Predicting probability density of extremal process from random state data

Predicting and assessing stochastic extremal processes is critical in many fields for tasks such as safety analysis, reliability-based control design, and extreme event forecasting. This paper proposes a novel data-driven method grounded in diffusion theory to predict the probability density of an extremal process directly from random state data. The approach consists of two main steps: first, a slowly varying process is identified from the random state data and approximated as a Markov diffusion process via stochastic averaging, and this process is augmented with its extremal process to form a Markov augmented vector. Second, using conditional moments from the Kramers-Moyal expansion and sparse regression algorithm, we extract the expressions for the drift and diffusion coefficients, leading to a Fokker-Planck-Kolmogorov (FPK) equation for the probability density of the Markov augmented vector. Solving the FPK equation and integrating the resultant yield the transient probability density of the extremal process. We demonstrated the method’s accuracy on three representative systems — a simple one-dimensional example, a Duffing oscillator, and a van der Pol system — each under Gaussian white noise excitation. The results show that the proposed diffusion-based data-driven approach enables effective long-term prediction of extremal process probability densities using only short-term historical state data.