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\subsubsection{Motion Prediction}  \label{motion-prediction}  While vehicle trajectories offer a rich set of observed behavioural data, they do not provide much collision data; this is by design of the proactive road safety approach: predicting collisions should be performed without observing them directly. In order to study collisions, they need to be extrapolated from traffic events with potential for collision. This potential is modeled by predicting future positions of vehicles using motion prediction at every instant in time and examining i) situations of particular risk (i.e. threshold) or ii) evolution of the risk. Several motion prediction models are proposed for study: study \cite{Mohamed_2013}:  \begin{itemize}  \item \textbf{Constant velocity} is the classic motion prediction model, wherin wherein  vehicles are projected along straight paths at a constant speed and heading using the velocity vector at that moment in time. This model is the simplest but also makes the most assumptions: only one movement is predicted at every instant, both users do not enter evasive action in the event of a collision course, and the natural (non-reacting) motion of a moving object is a straight path (not always true). These assumptions may be adequate for niche specific  applications of the methodology, e.g. highways \cite{St_Aubin_2013}. The current implementation is based off of \cite{Laureshyn_2010}. \item \textbf{Normal adaptation} uses constant the initial  velocity vector at the prediction moment  to project trajectories, but modifies the velocity vector to account for normal variation. This model benefits from a wider range of possible outcome velocity vectors, but otherwise suffers the same problems and makes the same assumptions as constant velocity. The implementation of normal adaption studied is based off of \cite{Mohamed_2013}, using a acceleration maxima maximum  $\alpha$ of \begin{equation} \label{eqn:norm-adapt-accel-maxima} \alpha = \pm \frac{2}{f^2} \end{equation}  and a mimimum maximum  steering paramter $\sigma$ of \begin{equation} \label{eqn:norm-adapt-steering-maxima} \sigma = \frac{0.2}{f} \end{equation}  where $f$ is the number of frames per second of the video.  \item \textbf{Motion patterns} are a family of models which use machine learning to calculate future position likelihoods from past behaviour. behaviour (\textbf{REF Saunier 2007 ITSC}).  This type of model is the most promising as motion prediction is probabilistic in nature and inherently models naturalistic behaviour. However, they may not be able to model erratic behaviour such as roadway departures. Motion patterns are also complex to implement and expensive to process. The type of motion pattern being studied for implementation is a discretised motion pattern \cite{St_Aubin_2014}. \end{itemize}  As illustrated in Figure~\ref{fig:prob-collision-space}, motion prediction is performed for each user pair over each interaction-instant $t_0$ for a number of timesteps of size $\Delta t$ between $t_0$ and some chosen timehorizon. Each motion prediction may generate a series or a matrix of collision points with a sum of probabilities equal to 1. This is significantly larger and more difficult to handle than trajectory data, and currently cannot be performed in real time.