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\section{WiFi Location Estimation}
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\label{sec:optimization}
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The \docWIFI{} sensor infers the pedestrian's current location based on a comparison between recent measurements
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(the smartphone continuously scans for nearby \docAP{}s) and reference measurements or
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The \docWIFI{} sensor infers the pedestrian's current location based on a comparison between live observations
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(the smartphone continuously scans for nearby \docAP{}s) and fingerprints or
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signal strength predictions for well known locations:
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\begin{equation}
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@@ -22,15 +22,12 @@
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In \refeq{eq:wifiProb}, $\mu_{i,\mPosVec}$ and $\sigma_{i,\mPosVec}$ denote the average signal strength
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and corresponding standard deviation for the \docAPshort{} identified by $i$,
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that should be measurable given the location $\mPosVec = (x,y,z)^T$. Those two value can be determined using various
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methods. Most common, as of today, seems fingerprinting, where hundreds of locations throughout the building
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are scanned beforehand. The received \docAP{}s including their average signal strength and deviation
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denote each location's fingerprint \cite{radar}.
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that should be measurable given the location $\mPosVec = (x,y,z)^T$. Those two values can be determined using various
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methods. Most common and accurate, as of today, is fingerprinting, where hundreds of locations throughout the building
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are scanned beforehand. The received \docAP{}s including their (average) signal strength and deviation
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denote each location's fingerprint.
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%
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While allowing for highly accurate location estimations, given enough fingerprints, such a setup is costly,
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as fingerprinting is a manual process.
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%
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We therefore use a model to predict the average signal strength for each location,
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To prevent the time-consuming setup process, we use a model to predict the average signal strength for each location,
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based on the \docAPshort{}'s position $\mPosAPVec{} = (x,y,z)^T$ and a few additional parameters.
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@@ -65,22 +62,23 @@
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Depending on the use case, this value describes the number and type of walls, ceilings, floors etc. between both positions.
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For obstacles, this requires an intersection-test of each obstacle with the line-of-sight, which is costly
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for larger buildings. For real-time use on a smartphone, a (discretized) model pre-computation might thus be necessary
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\cite{competition}. Furthermore this requires a detailed floorplan, that includes material information
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for walls, doors, floors and ceilings.
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\cite{competition2016}.
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%Furthermore this requires a detailed floorplan, that includes material information
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%for walls, doors, floors and ceilings.
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Throughout this work, we thus use a tradeoff between both models, where walls are ignored and only floors/ceilings are considered.
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Assuming buildings with even floor levels, the number of floors/ceilings between two position can be determined
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without costly intersection checks and thus allows for real-time use cases.
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without costly intersection checks and thus allows for real-time use cases running on smartphones.
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\begin{equation}
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x = \mTXP{} + 10 \mPLE{} + \log_{10} \frac{d}{d_0} + \numFloors{} \mWAF{} + \mGaussNoise{}
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\mRssi = \mTXP{} + 10 \mPLE{} + \log_{10} \frac{d}{d_0} + \numFloors{} \mWAF{} + \mGaussNoise{}
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\label{eq:logNormShadowModel}
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\end{equation}
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In \refeq{eq:logNormShadowModel}, those are included using a constant attenuation factor \mWAF{}
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multiplied by the number of floors/ceilings \numFloors{} between sender and the location in question.
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In \refeq{eq:logNormShadowModel}, a constant attenuation factor \mWAF{} is
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multiplied by the number \numFloors{} of floors/ceilings between sender and the location in question.
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The attenuation \mWAF{} (per element) depends on the building's architecture and for common,
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steel enforced concrete floors $\approx 8.0$ is a viable choice \cite{TODO}.
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steel enforced concrete floors $\approx 8.0$ is a viable choice \cite{ElectromagneticPropagation}.
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@@ -100,65 +98,100 @@
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\subsection{Model Parameter Optimization}
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\begin{figure}[t!]
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\input{gfx/wifiop_show_optfunc_params}
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\caption{
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The average error (in \SI{}{\decibel}) between all reference measurements and corresponding model predictions
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for one \docAPshort{} dependent on \docTXP{} \mTXP{} and \docEXP{} \mPLE{}
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[known position $\mPosAPVec{}$, fixed \mWAF{}] denotes a convex function.
