465 lines
24 KiB
TeX
Executable File
465 lines
24 KiB
TeX
Executable File
\section{Experiments}
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\todo{
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alles im FHWS gebäude [korrekte groesse fuer beide gebaeude!] mit nem nexus 6
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}
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Within all \docWIFI{} observations (offline and online) we only consider the \docAP{}s that are permanently installed
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within the building. Temporal and movable transmitters are ignored as they might cause estimation errors.
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modell direkt fuer den gelaufenen pfad optimiert (also wirklich jede wifi messung direkt auf den ground-truth)
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der fehler wird zwar kleiner, ist aber immernoch deutlich spürbar. das spricht dafür, dass das modell einfach nicht
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gut geeignet ist.
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optimierungs input: alle 4 walks samt ground-truth
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dann kommt fuer die 4 typen [fixed, all same par, each par, each par pos]
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log probability 50 75, meter 50, 75
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% -------------------------------- optimization -------------------------------- %
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\subsection{Model optimization}
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As the signal strength prediction model is the heart of the absolute positioning component
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described in \ref{sec:system} we start with the model parameter estimation (see \ref{sec:optimization}) for
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\mTXP, \mPLE and \mWAF based on some reference measurements and compare the results
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between various optimization strategies and a basic empiric choice of \mTXP = \SI{-40}{\decibel{}m} @ \SI{1}{\meter}
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(defined by the usual \docAPshort{} transmit power for europe), a path loss exponent $\mPLE \approx $ \SI{2.5} and
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$\mWAF \approx$ \SI{-8}{\decibel} per floor/ceiling (made of reinforced concrete) \todo{cite für werte}.
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Figure \ref{fig:referenceMeasurements} depicts the location of the used 121 reference measurements.
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Each location was scanned 30 times ($\approx$ \SI{25}{\second} scan time),
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non permanent \docAP{}s were removed, the values were grouped per physical transmitter (see \ref{sec:vap})
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and aggregated to form the average signal strength per transmitter.
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% used reference measurements
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\begin{figure}
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{
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\centering
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\input{gfx/all_fingerprints.tex}
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}
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\caption{
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Locations of the 121 reference measurements.
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The size of each square denotes the number of permanently installed \docAPshort{}s
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that are visible at this location,
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and ranges between 2 and 22 with an average of 9.
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}
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\label{fig:referenceMeasurements}
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\end{figure}
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% visible APs:
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% cnt(121) min(2.000000) max(22.000000) range(20.000000) med(8.000000) avg(9.322314) stdDev(4.386709)
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\begin{figure}[b]
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\centering
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\input{gfx/compare-wifi-in-out.tex}
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\caption{
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Measurable signal strengths of a testing \docAPshort{} (black dot).
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While the signal diminishes slowly along the corridor (upper rectangle)
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the metallised windows (dashed outline) attenuate the signal by over \SI{30}{\decibel} (lower rectangle).
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}
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\label{fig:wifiIndoorOutdoor}
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\end{figure}
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Figure \ref{fig:wifiIndoorOutdoor} depicts the to-be-expected issues by examining the signal strength
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values of the reference measurements for one \docAP{}.
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Even though the transmitter is only \SI{5}{\meter} away from the reference
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measurement (small box), the metallised windows attenuate the signal as much as \SI{50}{\meter}
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of corridor (wide box). The model described in section \ref{sec:sigStrengthModel} will not be able
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to match such situations, due to the lack of obstacle information.
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%
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We will thus look at various optimization strategies and the error between
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the resulting estimation model and our reference measurements:
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{\em\noOptEmpiric{}} uses the same three empiric parameters \mTXP{}, \mPLE{}, \mWAF{} for each \docAPshort{} in combination
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with its position, which is well known from the floorplan.
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{\em\optParamsAllAP{}} is the same as above, except that the three parameters are optimized
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using the reference measurements.
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{\em\optParamsEachAP{}} optimizes the three parameters per \docAP{} instead of using the same
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parameters for all.
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{\em\optParamsPosEachAP{}} does not need any prior knowledge and will optimize all six parameters
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(3D position, \mTXP, \mPLE, \mWAF) based on the reference measurements.
