\section{Filtering}
	
	\label{sec:filtering}

	\commentByToni{Bin mir nicht sicher ob wir diese Section überhaupt brauchen. Könnte man bestimmt auch einfach unter Section 3 packen. Aber dann können wir ungestört voneinander schreiben.}

\section{Evaluation}

	\subsection{Barometer}
	\label{sec:sensBaro}
		%
		The probability of currently residing on a given floor is evaluated using the smartphone's barometer.
		Environmental influences are circumvented by using relative pressure readings instead of absolute ones.
		To reduce the impact of noisy sensors, we calculate the average of several sensor reading, carried out
		while the pedestrian chooses his destination. This $\overline{\mObsPressure}$ serves as relative base.
		Likewise, we estimate the sensor's uncertainty $\sigma_\text{baro}$ for later use within the evaluation step.		
	
		In order to evaluate relative pressure readings, we need a prediction to compare them with. Therefore, each
		transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ estimates the state's relative pressure prediction
		$\mStatePressure$ by examining every height-change ($z$-axis):
		%		
		\begin{equation}
			\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b
			,\enskip
			\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
			,\enskip
			b \in \R
			\enspace .
			\label{eq:baroTransition}
		\end{equation}
		%
		In \refeq{eq:baroTransition}, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
		The evaluation step compares the predicted relative pressure with the observed
		one using a normal distribution with the previously estimated $\sigma_\text{baro}$:
		%
		\begin{equation}
			p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace.
			\label{eq:baroEval}
		\end{equation}
		%		
		%
		%
	\subsection{Wi-Fi \& iBeacons}
		%
		The smartphone's \docWIFI{} and \docIBeacon{} component provides absolute location estimation by
		measuring the signal-strengths of nearby transmitters. The positions of detected \docAP{}s  (\docAPshort{}) and \docIBeacon{}s
		are known beforehand. This allows a comparison of each measurement with a corresponding estimation
		using the wall-attenuation-factor signal strength prediction model \cite{Ebner-15}. This model uses the 3D distance $d$ and the
		number of floors $\Delta f$ between transmitter and the state-in-question:
		%	
		\begin{equation}
			P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace ,
		\end{equation}
		%
		As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is:
		%		
		\begin{equation}
			\mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} =
			\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace .
			\label{eq:wifiTotal}
		\end{equation}
		%
		The prediction model itself needs three parameters per \docAPshort{}: $\mTXP$ measured at a distance 
		$\mMdlDist_0$ (usually \SI{1}{\meter}), the path-loss exponent $\mPLE$ describing the environment
		and the attenuation per floor $\mWAF$.
		\commentByFrank{aufs andere paper beziehen zum kuerzen?}
		To reduce the system's setup time, we use the same values for all \docAP{}s at the cost of accuracy.
		All parameters are chosen empirically. Further details on how to determine this parameters exactly,
		can be found in \cite{PathLossPredictionModelsForIndoor}.
			
		The same holds for the \docIBeacon{} component, except $\mTXP$, which is broadcasted by each beacon.
		As \docIBeacon{}s cover only a small area, $\mPLE$ is usually much smaller compared to the one needed for \docWIFI{}.
	
		%
		

\section{Transition}

		The distribution $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is sampled via random walks on a graph
		$G=(V,E)$, which is generated from the buildings floorplan \todo{FUSION2016}.
		$p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is determined by walking along adjacent edges $\mEdgeAB$ connecting
		two vertices $\mVertexA, \mVertexB \in V$ until a certain distance $\gDist$ is reached.
		Thus, the position of any $\mStateVec$ is represented by the position $\fPos{\mVertexA}$ of the corresponding vertex.
		This approach draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
		
		While sampling, to-be-walked edges are not chosen uniformly, but depending on a probability $p(\mEdgeAB)$.
		The latter depends on several constraints and recent sensor-readings from the smartphone. Using sensors
		directly within the transition step provides a more robust posterior distribution. Adding them to the evaluation
		instead, would lead to sample impoverishment due to the used Monte Carlo methods.
				
		\subsection{Pedestrian's Destination}
			We assume the pedestrian's desired destination to be known beforehand. This prior knowledge is incorporated
			during the random walk using $p(\mEdgeAB)_\text{path}$, which is a simple heuristic, favouring movements (edges)
			approaching the chosen destination with a ratio of $0.9:0.1$ over those, departing from the destination
			\cite{FUSION2016}. The underlying shortest-path is based on a special distance metric, considering special
			architectural facts: 
			
		\subsection{Architectural Facts}
			To ensure realistic path estimations, we include additional architectural knowledge.
			Each vertex's distance from the nearest wall is used to prevent the shortest-path from clinging to walls.
			Likewise, his distance from the nearest door (favour ).
				
				
		\subsection{Step- \& Turn-Detection}
			Steps and turns are detected using the smartphone's IMU, implemented as described in \cite{Ebner-15}.
			The number of steps detected since the last transition is used to estimate the to-be-walked distance $\gDist$ 
			assuming a fixed step-size with some deviation:
			%
			\begin{equation}
				\gDist = \mObs_{t-1}^{\mObsSteps} \cdot \mStepSize	+ \mathcal{N}(0, \sigma_{\gDist}^2)
				\enspace .
			\end{equation}
			%
			Turn-Detection supplies the magnitude of the detected heading change by integrating	the gyroscope's change
			since the last transition. Together with some deviation and the state's previous heading, the magnitude is
			used to estimate the state's current heading:
			%
			\begin{equation}
				\gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading}	+ \mathcal{N}(0, \sigma_{\gHead}^2) \\
			\end{equation}
			%
			During the random walk, edges should satisfy the heading:
			%
			\begin{equation}
				p(\mEdgeAB)_\text{head} = p(\mEdgeAB \mid \gHead) = \mathcal{N} (\angle \mEdgeAB \mid \gHead, \sigma_\text{head}^2)
				\enspace .
				\label{eq:transHeading}
			\end{equation}
			%
			While the distribution \refeq{eq:transHeading} does not integrate to $1.0$ due to circularity of angular
			data, in our case, the normal distribution can be assumed as sufficient for small enough $\sigma^2$.
				
				
				
		
			
				
		
			
			
			
		\subsection{Activity-Detection}
			Additionally we perform a simple activity detection for the pedestrian, able to distinguish between 
			standing, walking, walking stairs upwards and downwards. Likewise, this knowledge
			is evaluated when walking the grid: Edges $\mEdgeAB$ matching the currently detected
			activity are favoured using $p(\mEdgeAB)_\text{act} = 0.8$ and $0.2$ otherwise.
			
			\begin{equation}
				p(\mEdgeAB)_\text{act} = 
				\begin{cases}
					0.8	& \text{stairs\_up}, \fPos{\mVertexB}_z > \fPos{\mVertexA}_z \\
					0.2	& \text{stairs\_up}, \fPos{\mVertexB}_z \le \fPos{\mVertexA}_z \\
					\cdots
				\end{cases}
			\end{equation}
			\commentByFrank{das switch ist wahrscheinlich unnoetig und der text reicht}
			
			\commentByFrank{hier passen die sachen vom lukas. kurze beschreibung der beiden geschaetzten verteilungen etc}