Miguel de Carvalho, Manuele Leonelli, Alex Rossi
In this paper we devise a statistical method for tracking and modeling change-points on the dependence structure of multivariate extremes. The methods are motivated by and illustrated on a case study on crypto-assets.
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Multivariate Poisson processes have many important applications in Insurance, Finance, and many other areas of Applied Probability. In this paper we study the backward simulation approach to modelling multivariate Poisson processes and analyze the connection to the extreme measures describing the joint distribution of the processes at the terminal simulation time.
The main approach to inference for multivariate extremes consists in approximating the joint upper tail of the observations by a parametric family arising in the limit for extreme events. The latter may be expressed in terms of componentwise maxima, high threshold exceedances or point processes, yielding different but related asymptotic characterizations and estimators. The present paper clarifies the connections between the main likelihood estimators, and assesses their practical performance. We investigate their ability to estimate the extremal dependence structure and to predict future extr…
Multivariate extreme-value analysis is concerned with the extremes in a multivariate random sample, that is, points of which at least some components have exceptionally large values. Mathematical theory suggests the use of max-stable models for univariate and multivariate extremes. A comprehensive account is given of the various ways in which max-stable models are described. Furthermore, a construction device is proposed for generating parametric families of max-stable distributions. Although the device is not new, its role as a model generator seems not yet to have been fully exploited.
We propose in this work a new method for estimating the main mode of multivariate distributions, with application to eye-tracking calibrations. When performing eye-tracking experiments with poorly cooperative subjects, such as infants or monkeys, the calibration data generally suffer from high contamination. Outliers are typically organized in clusters, corresponding to the time intervals when subjects were not looking at the calibration points. In this type of multimodal distributions, most central tendency measures fail at estimating the principal fixation coordinates (the first mode), resul…
Non-Parametric Simulation of Multivariate Extreme Events via Spectral Bootstrap
本文提出一种非参数模拟方法，通过多变量广义极值分布的频谱表示，生成额外的极值数据以提升极值推断的可靠性，适用于高维极值场景的风险评估。
We consider regularly varying random vectors. Our goal is to estimate in a non-parametric way some characteristics related to conditioning on an extreme event, like the tail dependence coefficient. We introduce a quasi-spectral decomposition that allow to improve efficiency of estimators. Asymptotic normality of estimators is based on weak convergence of tail empirical processes. Theoretical results are supported by simulation studies.
Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. (2016, Ann. Appl. Stat., to appear) use a Bayesian approach to fit a Brown--Resnick process to extreme tempe…
An utopic adventure in the modelling of conditional univariate and multivariate extremes
本文针对条件单变量和多变量极值建模中的四个挑战，提出新的交叉验证方法、验证指标及联合超出行概率公式，探讨模型验证与自定义估计策略。
Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-α- i β}$ for $α> 0$ and $β> 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-β/α})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/α}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $α< 1$.
The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable …
In environmental sciences, it is often of interest to assess whether the dependence between extreme measurements has changed during the observation period. The aim of this work is to propose a statistical test that is particularly sensitive to such changes. The resulting procedure is also extended to allow the detection of changes in the extreme-value dependence under the presence of known breaks in the marginal distributions. Simulations are carried out to study the finite-sample behavior of both versions of the proposed test. Illustrations on hydrological data sets conclude the work.
