251 lines
13 KiB
TeX
251 lines
13 KiB
TeX
\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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signal strength predictions for well known locations:
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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(\mRssiVecWiFi \mid \mPosVec) =
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\prod_{\mRssi_{i} \in \mRssiVec{}} 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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\label{eq:wifiObs}
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\end{equation}
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%
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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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\label{eq:wifiProb}
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\end{equation}
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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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%
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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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based on the \docAPshort{}'s position $\mPosAPVec{} = (x,y,z)^T$ and a few additional parameters.
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\subsection{Signal Strength Prediction Model}
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\begin{equation}
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\mRssi = \mTXP{} + 10 \mPLE{} + \log_{10} \frac{d}{d_0} + \mGaussNoise{}
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\label{eq:logDistModel}
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\end{equation}
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The log distance model \cite{TODO} in \refeq{eq:logDistModel} is a commonly used signal strength prediction model that
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is intended for line-of-sight predictions. However, depending on the surroundings, the model is versatile enough
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to also serve for indoor purposes.
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%
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It predicts an \docAP{}'s signal strength
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for an arbitrary location $\mPosVec{}$ given the distance between both and two environmental parameters:
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The \docAPshort{}'s signal strength \mTXP{} measurable at a known distance $d_0$ (usually \SI{1}{\meter}) and
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the signal's depletion over distance \mPLE{}, which depends on the \docAPshort{}'s surroundings like walls
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and other obstacles.
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\mGaussNoise{} is a zero-mean Gaussian noise and models the uncertainty.
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As \mPLE{} depends on the architecture around the transmitter, the model is bound to homogenous surroundings
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like one floor, solely divided by drywalls of the same thickness and material.
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%
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The log normal shadowing model is a slight modification, to adapt the log distance model to indoor use cases.
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It introduces an additional parameter, that considers obstacles between (line-of-sight) the \docAPshort{} and the
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location in question by attenuating the signal with a constant value.
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%
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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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\todo{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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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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\begin{equation}
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x = \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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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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\subsection {Model Parameters}
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As previously mentioned, for the prediction model to work, one needs to know the location $\mPosAPVec_i$ for every
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permanently installed \docAP{} $i$ within the building to derive the distance $d$, plus its environmental parameters
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\mTXP{}, \mPLE{} and \mWAF{}.
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While it is possible to use empiric values for those environmental parameters \cite{Ebner-15}, the positions are mandatory.
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For many buildings, there should be floorplans that include the locations of all installed transmitters.
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If so, a model setup takes only several minutes to (vaguely) position the \docAPshort{}s within a virtual
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map and assigning them some fixed, empirically chosen parameters for \mTXP{}, \mPLE{} and \mWAF{}.
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Depending on the building's architecture this might already provide enough accuracy for some use-cases,
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where a vague location information is sufficient.
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\subsection{Model Parameter Optimization}
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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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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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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 \cite{TODO} and (downhill) simpelx \cite{TODO}
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are well suited and will provide the global minima:
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\begin{equation}
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argmin_{bla} blub()
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TODO TODO TODO
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\end{equation}
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\begin{figure}
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\input{gfx/wifiop_show_optfunc_params}
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\label{fig:wifiOptFuncTXPEXP}
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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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\end{figure}
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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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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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\begin{figure}
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\input{gfx/wifiop_show_optfunc_pos_yz}
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\label{fig:wifiOptFuncPosYZ}
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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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\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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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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\subsection{Modified Signal Strength Model}
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%\todo{nicht: during initial eval, sondern gleich sagen, dass die vermutung nahe liegt, dass das modell
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%nicht gut klappen wird, weil waende und unser metall-glas nicht beruecksichtigt werden. deshalb
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%versuchen wir ein anderes modell das immernoch live arbeiten kann}
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%During the initial eval, some issues were discovered. While aforementioned optimization was able to
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%reduce the error between reference measurements and model estimations to \SI{50}{\percent},
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%the position estimation \ref{eq:wifiProb} did not benefit from improved model parameters.
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%To the contrary, there were several situations throughout the testing walks, where
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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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\subsection{\docWIFI{} quality factor}
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Past evaluations showed, that there are many situations where the \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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and the smartphone barely receives some \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 sensors and the movement
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model. However, if the error persists for a longer time period, the 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 not able 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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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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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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If the returned quality falls below a certain threshold, \docWIFI{} is ignored within
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the evaluation.
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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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\max(0,
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\min(
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\frac{
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\bar\mRssi - \leMin
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}{
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\leMax - \leMin
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},
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1
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)
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)
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\label{eq:wifiQuality}
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\end{equation}
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\subsection {VAP grouping}
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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
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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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%
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As those virtual networks normally share the same frequency, they are unable to transmit at the same 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 in time (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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wie wird optimiert
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a) bekannte pos + empirische params
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b) bekannte pos + opt params (fur alle APs gleich) [simplex]
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c) bekannte pos + opt params (eigene je AP) [simplex]
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d) alles opt: pos und params (je ap) [range-random]
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optimierung ist tricky. auch wegen dem WAF der ja sprunghaft dazu kommt, sobald messung und AP in zwei unterschiedlichen
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stockwerken liegen.. und das selbst wenn hier vlt sichtkontakt möglich wäre, da der test 2D ist und nicht 3D
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aps sind (statistisch) unaebhaengig. d.h., jeder AP kann fuer sich optimiert werden.
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optimierung des gesamtsystems ist nicht notwendig.
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pro AP also 6 params. pos x/y/z, txp, exp, waf
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