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}
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\label{fig:wifiOptFuncTXPEXP}
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\end{figure}
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%\begin{figure}
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% \input{gfx/wifiop_show_optfunc_params}
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% \caption{
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% The average error (in \SI{}{\decibel}) between all reference measurements and corresponding model predictions
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% for one \docAPshort{} dependent on \docTXP{} \mTXP{} and \docEXP{} \mPLE{}
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% [known position $\mPosAPVec{}$, fixed \mWAF{}] denotes a convex function.
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% }
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% \label{fig:wifiOptFuncTXPEXP}
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%\end{figure}
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For systems that demand a higher accuracy, one can choose a compromise between fingerprinting and
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pure empiric model parameters where (some) model parameters are optimized,
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aforementioned pure empiric model parameters by optimizing those parameters
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based on a few reference measurements throughout the building.
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Obviously, the more parameters are unknown ($\mPosAPVec{}, \mTXP{}, \mPLE{}, \mWAF{}$) the more
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reference measurements are necessary to provide a stable optimization.
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Obviously, the more parameters are staged for optimization ($\mPosAPVec{}, \mTXP{}, \mPLE{}, \mWAF{}$) the more
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reference measurements are necessary to provide a stable result.
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Depending on the desired accuracy, setup time and whether the transmitter positions are known or unknown,
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several optimization strategies arise, where not all 6 parameters are optimized, but only some of them.
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Just optimizing \mTXP{} and \mPLE{} with constant \mWAF{} and known transmitter position
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usually means optimizing a convex function as can be seen in figure \ref{fig:wifiOptFuncTXPEXP}.
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For such error functions, algorithms like gradient descent and simplex \cite{gradientDescent, downhillSimplex1, downhillSimplex2}
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are well suited and will provide the global minima:
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The target function \refeq{eq:optTarget} optimizes the model-parameters for one \docAP{} by reducing the squared error between
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reference measurements $s_{\mPosVec} \in \vec{s}$ with well-known location $\mPosVec$ and corresponding
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model predictions $\mu_{\mPosVec}$.
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\begin{equation}
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argmin_{bla} blub()
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TODO TODO TODO
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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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\label{eq:optTarget}
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\end{equation}
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Just optimizing \mTXP{} and \mPLE{} with constant \mWAF{} and known transmitter position
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usually means optimizing a convex function, as can be seen in figure \ref{fig:wifiOptFuncTXPEXP}.
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For such error functions, algorithms like gradient descent and simplex \cite{gradientDescent, downhillSimplex1, downhillSimplex2}
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are well suited and will provide the global minima.
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However, optimizing an unknown transmitter position usually means optimizing a non-convex, discontinuous
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function, especially when the $z$-coordinate, that influences the number of attenuating floors/ceilings,
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function, especially when the $z$-coordinate, that influences the number of attenuating floors / ceilings,
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is involved.
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While the latter can be mitigated by introducing a continuous function for the
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number $n$ of floors/ceilings, like a sigmoid, the function is not necessarily convex.
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As can be seen in figure \ref{fig:wifiOptFuncPosYZ}, there are two local minima and only one of
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both also is a global one.
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number $n$, e.g. a sigmoid, the function is not necessarily convex.
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Figure \ref{fig:wifiOptFuncPosYZ} depicts two local minima and only one of both also is a global one.
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\begin{figure}[t!]
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\begin{figure*}
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\centering
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\begin{subfigure}{0.48\textwidth}
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%\centering
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\input{gfx/wifiop_show_optfunc_params}
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\caption{
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Modifying \docTXP{} \mTXP{} and \docEXP{} \mPLE{}
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[known position $\mPosAPVec{}$, fixed \mWAF{}] denotes a convex function.
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}
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\label{fig:wifiOptFuncTXPEXP}
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\end{subfigure}%
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\enskip\enskip
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\begin{subfigure}{0.48\textwidth}
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%\centering
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\input{gfx/wifiop_show_optfunc_pos_yz}
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\caption{
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The average error (in \SI{}{\decibel}) between reference measurements and model predictions
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for one \docAPshort{} dependent on $y$- and $z$-position [fixed $x$, \mTXP{}, \mPLE{} and \mWAF{}]
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usually denotes a non-convex function with multiple [here: two] local minima.