Generative modelling of multivariate geometric extremes using normalising flows
本文提出基于归一化流和超球面几何方法的多元极值生成建模框架，实现全方向联合尾部外推和任意风险区域快速概率估计，并在风速极值与风电相关极端事件分析中验证其有效性。
A Bayesian multivariate extreme value mixture model
本文提出一种多元极值混合模型，用于评估自然灾害的影响，结合了尾部和主体的建模方法，并通过贝叶斯推断处理阈值选择的不确定性。
Modeling extremal dependence in multivariate and spatial problems: a practical perspective
本文从环境科学到金融领域，探讨了如何通过ExtremalDep包评估超出观测数据的极值风险，提供多变量和空间极值分析的实用方法和工具。
Parsimonious Factor Models for Asymmetric Dependence in Multivariate Extremes
本文提出了一种稀疏因子模型，用于捕捉多变量极值中的非对称依赖结构，通过极值理论推导并验证了其在实际数据中的有效性。
Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue we develop a novel statistical algorithm that describes extremal dependence taking advantage of the inherent hierarchical dependence structure of the max-stable nested logistic distribution and that identifies possible clusters of extreme variables using reversible jump Markov chain Monte Carlo techniques. Parsimonious representations are achieved when clusters of extreme variables are found to be completely independent. Moreover, we significantly decrease the computational comple…
Starting from the characterization of extreme-value copulas based on max-stability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data and a multiplier technique for obtaining approximate p-values for the derived statistics. The asymptotic validity of the multiplier approach is established, and the finite-sample performance of a large number of candidate test statistics is studied through extensive Monte Carlo experiments for data sets of dimension two to five. In the bivariate …
Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estima…
Principal Component Analysis for Multivariate Extremes
提出一种针对多元极值数据的降维方法，通过主成分分析保留极值相关信息，解决高维极值分析中的维度灾难问题。
Numerical climate models are complex and combine a large number of physical processes. They are key tools in quantifying the relative contribution of potential anthropogenic causes (e.g., the current increase in greenhouse gases) on high impact atmospheric variables like heavy rainfall. These so-called climate extreme event attribution problems are particularly challenging in a multivariate context, that is, when the atmospheric variables are measured on a possibly high-dimensional grid. In this paper, we leverage two statistical theories to assess causality in the context of multivariate extr…
We develop two models for the temporal evolution of extreme events of multivariate $k$th order Markov processes. The foundation of our methodology lies in the conditional extremes model of Heffernan & Tawn (2004), and it naturally extends the work of Winter & Tawn (2016,2017) and Tendijck et al. (2019) 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 n…
Multivariate Discrete Generalized Pareto Distributions: Theory, Simulation, and Applications to Dry spells
本文提出多元离散广义帕累托分布（MDGPD），扩展极值理论至离散数据，结合似然自由推断方法，用于干旱期等极端事件建模。
The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly exploring what happens when extremes of …
Causal tail coefficient for compound extremes in multivariate time series
本文提出复合因果尾部系数及bootstrap检验框架，用于检测多元时间序列中复合极端事件之间的因果关系，可处理非线性与潜在混杂变量，并通过理论、模拟和空间天气数据验证其有效性。
In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies with determining which subsets of variables can take their largest values simultaneously, while the others are of smaller order. Our approach to this problem exploits hidden regular variation properties on a collection of non-standard cones and provides a new set of indices that reveal aspects of the extremal dependence structure not available through existing measures of dependence. We derive theoretical propert…
Extreme value statistics provides accurate estimates for the small occurrence probabilities of rare events. While theory and statistical tools for univariate extremes are well-developed, methods for high-dimensional and complex data sets are still scarce. Appropriate notions of sparsity and connections to other fields such as machine learning, graphical models and high-dimensional statistics have only recently been established. This article reviews the new domain of research concerned with the detection and modeling of sparse patterns in rare events. We first describe the different forms of ex…
Statistical inference for multivariate extremes via a geometric approach
本文提出基于极限集形状的多元极值几何统计建模与参数估计方法，可外推联合尾部并估计小概率，模拟和环境数据应用均验证其竞争力与良好拟合。
Hypothesis testing for partial tail correlation in multivariate extremes
本文提出了一种用于多变量极值中部分尾相关性检验的方法，通过尾部回归和极值协方差矩阵分析，揭示极值依赖结构中的稀疏性。
