\section{Evaluation}

The probability density of the state evaluation in \eqref{equ:bayesInt} is given by 
%
\begin{equation}
	%\begin{split}
		p(\vec{o}_t \mid \vec{q}_t) = 
		p(\vec{o}_t \mid \vec{q}_t)_\text{wifi} 
		\,p(\vec{o}_t \mid \vec{q}_t)_\text{act}
		\enspace ,
	%\end{split}
	\label{eq:evalBayes}
\end{equation}
%
where every component refers to a probabilistic sensor model and are statistical independent.
The barometer readings are used to determine the current activity $\mObsActivity$, which is then evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{act}$. 
Absolute positioning information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. 

\subsection{\docWIFI{}}

As stated in section \ref{sec:relatedWork}, we use the smartphone's \docWIFI{} component to provide an absolute location estimation based on a comparison between recent RSSI measurements of nearby AP's and signal strength predictions. The probability given those measurements $\mRssiVec_\text{wifi}$ and a prediction, corresponding to a well-known location $\mPosVec = (x,y,z)^T$ provided by $\vec{q}_t$, can thus be written as

\begin{equation}
	p(\vec{o}_t \mid \vec{q}_t)_\text{wifi} = 
	p(\mRssiVec_\text{wifi} \mid \mPosVec) =
	\prod_{\mRssi_{i} \in \mRssiVec_\text{wifi}} p(\mRssi_{i} \mid \mPosVec),\enskip
	%\mPos = (x,y,z)^T
	\mPosVec \in \R^3
	\enskip .
	\label{eq:wifiObs}
\end{equation}

\noindent We assume a statistical independence between the respective AP's. 
The comparison between a single RSSI measurement $\mRssi_i$ and the reference is given by

\begin{equation}
	p(\mRssi_i \mid \mPosVec) = 
	\mathcal{N}(\mRssi_i \mid \mu_{i,\mPosVec}, \sigma_{i,\mPosVec}^2)
	\enskip ,
	\label{eq:wifiProb}
\end{equation}

\noindent where $\mu_{i,\mPosVec}$ denotes the (predicted) average signal strength and $\sigma_{i,\mPosVec}^2$ a corresponding standard deviation for the \docAPshort{} identified by $i$, regarding the location $\mPosVec$.
Within this work $\mu_{\mPosVec}$ is calculated by a modified version of the wall-attenuation-factor model as presented in \cite{Ebner-17}. Here, the prediction depends on the 3D distance $d$ from the \docAPshort{} and the number of floors $\Delta f$ between the \docAPshort{} and $\mPosVec$ of the particle-in-question:

\begin{equation}
	\mu_{\mPosVec} = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF 
	\label{eq:wallAtt}
\end{equation}

\noindent Here, $\mTXP$ is the \docAPshort{}'s signal strength measurable at a known distance $\mMdlDist_0$ (usually \SI{1}{\meter}) and $\mPLE$ denotes the signals depletion over distance, which depends on the \docAPshort{}'s surroundings like walls and other obstacles.
The attenuation per floor is given by $\mWAF$. 
For example, a viable choice for steel enforced concrete floors is $\mWAF \approx \SI{-8.0}{dB}$ \cite{Ebner-15}.

Of course, the environmental parameters $\mTXP$, $\mPLE$ and $\mWAF$ need to be known beforehand and often vary greatly between single \docAPshort{}'s. 
Nevertheless, for simplicity's sake it is common practice to use some fixed empirically chosen values, the same for every \docAPshort{}.
This might already provide enough accuracy for some use-cases and buildings, but fails in complex scenarios, as discussed in section \ref{sec:intro}. 
Therefore, instead of using a pure empiric model, we deploy a optimization scheme to find a well-suited set of parameters ($\mPosAPVec{}, \mTXP{}, \mPLE{}, \mWAF{}$) per \docAPshort{}, where $\mPosAPVec{} = (x,y,z)^T$ denotes the \docAPshort{}'s position. 
The optimization is based on a few reference measurements $s_{\mPosVec}$ throughout the building, e.g. every \SI{3}{} to \SI{5}{\meter} centred within a corridor and between \SI{1}{} and \SI{4}{} references per room, depending on the room's size. 
Compared to classical fingerprinting, where reference measurements are recorded on small grids between \SI{1}{} to \SI{2}{\meter}, this highly reduces their required number and thus the overall setup-time.

