recursive state estimation und paar kommentare.

tex/chapters/grid.tex

@@ -1,4 +1,5 @@
 \section{Transition Model}
+\label{sec:trans}
 
 	To sample only transitions $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ that are actually feasible
 	within the environment, we utilize a \SI{20}{\centimeter}-gridded graph

tex/chapters/sensors.tex

@@ -78,7 +78,9 @@
 	\subsection{Step- \& Turn-Detection}
 		
 		To prevent degradation within the particle-filter \cite{??} due to downvoting of particles with increased
-		heading deviation, we incorporate the turn-detection as control-data \commentByToni{ich würde es jetzt nicht unbedingt  controldata nennen. einfach nur das wir die observation in die transition integrieren. fertig.} directly into the transition
+		heading deviation, we incorporate the turn-detection as control-data 
+\commentByToni{ich würde es jetzt nicht unbedingt  controldata nennen. einfach nur das wir die observation in die transition integrieren. fertig. für den cite oben habe ich eine sehr sehr gute quelle. der beschreibt das ganz gut. hab das bei mir in der MA so begründet: Another big disadvantage is the high possibility of sample impoverishment due to a small measurement noise. This happens since accurate observations result in high peaks of the evaluation density and therefore the importance density is not able to sample outside that peak [IB98b].} 
+directly into the transition
 		$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$.
 		\cite{thrun?}\cite{lukas2014?} to get a more directed sampling instead of a truly random one.
 		

tex/chapters/system.tex

@@ -1,24 +1,50 @@
-\section{Recursive Density Estimation}
+\section{Recursive State Estimation}
+
+We consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem. 
+Using a recursive Bayes filter that satisfies the Markov property, the posterior distribution at time $t$ can be written as
+\begin{equation}
+\arraycolsep=1.2pt
+\begin{array}{ll}
+	&p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto\\ 
+		&\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}}
+		\int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t})}_{\text{transition}}
+		\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}} \enspace,
+\end{array}
+\label{equ:bayesInt}
+\end{equation}
+where $\mObsVec_{1:t} = \mObsVec_{1}, \mObsVec_{1}, ..., \mObsVec_{t}$ is a series of observations up to time $t$. 
+The hidden state $\mStateVec$ is given by
+\begin{equation}
+	\mStateVec = (x, y, z, \mObsHeading, \mStatePressure),\enskip
+	x,y,z,\mStatePressure \in \R \enspace,
+\end{equation}
+where $x, y, z$ represent the 3D position, $\mObsHeading$ the user's heading and $\mStatePressure$ the relative pressure prediction in hectopascal (hPa). 
+The recursive part of the density estimation contains all information up to time $t$.
+Further, the state transition models the pedestrian’s movement based upon random walks on graphs, which will be described in section \ref{sec:trans}. 
+It should be noted, that we also include the current observation $\mObsVec_{t}$ in it. 
+
+Containing all relevant sensor measurements to evaluate the current state, the observation vector is defined as follows:
+\begin{equation}
+	\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsPressure) \enspace,
+\end{equation}
+where $\mRssiVec_\text{wifi}$ is the Wi-Fi and $\mRssiVec_\text{ib}$ the iBeacon signal strength vector. 
+The information, if a step or turn was detected, is given as a Boolean value. 
+\commentByToni{Wie sieht die Observation nun genau aus? Fehlt da nicht Step und Turn?}
+Finally, $\mObsPressure$ is the relative barometric pressure referring to some fixed point in time. 
+For incorporating the highly different sensor types, one should refer to the process of probabilistic sensor fusion \cite{}.
+By assuming statistical independence of all sensor models, the probability density of the state evaluation is given by 
+\begin{equation}
+\begin{split}
+&p(\vec{o}_t \mid \vec{q}_t, \vec{q}_{t-1}) = \\
+&p(\vec{o}_t \mid \vec{q}_t, \vec{q}_{t-1})_\text{turn}
+\,p(\vec{o}_t \mid \vec{q}_t, \vec{q}_{t-1})_\text{step} \\
+&p(\vec{o}_t \mid \vec{q}_t)_\text{baro} 
+\,p(\vec{o}_t \mid \vec{q}_t)_\text{ib}
+\,p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}
+\end{split} \enspace.
+\label{eq:evalBayes}
+\end{equation}
+Here, every single component refers to a probabilistic sensor model. 
+The heading information is evaluated using $p(\vec{o}_t \mid \vec{q}_t, \vec{q}_{t-1})_\text{turn}$, the step length using a step detection process by $p(\vec{o}_t \mid \vec{q}_t, \vec{q}_{t-1})_\text{step}$, using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$ the barometer evaluates the current floor, whereby absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for iBeacons and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for Wi-Fi. 
+
 
-	\commentByFrank{particle-filter wie bei lukas mit $\vec{o}_{t}$ in transition und $\vec{q}_{t-1}$ in eval??}
-	\commentByFrank{brauchen wir in der observation ueberhaupt noch $q_{t-1}$??}
-	\commentByFrank{das ist die basis fuer unser system}
-	
-	
-	\begin{equation}
-		p(\mStateVec_{t} \mid \langle \mObsVec \rangle_{t}) = \\
-		p(\mObsVec_{t} \mid \mStateVec_{t})
-		\int
-			p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t})
-			p(\mStateVec_{t-1} \mid \langle \mObsVec \rangle_{t-1}
-			d\mStateVec_{t-1}
-	\end{equation}
-	
-	\begin{equation}
-		\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsPressure)
-	\end{equation}
-	
-	\begin{equation}
-		\mStateVec = (x, y, z, \mObsHeading, \mStatePressure),\enskip
-		x,y,z,\mStatePressure \in \R
-	\end{equation}

tex/gfx/baro/baro_setup_issue.aux

@@ -1,7 +1,7 @@
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