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\section{Evaluation}
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3/4 - 1 Seite
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The probability density of the state evaluation in \eqref{equ:bayesInt} is given by
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%
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\begin{equation}
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%\begin{split}
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p(\vec{o}_t \mid \vec{q}_t) =
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p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}
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\,p(\vec{o}_t \mid \vec{q}_t)_\text{act}
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\enspace ,
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%\end{split}
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\label{eq:evalBayes}
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\end{equation}
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%
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where every component refers to a probabilistic sensor model and are statistical independent.
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The barometer readings are used to determine the current activity $\mObsActivity$, which is then evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{act}$.
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Absolute positioning information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
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\subsection{Wifi}
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\subsection{\docWIFI{}}
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\begin{itemize}
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\item kleine Wifi Beacons
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\item optimimierung der ap positionen über schätzverfahren cite cite cite
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\item log dist model. können wir auch wieder viel citen.
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\item da vorher nie erwähnt, ggf. bisschen was über VAPGrouping.
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\end{itemize}
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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
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\subsection{Barometer}
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\begin{itemize}
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\item activity recognition über barometer
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\item rauf/runter/treppe/aufzug etc. pp.
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\end{itemize}
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\begin{equation}
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p(\vec{o}_t \mid \vec{q}_t)_\text{wifi} =
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p(\mRssiVec_\text{wifi} \mid \mPosVec) =
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\prod_{\mRssi_{i} \in \mRssiVec_\text{wifi}} p(\mRssi_{i} \mid \mPosVec),\enskip
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%\mPos = (x,y,z)^T
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\mPosVec \in \R^3
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\enskip .
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\label{eq:wifiObs}
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\end{equation}
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\noindent We assume a statistical independence between the respective AP's.
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The comparison between a single RSSI measurement $\mRssi_i$ and the reference is given by
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\begin{equation}
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p(\mRssi_i \mid \mPosVec) =
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\mathcal{N}(\mRssi_i \mid \mu_{i,\mPosVec}, \sigma_{i,\mPosVec}^2)
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\enskip ,
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\label{eq:wifiProb}
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\end{equation}
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\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 $\mPosVec$.
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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:
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\begin{equation}
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\mu_{\mPosVec} = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF
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\label{eq:wallAtt}
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\end{equation}
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\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.
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The attenuation per floor is given by $\mWAF$.
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For example, a viable choice for steel enforced concrete floors is $\mWAF \approx \SI{-8.0}{dB}$ \cite{Ebner-15}.
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Of course, the environmental parameters $\mTXP$, $\mPLE$ and $\mWAF$ need to be known beforehand and often vary greatly between single \docAPshort{}'s.
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Nevertheless, for simplicity's sake it is common practice to use some fixed empirically chosen values, the same for every \docAPshort{}.
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This might already provide enough accuracy for some use-cases and buildings, but fails in complex scenarios, as discussed in section \ref{sec:intro}.
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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.
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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.
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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.
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The target function to optimize the $6$ model parameters for one \docAPshort{} is given by
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\begin{equation}
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\epsilon^* =
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\argmin_{\mPosAPVec, \mTXP, \mPLE, \mWAF}
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\sum_{s_{\mPosVec} \in \vec{s}}
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(s_{\mPosVec} - \mu_{\mPosVec})^2
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\enskip,\enskip\enskip
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\mu_{\mPosVec} =
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\mTXP{} + 10 \mPLE{} \log_{10} \| \mPosVec-\mPosAPVec \| + \text{floors}(\mPosVec,\mPosAPVec) \mWAF{}
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\enspace .
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\label{eq:optTarget}
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\end{equation}
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\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}).
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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.
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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}.
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The here deployed algorithm starts with a initial population, that is uniformly sampled within predefined limits of the to-be-optimized parameters.
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The \docAPshort{}'s location $\mPosAPVec$ must be within the building and is therefore limited by its size.
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\mTXP{}, \mPLE{} and \mWAF{} are set within a sane interval around empirically chosen values.
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During each iteration, the best \SI{25}{\percent} of the population are kept.
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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.
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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.
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\todo{Wollen wir das mal genauer beschreiben? Also wie genau funktioniert das cooling. Das ist ja alles sehr wischi waschi gehalten}
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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.
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The reason for this comes from the assumptions made in eq. \eqref{eq:wallAtt}.
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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.
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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.
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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.
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%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.
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Basically, any kind of \docAPshort{} providing RSSI measurements can be used for the above.
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However, most buildings do not provide a satisfying and well covered \docWIFI{} infrastructure, e.g. staircases or hallways are often neglected for office spaces.
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This applies in particular to historical buildings, as discussed in section \ref{sec:intro}.
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To improve $\docWIFI$ coverage we are able to distribute a small number of simple and cheap \docWIFI{} beacons.
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As beacons we use a WEMOS D1 mini, which is based on the ESP-8266EX \docWIFI{} chip \cite{Wemos}.
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The building considered in this work has no \docWIFI{} infrastructure at all, not even a single \docAPshort{}.
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Nevertheless, our method also allows to distribute beacons in the whole building by just plugging them into available power outlets.
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\subsection{Activity Recognition}
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To enable continuous floor changes we use a simple activity recognition based on the smartphone's barometer and accelerometer.
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The method distinguishes between the following: standing, walking, walking up or walking down.
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\todo{fill me}
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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.
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If the current activity $\mObsActivity$ is recognized as "standing", a particle located on the floor receives a probability given by $\kappa$, otherwise $1 - \kappa$.
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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 $1 - \kappa$.
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The probability $\kappa$ is chosen empirically.
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It is useful to find a reasonable value that is not too restrictive.
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In most cases, $\kappa = 0.75$ provides good results by remaining enough room for erroneous classifications.
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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.
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\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. }
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toni