Estimating the Joint Probability of Scenario Parameters with Gaussian Mixture Copula Models

TL;DR

This work introduces Gaussian Mixture Copula Models (GMCMs) as a novel approach to estimating the joint probability distributions of driving-scenario parameters for scenario-based safety validation of automated driving systems. GMCMs combine the multimodal expressivity of Gaussian Mixture Models with the dependency-flexibility of copulas by decoupling marginals from dependencies, enabling explicit density estimation. Evaluated on two UN R157 scenarios using roughly 18 million real-world instances, GMCMs consistently outperform Gaussian Copulas and are competitive with or superior to GMMs in terms of log-likelihood and, notably, Sinkhorn distance. The results suggest GMCMs provide a promising probabilistic foundation for robust, risk-informed scenario-based validation in automated driving, with potential for broader application and future refinements.

Abstract

This paper presents the first application of Gaussian Mixture Copula Models to the statistical modeling of driving scenarios for the safety validation of automated driving systems. Knowledge of the joint probability distribution of scenario parameters is essential for scenario-based safety assessment, where risk quantification depends on the likelihood of concrete parameter combinations. Gaussian Mixture Copula Models bring together the multimodal expressivity of Gaussian Mixture Models and the flexibility of copulas, enabling separate modeling of marginal distributions and dependencies. We benchmark Gaussian Mixture Copula Models against previously proposed approaches - Gaussian Mixture Models and Gaussian Copula Models - using real-world driving data drawn from two scenarios defined in United Nations Regulation No. 157. Our evaluation on approximately 18 million instances of these two scenarios demonstrates that Gaussian Mixture Copula Models consistently surpass Gaussian Copula Models and perform better than, or at least comparably to, Gaussian Mixture Models, as measured by both log-likelihood and Sinkhorn distance. These results are promising for the adoption of Gaussian Mixture Copula Models as a statistical foundation for future scenario-based validation frameworks.

Figures (4)

• Figure 1: Illustration of the Deceleration scenario as defined in UN Regulation No. 157 (UN R157). For a detailed explanation of the scenario and the associated parameters, see Section \ref{['sec:experiment']}. Scenarios like this motivate the need for probabilistic modeling of parameter combinations in safety validation. Figure adapted from noauthor_uniform_2021.
• Figure 2: Visualization of the transformation process in a GMCM. Starting from samples $\textbf{z}$ drawn from a GMM, two successive bijective transformations are applied independently along each dimension. The first maps each component through the cumulative distribution functions (CDFs) of the GMM marginals, followed by the application of inverse CDFs (quantile functions) of the marginal distributions previously estimated from data. Figure concept and caption adapted from Tewari et al. tewari_estimation_2023.
• Figure 3: Marginal probability density functions (PDFs) for each parameter of the Follow‐Lead‐Vehicle Emergency Braking scenario, estimated via kernel density estimation. Real‐world data distributions are overlaid as histograms.
• Figure 4: Bivariate marginal distribution of the original data and samples from GMCM, GMM, and GCM, illustrated via density heatmaps (colour scale: turbo). Visual inspection indicates that the GMCM fits the data more faithfully than the GMM and the GCM.