# Deep learning joint extremes of metocean variables using the SPAR model

arXiv: https://arxiv.org/abs/2412.15808 · HTML: https://mlsanity.com/paper/2412.15808/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 73 references that are in the map.
- First posted on arXiv 20 Dec 2024.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)
- [Neural Networks for Extreme Quantile Regression with an Application to Forecasting of Flood Risk](https://mlsanity.com/paper/2208.07590/) (2022)
- [Recent advances in directional statistics](https://mlsanity.com/paper/2005.06889/) (2020)
- [Regression modelling of spatiotemporal extreme U.S. wildfires via partially-interpretable neural networks](https://mlsanity.com/paper/2208.07581/) (2022)
- [Deep graphical regression for jointly moderate and extreme Australian wildfires](https://mlsanity.com/paper/2308.14547/) (2023)
- [Modeling of spatial extremes in environmental data science: Time to move away from max-stable processes](https://mlsanity.com/paper/2401.17430/) (2024)
- [Deep Learning of Multivariate Extremes via a Geometric Representation](https://mlsanity.com/paper/2406.19936/) (2024)
- [Insights into the drivers and spatio-temporal trends of extreme Mediterranean wildfires with statistical deep-learning](https://mlsanity.com/paper/2212.01796/) (2022)
- [Modelling multivariate extremes through angular-radial decomposition of the density function](https://mlsanity.com/paper/2310.12711/) (2023)
- [Inference for bivariate extremes via a semi-parametric angular-radial model](https://mlsanity.com/paper/2401.07259/) (2024)
- [Extreme quantile regression with deep learning](https://mlsanity.com/paper/2404.09154/) (2024)
- [A penalised piecewise-linear model for non-stationary extreme value analysis of peaks over threshold](https://mlsanity.com/paper/2201.03915/) (2022)
- [Estimating the limiting shape of bivariate scaled sample clouds: with additional benefits of self-consistent inference for existing extremal dependence properties](https://mlsanity.com/paper/2207.02626/) (2022)
- [Temporal evolution of the extreme excursions of multivariate kth order Markov processes with application to oceanographic data](https://mlsanity.com/paper/2302.14501/) (2023)
- [Estimating Metocean Environments Associated with Extreme Structural Response to Demonstrate the Dangers of Environmental Contour Methods](https://mlsanity.com/paper/2404.16775/) (2024)

## Built on by (most cited first, 2 of 2)

- [Generative Machine Learning for Multivariate Angular Simulation](https://mlsanity.com/paper/2504.21505/) (2025)
- [Extrapolation in Statistical Learning with Extreme Value Theory](https://mlsanity.com/paper/2605.01909/) (2026)

## Abstract (arXiv)

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.