The target function to optimize the $6$ model parameters for one \docAPshort{} is given by

\begin{equation}
		\epsilon^* =
		\argmin_{\mPosAPVec, \mTXP, \mPLE, \mWAF}
		\sum_{s_{\mPosVec} \in \vec{s}}
			(s_{\mPosVec} - \mu_{\mPosVec})^2
		\enskip,\enskip\enskip
		\mu_{\mPosVec} = 
		\mTXP{} + 10 \mPLE{} \log_{10} \| \mPosVec-\mPosAPVec \| + \text{floors}(\mPosVec,\mPosAPVec) \mWAF{} 
\enspace .
	\label{eq:optTarget}
\end{equation}

\noindent Here, one reduces the squared error between reference measurements $s_{\mPosVec} \in \vec{s}$ with well-known location $\mPosVec$ and corresponding model predictions $\mu_{\mPosVec}$ (cf. eq. \eqref{eq:wallAtt}).
As discussed by \cite{Ebner-17}, optimizing all 6 parameters, especially the unknown \docAPshort{} position $\mPosAPVec$, usually results in optimizing a non-convex, discontinuous function.
A promising way to deal with non-convex functions is using a genetic algorithm, which is inspired by the process of natural selection \cite{goldberg89}. 

The here deployed algorithm starts with a initial population, that is uniformly sampled within predefined limits of the to-be-optimized parameters. 
The \docAPshort{}'s location $\mPosAPVec$ must be within the building and is therefore limited by its size. 
\mTXP{}, \mPLE{} and \mWAF{} are set within a sane interval around empirically chosen values.
During each iteration, the best \SI{25}{\percent} of the population are kept.
The remaining entries are then re-created by modifying the best entries with uniform random values within $\pm$\SI{10}{\percent} of the known limits.
Inspired by {\em cooling} known from simulated annealing \cite{Kirkpatrick83optimizationby},		 the result is stabilized by narrowing the allowed modification limits over time and thus decrease in the probability of accepting worse solutions.

\todo{Wollen wir das mal genauer beschreiben? Also wie genau funktioniert das cooling. Das ist ja alles sehr wischi waschi gehalten}

To further improve the results, we optimize a model for each floor of the building instead of a single global one, using only the reference measurements that belong to the corresponding floor. 
The reason for this comes from the assumptions made in eq. \eqref{eq:wallAtt}. 
Here, no walls are considered and thus we expect erroneous RSSI measurements for regions that are heavily shrouded, e.g. by steel-reenforced concrete or metallized glass.
During evaluation, the $z$-value from $\mPosVec$ in eq. \eqref{eq:wifiProb} is used to select the correct model for this location, what then provides the signal strength prediction.
For example, if a pedestrian walks on a staircase and thus is in-between multiple storeys, the average prediction of all corresponding models is calculated instead.

%man muss zwar messungen machen, dafür muss man aber die position der ap's nicht mehr kennen. daher kostet das jetzt nicht viel mehr zeit.

Basically, any kind of \docAPshort{} providing RSSI measurements can be used for the above. 
However, most buildings do not provide a satisfying and well covered \docWIFI{} infrastructure, e.g. staircases or hallways are often neglected for office spaces. 
This applies in particular to historical buildings, as discussed in section \ref{sec:intro}.
To improve $\docWIFI$ coverage we are able to distribute a small number of simple and cheap \docWIFI{} beacons.
As beacons we use a WEMOS D1 mini, which is based on the ESP-8266EX \docWIFI{} chip \cite{Wemos}. 
The building considered in this work has no \docWIFI{} infrastructure at all, not even a single \docAPshort{}.
Nevertheless, our method also allows to distribute beacons in the whole building by just plugging them into available power outlets.


\subsection{Activity Recognition}

To enable continuous floor changes we use a simple activity recognition based on the smartphone's barometer and accelerometer. 
The method distinguishes between the following: standing, walking, walking up or walking down.
\todo{fill me}

All this is evaluated within $p(\vec{o}_t \mid \vec{q}_t)_\text{act}$ by providing a probability based on whether a particle is located at a stair, an elevator or the floor. 
If the current activity $\mObsActivity$ is recognized as "standing", a particle located on the floor receives a probability given by $\kappa$, otherwise $1 - \kappa$.
The same applies to "walking up" and "walking down", here a particle located on one of the possible staircases or elevators receives $\kappa$ and those who remain on the floor $1 - \kappa$.
The probability $\kappa$ is chosen empirically.
It is useful to find a reasonable value that is not too restrictive.
In most cases, $\kappa = 0.75$ provides good results by remaining enough room for erroneous classifications.
A significant higher value like $\kappa = 0.99$ could cause the particle filter to be stuck on a staircase or to be unable to change floors.



\commentByToni{warum wir die große treeppe so schwer ist: wlan model zieht JEDE decke ab, nicht nur die sichtbaren, weil das model einfach so gebaut wurde. }