Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. For statistical inference it is often assumed that there is no temporal dependence, i.e., the observations at spatial locations are independent in time. We use two representations of stationary max-stable spatial random fields and extend the concepts to the space-time domain. In a first approach, we extend the idea of constructing max-stable random fields as limits of normalized and rescaled pointwise maxima of independent Gaussian random fields, which was introduced by Kabluchko, Schlather and de Haan [2009], who construct max-stable random fields associated to a class of variograms. We use a similar approach based on a well-known result by Hüsler and Reiss and apply specific spatio-temporal covariance models for the underlying Gaussian random field, which satisfy weak regularity assumptions. Furthermore, we extend Smith's storm profile model to a space-time setting and provide explicit expressions for the bivariate distribution functions.
The tail dependence coefficient is an important measure of extremal dependence. We show how the spatio-temporal covariance function underlying the Gaussian random field can be interpreted in terms of the tail dependence coefficient. Within this context, we examine different concepts for constructing spatio-temporal covariance models and analyse several specific examples, including Gneiting's class of nonseparable stationary covariance functions.
A new method is proposed for modelling the yearly maxima of sub-daily precipitation, with the aim of producing spatial maps of return level estimates. Yearly precipitation maxima are modelled using a Bayesian hierarchical model with a latent Gaussian field, with the blended generalised extreme value (bGEV) distribution used as a substitute for the more standard generalised extreme value (GEV) distribution. Inference is made less wasteful with a novel two-step procedure that performs separate modelling of the scale parameter of the bGEV distribution using peaks over threshold data. Fast inferen…
Modelling dependencies between climate extremes is important for climate risk assessment, for instance when allocating emergency management funds. In statistics, multivariate extreme value theory is often used to model spatial extremes. However, most commonly used approaches require strong assumptions and are either too simplistic or over-parameterized. From a machine learning perspective, Generative Adversarial Networks (GANs) are a powerful tool to model dependencies in high-dimensional spaces. Yet in the standard setting, GANs do not well represent dependencies in the extremes. Here we comb…
Models for extreme values accommodating non-stationarity have been amply studied and evaluated from a parametric perspective. Whilst these models are flexible, in the sense that many parametrizations can be explored, they assume an asymptotic distribution as the proper fit to observations from the tail. This paper provides a holistic approach to the modelling of non-stationary extreme events by iterating between parametric and semi-parametric approaches, thus providing an automatic procedure to estimate a moving threshold with respect to a periodic covariate in circular data. By exploiting adv…
Regression modelling of spatiotemporal extreme U.S. wildfires via partially-interpretable neural networks
School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学计算机、电气与数学科学与工程（CEMSE）分部统计项目）
Currently available models for spatial extremes suffer either from inflexibility in the dependence structures that they can capture, lack of scalability to high dimensions, or in most cases, both of these. We present an approach to spatial extreme value theory based on the conditional multivariate extreme value model, whereby the limit theory is formed through conditioning upon the value at a particular site being extreme. The ensuing methodology allows for a flexible class of dependence structures, as well as models that can be fitted in high dimensions. To overcome issues of conditioning on …
This paper considers nonlinear dynamic models where the main parameter of interest is a nonnegative matrix characterizing the network (contagion) effects. This network matrix is usually constrained either by assuming a limited number of nonzero elements (sparsity), or by considering a reduced rank approach for nonnegative matrix factorization (NMF). We follow the latter approach and develop a new probabilistic NMF method. We introduce a new Identifying Maximum Likelihood (IML) method for consistent estimation of the identified set of admissible NMF's and derive its asymptotic distribution. Mor…
Nonstationary POT modelling of air pollution concentrations: Statistical analysis of the traffic and meteorological impact
预测超过某一临界水平的高空气污染浓度的发生、水平和持续时间，使研究人员能够研究道路交通对当地空气质量的健康影响，并为公共政策行动提供信息。由于涉及复杂的物理和化学过程，对极端浓度的发生概率和水平进行精确估计是非常困难的。这就说明有必要开发复杂的极值模型，特别是考虑到环境时间序列的非平稳性。本文用两种模式研究了氮氧化物（NO）、二氧化氮（NO $_2$）和臭氧（O $_3$）浓度的极值。模型I基于A. C. Davison和R. L. Smith，由此将潜在的广义帕累托分布（GPD）的参数视为协变量的函数（即，交通和气象因素）。新模型II通过构造GPD参数的特殊函数形式解决了Davison-Smith模型中阈值稳定性的缺乏。对于每一个模型，利用马尔可夫链蒙特卡罗方法估计了交通和气象因素对极值频率和极值大小的影响。最后，适当的拟合优度测试和模型选择标准证实，模型II显着优于模型I的估计和预测的极端。
In this paper we extend the known methodology for fitting stable distributions to the multivariate case and apply the suggested method to the modelling of daily cryptocurrency-return data. The investigated time period is cut into 10 non-overlapping sections, thus the changes can also be observed. We apply bootstrap tests for checking the models and compare our approach to the more traditional extreme-value and copula models.
