Post Encroachment Time (PET)

• Post Encroachment Time (PET) is defined as the time gap between a leading user clearing a conflict point and a lagging user arriving, with near-zero values indicating near-misses.
• Measurement methodologies include UAV video analysis, multi-camera pixel-level computation, and deep learning-based predictive models to provide high-resolution, real-time safety monitoring.
• Statistical frameworks such as extreme value theory and Markov chains are applied to model PET distributions, optimize signal timing, and enhance proactive traffic intervention.

Post Encroachment Time (PET) is a microscopic surrogate safety measure quantifying the temporal separation between two road users traversing the same spatial conflict point in sequence. Unlike crash-based studies limited by data sparsity and latency, PET enables high-resolution, proactive traffic conflict analysis, supporting real-time monitoring and intervention for both vehicular and pedestrian interactions at intersections. Its operational definition, statistical treatment, and computational extraction frameworks have evolved to leverage computer vision, advanced statistical modeling, and machine learning, establishing PET as a central measure in contemporary intersection safety research (Hewett et al., 2023, Islam et al., 2022, Chaudhuri et al., 15 Nov 2025, Lin et al., 24 Apr 2024).

1. Formal Definition and Variants

PET is mathematically defined as the time difference between a leading road user clearing a conflict location and the subsequent arrival of a lagging user at the same location. If $t_1$ denotes the time at which the first user exits the conflict point and $t_2$ denotes the entry time of the second user, then

$\mathrm{PET} = t_2 - t_1$

Variants are distinguished for heterogeneous interactions. For a pedestrian–vehicle conflict, notation adapts based on clearance order:

$$\begin{aligned} \mathrm{PET}_{\mathrm{PF}} &= T_{V,\mathrm{enter}} - T_{P,\mathrm{leave}} \ \mathrm{PET}_{\mathrm{VF}} &= T_{P,\mathrm{enter}} - T_{V,\mathrm{leave}} \end{aligned}$$

where $T_{P,\cdot}$ and $T_{V,\cdot}$ mark pedestrian and vehicle entry/exit epochs, respectively. PET close to zero indicates a near-miss; PET = 0 corresponds to an actual collision. This orientation is fundamental for designing surrogate safety models and calibrating thresholds for risk stratification (Hewett et al., 2023, Lin et al., 24 Apr 2024).

2. Measurement Methodologies

2.1 Manual and UAV-Based Extraction

Classical studies computed PET from trajectory data or video annotation. Modern approaches employ UAVs with high-frame-rate recording (e.g., 30 fps), where vehicle bounding boxes are detected (Mask-RCNN, YOLO), trajectories smoothed, and sequence-aligned. Detection of the lagging user’s bounding box overlapping the last location of the leading user determines $t_2$ and $t_1$, respectively. Only PET intervals below a risk threshold (e.g., 5 s, 1 s) are typically retained for conflict severity analysis (Islam et al., 2022).

2.2 Multi-Camera Pixel-Level Computation

A contemporary advance replaces fixed spatial cells with a pixel-level “stopwatch” method. In this approach, a timer $s_{i,j}(t)$ associated with pixel $(i, j)$ resets to zero upon occupation and accumulates $\Delta t$ otherwise. Each occupancy transition yields a PET sample:

$$s_{i,j}(t) = \begin{cases} 0 & \text{if pixel occupied at } t\ s_{i,j}(t-\Delta t) + \Delta t & \text{else} \end{cases}$$

The empirical mean PET per pixel is then

$$\mathrm{PET}_{i, j} = \frac{1}{N} \sum_{k=1}^N t_{i, j}^{(k)}$$

This formulation—deployable at 3.3 cm² resolution in real time—enables precise spatial hazard mapping without discretization artifacts, supporting heatmaps and longitudinal monitoring (Chaudhuri et al., 15 Nov 2025).

