212 lines
10 KiB
TeX
212 lines
10 KiB
TeX
\section{WiFi Optimization}
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The WiFi sensor infers the pedestrian's current location based on a comparison between live measurements
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(the smartphone continuously scans for nearby \docAP{}s) and reference measurements / predictions
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with well known location.
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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 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} $\mu_{i,\mPosVec}$ denotes the average signal strength for the \docAPshort{} identified by $i$,
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that should be measurable given the location $\mPosVec = (x,y,z)^T$. This 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, and the received \docAP{}s including their signal strength denote the location's fingerprint.
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\todo{cite}
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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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We therefore use a model prediction instead, that just relies on the \docAPshort{}'s position
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$\mPosAPVec{} = (x,y,z)^T$
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and some 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 \todo{cite} 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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This model 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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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 models 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 locations.
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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}.
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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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Throughout this work, walls are ignored and only floors/ceilings are considered.
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In \refeq{eq:logNormShadowModel}, those
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are included using a constant attenuation factor \mWAF{} multiplied by the number of floors/ceilings \numFloors{}
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between sender and the location in question. Assuming \todo{passendes wort?} buildings, this number can be determined
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without costly intersection checks and thus allows for real-time use cases.
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The attenuation \mWAF{} per element depends on the building's architecture and for common, steel enforced concrete floors
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$\approx 8.0$ might be a viable choice \todo{cite}.
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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 plus its environmental parameters.
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%
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While it is possible to use empiric values for \mTXP{}, \mPLE{} and \mWAF{} \cite{Ebner-15}, the positions are mandatory.
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For many installations, 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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As a compromise between fingerprinting and pure empiric model parameters, one can optimize
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the model parameters 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 viable optimization.
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Just optimizing \mTXP{} and \mPLE{} usually means optimizing a convex function
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as can be seen in figure \ref{fig:wifiOptFuncTXPEXP}. For such functions,
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algorithms like gradient descent \todo{cite} and (downhill) simpelx \todo{cite}
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are well suited.
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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 reference measurements and model predictions
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for one \docAPshort{} dependent on \docTXP{} \mTXP{} and \docEXP{} \mPLE{}
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[fixed position $\mPosAPVec{}$ and \mWAF{}] denotes a convex function.
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}
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\end{figure}
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However, optimizing the transmitter's position usually means optimizing a non-convex function,
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especially when the $z$-coordinate, that influences the number of attenuating floors/ceilings,
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is involved. 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, this will still not work for all situations.
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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.
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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 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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\SI{10}{\percent} of the known range. To stabilize the result, the allowed modification range
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is adjusted over time, known as cooling \todo{cite}.
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\subsection{Modified Signal Strength Model}
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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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\subsection {VAP grouping}
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Assuming normal conditions, the received signal strength at one location will (strongly) vary
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due to environmental conditions like temperature, humidity, open/closed doors, 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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wenn man nur die fingerprints des floors nimmt in dem gelaufen wird, ist alles gut
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sobald man andere floors drueber/drunter dazu nimmt, ist es nicht mehr gnaz so gut, oder wird schlechter
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das spricht dafuer dass das modell nicht gut passt
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koennte man zeigen indem man den durchschnittlichen fehler je fingerprint plottet???
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the optimization-result also depends on the optimzation target:
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a) the (average) error between measurment and model prediction
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b) the (average) probability for the model's prediction given the fingerprint, ...
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c) ...
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probleme bei der optimierung beschreiben. convex usw..
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wo geht simplex gut, wo eher nicht
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harte WAF übergänge scheinen beim optimieren als auch beim matchen nicht so gut
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gleitende übergänge mittels sigmoid wirken besser
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war eine wichtige erkenntnis
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die vom AP bekannte position wird NICHT als input fuer die alles-OPT funktion benutzt
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die ist wirklich 'irgendwo'
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range-random algo
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domain bekannt [map groesse, txp/exp/waf in etwa]
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genetic refinement mit cooling [= erst grob, dann fein]
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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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