sample impoverishment weiter gemacht + bulli chapter in eigene datei

tex/chapters/estimation.tex

@@ -0,0 +1,10 @@
+\subsection{State Estimation}
+
+1/2 bis 3/4 Seite
+
+\begin{itemize}
+	\item weighted average
+	\item max particle
+	\item bulli methode (gleich citen :))
+\end{itemize}
+

tex/chapters/eval.tex

@@ -1,5 +1,6 @@
 \section{Evaluation}
 
+
 The probability density of the state evaluation in \eqref{equ:bayesInt} is given by 
 %
 \begin{equation}
@@ -17,6 +18,7 @@ The barometer readings are used to determine the current activity $\mObsActivity
 Absolute positioning information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. 
 
 \subsection{\docWIFI{}}
+\label{sec:wifi}
 
 As stated in section \ref{sec:relatedWork}, we use the smartphone's \docWIFI{} component to provide an absolute location estimation based on a comparison between recent RSSI measurements of nearby AP's and signal strength predictions. The probability given those measurements $\mRssiVec_\text{wifi}$ and a prediction, corresponding to a well-known location $\mPosVec = (x,y,z)^T$ provided by $\vec{q}_t$, can thus be written as
 

tex/chapters/misc.tex

@@ -14,6 +14,8 @@ p(\mStateVec_{t} \mid \mObsVec_{1:t}) \approx \sum^N_{i=1} w^i_t \delta_{\vec{X}
 As one can imagine, after a few iterations with continuously reweighting particles, the weight will concentrate on a few particles only.
 To handle this phenomenon of weight degeneracy, a resampling procedure is performed after every filter step \cite{robotics}. 
 
+\input{chapters/estimation}
+
 \subsection{Sample Impoverishment}
 As we have extensively discussed in \cite{Fetzer-17}, besides sample degeneracy, particle filters (and nearly all of its modifications) continue to suffer from another notorious problem: sample impoverishment. 
 It refers to a situation, in which the filter is unable to sample enough particles into proper regions of the building, caused by a high concentration of misplaced particles.
@@ -50,9 +52,19 @@ This allows the primary filter to recover, while retaining prior knowledge.
 However, we believe that such a combination of two independent filters is not necessary for most scenarios and thus the resulting overhead can be avoided.
 
 %neue methode:
+For the simplified version we distribute \SI{10000}{} samples uniformly within the complete building to approximate $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ as presented in section \ref{sec:wifi}.
+From the resulting probability grid $\probGrid_{t, \text{wifi}}$ we are able to identify the areas where the \docWIFI{} model assumes the pedestrian is most likely located. 
+Of course, this often results in a multimodal representation of the probability density and thus multiple possible whereabouts.  
+However, compared to the used particle filter, this representation enables us to monitor the complete building without any environmental restrictions and can thus be deployed as an indicator to detect sample impoverishment. 
+If $\probGrid_{t, \text{wifi}}$ and the current posterior $p(\mStateVec_{t} \mid \mObsVec_{1:t})$ show a significant difference, we can assume that either the posterior got stuck and suffers from impoverishment or the \docWIFI{} quality is low due to factors like attenuation or bad coverage. 
+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.
+
+
+We compare the particles provided by the posterior and the samples of $\probGrid_{t, \text{wifi}}$
 
 
-we sample 10k samples uniformaly within the whole building..
 
 \begin{itemize}
 	\item zufällig einen partikel streuen
@@ -60,15 +72,7 @@ we sample 10k samples uniformaly within the whole building..
 	\item KLD zwischen wifi und aktuellen particeln des filters. 
 \end{itemize}
 
-\subsection{State Estimation}
 
-1/2 bis 3/4 Seite
-
-\begin{itemize}
-	\item weighted average
-	\item max particle
-	\item bulli methode (gleich citen :))
-\end{itemize}
 
 
 

tex/misc/functions.tex

@@ -27,6 +27,8 @@
 \newcommand{\mParticle}{X}	
 \newcommand{\mParticleVec}{\vec{X}}
 
+\newcommand{\probGrid}{\vec{Q}}
+
 \newcommand{\mProb}{p}									% char for probability
 
 \newcommand{\mMovingAvgWithSize}[1]{\ensuremath{\text{avg}_{#1}}}