Data-driven modeling of multivariate stochastic trajectories -- Application to water waves

A data-driven methodology is proposed to model the distribution of multivariate stochastic trajectories from an observed sample. As a first step, each trajectory in the sample is reduced to a vector of features by means of Functional Principal Component Analysis. Next, the joint distribution of features is modeled using (i) a non-parametric vine copula approach for the bulk of the distribution, and (ii) the conditional modeling framework of Heffernan and Tawn (2004) for the multivariate tail. The method is applied to the modeling of water waves. The dataset used is the DeRisk database, which consists of numerical simulations of water waves. The analysis is restricted to the portion of the wave period between the free-surface zero-upcrossing and the wave crest. The kinematic variables considered are the free-surface slope, the normal component of the fluid velocity at the free surface, and the vertical Lagrangian acceleration of the fluid at the free surface. The stochastic trajectories of these three variables are modeled jointly. The vertical Lagrangian acceleration of the fluid is employed to enforce a wave-breaking filter in the stochastic model. The number of hyperparameters in the stochastic framework is reduced to three, and a stepwise calibration strategy is proposed for their adjustment. The capabilities of the model are illustrated by predicting the distributions of selected response variables and by generating synthetic trajectories.