\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)_\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 barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$, 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. 

\todo{art unseres particle filters hier einfuehren. transition als proposal. dann kann man spaeter bei step und turn besser begruenden warum wir es in die transition ziehen.}


