurements using a nearest-neighbour procedure.
-In fig. \ref{fig:paths} the resulting colourized floorplan is illustrated.
+To get an idea about the quality, we interpolated the \docWIFI{} quality factor $q(\mObsVec_t^{\mRssiVec_\text{wifi}})$, with $l_{\text{max}} = \SI{-75}{\dBm}$ and $l_{\text{min}} = \SI{-90}{\dBm}$, between some measurements using a nearest-neighbour procedure.
+In fig. \ref{fig:paths} the resulting colourized floorplan is shown.
 Here, the intensity of red indicates a low signal strength and thus a bad quality for localising a pedestrian within this area using \docWIFI{}.
 
 % bild: mit pfaden drauf und eventl. wifi qualität in jeweiligen bereichen? (kann frank das)
@@ -36,9 +36,8 @@ Due to legal terms, we are not allowed to depict their positions and therefore o
 To model the uncertainty of the \docWIFI{}, a white Gaussian noise with $\sigma_\text{wifi} = \SI{8}{\dBm}$, growing with each measurement's age, was chosen.
 The pressure change was assumed to be \SI{0.105}{$\frac{\text{\hpa}}{\text{\meter}}$}, all other barometer-parameters are determined automatically as described in \cite{Fetzer2016OMC}. 
 
-
 As expected before, the kernel density estimation in \eqref{eq:KDE} and the much simpler multivariate normal distribution estimation are both producing similar results in context of the divergence measure given in \eqref{equ:KLD}. 
-For this reason and because of the faster processing time, especially in context of calculating on a smartphone, we estimated the posteriors of the modes by approximating a multivariate normal distribution (cf. section \ref{sec:divergence}).
+For this reason and because of the faster processing time, especially in context of smartphones, we estimated the posteriors of the modes by approximating a multivariate normal distribution (cf. section \ref{sec:divergence}).
 For calculating \eqref{equ:KLD} we set $\lambda = 0.03$ based on best practices. 
 Finally, the pedestrian's most likely position (state) was then estimated using the weighted arithmetic mean of the particle set.
 
@@ -76,13 +75,15 @@ As mentioned above, every run of a walk starts with a uniform distribution as pr
 Due to a low \docWIFI{} coverage at the starting point in seg. 1, the pedestrian's position is falsely estimated into a room instead of the corridor.   
 All three filters are able to overcome this false detection. 
 However, the graph-based filter (red) gets then indissoluble captured within a room, because of its restrictive behaviour and the aftereffects of the initial \docWIFI{} readings (cf. fig. \ref{fig:path3} seg. 2). 
-It provides an \SI{75}{\percent}-quantil of $\tilde{x}_{75} = \SI{3884}{\centimeter}$ and got stuck in \SI{100}{\percent} of all runs.
+It provides a \SI{75}{\percent}-quantil of $\tilde{x}_{75} = \SI{3884}{\centimeter}$ and got stuck in \SI{100}{\percent} of all runs.
 As expected and discussed earlier, the simple filter (blue) is less prone to bad observations and provides not accurate, but very robust results of $\tilde{x}_{75}= \SI{809}{\centimeter}$ and a standard deviation over all results of $\bar{\sigma} = \SI{529}{\centimeter}$.
 Looking at $D_{\text{KL}}$ over time confirms our assumption made in section \ref{sec:divergence}. 
 The graph-based filter (red) gets stuck and is not able to recover. 
 Starting on from this point, the Kullback-Leibler divergence $D_{\text{KL}}$ (purple) further increases due to the growing distance between both filters (blue and red) starting at seg. 2 until the end of seg. 4.
 It is clearly visible, that this divergence between both filters is a very good indicator to observe, if a filter gets stuck or loses track.
 Following, the IMMPF (green) results in a very natural and straight path estimation and a low $D_{\text{KL}}$ between modes.
+Of course, the low $D_{\text{KL}}$ of the IMMPF results from the mixing between the modes caused by $\Pi_t$.
+Therefore, the matrix $\Pi_t$ can also be seen as some kind of limiter, paying attention that the modes do not diverge to much from each other.
 The benefits of mixing both filtering schemes within the scenario of path 3 are thus obvious.
 The IMMPF filters with an error of $\tilde{x}_{75} = \SI{667}{\centimeter}$ and $\bar{\sigma} = \SI{558}{\centimeter}$.
 
