The time-to-collision calculations for constant velocity \cite{Laureshyn_2010} and motion patterns \cite{St_Aubin_2014} are performed with no need for additional constants or parameters. For normal adaptation, the empirical constants used in \cite{Mohamed_2013} are re-used, namely triangular distributions for acceleration and steering with an acceleration range \(\alpha\) of \(\pm 2\) m/s\(^2\) and a maximum steering range \(\sigma\) of \(\pm 0.2\) rad/s (these ranges are empirical \cite{Mohamed_2013}). In all cases of motion prediction, predictions are performed into the future no more than a chosen time horizon. Indicators derived from motion prediction using time horizons above 10 seconds are generally ignored in the literature for two reasons: i) they produce indicators corresponding to mostly uniform noise, and ii) are significantly larger than reaction times of all drivers and so are of little value. Also, because motion-pattern prediction is based on observed behaviour, it can only predict motion that falls within the space of observed behaviour. This adds a practical time horizon constraint.
Indicators computed continuously for all interaction instants may be aggregated in various ways at the user-pair level (over all interaction instants).
The all indicators method treats every single instantaneous observation as an indicator of safety. It has the advantage of generating large datasets and capturing continuous behaviour to reduce errors from noise, but suffers from sampling bias and issues interpreting conditional probability. The sampling bias stems from oversampling of objects moving at slower speeds.
The minimum unique method uses the most severe observation in the time series of indicators, usually the lowest for temporal indicators. This approach solves the problems with the previous approach, but assumes that dangerous traffic events occur only once per user pair and is prone to outlier effects from noisy data and instantaneous tracking errors. This technique is identical to the principle of \(TTC_{min}\) commonly used in the literature \cite{Hayward_1971, vanderHorst_1990}.
The 15\(^{th}\) centile unique method is identical to the minimum unique method but proposes using a centile of the indicator values over time instead of a minimum in order to be more robust to the effects of noise and instantaneous tracking errors.
The classic TCT method is examined by comparing consistency of site risk ranking of traffic events corresponding to threshold criteria for various motion prediction and indicator aggregation methods.
The criteria examined in this paper are based on the traditional time-to-collision interpretation of the 1.5 s human reaction time \cite{Hyden_1987}. In addition to indicator aggregation, events can be reported by probability
\[\label{eqn:event_frequency} P(U_{i,j,ind_{agg} < ind_{threshold}}) = \frac{\sum [U_{i,j,ind_{agg} < ind_{threshold}}]}{\sum [U_{i,j}]}\]
where \(U_{i,j}\) is a time series of interaction instants between the same two road users \(i\) and \(j\), \(ind_{agg}\) is the representative indicator (\(TTC_{min}\) or \(TTC_{15th}\) in this case) of the user pair’s time series according to the chosen indicator aggregation method and \(ind_{threshold}\) is a chosen maximum indicator criterion, in this case \(ind_{threshold} = 1.5\) s. This may be useful for comparison with accident rates. Events may also be reported simply by hourly counts. This approach is sensitive to exposure, and may be a better predictor of expected accidents over time.
A secondary constraint for motion-pattern prediction is also put in place during modelling. Motion pattern prediction creates a probabilistic spectrum of collision events. The indicator alone isn’t sufficient to describe the event. The event criteria need to be expanded to take into account the modelled collision probability criteria because motion patterns will generate many more collision points than constant velocity motion prediction, only with smaller collision probabilities. Equation \ref{eqn:event_frequency_mp} is adapted accordingly for motion prediction using a prediction continuum:
\[\label{eqn:event_frequency_mp} P(U_{i,j,ind_{agg} < ind_{threshold}}) = \frac{\sum [U_{i,j,ind_{agg} < ind_{threshold} \bigcap P(collision) > p_{threshold}}]}{\sum [U_{i,j}]}\]
where \(P(collision)\) is the total probability of predicted collision at the indicator’s instant determined by the aggregation method and where \(p_{threshold}\) is the desired minimum probability of collision at that instant in time. For this work, we choose \(p_{threshold} = 0.001\) as a control case arbitrarily and compare its performance with \(p_{threshold} = 0.01\). An attempt at accident-rate modelling should calibrate this parameter accordingly or remove it entirely by incorporating a calibrated collision prediction.
\label{indicator_distributions}
Assuming some association between TTC and collision probability, qualitative safety comparisons can be made between density functions of indicators when the direction of shift in weight (safety impact) of a density function cannot be governed by the individual weight (safety impact) of that indicator (e.g. TTC). This occurs when probability density functions intersect exactly once, or their corresponding cumulative distribution functions do not intersect at all. For example, if TTC follows a gamma distribution, this occurs when either the scale or the shape parameters are conserved.
The principle of this analysis is demonstrated in Figure \ref{fig:distro_comp_v2}. Two different control cases for two options are presented, each with their corresponding probability and cumulative distribution functions. In the first case, all high-risk behaviour (low TTCs) is shifted towards low-risk behaviour (high TTCs), resulting in a non-ambiguous gain in safety for one of the two options, assuming TTC is associated with safety. The gain is however not quantified. In the second control case, some high-risk behaviour (low TTCs) and low-risk behaviour (high TTCs) is shifted towards the middle. Results are therefore inconclusive without quantifying collision probability of individual TTC observations.