\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-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). 
	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) \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.	
	Finally, $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference. 
	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. 
	Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ is used as proposal distribution,
	what is also known as CONDENSATION algorithm \cite{Isard98:CCD}.