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{\em\optPerFloor{}} and {\em\optPerRegion{}} are just like \optParamsPosEachAP{} except that
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there are several sub-models that are optimized for one floor / region instead of the whole building.
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\todo{grafik, die die regionen zeigt???}
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Figure \ref{fig:wifiModelError} shows the optimization results for all strategies, which are as expected:
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The estimation error is indirectly proportional to the number of optimized parameters.
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However, even with {\em \optPerRegion{}} the maximal error is relatively high due to some locations that do
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not fit the model at all. Looking at the optimization results for \mTXP{}, \mPLE{} and \mWAF{} supports
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this finding. While the median for those values based on all optimized transmitters is totally sane
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(\SI{-42}{\decibel{}m}, \SI{2.4}, \SI{-6.0}{\decibel}), the minimum and maximum values are clearly outside of the physically possible range.
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The same holds for the estimated transmitter position when using {\em \optParamsPosEachAP{}}: The median
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distance between estimated and real position is $\sim$\SI{8}{\meter} and the maximum $\sim$\SI{27}{\meter}.
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For \SI{68}{\percent} of all installed transmitters, the estimated floor-number matched the real location.
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\begin{figure}
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\input{gfx/wifi_model_error_0_95.tex}
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%\input{gfx/wifi_model_error_95_100.tex}
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\caption{
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Comparison between different optimization strategies by examining the error (in \decibel) at each reference measurement.
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The higher the number of variable parameters, the better the model resembles real world conditions.
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}
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\label{fig:wifiModelError}
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\end{figure}
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% statds:
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%TXP: cnt(34) min(-67.698959) max(4.299183) range(71.998146) med(-41.961170) avg(-41.659286) stdDev(17.742294)
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%EXP: cnt(34) min(0.932817) max(4.699000) range(3.766183) med(2.380410) avg(2.546959) stdDev(1.074687)
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%WAF: cnt(34) min(-27.764957) max(5.217187) range(32.982143) med(-5.921916) avg(-7.579522) stdDev(5.840527)
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%Pos: cnt(34) min(3.032438) max(26.767128) range(23.734690) med(7.342710) avg(8.571227) stdDev(4.801449)
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While {\em \optPerRegion{}} is able to overcome the indoor vs. outdoor issues depicted in
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figure \ref{fig:wifiIndoorOutdoor} e.g. by using a separate bounding box just for the outdoor area,
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it obviously requires a profound prior knowledge when selecting the individual regions for the sub-model.
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%Such issues can only be fixed using more appropriate models that consider walls and other obstacles.
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% das ist wohl zu viel
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%\begin{figure}
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% \centering
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% \input{gfx/wifiOptApPosDifference.tex}
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% \caption{zu viel, oder?}
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%\end{figure}
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% -------------------------------- number of fingerprints -------------------------------- %
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As we try to minimize the system's setup time as much as possible, we need to determine
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the amount of necessary reference measurements for the optimization to produce viable model parameters.
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Depending on the chosen model and thus the number of to-be-optimized parameters, more measurements are required.
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While there was almost no difference between using 121 or 30 reference measurements for
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{\em \optParamsAllAP{}} and {\em \optParamsEachAP{}}
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(average \SIrange{5.3}{5.4}{\decibel} and \SIrange{4.5}{5.0}{\decibel}),
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{\em \optPerRegion{}} is highly affected
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(average \SIrange{2.0}{6.2}{\decibel}), as it needs at least a certain number of measurements for each
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of its regions for the optimization to converge.
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\begin{figure}[b]
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\input{gfx/wifi_model_error_num_fingerprints_method_5_0_90.tex}
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\input{gfx/wifi_model_error_num_fingerprints_method_5_90_100.tex}
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\caption{%
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Impact of reducing the number of reference measurements during optimization on {\em \optPerRegion{}}.
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The model's cumulative error distribution is determined by comparing the its signal strength prediction against all 121 measurements.
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While using only \SI{50}{\percent} of the 121 scans has barely an impact on the error,
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30 measurements (\SI{25}{\percent}) are clearly insufficient.
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}%
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\label{fig:wifiNumFingerprints}%
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\end{figure}
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Figure \ref{fig:wifiNumFingerprints} depicts the impact of reducing the number of reference measurements
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during the optimization process for the {\em \optPerRegion{}} strategy.
