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@@ -37,77 +37,81 @@ The comparison between a single RSSI measurement $\mRssi_i$ and the reference is
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\begin{equation}
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\begin{equation}
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p(\mRssi_i \mid \mPosVec) =
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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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\mathcal{N}(\mRssi_i \mid \mu_{i,\mPosVec}, \sigma_{\text{wifi}}^2)
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\enskip ,
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\enskip ,
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\label{eq:wifiProb}
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\label{eq:wifiProb}
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\end{equation}
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\end{equation}
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\commentByFrank{ich wuerde einfach $\sigma_\text{wifi}$ nehmen. es haengt nicht von der pos $\mPosVec$ ab, und wir hatten immer fuer jeden AP das gleiche}
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%\commentByFrank{ich wuerde einfach $\sigma_\text{wifi}$ nehmen. es haengt nicht von der pos $\mPosVec$ ab, und wir hatten immer fuer jeden AP das gleiche}
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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 state-in-question:
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\noindent where $\mu_{i,\mPosVec}$ denotes the (predicted) signal strength for the \docAPshort{} identified by $i$, regarding the location $\mPosVec$.
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A certain noise is allowed by the corresponding standard deviation $\sigma_{\text{wifi}}$.
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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}.
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Here, the prediction depends on the 3D distance $d$ between the \docAPshort{} in question and the location $\mPosVec$ as well as the number of floors $\Delta f$ between them:
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\begin{equation}
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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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\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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\label{eq:wallAtt}
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\end{equation}
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\end{equation}
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\commentByFrank{
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%\commentByFrank{
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hier sollte das $i$, das du vorher hattest, wohl wieder mit rein?
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% hier sollte das $i$, das du vorher hattest, wohl wieder mit rein?
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was genau $d$ bzw $d_i$ oder $d_{i,\mPosVec}$ ist, muessten wir vermutlich auch kurz erklären.
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% was genau $d$ bzw $d_i$ oder $d_{i,\mPosVec}$ ist, muessten wir vermutlich auch kurz erklären.
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Ich hatte auch immer unterschieden zwischen der fraglichen position (z.B. $\vec{\rho}$)
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% Ich hatte auch immer unterschieden zwischen der fraglichen position (z.B. $\vec{\rho}$)
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und der position des access points (z.B. $\mPosVec_i$). also, zwei verschiedene zeichen, dass das klar wird.
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% und der position des access points (z.B. $\mPosVec_i$). also, zwei verschiedene zeichen, dass das klar wird.
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ich weis aber nicht, ob $\vec{\rho}$ noch frei ist, bzw was auf den folgenden seiten nocht kommt.
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% ich weis aber nicht, ob $\vec{\rho}$ noch frei ist, bzw was auf den folgenden seiten nocht kommt.
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eigentlich gehoert das $i$ dann auch noch ans $P_0$ und $\gamma$ und $d_0$.. aber der einfachheit halber, reicht das ja im text.
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% eigentlich gehoert das $i$ dann auch noch ans $P_0$ und $\gamma$ und $d_0$.. aber der einfachheit halber, reicht das ja im text.
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vorschlag wäre etwas wie:
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% vorschlag wäre etwas wie:
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}
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%}
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\begin{equation}
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%\begin{equation}
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\mu(i,\vec{\rho}) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF
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% \mu(i,\vec{\rho}) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF
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,\enskip
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% ,\enskip
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d = \| \vec{\rho} - \mPosVec_i \|
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% d = \| \vec{\rho} - \mPosVec_i \|
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\label{eq:wallAtt}
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% \label{eq:wallAtt}
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\end{equation}
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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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\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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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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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, eq. \eqref{eq:wallAtt} needs to be calculated separately for every $i$ and thus available \docAPshort{}.
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It should be noted, that we omitted the index $i$ in eq. \eqref{eq:wallAtt} for the sake of clarity and consistency with other literature.
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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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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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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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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 an 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 estimated position.
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Therefore, instead of using a pure empiric model, we deploy an 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 estimated 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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The optimization is based on a few reference measurements $\vec{s_{\text{opt}}}$ 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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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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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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\begin{equation}
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\epsilon^* =
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(\mPosAPVec, \mTXP, \mPLE, \mWAF) =
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\min_{\mPosAPVec, \mTXP, \mPLE, \mWAF}
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\argmin_{\mPosAPVec, \mTXP, \mPLE, \mWAF}
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\sum_{s_{\mPosVec} \in \vec{s}}
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\sum_{s_{i} \in \vec{s_{\text{mac}}}}
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(s_{\mPosVec} - \mu_{\mPosVec})^2
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(s_{i} - \mu_{\mPosVec})^2
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\enskip,\enskip\enskip
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\enskip,\enskip\enskip
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\mu_{\mPosVec} =
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\mu_{\mPosVec} =
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\mTXP{} + 10 \mPLE{} \log_{10} \| \mPosVec-\mPosAPVec \| + \Delta f \mWAF{}
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\mTXP{} + 10 \mPLE{} \log_{10} \| \mPosVec-\mPosAPVec \| + \Delta f \mWAF{}
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\enspace .
