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tex fixes

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This commit is contained in:
FrankE
2016-02-29 12:13:00 +01:00
parent 6ef06459cb
commit 020cbc5d94
1 changed files with 11 additions and 8 deletions
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19
tex/chapters/grid.tex
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@@ -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 $\sigma^2$.