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

TL;DR

This paper introduces Gaussian Mixture Copula Models for modeling joint distributions of driving scenario parameters, improving accuracy over previous methods and aiding automated driving safety validation.

Contribution

It is the first application of Gaussian Mixture Copula Models to driving scenario modeling, combining multimodal expressivity and flexible dependence modeling.

Findings

Gaussian Mixture Copula Models outperform Gaussian Copula Models in log-likelihood and Sinkhorn distance.

They perform comparably to Gaussian Mixture Models, with scenario-dependent relative performance.

Results support using Gaussian Mixture Copula Models for scenario-based safety validation.

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 dependence. 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 competitively with Gaussian Mixture Models, as measured by both log-likelihood and Sinkhorn distance, with relative performance depending on the scenario. The results are promising for the adoption of Gaussian Mixture Copula Models as a statistical foundation for future scenario-based validation frameworks.