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Modifying $y$- and $z$-position [fixed $x$, \mTXP{}, \mPLE{} and \mWAF{}]
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denotes a non-convex function with multiple local minima.
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}
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\label{fig:wifiOptFuncPosYZ}
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\end{figure}
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\end{subfigure}
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\caption{
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Average error (in \SI{}{\decibel}) between all reference measurements and corresponding model predictions
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for one \docAPshort{}.
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}
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\end{figure*}
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%\begin{figure}
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% \input{gfx/wifiop_show_optfunc_pos_yz}
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% \caption{
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% The average error (in \SI{}{\decibel}) between reference measurements and model predictions
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% for one \docAPshort{} dependent on $y$- and $z$-position [fixed $x$, \mTXP{}, \mPLE{} and \mWAF{}]
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% usually denotes a non-convex function with multiple [here: two] local minima.
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% }
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% \label{fig:wifiOptFuncPosYZ}
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%\end{figure}
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Such functions demand for optimization algorithms, that are able to deal with non-convex functions,
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like genetic approaches. However, initial tests indicated that while being superior to simplex
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and similar algorithms, the results were not satisfactorily and the optimization often did not converge.
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As the Range of the six to-be-optimized parameters is known ($\mPosAPVec{}$ within the building,
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\mTXP{}, \mPLE{}, \mWAF{} within a sane interval), we used some modifications.
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\mTXP{}, \mPLE{}, \mWAF{} within a sane interval around empiric values), we used some modifications.
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The algorithms initial population is uniformly sampled from the known range. During each iteration
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the best \SI{25}{\percent} of the population are kept and the remaining entries are
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re-created by modifying the best entries with uniform random values within
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$\pm$\SI{10}{\percent} of the known range. To stabilize the result, the allowed modification range
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(starting at \SI{10}{\percent}) is reduced over time, known as cooling \cite{todo}.
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%(starting at \SI{10}{\percent})
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is reduced over time, often referred to as {\em cooling} \cite{Kirkpatrick83optimizationby}.
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\subsection{Modified Signal Strength Model}
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@@ -174,43 +207,56 @@
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%the inferred location was more erroneous than before.
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As the used model tradeoff does not consider walls, it is expected to provide erroneous values
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for regions that are heavily shrouded by e.g. steel-enforced concrete or metallised glass.
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for regions that are heavily shrouded, e.g. by steel-enforced concrete or metallised glass.
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Instead of using only one optimized model per \docAP{}, we use several instances with different
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parameters that are limited to some region within the building:
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parameters that are limited to some region within the building. By reducing the area
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that the model has to describe, we expect the limited number of model parameters to
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provide better (local) results.
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{\bf \optPerFloor{}} will use one model for each story, that is optimized using
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{\em \optPerFloor{}} will use one model for each story, that is optimized using
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only the fingerprints that belong to the corresponding floor. During evaluation,
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the $z$-value from $\mPosVec{}$ in \refeq{eq:wifiProb} is used to select the model
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the $z$-value from $\mPosVec{}$ in \refeq{eq:wifiProb} is used to select the correct model
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for this location's signal strength estimation.
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{\bf \optPerRegion{}} works similar, except that the model is limited to a predefined,
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axis-aligned bounding box. This approach allows a distinction between in- and outdoor-regions
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or locations that are expected to highly differ from their surroundings.
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{\em \optPerRegion{}} works similar, except that each model is limited to a predefined,
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axis-aligned bounding box. This approach allows for an even more refined distinction between
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several areas like in- and outdoor-regions or locations that are expected to highly differ
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from their surroundings.
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\todo{AP wird in einer region nur dann beruecksichtigt, wenn mindestanzahl an messungen vorhanden ist!}
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Especially the second model imposes a potential issue we need to address:
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If an \docAPshort{} is seen only once or twice within such a bounding box, it is impossible
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to optimize its parameters, just like a line can not be defined using one single point.
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However, due to \refeq{eq:wifiProb}, we need each model to provide the same number of
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\docAP{}s. Otherwise regions with less known transmitters would automatically be more
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likely than others. We therefore use fixed model parameters,
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\mTXP = \SI{-100}{\decibel{}m}, \mPLE = 0 and \mWAF = \SI{0}{\decibel}. This yields
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a model that always returns \SI{-100}{\decibel{}m}, independent of the distance from the transmitter.
