deletions | additions       diff --git a/Methodology Complementary Data.tex b/Methodology Complementary Data.tex  index 58f2771..5737cda 100644  --- a/Methodology Complementary Data.tex  +++ b/Methodology Complementary Data.tex 
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Trajectory clustering is the first step in behavioural analysis. Trajectory clustering is an abstract representation of movements along prototypical paths through a scene, called alignments. This is the foundation for relating spatial position of trajectories with elements of road geometry and, in particular, the position of moving objects in relation to traffic lanes, bike paths, and side walks.  Many approaches to trajectory clustering have been explored. While some methods are supervised \cite{Schreck_2008}, many more are unsupervised (e.g. k-means \cite{MacQueen_1967} or hidden Markov models \cite{Rodr_guez_Serrano_2012}). Manual trajectory clustering is labour intensive and may be a source of bias, but it allows for tight control of scene description and analysis oversight. Unsupervised clustering is systematic but naive, as this form of clustering can only make use of trajectory data to infer spatial relationship. Manual clustering along a series of splines, called alignments, is chosen for its simple implementation and tight control over interpretation. A hybrid approach, which automatically refines tthe the  spatial positioning of manually defined alignments through traditional unsupervised clustering approaches, is considered for future improvements \cite{morris08survey, Schreck_2008}. The alignment is represented as a simple series of points with a beginning and an end,typically  in the same direction of travel  as the majority of flows movements  along this path. This process introduces a new coordinate system which maps a position of a moving object in Cartesian space to a position in curvilinear space: \begin{equation} \label{eqn:coordinate-transform} (x,y)\to(l,s,\gamma). \end{equation}  where a point located at $(x,y)$ in Cartesian space is snapped orthogonally to the nearest position on the nearest alignment $l$, and is represented by the curvilinear distance $s$ along this alignment from its beginning and the offset $\gamma$, $\gamma$ (positive to the right of the vector),  orthogonal to this alignment, measuring the distance between the original point and its position snapped to the alignment. A second pass may be performed over a moving window in less than the time users take to perform real lane changes to correct any localized lane "jumping" errors which frequently appear near converging or diverging alignments. These coordinates are useful for studying following behaviour, lane changes, and lane deflection. \subsubsection{Network Topology}