Probabilistic modeling of car traffic accidents

TL;DR

This work develops a self-exciting counting model for car traffic accidents with intensity $\lambda^*(t) = \lambda(t) + \mu(t) N_t$, and analyzes its distributional dynamics via a kinetic-theory-inspired framework. It derives the evolution of the distribution $f(n,t)$, its moments, and large-time behavior, and provides explicit expressions for accident times $T_k$ and interaccident times $\Delta T_k$, including special results in the constant-intensity case. Tail bounds and uniform moment results are established under $\lambda,\mu \in L^1(\mathbb{R}_+)$, and exact formulas are obtained when $\lambda,\mu$ are constant. Calibration against UK weekly data suggests that a sinusoidal background intensity with a rational time-varying self-excitation best reproduces observed weekly evolution and variability, demonstrating the approach’s potential for risk assessment and planning in traffic safety.

Abstract

We introduce a counting process to model the random occurrence in time of car traffic accidents, taking into account some aspects of the self-excitation typical of this phenomenon. By combining methods from probability and differential equations, we study this stochastic process in terms of its statistical moments and large-time trend. Moreover, we derive analytically the probability density functions of the times of occurrence of traffic accidents and of the time elapsing between two consecutive accidents. Finally, we demonstrate the suitability of our modelling approach by means of numerical simulations, which address also a comparison with real data of weekly trends of traffic accidents.

Figures (11)

• Figure 1: Weekly evolution in time of the number of accidents in an area in the UK for all weeks of 2020. For further details on the data, see Section \ref{['sec:realData']}.
• Figure 2: A prototypical trajectory of the process $\{N_t,\,t\geq 0\}$.
• Figure 3: Real data study for $\mu_1$, the time-constant excitation function. Comparison of temporal evolution of $N_{t}$ and the histograms for $N_{10080}$ with different choices of the weights to real data.
• Figure 4: Real data study for $\mu_2$, the exponentially decreasing excitation function. Comparison of temporal evolution of $N_{t}$ and the histograms for $N_{10080}$ with different choices of the weights to real data.
• Figure 5: Real data study for $\mu_3$, the polynomially decreasing excitation function. Comparison of temporal evolution of $N_{t}$ and the histograms for $N_{10080}$ with different choices of the weights to real data.

Theorems & Definitions (21)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.5
• proof
• Remark 2.6
• Theorem 2.7
• proof
• Theorem 2.8
• proof