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tex/chapters/filtering.tex
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@@ -1,84 +1,82 @@
%\section{Filtering}
%
% \label{sec:filtering}
%
% \commentByToni{Bin mir nicht sicher ob wir diese Section überhaupt brauchen. Könnte man bestimmt auch einfach unter Section 3 packen. Aber dann können wir ungestört voneinander schreiben.}
%
\section{Evaluation}
\section{Filtering}
\commentByFrank{brauchen wir hier noch was (kurze einleitung) oder passt das so?}
\commentByFrank{eval und transition tauschen von der reihenfolge?}
\subsection{Barometer}
\label{sec:sensBaro}
%
The probability of currently residing on a floor is evaluated using the smartphone's barometer.
Environmental influences are circumvented by using relative pressure values instead of absolute ones.
To reduce the impact of noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several
sensor readings, carried out while the pedestrian chooses his destination. This average serves as relative base
for all future measurements. Likewise, we estimate the sensor's uncertainty $\sigma_\text{baro}$ for later use
within the evaluation step.
In order to evaluate the relative pressure readings, we need a prediction to compare them with. Therefore, each
transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ estimates the state's relative pressure prediction
$\mStatePressure$ by tracking every height-change ($z$-axis):
%
\begin{equation}
\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b
,\enskip
\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
,\enskip
b \in \R
\enspace .
\label{eq:baroTransition}
\end{equation}
%
In \refeq{eq:baroTransition}, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
The evaluation step for time $t$ compares every predicted relative pressure $\mState_t^{\mStatePressure}$ with the observed
one $\mObs_t^{\mObsPressure}$ using a normal distribution with the previously estimated $\sigma_\text{baro}$:
%
\begin{equation}
p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace.
\label{eq:baroEval}
\end{equation}
%
%
%
\subsection{Wi-Fi \& iBeacons}
%
The smartphone's \docWIFI{} and \docIBeacon{} component provides an absolute location estimation by
measuring the signal-strengths of nearby transmitters. The positions of detected \docAP{}s (\docAPshort{}) and \docIBeacon{}s
are known beforehand. Using the wall-attenuation-factor signal strength prediction model \cite{Ebner-15}, we are able to
compare each measurement with a corresponding estimation. To infer this estimation, the prediction model
uses the 3D distance $d$ and the number of floors $\Delta f$ between transmitter and the state-in-question $\mStateVec$:
%
\begin{equation}
P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace ,
\label{eq:waf}
\end{equation}
%
In \refeq{eq:waf}, there are three more parameters per \docAPshort{}. The signal-strength $\mTXP$ measurable at a distance
$\mMdlDist_0$ (usually \SI{1}{\meter}), a path-loss exponent $\mPLE$ describing the transmitter's environment and the attenuation
per floor $\mWAF$.
To reduce the system's setup time, we use the same three values for all \docAP{}s at the cost of accuracy.
All parameters are chosen empirically. Further details on how to determine this parameters exactly,
can be found in \cite{PathLossPredictionModelsForIndoor}.
\subsection{Evaluation}
\commentByFrank{brauchen wir hier noch was (kurze einleitung) oder passt das so?}
\subsubsection{Barometer}
\label{sec:sensBaro}
%
The probability of currently residing on a floor is evaluated using the smartphone's barometer.
Environmental influences are circumvented by using relative pressure values instead of absolute ones.
To reduce the impact of noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several
sensor readings, carried out while the pedestrian chooses his destination. This average serves as relative base
for all future measurements. Likewise, we estimate the sensor's uncertainty $\sigma_\text{baro}$ for later use
within the evaluation step.
The same holds for the \docIBeacon{} component, except $\mTXP$,
which is broadcasted by each beacon. However, as \docIBeacon{}s cover only a small area, $\mPLE$ is usually much smaller compared
to the one needed for \docWIFI{}.
In order to evaluate the relative pressure readings, we need a prediction to compare them with. Therefore, each
transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ estimates the state's relative pressure prediction
$\mStatePressure$ by tracking every height-change ($z$-axis):
%
\begin{equation}
\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b
,\enskip
\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
,\enskip
b \in \R
\enspace .
