deletions | additions       diff --git a/Methodology Complementary Data.tex b/Methodology Complementary Data.tex  index 2804b86..cb89f64 100644  --- a/Methodology Complementary Data.tex  +++ b/Methodology Complementary Data.tex 
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\subsubsection{Alignments}  \label{alignments}  Trajectory clustering is the first step in scene interpretation. Trajectory clustering is an abstract representation of movements along prototypical paths through a scene, called alignments. This is the foundation for relating spatial position with road geometry and, in particular, position of moving objects in relation to lanes and sidewalks. The alignment is represented as a simple series of points with a begining and end, typically in the same direction is the majority of flows along this path. This process introduces a new coordinate system which maps a position of a moving object in cartesian 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 begining beginning  and the offset $\gamma$, 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 window of time less than the time users take to perform real lane changes to correct any localised lane "jumping" errors which frequently appear near converging or diverging alignments. These coordinates are useful for measuring following behaviour, lane changes, and lane deflection. Many approaches exist to trajectory clustering, some methods are supervised, many more are unsupervised (e.g. k-means \cite{MacQueen_1967}). Supervised trajectory clustering is labour intensive and potentially a source of bias, but 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. Supervised clustering along a series of splines, called alignments, is chosen for its simple implementation and tight control over interpretation. A hybrid approach, which refines spatial positioning of manually defined alignments through traditional unsupervised clustering approaches, is proposed as future work. 
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Finally, a traditional inventory of testable factors needs to be constructed and associated with each site. These include:  \begin{itemize}  \item \textbf{TODO}  \end{itemize}