deletions | additions       diff --git a/Methodology Measurements TTC.tex b/Methodology Measurements TTC.tex  index 0655250..ab4f127 100644  --- a/Methodology Measurements TTC.tex  +++ b/Methodology Measurements TTC.tex 
...
\subsubsection{Time-to-collision}  Time-to-collision (TTC) 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 interaction-instant instant  $t_0$, until two road users collide, if they collide, based on the motion prediction model. In the simplest form, e.g. constant velocity, time-to-collision is the ratio of differential velocity 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 and breaking time. Larger values of observed TTC thus provide greater factors of safety for these driving tasks. Time-to-collision is measured instantaneously: a new value of TTC may be computed for every interaction-instant. 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}. Other approaches have focused on quantile or threshold observations \cite{Svensson_2006}, while others still attempt (i.e. counting the number of interactions with minimum TTC below a threshold as in classical traffic contlict techniques \cite{Svensson_2006}), or even  to examine instantaneous risk and significance of TTC. TTC \cite{St_Aubin_2013}.  A sample pair of road user trajectories (\#303 and \#304, Figure~\ref{fig:conflict-video}) and spatial relationships at a single site simultaneously existing  over a time series of 64 interaction-instants, interval  lasting 64 instants or  just over 4 seconds is presented in Figure~\ref{fig:conflict-series}. In this scenario, vehicle \#304 is approaching at high velocity vehicle \#303 which is engaged in an illegal U-turn (in a right-hand roundabout, users are supposed to travel counter-clockwise around the centre island at all times). The differential velocity $\Delta v$, relative distance $d$, and corresponding time $t$ is measured for every interaction-instant. 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. These predicted collisions and associated TTC measures are presented in Figure~\ref{fig:ttc-timeseries}. Motion pattern prediction generates many more possible collision points than constant velocity prediction, though each of these points has a significantly lower associated probability. For better instant-to-instant comparison, When several potential collision points are predicted,  the expected TTC  $TTC'_i$ at time $t_i$ is calculated as the probability-weighted TTC average \begin{equation} \label{eqn:TTC-mp-weight-avg} TTC'_i = \frac{\sum_{j=1}^{n}{TTC_{ij} Prob(collision)_{ij}} }{n} \end{equation}  of all possible collision points indexed  $j=1..m$ with $Prob(collision)$ $Prob(collision)_{ij}$  \cite{St_Aubin_2014}. It isquite  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 they only share similar headings for a brief period of time with these prediction methods.