\label{eq:baroTransition}
\end{equation}
%
In \refeq{eq:baroTransition}, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
The evaluation step for time $t$ compares every predicted relative pressure $\mState_t^{\mStatePressure}$ with the observed
one $\mObs_t^{\mObsPressure}$ using a normal distribution with the previously estimated $\sigma_\text{baro}$:
%
\begin{equation}
p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace.
\label{eq:baroEval}
\end{equation}
%
%
%
\subsubsection{Wi-Fi \& iBeacons}
%
The smartphone's \docWIFI{} and \docIBeacon{} component provides an absolute location estimation by
measuring the signal-strengths of nearby transmitters. The positions of detected \docAP{}s (\docAPshort{}) and \docIBeacon{}s
are known beforehand. Using the wall-attenuation-factor signal strength prediction model \cite{Ebner-15}, we are able to
compare each measurement with a corresponding estimation. To infer this estimation, the prediction model
uses the 3D distance $d$ and the number of floors $\Delta f$ between transmitter and the state-in-question $\mStateVec$:
%
\begin{equation}
P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace ,
\label{eq:waf}
\end{equation}
%
In \refeq{eq:waf}, there are three more parameters per \docAPshort{}. The signal-strength $\mTXP$ measurable at a distance
$\mMdlDist_0$ (usually \SI{1}{\meter}), a path-loss exponent $\mPLE$ describing the transmitter's environment and the attenuation
per floor $\mWAF$.
To reduce the system's setup time, we use the same three values for all \docAP{}s at the cost of accuracy.
All parameters are chosen empirically. Further details on how to determine this parameters exactly,
can be found in \cite{PathLossPredictionModelsForIndoor}.
The same holds for the \docIBeacon{} component, except $\mTXP$,
which is broadcasted by each beacon. However, as \docIBeacon{}s cover only a small area, $\mPLE$ is usually much smaller compared
to the one needed for \docWIFI{}.
As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is:
%
\begin{equation}
\mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} =
\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace .
\label{eq:wifiTotal}
\end{equation}
As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is:
%
\begin{equation}
\mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} =
\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace .
\label{eq:wifiTotal}
\end{equation}
\section{Transition}
\label{sec:transition}
\subsection{Transition}
\label{sec:transition}
The transition-distribution $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is sampled via random walks on a graph
$G=(V,E)$, which is generated from the buildings floorplan \cite{Ebner-16}.
@@ -94,14 +92,14 @@
\commentByFrank{ist das verstaendlich oder schon zu kurz?}
\subsection{Pedestrian's Destination}
\subsubsection{Pedestrian's Destination}
We assume the pedestrian's desired destination to be known beforehand. This prior knowledge is incorporated
during the random walk using $p(\mEdgeAB)_\text{path}$, which is a simple heuristic, favouring movements (edges)
approaching his chosen destination with a ratio of $0.9:0.1$ over those, departing from the destination
\cite{Ebner-16}. The underlying shortest-path uses Dijkstra's algorithm with special weight (distance) metric,
considering special architectural facts:
\subsection{Architectural Facts}
\subsubsection{Architectural Facts}
Normally, the shortest-path calculated for a narrow grid would stick unnaturally close to obstacles like walls.
To ensure realistic (human like) path estimations, we include architectural knowledge within Dijkstra's edge-weight function \cite{Ebner-16}:
Each vertex's distance from the nearest wall is used to artificially increase the edge-weight and thus prevent the shortest-path
@@ -109,7 +107,7 @@
and favoured by decreasing their edge-weight.
\subsection{Step- \& Turn-Detection}
\subsubsection{Step- \& Turn-Detection}
Steps and turns are detected using the smartphone's IMU, implemented as described in \cite{Ebner-15}.
The number of steps detected since the last transition is used to estimate the to-be-walked distance $\gDist$
by assuming a fixed step-size with some deviation:
@@ -138,7 +136,7 @@
While the distribution \refeq{eq:transHeading} does not integrate to $1.0$ due to circularity of angular
data, in our case, the normal distribution can be assumed as sufficient for small enough $\sigma^2$.
\subsection{Activity-Detection}
\subsubsection{Activity-Detection}
Additionally we perform a simple activity detection for the pedestrian, able to distinguish between several actions
$\mObsActivity \in \{ \text{unknown}, \text{standing}, \text{walking}, \text{stairs\_up}, \text{stairs\_down} \}$.