\section{Recursive State Estimation}

	\commentByFrank{schon mal kopiert, dass es da ist.}
	\commentByFrank{die neue activity in die observation eingebaut}
	\commentByFrank{magst du hier auch gleich smoothing ansprechen? denke es würde sinn machen weils ja zum kompletten systemablauf gehört und den hatten wir hier ja immer drin. oder was meinst du?}

	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-1})}_{\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, \mStateHeading, \mStatePressure),\enskip
		x, y, z, \mStateHeading, \mStatePressure \in \R \enspace,
	\end{equation}
	%
	where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the
	relative atmospheric pressure prediction in hectopascal (hPa).
	\commentByFrank{hier einfach kuerzen und aufs fusion paper verweisen? auch wenn das noch ned durch ist?}
	The recursive part of the density estimation contains all information up to time $t-1$.
	Furthermore, the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ models the pedestrian's movement as described in section \ref{sec:trans}. 
	%It should be noted, that we also include the current observation $\mObsVec_{t}$ in it.
	As proven in \cite{Koeping14-PSA}, we may include the observation $\mObsVec_{t-1}$ into the state transition. 

	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, \mObsSteps, \mObsPressure, \mObsActivity) \enspace,
	\end{equation}
	%
	where $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{})
	and \docIBeacon{}s, respectively. $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number
	of steps detected for the pedestrian. $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference. 
	Finally, $\mObsActivity$ contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or
	walking stairs. 
	%For further information on how to incorporate such highly different sensor types,
	%one should refer to the process of probabilistic sensor fusion \cite{Khaleghi2013}.
	By assuming statistical independence of all sensors, the probability density of the state evaluation is given by 
	%
	\begin{equation}
		%\begin{split}
			p(\vec{o}_t \mid \vec{q}_t) = 
			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}
			\enspace.
		%\end{split}
		\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
	\docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. 

	%It is well known that finding analytic solutions for densities is very difficult and only possible in rare cases.
	%Therefore, numerical solutions like Gaussian filters or the broad class of Monte Carlo methods are deployed \cite{sarkka2013bayesian}. 
	Since we assume indoor localisation to be a time-sequential, non-linear and non-Gaussian process,
	a particle filter is chosen as approximation of the posterior distribution. 
	\commentByFrank{smoothing?}
	%Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ is used as proposal distribution,
	%also known as CONDENSATION algorithm \cite{Isard98:CCD}.