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The error is determined by using the (absolute) difference between expected signal strength and
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the optimized model's corresponding prediction for all of the 121 reference measurements.
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%
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Considering only 60 of the 121 scans (\SI{50}{\percent}) yields a slightly increasing model error and still provides good results.
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While using only \SI{25}{\percent} of the reference measurements increases the error rapidly,
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for \SI{75}{\percent} of the 121 considered cases the estimation is still better than using just empiric values without optimization.
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The extremely large outlier depicted in the lower half of figure \ref{fig:wifiNumFingerprints} (red line) relates to one
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sub-model with only one assigned reference measurement, where the optimized result is unable to predict values
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for the rest of the sub-model's region. \todo{versteht man das?}
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Additionally we examined the impact of skipping reference measurements for difficult locations
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like staircases, surrounded by steel-enforced concrete. While this slightly decreases the
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estimation error for all other positions (hallway, etc) as expected, the error within the skipped locations is dramatically
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increasing (see lower half of figure \ref{fig:wifiNumFingerprints}). It is thus highly recommended
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to also perform reference measurements for locations, that are expected to strongly deviate (signal strength)
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from their surroundings.
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%leaving out fingerprints for model 1
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% 25%: cnt(1128) min(0.007439) max(27.804710) range(27.797272) med(4.404236) avg(5.449720) stdDev(4.470373)
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% 50%: cnt(1128) min(0.006027) max(27.732193) range(27.726166) med(4.367859) avg(5.437861) stdDev(4.475426)
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% 100%: cnt(1128) min(0.000282) max(27.705376) range(27.705093) med(4.272881) avg(5.411202) stdDev(4.493495)
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% noStair%: cnt(1128) min(0.000801) max(27.209221) range(27.208420) med(4.333328) avg(5.459918) stdDev(4.459484)
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%leaving out fingerprints for model 2
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% 25%: cnt(1128) min(0.000320) max(29.752560) range(29.752239) med(3.837357) avg(5.027578) stdDev(4.617191)
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% 50%: cnt(1128) min(0.015305) max(34.152130) range(34.136826) med(3.627090) avg(4.635868) stdDev(4.135866)
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% 100%: cnt(1128) min(0.000488) max(25.687740) range(25.687252) med(3.319756) avg(4.441193) stdDev(3.912525)
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% noStair%: cnt(1128) min(0.017693) max(25.687740) range(25.670048) med(3.304321) avg(4.507620) stdDev(3.957071)
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%leaving out fingerprints for model 3
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% 25%: cnt(1128) min(0.003242) max(39.470978) range(39.467735) med(3.371758) avg(4.977330) stdDev(5.213937)
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% 50%: cnt(1128) min(0.002808) max(30.113415) range(30.110607) med(2.941238) avg(4.015042) stdDev(3.696969)
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% 100%: cnt(1128) min(0.000557) max(16.813850) range(16.813293) med(3.056915) avg(3.813013) stdDev(3.062580)
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% noStair%: cnt(1128) min(0.002518) max(30.370636) range(30.368118) med(3.016884) avg(3.983101) stdDev(3.508327)
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%leaving out fingerprints for model 4
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% 25%: cnt(1128) min(0.000000) max(62.233345) range(62.233345) med(2.502831) avg(5.432897) stdDev(8.664582)
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% 50%: cnt(1128) min(0.000000) max(56.843803) range(56.843803) med(1.543137) avg(2.937506) stdDev(4.417061)
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% 100%: cnt(1128) min(0.000046) max(33.175812) range(33.175766) med(1.537933) avg(2.441976) stdDev(2.793499)
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% noStair%: cnt(1128) min(0.000000) max(62.233345) range(62.233345) med(1.493668) avg(2.744918) stdDev(4.428092)
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%leaving out fingerprints for model 5
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% 25%: cnt(1128) min(0.000000) max(62.620842) range(62.620842) med(2.140709) avg(6.257105) stdDev(11.638572)
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% 50%: cnt(1128) min(0.000000) max(57.371948) range(57.371948) med(1.357452) avg(2.982217) stdDev(5.877471)
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% 100%: cnt(1128) min(0.000000) max(14.837151) range(14.837151) med(1.251358) avg(1.989277) stdDev(2.189072)
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% noStair%: cnt(1128) min(0.000000) max(62.233345) range(62.233345) med(1.143669) avg(2.316189) stdDev(4.164822)
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% -------------------------------- wifi walk error -------------------------------- %
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\subsection{Location estimation error}
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\todo{übergang holprig}
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%Using the optimized model setups and the measurements $\mRssiVec$ determined by scanning for nearby \docAPshort{}s,
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%we can directly perform a location estimation by rewriting \refeq{eq:wifiProb}:
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For each of the discussed optimization strategies we can now determine the resulting localization accuracy.