National University of Singapore（新加坡国立大学）；IIT Bombay（印度理工学院孟买分校）
Mitigating the risk arising from extreme events is a fundamental goal with many applications, such as the modelling of natural disasters, financial crashes, epidemics, and many others. To manage this risk, a vital step is to be able to understand or generate a wide range of extreme scenarios. Existing approaches based on Generative Adversarial Networks (GANs) excel at generating realistic samples, but seek to generate typical samples, rather than extreme samples. Hence, in this work, we propose ExGAN, a GAN-based approach to generate realistic and extreme samples. To model the extremes of the …
Understanding multivariate extreme events play a crucial role in managing the risks of complex systems since extremes are governed by their own mechanisms. Conditional on a given variable exceeding a high threshold (e.g.\ traffic intensity), knowing which high-impact quantities (e.g\ air pollutant levels) are the most likely to be extreme in the future is key. This article investigates the contribution of marginal extreme events on future extreme events of related quantities. We propose an Extreme Event Propagation framework to maximise counterfactual causation probabilities between a known ca…
The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory and is in common use for analyzing the far tail of observed phenomena. Curiously, important asymptotic properties of likelihood-based estimation under this standard model have yet to be established. In this paper, we formally prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighborhood of the shape parameter.
Deep learning joint extremes of metocean variables using the SPAR model
University of Exeter, UK（埃克塞特大学）；Technische Universität Dresden, Germany（德累斯顿技术大学）；ScaDS.AI, Germany（数学学院和麦克斯韦数学科学研究所，爱丁堡大学）；School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, UK（兰卡斯特大学）；Lancaster University, UK
Count data are omnipresent in many applied fields, often with overdispersion due to an excess of zeroes or extreme values. With mixtures of Poisson distributions representing an elegant and appealing modelling strategy, we focus here on the challenging problem of identifying a suitable mixing distribution and study how extreme value theory can be used. We propose an original strategy to select the most appropriate candidate among three categories: Fr{é}chet, Gumbel and pseudo-Gumbel. Such an approach is presented with the aid of a decision tree and evaluated with numerical simulations.
A successful model for high-dimensional spatial extremes should, in principle, be able to describe both weakening extremal dependence at increasing levels and changes in the type of extremal dependence class as a function of the distance between locations. Furthermore, the model should allow for computationally tractable inference using inference methods that efficiently extract information from data and that are robust to model misspecification. In this paper, we demonstrate how to fulfil all these requirements by developing a comprehensive methodological workflow for efficient Bayesian model…
Although most models for rainfall extremes focus on point-wise values, it is aggregated precipitation over areas up to river catchment scale that is of the most interest. To capture the joint behaviour of precipitation aggregates evaluated at different spatial scales, parsimonious and effective models must be built with knowledge of the underlying spatial process. Precipitation is driven by a mixture of processes acting at different scales and intensities, e.g., convective and frontal, with extremes of aggregates for typical catchment sizes arising from extremes of only one of these processes,…
Recent extreme value theory literature has seen significant emphasis on the modelling of spatial extremes, with comparatively little consideration of spatio-temporal extensions. This neglects an important feature of extreme events: their evolution over time. Many existing models for the spatial case are limited by the number of locations they can handle; this impedes extension to space-time settings, where models for higher dimensions are required. Moreover, the spatio-temporal models that do exist are restrictive in terms of the range of extremal dependence types they can capture. Recently, c…
Having reliable estimates of the occurrence rates of extreme events is highly important for insurance companies, government agencies and the general public. The rarity of an extreme event is typically expressed through its return period, i.e., the expected waiting time between events of the observed size if the extreme events of the processes are independent and identically distributed. A major limitation with this measure is when an unexpectedly high number of events occur within the next few months immediately after a \textit{T} year event, with \textit{T} large. Such events undermine the tr…
Inference on the extremal behaviour of spatial aggregates of precipitation is important for quantifying river flood risk. There are two classes of previous approach, with one failing to ensure self-consistency in inference across different regions of aggregation and the other imposing highly restrictive assumptions. To overcome these issues, we propose a model for high-resolution precipitation data, from which we can simulate realistic fields and explore the behaviour of spatial aggregates. Recent developments have seen spatial extensions of the Heffernan and Tawn (2004) model for conditional …
Viruses causing flu or milder coronavirus colds are often referred to as "seasonal viruses" as they tend to subside in warmer months. In other words, meteorological conditions tend to impact the activity of viruses, and this information can be exploited for the operational management of hospitals. In this study, we use three years of daily data from one of the biggest hospitals in Switzerland and focus on modelling the extremes of hospital visits from patients showing flu-like symptoms and the number of positive cases of flu. We propose employing a discrete Generalized Pareto distribution for …
Determining the causes of extreme events is a fundamental question in many scientific fields. An important aspect when modelling multivariate extremes is the tail dependence. In application, the extreme dependence structure may significantly depend on covariates. As for the general case of modelling including covariates, only some of the covariates are causal. In this paper, we propose a methodology to discover the causal covariates explaining the tail dependence structure between two variables. The proposed methodology for discovering causal variables is based on comparing observations from d…
