Deep learning joint extremes of metocean variables using the SPAR model

TL;DR

This work advances multivariate extreme value analysis for metocean variables by adopting a Semi-Parametric Angular-Radial (SPAR) framework that decomposes a d-dimensional vector $\mathbf{X}$ into polar coordinates $(R,\mathbf{W})$ with $R=\|\mathbf{X}\|_2$ and $\mathbf{W}=\mathbf{X}/R$. It jointly models the angular density $f_{\mathbf{W}}$ on the sphere and the GP tail of $R|\mathbf{W}$ above a threshold $u(\mathbf{w})$, with angle-dependent GP parameters $(\xi(\mathbf{w}),\sigma(\mathbf{w}))$. The angular density is estimated via kernel density on the sphere using a Power Spherical kernel, while the angle-conditioned GP parameters are represented by neural networks (MLPs), enabling scalable inference in five dimensions. The framework is demonstrated on a 5D metocean hindcast dataset, showing good diagnostic agreement for both the angular and radial components, and providing a principled method for extrapolating beyond observed data to generate large sets of extreme events for robust coastal and offshore design risk assessment. Overall, the method relaxes margin and dependence assumptions and yields asymptotically justified extrapolation in high dimensions, with practical applicability to environmental contour estimation and extreme-event simulation.

Abstract

This paper presents a novel deep learning framework for estimating multivariate joint extremes of metocean variables, based on the Semi-Parametric Angular-Radial (SPAR) model. When considered in polar coordinates, the problem of modelling multivariate extremes is transformed to one of modelling an angular density, and the tail of a univariate radial variable conditioned on angle. In the SPAR approach, the tail of the radial variable is modelled using a generalised Pareto (GP) distribution, providing a natural extension of univariate extreme value theory to the multivariate setting. In this work, we show how the method can be applied in higher dimensions, using a case study for five metocean variables: wind speed, wind direction, wave height, wave period, and wave direction. The angular variable is modelled using a kernel density method, while the parameters of the GP model are approximated using fully-connected deep neural networks. Our approach provides great flexibility in the dependence structures that can be represented, together with computationally efficient routines for training the model. Furthermore, the application of the method requires fewer assumptions about the underlying distribution(s) compared to existing approaches, and an asymptotically justified means for extrapolating outside the range of observations. Using various diagnostic plots, we show that the fitted models provide a good description of the joint extremes of the metocean variables considered.

Figures (16)

• Figure 1: Optimisation of bandwidth parameter $\kappa$ for kernel density estimate of the angular distribution, in terms of the minimum predictive negative log-likelihood.
• Figure 2: Example schematic of a multi-layered perceptron (MLP) model for the GP parameters, with $L=2$ hidden layers. The inputs are the components of the angle $\mathbf{w} = (w_1,w_2,...,w_d)$ and the outputs are the GP parameter functions, $(\nu(\mathbf{w}), \xi(\mathbf{w}))$.
• Figure 3: Box-plots of GP shape parameter estimates (left) and conditional radial quantile at exceedance level $10^{-6}$ (right) at observed angles for various threshold non-exceedance probabilities, $1-\zeta$.
• Figure 4: Upper right plots: Empirical joint densities of pairs of normalised observations. Dashed red lines in plots of $(x_3,x_5)$ and $(x_4,x_5)$ are lines of constant wave steepness $s=0.08$. Plots on diagonal are histograms of each normalised variable. Lower left plots: Empirical density of pairs of angular components of observed data (yellow = high density, blue = low density). The normalised variables $(X_1 , ..., X_5)$ correspond to normalised $(U_x,U_y,H_x,H_y,\log(T_m))$.
• Figure 5: Empirical joint density of wind direction and wave direction. (Blue = low density, yellow = high density).