Merge branch 'master' of https://git.frank-ebner.de/toni/IPIN2016

tex/chapters/filtering.tex

@@ -4,10 +4,9 @@
 
 	\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.}
 
-	\subsection{Evaluation}
+\section{Evaluation}
-
-\section{Barometer}
 
+	\subsection{Barometer}
 	\label{sec:sensBaro}
 		%
 		The probability of currently residing on a given floor is evaluated using the smartphone's barometer.
@@ -75,23 +74,104 @@
 		%
 		
 
-	\subsection{Transition}
+\section{Transition}
 
-		The transition step depends on random walks on a graph, generated from the buildings floorplan
+		The distribution $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is sampled via random walks on a graph
-		\todo{cite}. This setup allows only valid movements, as ambient conditions (walls, doors, etc.) are considered.
+		$G=(V,E)$, which is generated from the buildings floorplan \todo{FUSION2016}.
+		$p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is determined by walking along adjacent edges $\mEdgeAB$ connecting
+		two vertices $\mVertexA, \mVertexB \in V$ until a certain distance $\gDist$ is reached.
+		Thus, the position of any $\mStateVec$ is represented by the position $\fPos{\mVertexA}$ of the corresponding vertex.
+		This approach draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
 		
-		Furthermore, we assume the pedestrian's desired destination to be known beforehand. This prior knowledge is evaluated
+		While sampling, to-be-walked edges are not chosen uniformly, but depending on a probability $p(\mEdgeAB)$.
-		during the random walk, to favour movements approaching the chosen destination.
+		The latter depends on several constraints and recent sensor-readings from the smartphone. Using sensors
+		directly within the transition step provides a more robust posterior distribution. Adding them to the evaluation
+		instead, would lead to sample impoverishment due to the used Monte Carlo methods.
+				
+		\subsection{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 the chosen destination with a ratio of $0.9:0.1$ over those, departing from the destination
+			\cite{FUSION2016}. The underlying shortest-path is based on a special distance metric, considering special
+			architectural facts: 
+			
+		\subsection{Architectural Facts}
+			To ensure realistic path estimations, we include additional architectural knowledge.
+			Each vertex's distance from the nearest wall is used to prevent the shortest-path from clinging to walls.
+			Likewise, his distance from the nearest door (favour ).
 				
-		To ensure the transition step provides a viable posterior distribution, we include some sensors directly into the transition step.
-		Adding them to the evaluation instead, would lead to sample impoverishment when using Monte Carlo methods.
 				
 		\subsection{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 $\gDist$ 
+			assuming a fixed step-size with some deviation:
 			%
-			Steps and turns are detected using the smartphone's IMU and are implemented as described in \cite{Ebner-15}.
+			\begin{equation}
+				\gDist = \mObs_{t-1}^{\mObsSteps} \cdot \mStepSize	+ \mathcal{N}(0, \sigma_{\gDist}^2)
+				\enspace .
+			\end{equation}
+			%
+			Turn-Detection supplies the magnitude of the detected heading change by integrating	the gyroscope's change
+			since the last transition. Together with some deviation and the state's previous heading, the magnitude is
+			used to estimate the state's current heading:
+			%
+			\begin{equation}
+				\gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading}	+ \mathcal{N}(0, \sigma_{\gHead}^2) \\
+			\end{equation}
+			%
+			During the random walk, edges should satisfy the heading:
+			%
+			\begin{equation}
+				p(\mEdgeAB)_\text{head} = p(\mEdgeAB \mid \gHead) = \mathcal{N} (\angle \mEdgeAB \mid \gHead, \sigma_\text{head}^2)
+				\enspace .
+				\label{eq:transHeading}
+			\end{equation}
+			%
+			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}
-			\todo{write}
+
+			Additionally we perform a simple activity detection for the pedestrian, able to distinguish between 
+			standing, walking, walking stairs upwards and downwards. Likewise, this knowledge
+			is evaluated when walking the grid: Edges $\mEdgeAB$ matching the currently detected
+			activity are favoured using $p(\mEdgeAB)_\text{act} = 0.8$ and $0.2$ otherwise.
+			
+			\begin{equation}
+				p(\mEdgeAB)_\text{act} = 
+				\begin{cases}
+					0.8	& \text{stairs\_up}, \fPos{\mVertexB}_z > \fPos{\mVertexA}_z \\
+					0.2	& \text{stairs\_up}, \fPos{\mVertexB}_z \le \fPos{\mVertexA}_z \\
+					\cdots
+				\end{cases}
+			\end{equation}
+			\commentByFrank{das switch ist wahrscheinlich unnoetig und der text reicht}
+			
+			\commentByFrank{hier passen die sachen vom lukas. kurze beschreibung der beiden geschaetzten verteilungen etc}
+			
+		
+
+			% Activity Recognition
+			% Naives Bayes als Klassifikator
+			% Features -> 1: Variance of mean 2: Differenz zwischen Barometer
+			% Zeitintervall für das die Merkmale berechnet werden
+			
+			The transition model includes a simple recognizer of different locomotion modes like normal walking or ascending/descending stairs. The reasoning behind this is to favour paths that correspond with the detected locomotion mode. 
+			
+			We use a Naives Bayes classifier with two features. For this, the sensor signals are split in sliding windows. Each window has a length of one second and overlaps 500 ms with its prior window.  
+			The first feature is the variance of the accelerometer's magnitude during a window and the second feature is the difference between the last and first barometer measurement of the particular window.
+			Based on these features the classifier assigns an activity to each sliding window.
+			  
+			
+			\todo{Was passiert wenn ein überlappendes Fenster zwei verschiedene Aktivitäten zugewiesen bekommt? Sliding windows evtl. weglassen?}
+
 

tex/misc/functions.tex

@@ -57,6 +57,9 @@
 \newcommand{\mVertexB}{v_j}
 \newcommand{\mEdgeAB}{e_{i,j}}
 \newcommand{\mVertexDest}{v_\text{dest}}
+\newcommand{\gDist}{d_\text{step}}
+\newcommand{\gHead}{\theta_\text{walk}}
+
 
 \newcommand{\mUsePath}{\kappa}