The notion of expectiles, originally introduced in the context of testing for homoscedasticity and conditional symmetry of the error distribution in linear regression, induces a law-invariant, coherent and elicitable risk measure that has received a significant amount of attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk management. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference of a …
Modeling Extreme Events: Univariate and Multivariate Data-Driven Approaches
本文总结团队genEVA在EVA 2023数据挑战赛中的工作，针对单变量与多变量极值任务，分别采用神经网络分位数回归、损失函数微调、copula模型及聚类降维方法，在单变量任务中取得了最准确的分位数估计。
Joint modeling of low and high extremes using a multivariate extended generalized Pareto distribution
本研究提出多元扩展广义帕累托分布，联合建模低高极端值，保留单变量特性且上下尾符合极值理论，并用神经网络快速估计，在荷兰降雨数据上验证有效。
Testing the Regular Variation Model for Multivariate Extremes with Flexible Circular and Spherical Distributions
本文提出基于非负三角和的圆形与球形分布作为谱测度的灵活概率模型，用于多元极值正则变化模型，并应用统计检验推断极值依赖结构，在金融数据上验证。
The joint extremal spatial dependence of wind speed and significant wave height in the North East Atlantic is quantified using Metop satellite scatterometer and hindcast observations for the period 2007-2018, and a multivariate spatial conditional extremes (MSCE) model, ultimately motivated by the work of Heffernan and Tawn (2004). The analysis involves (a) registering individual satellite swaths and corresponding hindcast data onto a template transect (running approximately north-east to south-west, between the British Isles and Iceland), (b) non-stationary directional-seasonal marginal extre…
We revisit multivariate extreme value theory modeling by emphasizing multivariate regular variations and the multivariate Breiman Lemma. This allows us to recover in a simple framework the most popular multivariate extreme value distributions, such as the logistic, negative logistic, Dirichlet, extremal-$t$ and Hüsler-Reiss models. In a second part of the paper, we focus on the Hüsler-Reiss Pareto model and its surprising exponential family property. After a thorough study of this exponential family structure, we focus on maximum likelihood estimation. We also consider the generalized Hüsler-R…
We propose kernel PCA as a method for analyzing the dependence structure of multivariate extremes and demonstrate that it can be a powerful tool for clustering and dimension reduction. Our work provides some theoretical insight into the preimages obtained by kernel PCA, demonstrating that under certain conditions they can effectively identify clusters in the data. We build on these new insights to characterize rigorously the performance of kernel PCA based on an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. More specifically, we focus…
Wasserstein-based Minimax Estimation of Dependence in Multivariate Regularly Varying Extremes
本文首次为多元线性因子模型中谱测度估计建立极小极大风险界，以Wasserstein距离导出非渐近收敛速率并证明最优性，提出两步估计量达到下界，弥合极值理论中偏差-方差权衡的空白。
Causal Discovery in Multivariate Extremes with a Hydrological Analysis of Swiss River Discharges
本文提出一种基于最大吸引域和广义帕累托向量表示的模型无关因果得分，通过Wasserstein距离比较方向，用于多元极值因果发现，并在瑞士水文数据上验证。
Assessing Extreme Risk using Stochastic Simulation of Extremes
本文提出基于多变量极值理论的非参数模拟算法，用于评估极端事件风险，通过随机模拟和广义极值分布提升极值风险估计的准确性。
Modelling multivariate extreme value distributions via Markov trees
针对高维多元极值建模难题，提出将二元 max-stable 分布组合成马尔可夫树，利用 Prim 算法学习树结构，实现对未知 max-stable 分布的近似及罕见事件概率推断。
In many applications of time series models, such as climate analysis and social media analysis, we are often interested in extreme events, such as heatwave, wind gust, and burst of topics. These time series data usually exhibit a heavy-tailed distribution rather than a Gaussian distribution. This poses great challenges to existing approaches due to the significantly different assumptions on the data distributions and the lack of sufficient past data on extreme events. In this paper, we propose the Sparse-GEV model, a latent state model based on the theory of extreme value modeling to automatic…
Piecewise-linear modeling of multivariate geometric extremes
本文提出一种分段线性半参数规范函数，用于多元几何极值建模，兼具灵活性与计算效率，并扩展了高径向分位数估计工具，在空气污染数据上验证了其对复杂极值依赖结构的建模能力。
A Proxy-likelihood Estimator for Multivariate Extremes Models with Intractable Likelihoods
针对似然不可处理的多元极值模型，提出基于尾部成对依赖性的代理似然估计器，采用Hüsler-Reiss分布，在TLETS模型上优于现有方法，并应用于野火天气数据。
Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead…
Conditional Extremes with Graphical Models
研究多元极端事件概率和幅度，针对经典模型计算成本高及CMEVM半参数推理和高维预测差的问题，提出CMEVM扩展模型及高效推理程序，模拟研究和实际应用验证了模型灵活性和有效性。
Bayesian model averaging of risk set probabilities using a geometric representation of multivariate extremes
本研究针对多元极值建模，提出结合删失似然的贝叶斯模型平均方法，在小样本下提升尾风险概率预测性能，并通过火灾气象指数验证其环境应用价值。