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The position within the building that best fits some signal strength measurements $\mRssiVec$ received by the smartphone
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is the one that maximizes $p(\mPosVec \mid \mRssiVec)$ and can be rewritten as:
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\begin{equation}
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p(\mPosVec \mid \mRssiVec) =
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\frac{p(\mRssiVec \mid \mPosVec) p(\mPosVec)}{p(\mRssiVec)}
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\propto p(\mRssiVec \mid \mPosVec),\enskip
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p(\mPosVec) = p(\mRssiVec) = \text{const}
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.
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\label{eq:wifiBayes}
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\end{equation}
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Following \refeq{eq:wifiObs} and \refeq{eq:wifiProb}, the best
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location $\mPosVec^*$ given $\mRssiVec$ is the one that satisfies
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\begin{equation}
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\mPosVec^* = \argmax_{\mPosVec}
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\prod_{\mRssi_{i} \in \mRssiVec{}}
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\mathcal{N}(\mRssi_i \mid \mu_{i,\mPosVec}, \sigma^2)
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\label{eq:bestWiFiPos}
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\end{equation}
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where $\mu_{i,\mPosVec}$ is the signal strength for \docAP{} $i$
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at location $\mPosVec$ returned from the to-be-examined prediction model.
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For all comparisons we use a constant uncertainty $\sigma = $\SI{8}{\decibel}.
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The quality of the estimated location is determined by using the Euclidean distance between estimation
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$\mPosVec^*$ and the pedestrian's ground truth position at the time the scan $\mRssiVec$
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has been received.
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We therefore conducted 10 walks on 5 different paths within our building,
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each of which is defined by connecting marker points at well known positions
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(see figure \ref{fig:allWalks}).
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Whenever the pedestrian reached such a marker, the current time was recorded.
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Due to constant walking speeds, the ground-truth for any timestamp can be approximated
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using linear interpolation between adjacent markers.
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% walked paths
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\begin{figure}[t]
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{
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\centering
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\input{gfx/all_walks.tex}
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}
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\label{fig:allWalks}
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\caption{
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Overview of all conducted paths.
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Outdoor areas are marked in green.
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}
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\end{figure}
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\begin{figure}[b]
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\input{gfx/modelPerformance_meter.tex}
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\caption{
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Error between ground truth and estimation using \refeq{eq:bestWiFiPos} depending
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on the underlying signal strength prediction model.
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Extremely high errors between the \SIrange{90}{100}{\percent} quartile are related to bad \docWIFI{}
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coverage within outdoor areas (see figure \ref{fig:wifiIndoorOutdoor}).
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}
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\label{fig:modelPerformance}
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\end{figure}
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%To estimate the overall performance of the prediction models, we compare the position estimation
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%for each \docWIFI{} measurement within the recorded paths (3756 \docAPshort{} scans in total)
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%against the corresponding ground-truth, which indicates the absolute 3D error in meter.
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The position estimation for each \docWIFI{} measurement within the recorded walks (3756 scans in total)
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is compared against its corresponding ground-truth, indicating the 3D error.
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The resulting cumulative error distribution can be seen in figure \ref{fig:modelPerformance}.
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The quality of the location estimation directly scales with the quality of the signal strength prediction model.
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However, as discussed earlier, the maximal estimation error might increase for some setups.
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%
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This is either due to multimodalities, where more than one area is possible based on the recent
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\docWIFI{} observation, or optimization yielded an overadaption where the average signal
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strength prediction error is small, but the maximum error is dramatically increased for some regions.
