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

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 scenarios defined in United Nations Regulation No. 157. Our evaluation across 18 million scenario instances demonstrates that Gaussian Mixture Copula Models provide a better fit to the data in terms of both likelihood and Sinkhorn distance. These results suggest that Gaussian Mixture Copula Models are a compelling foundation for future scenario-based validation frameworks.