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While this most probably is not the correct reading for all locations, it works
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for most cases, as usual smartphones are unable to measure signals below this threshold.
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\todo{das heißt aber, dass an unterschiedlichen stellen unterschiedlich viele APs verglichen werden. das geht ned. deshalb feste -100}
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%\todo{AP wird in einer region nur dann beruecksichtigt, wenn mindestanzahl an messungen vorhanden ist!}
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%\todo{das heißt aber, dass an unterschiedlichen stellen unterschiedlich viele APs verglichen werden. das geht ned. deshalb feste -100}
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\subsection{\docWIFI{} quality factor}
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Evaluations within previous works showed, that there are many situations where the \docWIFI{} location estimation
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Evaluations within previous works showed, that there are many situations where the overall \docWIFI{} location estimation
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is highly erroneous. Either when the signal strength prediction model does not match real world
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conditions or the received measurements are ambiguous and there is more than one location
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within the building that matches those readings. Both cases can occur e.g. in areas surrounded by
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concrete walls where the model does not match the real world conditions as those walls are not considered,
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concrete walls, where the model does not match the real world conditions as those walls are not considered,
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and the smartphone barely receives \docAPshort{}s due to the high attenuation.
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If such a sensor error occurs only for a short time period, the recursive density estimation
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\refeq{eq:recursiveDensity} is able to compensate those errors using other observations and the transition
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If such a sensor error occurs only for a short time period, the recursive density estimation from
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\refeq{eq:recursiveDensity} is able to compensate using other observations and the transition
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model. However, if the sensor-fault persists for a longer time period, such an error will slowly distort
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the posterior distribution. As our movement model depends on the actual floorplan, the density
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might get trapped e.g. within a room if the other sensors are unable to compensate for
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the \docWIFI{} error.
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Thus, we try to determine the quality of received \docWIFI{} measurements, which allows for
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Thus, we try to determine the quality of received measurements, which allows for
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temporarily disabling \docWIFI{}'s contribution within the evaluation \refeq{eq:evalDensity}
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for situations where the quality is insufficient.
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if the quality is insufficient.
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In \refeq{eq:wifiQuality} we use the average signal strength of all \docAP{}s seen within one measurement
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and scale this value to match a region of $[0, 1]$ depending on an upper- and lower bound.
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@@ -219,7 +265,7 @@
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\begin{equation}
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\newcommand{\leMin}{l_\text{min}}
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\newcommand{\leMax}{l_\text{max}}
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q(\mRssiVec) =
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\text{quality}(\mRssiVec) =
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\max(0,
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\min(
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\frac{
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@@ -237,21 +283,21 @@
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\label{sec:vap}
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Assuming normal conditions, the received signal strength at one location will also (strongly) vary over time
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due to environmental conditions like temperature, humidity, open/closed doors and RF interference.
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due to environmental conditions like temperature, humidity, open / closed doors and RF interference.
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Fast variations can be addressed by averaging several consecutive measurements at the expense
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of a delay in time.
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To prevent this delay we use the fact, that many buildings use so called virtual access points
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where one physical hardware \docAP{} provides more than one virtual network to connect to.
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They can usually be identified, as only the last digit of the MAC-address is altered among the virtual networks.
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They can usually be identified, as only the last digit of the MAC address is altered among the virtual networks.
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%
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As those virtual networks normally share the same frequency, they are unable to transmit at the same instant in time.
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As those normally share the same frequency, they are unable to transmit at the same instant in time.
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When scanning for \docAPshort{}s one will thus receive several responses from the same hardware, all with
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a very small delay (micro- to milliseconds). Such measurements may be grouped using some aggregate
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function like average, median or maximum.
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function like average, median or maximum instead of using each single measurement.
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Furthermore, VAP grouping can be used to suppress unlikely observations: If a physical hardware is known
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to provide six virtual networks, it is unlikely to only see one of those networks. This is likely due to
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temporal effects and/or multipath signal propagation and the received signal strength will often be far from
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to provide six virtual networks, it is unlikely for the smartphone to only see one of those networks.
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This is due to temporal effects or multipath signal propagation and the received signal strength will often be far from
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the normal average. It thus makes sense to just omit such unlikely observations, focusing on the remaining, stable ones.
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