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% -------------------------------- plots indicating walk issues -------------------------------- %
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\begin{figure}[t]
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\input{gfx/wifiMultimodality.tex}
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\caption{
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Location probability \refeq{eq:bestWiFiPos} for three scans. Higher color intensities are more likely.
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Ideally, places near the ground truth (black) are highly highly probable (green).
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Often, other locations are just as likely as the ground truth (blue),
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or the location with the highest probability does not match at all (red).
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}
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\label{fig:wifiMultimodality}
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\end{figure}
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Figure \ref{fig:wifiMultimodality} depicts aforementioned issues of multimodal (blue) or wrong (red) location
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estimations. Filtering (\refeq{eq:recursiveDensity}) thus is highly recommended, as minor errors are compensated
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using other sensors and/or a movement model that prevents the estimation from leaping within the building.
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However, if wrong sensor values (red) are observed for longer time periods, even filtering will produce erroneous
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results and might get stranded (density is trapped e.g. within a room),
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as the movement model is constrained by the actual floorplan.
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% -------------------------------- other distributions, unseen APs, etc -------------------------------- %
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To reduce the amount such of misclassifications, where other locations within the building are
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as likely as the pedestrians actual location, we examined various approaches.
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Unfortunately, none of which provided a viable enhancement under all conditions for the performed walks.
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The misclassification-rate is determined by counting the amount of (random) locations within
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the building that produce a similar probability \refeq{eq:wifiProb} as the actual ground-truth
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position.
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One possibility to dissolve such an equal \docWIFI{}-likelihood between two (or more) locations is,
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to not only consider the \docAPshort{}s seen by the Smartphone, but also the \docAPshort{}s not seen
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by the Smartphone. This additional information can be used to rule out all locations where this
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\docAP{} should be received (high signal strength from the prediction model).
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% There might be an \docAP{} that should be visible at the other locations. However,
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%as the Smartphone did not see this \docAPshort{} the other location can be ruled out.
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While this works in theory, evaluations revealed several issues:
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There is a chance that even a nearby \docAPshort{} is unseen during a scan due to packet collisions or
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temporal effects within the surrounding. It thus might make sense to opt-out other locations
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only, if at least two \docAPshort{}s are missing. On the other hand, this obviously requires (at least)
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two \docAPshort{}s to actually be different between the two locations, and requires a lot of permanently
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installed transmitters to work out.
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Furthermore, this requires the signal strength prediction model to be fairly accurate. Within our testing
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walks, several places are surrounded by concrete walls, which cause a harsh, local drop in signal strength.
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The models used within this work will not accurately predict the signal strength for such locations.
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%%Including \docAPshort{}s unseen by the Smartphone thus often increases the estimation error instead
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%%of fixing the multimodality.
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To sum up, while some situations, e.g. outdoors, could greatly be improved,
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many other situations are deteriorated, especially when some transmitters are (temporarily)
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attenuated by ambient conditions like concrete walls.
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We therefore examined variations of the probability calculation from \refeq{eq:wifiProb}.
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Removing the strongest/weakest \docAPshort{} from $\mRssiVec{}$ yielded similar results.
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While some estimations were improved, the overall estimation error increased for our walks,
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as there are many situations where only a handful \docAP{}s can be seen. Removing (valid)
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information will highly increase the error for such situations.
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Using a more strict exponential distribution for the model vs. scan comparison in \refeq{eq:wifiProb}
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had a positive effect on the misclassification error for some of the walks, but slightly increased
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the estimation error (see figure \ref{fig:normalVsExponential}) and thus produced negative side effects.
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\begin{figure}
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\input{gfx/wifiCompare_normalVsExp_cross.tex}
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\input{gfx/wifiCompare_normalVsExp_meter.tex}
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\caption{
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Comparison between normal- (black) and exponential-distribution (red) for \refeq{eq:wifiProb}.
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While misclassifications are slightly reduced (upper chart),
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the median error between ground-truth and estimation (lower chart) increases by
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about \SI{1}{\meter}.
