recursive state estimation und paar kommentare.

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}