2.3 Predictive PET (P-PET)

For proactive safety intervention, Predicted Post-Encroachment Time (P-PET) replaces observed crossing times with predicted arrival/leaving times, computed using deep learning trajectory prediction (e.g., LSTM/GRU/Transformer). P-PET thus anticipates future conflicts:

$$\begin{aligned} \mathrm{P\!-\!PET}_{\mathrm{PF}} &= \hat T_{P,\mathrm{leave}} - \hat T_{V,\mathrm{enter}} \ \mathrm{P\!-\!PET}_{\mathrm{VF}} &= \hat T_{V,\mathrm{leave}} - \hat T_{P,\mathrm{enter}} \end{aligned}$$

This predictive extension enables real-time, personalized risk evaluation before conflict materializes (Lin et al., 24 Apr 2024).

3. Statistical and Modeling Frameworks

3.1 Extreme Value Theory for PET Distribution Tails

PET distributions are heavily skewed toward larger safe intervals, with only the lower tail comprising risk-relevant near-misses. Extreme value theory (EVT), particularly the peaks-over-threshold (POT) method, is employed to efficiently model this tail. After negating PETs ($Y_i = -\mathrm{PET}_i$) for compatibility with software and conventions, threshold $u$ selection via mean residual life plots and threshold-stability assessments ensures that the excesses $y_i = Y_i - u$ conform to a Generalized Pareto Distribution (GPD):

$$f(y; \sigma, \xi) = \frac{1}{\sigma}\left(1 + \xi \frac{y}{\sigma}\right)^{-1/\xi - 1},\quad y > 0,\quad 1 + \xi \frac{y}{\sigma} > 0$$

Parameters: $\sigma$ (scale—tail spread), $\xi$ (shape—tail heaviness).

Bayesian and frequentist paradigms are both used; the Bayesian approach (uninformative Gaussian priors, MCMC inference) captures full posterior uncertainty (Hewett et al., 2023).

3.2 Adjusting for Temporal Dependence

Successive PET exceedances are often serially dependent due to traffic flow. Two strategies are established:

• Declustering: Clusters exceedances within a run parameter $\kappa$, retaining only cluster maxima. Removes dependency but can discard >99% of data if exceedance density is high.
• Markov Chain Modeling: Models $Y_1, \ldots, Y_n$ as a first-order extreme-value Markov chain; serial dependence is parameterized by $\alpha \in (0, 1]$. The joint likelihood incorporates the dependence structure, enabling use of the full exceedance set while accounting for temporal clustering.

The Markov approach provides efficiency and sharper, but realistic, credible intervals when inferring treatment effects (Hewett et al., 2023).

3.3 Incorporating Covariates

Effectiveness of safety interventions (e.g., Leading Pedestrian Interval, LPI) is captured by modeling the GPD scale as

$\sigma_i = \exp(\beta_0 + \beta_1 T_i)$

where $T_i=0$ for pre-treatment (“before”), $T_i=1$ for post-treatment (“after”). Significant negative $\beta_1$ values (in the negated PET model) indicate increased, and thus safer, PETs following intervention (Hewett et al., 2023).

Advanced logit modeling and random parameters capture PET’s dependence on dynamic signal phasing and driver heterogeneity (Islam et al., 2022).

4. Applications in Traffic Engineering

4.1 Signal Timing Optimization

Empirical modeling demonstrates direct relationships between specific signal timings and PET outcomes. Increased yellow time (by 1 s) yields a 16% improvement in PET odds (β ≈ 0.151, p < 0.001). Similarly, extending red clearance by 1 s adds ≈3% PET improvement, while longer all‐red periods may counterintuitively decrease PET due to driver anticipation and movement clustering (Islam et al., 2022). PET analysis thus underpins data-driven signal retiming:

Parameter	Effect per +1s change	Odds ratio	Significance
Yellow time	+16% PET	1.163	p < 0.001
Red clearance	+3% PET	1.033	p ≈ 0.15 (agg.), higher in phases
All-red	–43% PET	0.572	Significant

Intersection phases with systematic low PET values (particularly protected left turns) are primary candidates for adjustment (Islam et al., 2022).

