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We therefore examined variations of the probability calculation from \refeq{eq:wifiProb}.
Removing the strongest/weakest \docAPshort{} from $\mRssiVec{}$ yielded similar results.
While some estimations were improved, the overall estimation error increased for our walks,
as there are many situations where only a handful \docAP{}s can be seen. Removing (valid)
information will highly increase the error for such situations.
Despite the results show in \cite{PotentialRisks}, removing weak \docAPshort{}s from $\mRssiVec{}$
yielded similar results. While some estimations were improved, the overall estimation error increased
for our walks, as there are many situations where only a handful \docAP{}s can be seen.
Removing this (valid) information will highly increase the error for such situations.
Incorporating additional knowledge provided by virtual \docAP{}s (see section \ref{sec:vap}) mitigated this issues.
If only one out of six virtual networks is observed, this observation is likely to be erroneous, no matter

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relatedwork
\section{Related Work}
wifi anfänge von radar (microsoft) etc
\cite{radar} \cite{horus} \cite{secureAndRobust}
Indoor localization based on \docWIFI{} signal strengths dates back to the year
2000 and the work of Bahl and Padmanabhan \cite{radar}. During an offline-phase, a
multitude of reference measurements are conducted once. Those measurements are compared
against live readings during an online-phase. The pedestrian's location is inferred
using the $k$-nearest neighbor(s) based on the Euclidean distance between currently
received signal strengths and the readings during the offline phase.
Inspired by this initial work, Youssef et al. \cite{horus} proposed a more robust, probabilistic
approach. Fingerprints were placed every \SI{1.52}{\meter} and estimated by scanning each location
100 times. The resulting signal strength propagation for one location is hereafter denoted by a histogram.
The latter can be compared against live measurements to infer its matching-probability. The center
of mass among the $k$ highest probabilities, including their weight, describes the pedestrian's current location.
%
In \cite{ProbabilisticWlan}, a similar approach is used and compared against nearest neighbor and machine learning.
Furthermore, they mention potential issues of unseen transmitters and describe a simple heuristic of how to handle such cases.
Meng et al \cite{secureAndRobust} further discuss several fingerprinting issues like environmental changes
after the fingerprints were recorded. They propose an outlier detected based on RANSAC to remove potentially
distorted measurements and thus improve the matching process.
Despite a very high accuracy due to real-world comparisons, all approaches suffer from tremendous setup-
and maintainance times.
Therefore it makes sense to replace those time consuming fingerprints by model predictions.
Those are a well established research field, mainly used to determine the \docWIFI{}-coverage
for new installations. \cite{ANewPathLossPrediction, PredictingRFCoverage, empiricalPathLossModel}
einfach messen, ab und zu einen GPS fix und danach genetisch alles zuusammenoptimieren. also kein vorwissen
\cite{WithoutThePain}
das muesste noch was aehnliches sein:
\cite{crowdinside}
neben signalstärke gibt es noch viele andere methoden über laufzeiten wie beim gps etc.
diese erfordern meist aber spezial-hardware und laufen deshalb nicht so einfach auf dem smartphone [= ueberleitung!]
\cite{secureAndRobust}
andere methoden neben signalstärke

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Just optimizing \mTXP{} and \mPLE{} with constant \mWAF{} and known transmitter position
usually means optimizing a convex function as can be seen in figure \ref{fig:wifiOptFuncTXPEXP}.
For such error functions, algorithms like gradient descent \cite{TODO} and (downhill) simpelx \cite{TODO}
For such error functions, algorithms like gradient descent and simplex \cite{gradientDescent, downhillSimplex1, downhillSimplex2}
are well suited and will provide the global minima:
\begin{equation}