@@ -97,17 +98,17 @@ The IMMPF filters with an error of $\tilde{x}_{75} = \SI{667}{\centimeter}$ and
 	\end{figure}
 	%
 Next, we investigate the performance of our approach by considering the scenario in path 2. 
-Here, the overall \docWIFI{} quality is rather low, especially in the zig-zag stairwell on the buildings back and the small entrance area at floor 1 (cf. fig. \ref{fig:paths} and fig. \ref{fig:path2} seg. 3). 
-Path 2 starts in the second floor, walking down the centred stairs into the first floor, then making a right turn and walking the stairs down to zeroth floor, from there we walk back to second floor using the zig-zag stairwell and after finally crossing a room we are back at the start. 
+Here, the overall \docWIFI{} quality is rather low, especially in the zig-zag stairwell on the building's back and the small entrance area at floor 1 (cf. fig. \ref{fig:paths} and fig. \ref{fig:path2} seg. 3). 
+Path 2 starts in the second floor, walking down the centred stairs into the first floor, then making a right turn and walking the stairs down to zeroth floor. From there we walk back to the second floor using the zig-zag stairwell and after finally crossing a room we are back at the start. 
 This is a very challenging scenario, at first the estimation got stuck on the first floor in a room's corner for \SI{20}{\second} (see fig. \ref{fig:path2} seg. 2) and after that the \docWIFI{} is highly attenuated at the beginning and end of seg. 3.
 Looking at fig. \ref{fig:path2}, one can observe the impact of the \docWIFI{} quality factor within the Markov transition matrix. 
-Without it, the position estimation (red) is drifting in the courtyard, missing the stairwell and producing high errors between \SIrange{80}{130}{\second} in seg. 3. 
+Without it, the position estimation (red) is drifting into the courtyard, missing the stairwell and producing high errors between \SIrange{80}{130}{\second} in seg. 3. 
 As described earlier, the bad \docWIFI{} readings are causing $D_{\text{KL}}$ to grow.
-It follows that the accurate dominant filter draws new particles from the uncertain support and therefore worsen the position estimation.
+It follows that the accurate dominant filter draws new particles from the uncertain support and therefore worsens the position estimation.
 Given this outcome, it is not very satisfying that the system is able to recover thanks to its high diversity during situations with uncertain measurements.
 By adding the \docWIFI{} quality factor, the system is able to improve the approximated path (green) and the overall estimation results from $\tilde{x}_{75} = \SI{1278}{\centimeter}$ with $\bar{\sigma} = \SI{948}{\centimeter}$ to $\tilde{x}_{75} = \SI{953}{\centimeter}$ with $\bar{\sigma} = \SI{543}{\centimeter}$.
-However, this is far from perfect and in some cases ($\sim \SI{9}{\percent}$) the quality factor was not able to prevent the estimation to drift in the courtyard. 
-This solely happened when particles were sampled directly onto the courtyard while changing from first to zeroth floor (cf. fig. \ref{fig:path2} seg. 3). 
+However, this is far from perfect and in some cases ($\sim \SI{9}{\percent}$) the quality factor was not able to prevent the estimation from drifting into the courtyard. 
+This solely happened when particles were sampled directly onto the courtyard while changing from first to zeroth floor (cf. fig. \ref{fig:path2} seg. 3), instead of within the building first. 
 Those particles then received a high weight due to the attenuated measurements, causing a weight degeneracy.
 Adapting the bounds $l_{\text{max}}$ and $l_{\text{min}}$ of the quality factor or optimizing the access-point's parameters can resolve this problem \cite{Ebner-17}.
 
@@ -125,17 +126,17 @@ An exemplary result for path 1 is illustrated in fig. \ref{fig:path1}.
 The path starts on the first floor and finishes on the third after walking two straight stairs. 
 Using the graph-based filter for localisation, we were able to obtain a very accurate path (blue) in \SI{80}{\percent} of the runs providing $\tilde{x}_{75} = \SI{526}{\centimeter}$ with $\bar{\sigma} = \SI{316}{\centimeter}$. 
 Due to a lack of particles near the stairs, the other \SI{20}{\percent} of the estimations given by the graph-based filter failed to detect the first floor change (red). 
-Using our approach (green), we were able detect all floor changes and thus never lost track. 
+Using our approach (green), we were able to detect all floor changes and thus never lost track. 
 Fig. \ref{fig:path1} shows an example, where the dominant filter also failed to change floors within seg. 2. 
-By looking at seg. 2 of the error plot, we can observe a more or less constant error of \SIrange{5}{6}{\meter}, which then drops rapidly at the beginning of seg. 3. 
-This reduction of the error is caused by the growing importance of the mixing stage, where more and more particles from the support filter are incorporated into the dominant filter.
+By looking at this segment of the error plot, we can observe a more or less constant error of \SIrange{5}{6}{\meter}, which then drops rapidly at the beginning of seg. 3. 
+This is due to the growing importance of the mixing stage, where more and more particles from the support filter are incorporated into the dominant filter.
 Going on in seg. 3, the \docWIFI{} measurements suffer from an attenuation directly after leaving the stairs, what leads to a high error using the graph-based filter (blue and red). 
 In contrast, the IMMPF is able to compensate the false detection due to an decreasing \docWIFI{} quality and thus a highly focused posterior provided by the dominant filter. 
 The IMMPF then performs with $\tilde{x}_{75} = \SI{544}{\centimeter}$ and $\bar{\sigma} = \SI{281}{\centimeter}$. 
 Those very similar estimation results between IMMPF (green) and the graph-based filter (blue) confirm the efficiency of the mixing and how it is able to keep the accuracy while providing a higher robustness against failures.
 
 %allgemeines abschließendes blabla?
-To summarize, the here presented approach was able to recover in all situations and thus never got completely stuck within a demarcated area.
+To summarize, the here presented approach was able to recover in all situations and thus never got stuck within a demarcated area.
 The above deployed experiments have shown, that the Markov switching process, as presented in sec. \ref{sec:immpf}, enables a reasonable mixing between two particle filters with different transition schemes.
 Therefore, the quality and success of the results depend highly on the parameters used within the Markov matrix $\Pi_t$. 
 Those values are very sensitive and should be chosen carefully in regard to the specific system and use case.