Temporal evolution of the extreme excursions of multivariate k k th order Markov processes with application to oceanographic data

We develop two models for the temporal evolution of extreme events of multivariate th order Markov processes. The foundation of our methodology lies in the conditional extremes model of Heffernan and Tawn (Journal of the Royal Statistical Society: Series B (Methodology), 2014, 66, 497–546), and it naturally extends the work of Winter and Tawn (Journal of the Royal Statistical Society: Series C (Applied Statistics), 2016, 65, 345–365; Extremes, 2017, 20, 393–415) and Tendijck et al. (Environmetrics 2019, 30, e2541) to include multivariate random variables. We use cross‑validation‑type techniques to develop a model order selection procedure, and we test our models on two‑dimensional meteorological‑oceanographic data with directional covariates for a location in the northern North Sea. We conclude that the newly‑developed models perform better than the widely used historical matching methodology for these data.