Enforcing tail calibration when training probabilistic forecast models

Probabilistic forecasts are typically obtained using state-of-the-art statistical and machine learning models, with model parameters estimated by optimizing a proper scoring rule over a set of training data. If the model class is not correctly specified, then the learned model will not necessarily issue forecasts that are calibrated. Calibrated forecasts allow users to appropriately balance risks in decision making, and it is particularly important that forecast models issue calibrated predictions for extreme events, since such outcomes often generate large socio-economic impacts. In this work, we study how the loss function used to train probabilistic forecast models can be adapted to improve the reliability of forecasts made for extreme events. We investigate loss functions based on weighted scoring rules, and additionally propose regularizing loss functions using a measure of tail miscalibration. We apply these approaches to a hierarchy of increasingly flexible forecast models for UK wind speeds, including simple parametric models, distributional regression networks, and conditional generative models. We demonstrate that state-of-the-art models do not issue calibrated forecasts for extreme wind speeds, and that the calibration of forecasts for extreme events can be improved by suitable adaptations to the loss function during model training. This, however, introduces a trade-off between calibrated forecasts for extreme events and calibrated forecasts for more common outcomes.

A comparison of generative deep learning methods for multivariate angular simulation

With the recent development of new geometric and angular-radial frameworks for multivariate extremes, reliably simulating from angular variables in moderate-to-high dimensions is of increasing importance. Empirical approaches have the benefit of simplicity, and work reasonably well in low dimensions, but as the number of variables increases, they can lack the required flexibility and scalability. Classical parametric models for angular variables, such as the von Mises-Fisher (vMF) distribution, provide an alternative. Exploiting mixtures of vMF distributions increases their flexibility, but there are cases where even this is not sufficient to capture the intricate features that can arise in data. Owing to their flexibility, generative deep learning methods are able to capture complex data structures; they therefore have the potential to be useful in the simulation of angular variables. In this paper, we explore a range of deep learning approaches for this task, including generative adversarial networks, normalizing flows and flow matching. We assess their performance via a range of metrics and make comparisons to the more classical approach of using a mixture of vMF distributions. The methods are also applied to a metocean data set, demonstrating their applicability to real-world, complex data structures.

Improving probabilistic forecasts of extreme wind speeds by training statistical post-processing models with weighted scoring rules

Accurate forecasts of extreme wind speeds are of high importance for many applications. Such forecasts are usually generated by ensembles of numerical weather prediction (NWP) models, which however can be biased and have errors in dispersion, thus necessitating the application of statistical post-processing techniques. In this work we aim to improve statistical post-processing models for probabilistic predictions of extreme wind speeds. We do this by adjusting the training procedure used to fit ensemble model output statistics (EMOS) models - a commonly applied post-processing technique - and propose estimating parameters using the so-called threshold-weighted continuous ranked probability score (twCRPS), a proper scoring rule that places special emphasis on predictions over a threshold. We show that training using the twCRPS leads to improved extreme event performance of post-processing models for a variety of thresholds. We find a distribution body-tail trade-off where improved performance for probabilistic predictions of extreme events comes with worse performance for predictions of the distribution body. However, we introduce strategies to mitigate this trade-off based on weighted training and linear pooling. Finally, we consider some synthetic experiments to explain the training impact of the twCRPS and derive closed-form expressions of the twCRPS for a number of distributions, giving the first such collection in the literature. The results will enable researchers and practitioners alike to improve the performance of probabilistic forecasting models for extremes and other events of interest.