\subsubsection{Time-to-collision}

Time-to-collision (TTC), first proposed by \cite{Hayward_1971}, is one of the most popular surrogate safety measures. It is a method of quantifying proximity to danger. Time-to-collision measures the time, at a given instant $t_0$, until two road users on a collision course collide, based on the motion prediction model. In its simplest form, e.g. motion prediction at constant velocity or in a car-following situation, time-to-collision is the ratio of differential velocity or speed and differential position. A TTC value of 0 seconds is, by definition, a collision. TTC is particularly useful as it has the same dimensions as some important traffic accident factors such as user perception and reaction time, typically set at the critical value of 1.5 seconds \cite{Hyden_1987, Green_2000} in the literature, and breaking time. Larger values of observed TTC thus provide greater factors of safety for particular driving tasks.

Time-to-collision is measured instantaneously: a new value of TTC may be computed at every instant. Thus, a pair of users may have a time series of TTC observations evolving over time. Some efforts have been made to study these evolutions \cite{Mohamed_2013} as a form of safety continuum \cite{Zheng_2014}. Other approaches have focused counting discrete events using a threshold similar to the TCTs \cite{Hyden_1987, Svensson_2006}, although care must be taken because results can be shown to vary with prediction methodology and threshold level \cite{St_Aubin_2015_TRBa}.

A sample pair of road user trajectories (\#303 and \#304, Figure~\ref{fig:conflict-video}) and spatial relationships simultaneously existing over a time interval lasting 64 instants, or just over 4 seconds, is presented in Figure~\ref{fig:conflict-series}. In this scenario, vehicle \#304 is approaching vehicle \#303, which is engaged in an illegal U-turn, at high velocity. The norm of the differential velocity $\Delta v$, relative distance $d$, and corresponding time $t$ are measured for every instant. In a matter of just under 4 seconds, the differential velocity changes from 9.63 to 2.26~m/s while the relative distance changes from 28.57 to 9.57~m. For every interaction instant of this user pair, motion prediction is used to calculate resulting TTC under each motion prediction method. The time series of the predicted collisions and associated TTC measures for this pair of users and for each different motion prediction method is presented in Figure~\ref{fig:ttc-timeseries}. At each instant in time, normal adaptation and motion pattern prediction generate multiple possible collisions, while constant velocity can only ever produce one possible point of collision. The existence of more predicted collision instants using motion patterns demonstrates the potential for predicting many more complex situations. When several potential collision points are predicted, the expected time-to-collision, defined as $TTC'_i$, at time $t_i$ (for instant $i$) is calculated as the probability-weighted TTC average of all possible collision points indexed $j=1..n$ with $Prob(collision)_{ij}$ as defined and measured in \cite{St_Aubin_2014}.

\begin{equation} \label{eqn:TTC-mp-weight-avg} TTC'_i = \frac{\sum_{j=1}^{n}{TTC_{ij} Prob(collision)_{ij}} }{n} \end{equation}

It is clear from both this figure and the trajectories themselves that constant velocity and normal adaptation motion predictions are inadequate for roundabout conflict analysis: the trajectories share the same destination yet their headings lead them to a potential collision point only for a brief period of time with these prediction methods.