
\subsection{Measurement Definitions}

\subsubsection{General versus Specific Analysis Measures}

Some measures are generalizable for all traffic studies using alignments, while others are not \cite{Gettman_2003}. General traffic measures include speed profiles, counts, lane changes, origin-destination matrices, and basic spatial relationships including conflicts. Other measures may be specific to the study and generally require high-level interpretation (HLI). This interpretation makes use of study-specific geometric information to generate custom measures. As such, this section will not cover these custom measures, but will instead focus on generalizable measures. However, an application of HLI calculations will be briefly presented in section \ref{sample_HLI_analysis}.

\subsubsection{Interactions}

An interaction quantifies the spatial relationship between moving objects in a scene, as is depicted in Figure~\ref{fig:conflict-video}. At the most fundamental level, an interaction is defined as a pair of moving objects simultaneously present in a scene over a common time interval (also referred to as a user pair). We further define an instantaneous observation (i.e. in a given video frame) within this time interval as an interaction instant \cite{St_Aubin_2015_TRBa}.

This interaction definition is generic, if not naive, as the quality depends largely on how the scene is constructed. For example, the significance of an interaction between two vehicles separated from each other physically (e.g. via a median or a building) may not be comparable to an interaction between two vehicles merely separated by lane markings because of the implication that they may cross the lane marking intentionally or inadvertently very easily. This may interfere with collision prediction attempts, particularly if scenes are not consistently selected and geometry is not controlled.

One solution is to perform a triage of user pairs based on physical access and proximity. A network topology coupled with a driving distance horizon is proposed. This is not a perfect solution, however, as physical access may not necessarily be a discrete choice. For the example of the median, it is still physically possible, although much less likely, for a vehicle to cross over into oncoming traffic.

\subsubsection{Motion Prediction}
\label{motion-prediction}

While vehicle trajectories offer a rich set of observed behavioural data, they do not provide much collision data. The proactive road safety approach requires that predicting collisions should be performed without observing them directly. To be studied, collisions must be extrapolated from traffic events with potential for collision. This potential is modelled by predicting future possible positions between each pair of road users at every instant in time. Several motion prediction models are proposed for study, including constant velocity \cite{Laureshyn_2010}, normal adaptation \cite{Mohamed_2013}, and motion patterns \cite{saunier07probabilistic,morris08survey,Sivaraman_2013}, particularly the implementation of a discretized motion pattern \cite{St_Aubin_2014}. Specific details of implementation of each method can be found in \cite{St_Aubin_2015_TRBa}.

%\begin{itemize}
%\item \textbf{Constant velocity} is the classic motion prediction model, 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 specific applications of the methodology, e.g. highways \cite{St_Aubin_2013}. The current implementation is based on \cite{Laureshyn_2010}.
%\item \textbf{Normal adaptation} uses 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 adaptation being studied is based on \cite{Mohamed_2013}, using a acceleration maximum $\alpha$ of

%\begin{equation} \label{eqn:norm-adapt-accel-maxima} \alpha =  \pm \frac{2}{f^2}  \end{equation}

%and a maximum steering parameter $\sigma$ 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 \cite{saunier07probabilistic,morris08survey,Sivaraman_2013}. This type of model is the most promising, as motion prediction is probabilistic in nature and inherently models naturalistic behaviour. However, it may not be possible 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 discretized 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 instant $t_0$ for a number of time steps of size $\Delta t$ between $t_0$ and some chosen time horizon. Each motion prediction may generate for two road users a series or a matrix of collision points with a sum of probabilities inferior or equal to 1.

%This is significantly larger and more difficult to handle than the trajectory data, and currently cannot be performed in real time.