We develop a novel multi-factor copula model for multivariate spatial extremes, which is designed to capture the different combinations of marginal and cross-extremal dependence structures within and across different spatial random fields. Our proposed model, which can be seen as a multi-factor copula model, can capture all possible distinct combinations of extremal dependence structures within each individual spatial process while allowing flexible cross-process extremal dependence structures for both upper and lower tails. We show how to perform Bayesian inference for the proposed model usin…
Modelling multivariate extremes through angular-radial decomposition of the density function
提出一种基于密度函数角-径向分解的SPAR模型，用于多元极值建模，该模型将极值建模转化为角度密度和条件径向尾部建模，并证明其比现有方法更通用，且涵盖多种常用方法。
There are many data sources available that report related variables of interest that are also referenced over geographic regions and time; however, there are relatively few general statistical methods that one can readily use that incorporate these multivariate-spatio-temporal dependencies. As such, we introduce the multivariate-spatio-temporal mixed effects model (MSTM) to analyze areal data with multivariate-spatio-temporal dependencies. The proposed MSTM extends the notion of Moran's I basis functions to the multivariate-spatio-temporal setting. This extension leads to several methodologica…
Deep Learning of Multivariate Extremes via a Geometric Representation
针对现有几何极值推断方法仅适用于低维、假设严格的问题，本文提出新理论结果并首次用深度学习建模极限集，构建灵活的高维极值依赖半参数模型，在北海气象海洋数据上验证了有效性。
In a wide variety of situations, anomalies in the behaviour of a complex system, whose health is monitored through the observation of a random vector X = (X1,. .. , X d) valued in R d , correspond to the simultaneous occurrence of extreme values for certain subgroups $α$ $\subset$ {1,. .. , d} of variables Xj. Under the heavy-tail assumption, which is precisely appropriate for modeling these phenomena, statistical methods relying on multivariate extreme value theory have been developed in the past few years for identifying such events/subgroups. This paper exploits this approach much further b…
Regression modeling of multivariate precipitation extremes under regular variation
本文提出了一种基于正则变化的回归方法，用于估计多变量降水极值的发生和强度，通过简化模型和高效计算在EVA2025挑战中取得第二名。
On estimation and order selection for multivariate extremes via clustering
本文提出带惩罚的简化平均轮廓宽度来一致选择多元极值谱测度的聚类阶，并分析聚类估计的大偏差性质及重尾因子模型参数估计。
This paper studies a basic notion of distributional shape known as orthounimodality (OU) and its use in shape-constrained distributionally robust optimization (DRO). As a key motivation, we argue how such type of DRO is well-suited to tackle multivariate extreme event estimation by giving statistically valid confidence bounds on target extremal probabilities. In particular, we explain how DRO can be used as a nonparametric alternative to conventional extreme value theory that extrapolates tails based on theoretical limiting distributions, which could face challenges in bias-variance control an…
Total positivity in multivariate extremes
本研究针对多元极值中的正相关性问题，提出极值版二阶多元全正性EMTP₂，证明Hüsler–Reiss分布满足该性质的充要条件，给出带拉普拉斯约束的凸优化估计量，验证其一致性与稀疏性，在多瑙河流量数据上表现优于现有方法。
Graphical models for multivariate extremes
本文综述极值图模型的基本概念与最新进展，统一了指数测度视角，涵盖非参数与Hüsler--Reiss模型，并应用于美国机场航班延误数据分析。
A Dimension-Reduced Multivariate Spatial Model for Extreme Events: Balancing Flexibility and Scalability
本文提出了一种降维的多变量空间模型，用于更灵活且可扩展地建模空间极端事件，适用于环境和跨学科研究。
Directional variograms for multivariate extremes
研究多元广义帕累托分布，通过将多元广义帕累托随机向量Y限制在任意半空间来推广现有推断方法，引入v-变差函数，建立分解关系，得出相关随机变量结果，并在模拟研究中展示了方向选择对偏差-方差权衡的影响。
Full likelihood-based inference for high-dimensional multivariate extreme value distributions, or max-stable processes, is feasible when incorporating occurrence times of the maxima; without this information, $d$-dimensional likelihood inference is usually precluded due to the large number of terms in the likelihood. However, some studies have noted bias when performing high-dimensional inference that incorporates such event information, particularly when dependence is weak. We elucidate this phenomenon, showing that for unbiased inference in moderate dimensions, dimension $d$ should be of a m…
The estimation of extreme flood quantiles is challenging due to the relative scarcity of extreme data compared to typical target return periods. Several approaches have been developed over the years to face this challenge, including regional estimation and the use of historical flood data. This paper investigates the combination of both approaches using a multivariate peaks-over-threshold model, that allows estimating altogether the intersite dependence structure and the marginal distributions at each site. The joint distribution of extremes at several sites is constructed using a semi-paramet…
Clustering of multivariate tail dependence using conditional methods
针对多元极值聚类难解释比较的问题，提出基于条件极值框架的新方法，利用偏几何詹森 - 香农散度构建差异度量，经模拟研究和实际应用验证，该方法优于现有方法且能扩展到多元场景。
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