4.2 Real-Time Monitoring and Hazard Mapping

Pixel-level PET heatmaps provide sub-second, centimeter-resolution visualization of spatial hazard density, supporting dynamic, automated identification of high-risk regions. Novel multi-camera pipelines using YOLOv11 segmentation, homography-based localization, and decentralized/edge computation enable real-time throughput (2.68 FPS at 3.3 cm² per pixel, ~350 ms synchronization lag) (Chaudhuri et al., 15 Nov 2025).

4.3 Pedestrian Risk Prediction

Leveraging computer vision and sequence prediction, predictive PET frameworks (P-PET) allow for real-time, personalized crossing risk categorization. Classification by pedestrian type (adult, child, cyclist) and context-specific thresholds (e.g., $\mathrm{P\!-\!PET}_{\mathrm{PF}}\in[-0.7,0.1]$ for adults) enables targeted alerts and interventions (e.g., active braking, warning signals) (Lin et al., 24 Apr 2024).

5. System Implementations and Computational Considerations

Modern systems for PET computation integrate multi-camera, deep network inference and high-frequency data synchronization. Architectures typically deploy dedicated edge inference nodes (e.g., NVIDIA Jetson AGX Xavier for each camera), a local aggregator (x86 PC for spatial fusion and SQL logging), and a central backend (multi-GPU for heatmap generation and analytics), achieving high spatial-temporal resolution with quantifiable resource utilization and latency (Chaudhuri et al., 15 Nov 2025). Real-time predictive risk frameworks, mostly based on YOLOv7 + Deep SORT + segmentation, demonstrate per-frame compute times of ≈535 ms on RTX 3090 Ti—sufficient for actionable traffic interventions (Lin et al., 24 Apr 2024).

6. Impact, Limitations, and Extensions

PET is now established as a primary surrogate safety indicator, enabling detection of near-misses orders of magnitude more frequent than reported crashes. It offers robust evidence for intervention efficacy (e.g., LPI’s impact on safety), enables data-driven engineering countermeasures, and supports scalable, real-time intersection monitoring (Hewett et al., 2023, Islam et al., 2022, Chaudhuri et al., 15 Nov 2025, Lin et al., 24 Apr 2024).

Limitations include manual calibration requirements (homography setup), edge-case handling (sparse coverage, occlusions), and computational bottlenecks (segmentation latency on CPU). Recent research points toward automation of calibration (checkerboard/SLAM), enhanced tracking (DeepSORT, object-specific segmentation), grid-free spatial representation, and deployment on edge NPUs for fully decentralized operation (Chaudhuri et al., 15 Nov 2025, Lin et al., 24 Apr 2024).

Proactive extensions, including P-PET and agent-specific risk scoring, represent a paradigm shift from reactive to anticipatory traffic safety assessment, with increasing use of explainable machine learning, adaptive thresholds, and category-specific rule sets for different road user typologies (Lin et al., 24 Apr 2024).

Study Context	PET Variant	Modality	Key Methodologies	arXiv ID
Signal timing & vehicle	PET	UAV video, Mask-RCNN	Random param. ordered logit	(Islam et al., 2022)
Pedestrian–vehicle safety	PET	Intersection sensors	EVT, Markov chain, GPD	(Hewett et al., 2023)
Intersection vision	PET (pixel)	YOLOv11, multi-camera	Stopwatch/pixel-level, heatmap	(Chaudhuri et al., 15 Nov 2025)
Pedestrian risk (predict)	P-PET	YOLOv7, LSTM/GRU/TF, CV	Deep CV, trajectory prediction	(Lin et al., 24 Apr 2024)

PET’s evolution from a manually coded, post hoc metric to a high-resolution, proactively computed safety surrogate underpins the transition to intelligent, scalable, and explainable transportation safety analytics.