first draft impoverishment finished

tex/chapters/misc.tex

@@ -60,25 +60,57 @@ If $\probGrid_{t, \text{wifi}}$ and the current posterior $p(\mStateVec_{t} \mid
 A good measure of how one probability distribution differs from a second is the well-established Kullback-Leibler divergence $D_\text{KL}$ \cite{Fetzer-17}.
 To calculate $D_\text{KL}$, we need to sample densities from both probability density functions likewise. 
 For the posterior we use the results provided by our rapid kernel density estimation performed in the state estimation procedure, while $\probGrid_{t, \text{wifi}}$ is already in the desired form.
+%To handle $D_\text{KL}$ as probability, we use a positive exponential distribution 
+
+%	\begin{equation}
+%		f(D_{\text{KL}}, \lambda) = e^{-\lambda D_{\text{KL}}}
+%		\enspace .
+%	\label{equ:KLD}
+%	\end{equation}
+	%
+
+Using $D_\text{KL}$, we are now able to take countermeasures against sample impoverishment, depending on its size. 
+However, those countermeasures will only work reliable if the \docWIFI{} measurement noise is within reasonable limits. 
+Attenuated or bad \docWIFI{} readings are leading $D_\text{KL}$ to grow, even if the posterior provides good results. 
+For this, we introduce a \docWIFI{} quality factor, enabling us to identify such situations. 
+The quality factor is defined by
+
+		\begin{equation}
+            \newcommand{\leMin}{l_\text{min}}
+            \newcommand{\leMax}{l_\text{max}}
+            q(\mObsVec_t^{\mRssiVec_\text{wifi}}) = 
+                \max \left(0,
+                    \min \left(
+                        \frac{
+                            \bar\mRssi_\text{wifi} - \leMin
+                        }{
+                            \leMax - \leMin
+                        },
+                        1
+                    \right)
+                \right)
+                %,\enskip
+				%\bar\mRssi_\text{wifi} = \frac{1}{n} \sum_{i = 1}^{n} \mRssi_i
+            \label{eq:wifiQuality}
+        \end{equation}
+
+\noindent where $\bar\mRssi_\text{wifi}$ is the average of all signal strength measurements received from the observation $\mObsVec_t^{\mRssiVec_\text{wifi}}$. An upper and lower bound is given by $l_\text{max}$ and $l_\text{min}$. 
+The quality factor is extensively discussed within \cite{Ebner-17} and \cite{Fetzer-17}.
+
+Finally, we have all necessary tools to introduce a second method to prevent impoverishment. 
+For this, the state transition model is extended. 
+Compared to the resampling step, as used by the first method, the transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ enables us to use prior measurements, which is obviously necessary for all \docWIFI{} related calculations.
+As described in chapter \ref{sec:transition}, our transition method only allows to sample particles at positions, that are actual feasible for a humans within a building e.g. no walking trough walls. 
+If a particle targets a position which is not walk-able e.g. behind a wall, we draw a new position within a very small, but reachable area around it. 
+%To prevent sample impoverishment we extend our transition method.
+Instead of drawing particles like this or even the complete building, as suggested in method one, we define a sphere.
+The radius is given by $D_\text{KL} \cdot q(\mObsVec_t^{\mRssiVec_\text{wifi}})$.
+This allows to increase the diversity of particles by the means of \docWIFI{}. 
+The subsequent evaluation of the particle filter then reweights the particles, so that only those in proper regions will survive the resampling. 
+ 
+To further improve the method we give particles a chance of \SI{0.01}{\percent} to walk trough a nearby wall, if the destination is not outside. 
+This enables to handle sample impoverishment more quickly in situations caused by environmental restrictions, even when the \docWIFI{} quality is low.
+Especially in areas full of nooks an crannies, the vulnerability to errors should be decreased.
+%In most cases $\lambda$ tends to be somewhere between \SI{0.01}{} and \SI{0.10}{}.
 
 
-We compare the particles provided by the posterior and the samples of $\probGrid_{t, \text{wifi}}$
-
-
-
-\begin{itemize}
-	\item zufällig einen partikel streuen
-	\item partikel bekommen eine kleine chance durch wände zu laufen
-	\item KLD zwischen wifi und aktuellen particeln des filters. 
-\end{itemize}
-
-
-
-
-
-
-
-
-%fürs verständnis, diesen satz hier nicht vergessen.
-as described in chapter \ref{sec:rse}, a particle is a weighted representation of one possible system state $\mStateVec$.... 
-

tex/chapters/transition.tex

@@ -1,4 +1,5 @@
 \section{Transition}
+\label{sec:transition}
 
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