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\enspace .
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\label{eq:optTarget}
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\label{eq:optTarget}
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\end{equation}
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\end{equation}
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\commentByFrank{hier muesste dann auch das $i$ rein, bzw die funktion $\mu()$. vorschlag waere dann:}
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%\commentByFrank{hier muesste dann auch das $i$ rein, bzw die funktion $\mu()$. vorschlag waere dann:}
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\begin{equation}
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%\begin{equation}
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(\mPosAPVec, \mTXP, \mPLE, \mWAF)_i =
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% (\mPosAPVec, \mTXP, \mPLE, \mWAF)_i =
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\argmin_{\mPosVec, \mTXP, \mPLE, \mWAF}
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% \argmin_{\mPosVec, \mTXP, \mPLE, \mWAF}
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\sum_{s_{i,\vec{\rho}} \in \vec{s}_i}
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% \sum_{s_{i,\vec{\rho}} \in \vec{s}_i}
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\big(s_{i,\vec{\rho}} - \mu(i,\mPosVec) \big)^2
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% \big(s_{i,\vec{\rho}} - \mu(i,\mPosVec) \big)^2
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\enspace .
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%\enspace .
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\label{eq:optTarget}
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% \label{eq:optTarget}
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\end{equation}
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%\end{equation}
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%\commentByFrank{argmin liefert die argumente, nicht den fehler. da muesste nur min stehen}
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%\commentByFrank{argmin liefert die argumente, nicht den fehler. da muesste nur min stehen}
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\commentByFrank{
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%\commentByFrank{
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hier braucht es drigend eine unterscheidung zwischen den beiden positionen. der vom fingerprint und der vom ap
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% hier braucht es drigend eine unterscheidung zwischen den beiden positionen. der vom fingerprint und der vom ap
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$\mPosAPVec$ ist, wegen dem $\hat{ }$ einfach nur die \emph{beste}. aber sie ist halt generell anders als der fingerprint.
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% $\mPosAPVec$ ist, wegen dem $\hat{ }$ einfach nur die \emph{beste}. aber sie ist halt generell anders als der fingerprint.
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deshalb brauchen wir da zwei formel zeichen.
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% deshalb brauchen wir da zwei formel zeichen.
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und wir muessen einheitlich machen, ob wir das $i$ jetzt mitnehmen, oder nicht. sonst wirkt es verwirrend
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% und wir muessen einheitlich machen, ob wir das $i$ jetzt mitnehmen, oder nicht. sonst wirkt es verwirrend
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}
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%}
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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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\noindent Here, one reduces the squared error between reference measurements $s_{i} \in \vec{s_{\text{mac}}}$ with well-known location $\mPosVec$ and corresponding model predictions $\mu_{\mPosVec}$ (cf. eq. \eqref{eq:wallAtt}).
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Whereas $\vec{s_{\text{mac}}}$ is the subset of $\vec{s_{\text{opt}}}$ for the \docAPshort{} in question, identified by its MAC-adress.
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The number of floors between $\mPosVec$ and $\mPosAPVec$ is again given by $\Delta f$.
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The number of floors between $\mPosVec$ and $\mPosAPVec$ is again given by $\Delta f$.
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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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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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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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@@ -119,8 +123,8 @@ 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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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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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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\commentByToni{Wollen wir das mal genauer beschreiben? Also wie genau funktioniert das cooling. Das ist ja alles sehr wischi waschi gehalten}
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%\commentByToni{Wollen wir das mal genauer beschreiben? Also wie genau funktioniert das cooling. Das ist ja alles sehr wischi waschi gehalten}
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\commentByFrank{ich wuerde es so lassen. da gibts genug in der literatur ueber ideen und potentielle ansaetze}
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%\commentByFrank{ich wuerde es so lassen. da gibts genug in der literatur ueber ideen und potentielle ansaetze}
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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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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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The reason for this comes from the assumptions made in eq. \eqref{eq:wallAtt}.
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@@ -130,8 +134,8 @@ For example, if a pedestrian walks on a staircase and thus is in-between multipl
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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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%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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Basically, any kind of wireless network which allows to measure RSSI can be used for the above.
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\commentByMarkus{Provieded der AP die RSSI? Misst nicht das Smartphone an seiner Antenne?}
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%\commentByMarkus{Provieded der AP die RSSI? Misst nicht das Smartphone an seiner Antenne?}
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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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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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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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To improve $\docWIFI$ coverage we are able to distribute a small number of simple and cheap \docWIFI{} beacons.
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toni