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}
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\label{fig:normalVsExponential}
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\end{figure}
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\todo{
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erwähnen??? sigma je nach signalstärke anpassen bringt leider auch nichts. wenn man das aber macht,
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dann: fuer grosse signalstaerken ein grosses sigma! andersrum gehts nach hinten los!
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}
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% -------------------------------- final system -------------------------------- %
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% REAL WALKS
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\todo{obwohl das angepasste modell doch recht gut laeuft und der fehler recht klein wird, sind immernoch stellen dabei,
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wo es einfach nicht gut passt, unguenstige mehrdeutigkeiten vorliegen, oder regionen einfach nicht passen wie sie sollten.
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das liegt teils auch daran, dass die fingerprints drehend aufgenommen wurden und beim laufen nach hinten durch den
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menschen abgeschottet wird. auch zeitlicher verzug kann ein problem darstellen.}
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\todo{GPS ist leider kaum eine hilfe. entweder kein empfang wegen ueberdachung oder abschattung, oder
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zu kurz draußen um einen guten gps-fix zu bekommen.}
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\todo{
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walk1 hat eine issue kurz bevor man zur tuer zum hoersaalgebaude reingeht
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je nach resampling killt dieser wlan error evtl alle partikel!
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}
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\todo{
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das bbox modell hat probleme an den uebergängen zwischen bboxes da dort teils starke spruenge sind
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die nicht immer in der realität so auch vorliegen. z.B. z-wechsel machen teils probleme.
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hier wäre ein kontinuierliches modell hilfreich bzw interpolation in randbereichen
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}
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\todo{
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wenn ich beim fingerprinten einen AP an einer stelle NICHT gesehen habe,
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ist das auch eine aussage für die model optimierung.. da kann dann sicher keine signatlstaerke > -90 an der stelle raus kommen
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}
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\todo{gps wird so schnell nicht warm, versagt denn auf dem hof als hilfestellung}
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ware das grid-model nicht da, wuerde der outdoor teil richtig schlecht laufen,
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weil das wlan hier absolut ungenau ist.. da die partikel aber aufgrund des vorherigen
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walks schon recht dicht beisamen sind, kittet das das ganze sehr gut.
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kann man testen, indem man z.B. weniger resampling macht und mehr alte partikel aufhebt.
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geht sofort kaputt sobald man aus dem gebäude raus kommt
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signalstaerke limitieren, wie : alles was im model oder scan < -90 ist, wird auf -90 abgeschnitten hilft
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zwar an manchen stellen, im groben und ganzen führt es aber eher zu fehlern als zu verbesserungen.
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zudem ist zu erwarten, dass diese zahl stark vom geraet/hardware abhaengt
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jeweils beim weighting die niedrigste wifi probability weglassen [je nach particle also ein anderer AP]
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bringt auch nicht immer was.. killt gelegentlich floor-changes. zudem stehen am ende nur sehr wenige
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APs zur verfügung. da einen zu ignorieren, macht noch mehr kaputt
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auch ein versuch wie werfe alle APs aus dem handy-scan weg, die kleiner -90 sind, birgt die selben risiken
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es scheint wirklich am sinnvollsten, die scan-daten einfach 1:1 zu nehmen wie sie sind
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kurz vor ende von path 2 will die estimation nicht in die cafeteria, weil ein paar particle
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die treppe richtung h.1.5 hochgehen und durch das wlan sehr sehr hoch gewichtet werden.
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die mittelwert-estimation versagt hier
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% was ist das??
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%\input{gfx/wifi-opt-error-hist-methods.tex}
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%\input{gfx/wifi-opt-error-hist-stair-outdoor.tex}
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%outdoor hat insgesamt nicht all zu viel einfluss, da die meisten APs
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%an den outdoor punkten kaum gesehen werden. auf einzelne APs kann
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%der einfluss jedoch recht groß sein, siehe den fingerprint plot von
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%dem einen ausgewählten AP
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\todo{anfaenglich falsches heading ist gift, wegen rel. heading, weil sich dann alles verlaeuft. fix: anfaenglich große heading variation erlauben}
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\todo{NICHT MEHR AKTUELL: abs-head ist in der observation besser, weil es beim resampling mehr bringt und dafuer srogt, dass die richtigen geloescht werden!}
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\todo{ deutlich machen
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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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}
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