From 020cbc5d94b6cdab3ca829cd5c22e99a7ff4874b Mon Sep 17 00:00:00 2001
From: FrankE <support@frank-ebner.de>
Date: Mon, 29 Feb 2016 12:13:00 +0100
Subject: [PATCH] tex fixes

---
 tex/chapters/grid.tex | 19 +++++++++++--------
 1 file changed, 11 insertions(+), 8 deletions(-)

diff --git a/tex/chapters/grid.tex b/tex/chapters/grid.tex
index e9321d9..e3f00b5 100644
--- a/tex/chapters/grid.tex
+++ b/tex/chapters/grid.tex
@@ -271,24 +271,25 @@
 			%
 			As new states $\mStateVec_{t}$ should approach the pedestrian's destination
 			we use a reference $\pathRef$ all states try to reach. This references must
-			be both part of the shortest path and located somewhere outside of the sample-set.
+			be a) part of the shortest path and b) located somewhere outside of the sample-set.
 			%			
-			We thus calculate the standard deviation of the distance of all sample-positions
-			$\fPos{\mStateVec_{t-1}}$ from aforementioned centre $\pathCentroid$.
+			For b) we need to know the current position distribution and therefore calculate
+			the standard deviation of the distance of all sample-positions $\fPos{\mStateVec_{t-1}}$
+			from aforementioned centre $\pathCentroid$.
 			%\commentByFrank{so klarer? platz fuer groese Eq. fehlt und Notation zum ansprechen jedes einzelnen Particles vermeide ich lieber...}
 			%
 			%\begin{equation}
 			%	d_\text{cen} = \| pos(q_{t-1}) - \pathCentroid \|
 			%	\sigma_\text{cen} = stdDev(distance)
 			%\end{equation}
-			
+			%
 			After advancing the starting-vertex by three times this deviation
 			we get the new point $\pathRef$ that is: part of the shortest path, outside of the sample-set
 			and closer (but not too close) to the desired destination.
-			%
+			
 			Hereafter, the simple transition \refeq{eq:transSimple} is combined with a second probability,
 			downvoting all grid-steps that depart from $\pathRef$.
-			To still allow leaving the shortest path, the intensity of the downvoting is controlled via $\mUsePath$,
+			To still allow leaving the shortest path, the intensity of downvoting is controlled via $\mUsePath$,
 			with $0 < \mUsePath < 1$.
 			Finally, \refeq{eq:transShortestPath} provides a metric tending towards the reference while
 			still allowing the pedestrian to leave the shortest path:
@@ -335,7 +336,9 @@
 			\end{equation}
 			%
 			Fig. \ref{fig:multiHeatMap} shows a heat-map of how often vertices were visited after several
-			\SI{125}{\meter} walks. The colours from cold to hot indicate that both possible paths
+			random walks from $\mVertexA$ towards $\mVertexDest$.
+			%\SI{125}{\meter} walks.
+			The colours from cold to hot indicate that both possible paths
 			are covered and slight deviations from the shortest version are possible.
 			%
 			\begin{figure}
@@ -348,7 +351,7 @@
 				%\commentByFrank{so besser?}
 				\label{fig:multiHeatMap}
 			\end{figure}
-%
+
 			It should be noted that the normal distribution in \eqref{eq:transShortestPath} and \eqref{eq:transMultiPath} does not integrate to \SI{1}{} due to circularity of angular data. 
 This normally requires wrapped distributions like the von Mises distribution. 
 However, in our case, the normal distribution can be assumed as sufficient for small enough $\sigma^2$.