Stochastic fundamental diagram modeling using asymmetric vine and nested Archimedean copulas

Traditional deterministic fundamental diagrams (FDs) often fail to capture the extensive scatter observed in empirical traffic data. Previous stochastic fundamental diagram (SFD) models typically focus only on speed-density (v-k) scatter, neglecting flow-density (q-k) scatter and their interdependence. Since the distribution of traffic density varies across different traffic states, flow distributions are determined by the joint distribution of speed and density, making linear derivation from v-k scatter inadequate. Such limitations can bias capacity estimation and traffic control under extreme conditions. To address this, we introduce a multivariate copula-based approach to extend two-dimensional SFD to three-dimensional SFD, simultaneously modeling the scatter and dependence of speed, density, and flow. Copulas separate marginal distributions from dependence structures, flexibly capturing heterogeneous scatter in empirical FDs. Vine copulas and nested Archimedean copulas are used due to their ability to model multidimensional asymmetry and tail dependence, improving scatter representation accuracy. The proposed framework includes four components: (1) abstraction of stochastic residuals from deterministic v-k and q-k diagrams to form a ternary random variable set; (2) modeling marginal distributions using normal, log-normal, and logistic distributions; (3) modeling dependence structures via nested Archimedean and Vine copulas; and (4) parameter estimation using real-world empirical datasets. Results demonstrate that the framework is applicable to various classical FDs, with the five-parameter logistic v-k model achieving the most accurate v-k reproduction and the S-shaped three-parameter (S3) model performing best for q-k and v-q relationships. The method also shows consistent performance across datasets of different sizes and temporal spans. Compared with existing SFD models, it can better capture v-k variability, particularly under low-density free-flow and high-density congested states. In practice, the proposed method enhances the robustness of traffic control decision-making under extreme conditions by providing probabilistic estimates of key traffic variables, thereby supporting more reliable and resilient traffic management.