arXiv subjects

Sebastian Engelke

Publications and source records attributed to Sebastian Engelke.

At least 19 recordsLinked to original sources

arXiv2026

An Explicit Link between Extreme Value Theory and Compositional Data Analysis

Extreme value theory and compositional data analysis both study settings where relative information plays a central role. In multivariate extreme value theory, threshold exceedance limits satisfy homogeneity properties that separate the radial size of an extreme event from its relative profile. In compositional data analysis, positive vectors are analysed up to multiplicative scale, and inference is based on ratios or log-ratios between components. Consequently, both fields have developed several covariance and dependence representations of the underlying relative structure. In the H\"usler-Reiss model for extremes, these include variogram, covariance, and precision parametrizations. In compositional data analysis, analogous representations arise from pairwise log-ratios, centred log-ratios, and additive log-ratios. We establish an explicit link between the two fields that relates these different representations by a small set of simple transformations, including oblique projections, H\"usler-Reiss inverses, and the variogram map. From a methodological perspective, leveraging this algebraic connection enables the transfer of statistical approaches from one field to the other. For instance, we introduce intrinsic logistic-normal graphical models for compositional data, which are based on H\"usler-Reiss graphical models for extremes. Conversely, we explore how dimensionality reduction methods from compositional data analysis can be applied to the analysis of multivariate extremes.

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arXiv2026

Extrapolation of extreme covariates in generalized additive regression using extreme-value theory

We propose methods to enhance the predictive performance of generalized additive models (GAMs) in the context of covariate extrapolation, where predictions rely on covariates beyond their observed range. When using predictive models such as GAMs, shifts in the covariate distribution between training and prediction datasets can occur. Ignoring this issue may lead to inaccurate predictions in the tail of the covariate distributions. For example, this problem is particularly critical in climate-change scenarios, where covariates simulated from future climate scenarios are likely to contain more extreme conditions. Our approach integrates GAMs for the bulk of covariate distributions with asymptotic models from multivariate extreme-value theory at high covariate values. We consider binary responses based on a latent variable assumption, and also continuous responses. For large values of the covariates, on a specific marginal scale motivated by extreme-value theory the latent variable or continuous response is assumed to depend linearly on the covariates with an additive error term, when using an appropriate link function. In an application to wildfires in Europe, we explore how the new method can improve predictions, using environmental and meteorological covariates.

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arXiv2026

Directional variograms for multivariate extremes

Multivariate generalized Pareto distributions arise as limits of threshold exceedances and form a central model class for multivariate extremes. Existing inference methods based on the extremal variogram condition on the value of a single component, which can be statistically suboptimal. We generalize this approach by conditioning the multivariate generalized Pareto random vector $Y$ to lie on arbitrary half-spaces. Specifically, for a direction vector $v$, we introduce the random vector $Y^v = (Y \mid v^\top Y > 0)$ and define the associated $v$-variogram $\Gamma_{ij}^v=\mathrm{Var}(Y_i^v-Y_j^v)$. We establish the decomposition $Y^v \stackrel{d}{=} W^v+E\mathbf{1}$ into the so-called $v$-extremal function $W^v$ and an independent exponential random variable $E$, and derive several results relating these random variables to each other. For logistic, Dirichlet, and H\"usler-Reiss multivariate generalized Pareto models, we derive closed-form expressions for $\Gamma^v$. In the H\"usler-Reiss case, we further derive new density representations and identify a distinguished resistance-curvature vector $v_0$ that uniquely centers the Gaussian law of $W^{v_0}$ while characterizing the least-mass half-space. On the statistical side, we introduce empirical $v$-variograms and show in a simulation study that the choice of $v$ induces a pronounced bias-variance trade-off that is strongly related to the mass of the conditioning half-space. Moreover, combining information across multiple directions $v$ can substantially reduce estimation variance relative to methods based on a single vector.

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arXiv2026

Extrapolation in Statistical Learning with Extreme Value Theory

Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from these advances, including regression and classification beyond the training data, extreme quantile regression, supervised and unsupervised dimension reduction, generative artificial intelligence and anomaly detection. This review synthesizes recent developments in these fields at the intersection of statistical learning and extreme value theory, with a focus on principled methods based on asymptotically motivated representations of the tail of univariate and multivariate distributions. We consider different theoretical frameworks for both asymptotically dependent and independent data and discuss how they translate into efficient statistical methods for extrapolation to extreme regions. By addressing both theoretical and practical aspects, we offer a comprehensive overview of the state-of-the-art in this quickly evolving field, and identify promising directions for future research.

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arXiv2026

A Kullback-Leibler divergence test for multivariate extremes: theory and practice

Testing whether two multivariate samples exhibit the same extremal behavior is an important problem in various fields including environmental and climate sciences. While several ad-hoc approaches exist in the literature, they often lack theoretical justification and statistical guarantees. On the other hand, extreme value theory provides the theoretical foundation for constructing asymptotically justified tests. We combine this theory with Kullback-Leibler divergence, a fundamental concept in information theory and statistics, to propose a test for equality of extremal dependence structures in practically relevant directions. Under suitable assumptions, we derive the limiting distributions of the proposed statistic under null and alternative hypotheses. Importantly, our test is fast to compute and easy to interpret by practitioners, making it attractive in applications. Simulations provide evidence of the power of our test. In a case study, we apply our method to show the strong impact of seasons on the strength of dependence between different aggregation periods (daily versus hourly) of heavy rainfall in France.

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arXiv2026

Graph structure learning for stable processes

We introduce Ising-H\"usler-Reiss processes, a new class of multivariate L\'evy processes that allows for sparse modeling of the path-wise conditional independence structure between marginal stable processes with different stability indices. The underlying conditional independence graph is encoded as zeroes in a suitable precision matrix. An Ising-type parametrization of the weights for each orthant of the L\'evy measure allows for data-driven modeling of asymmetry of the jumps while retaining an arbitrary sparse graph. We develop consistent estimators for the graphical structure and asymmetry parameters, relying on a new uniform small-time approximation for L\'evy processes. The methodology is illustrated in simulations and a real data application to modeling dependence of stock returns.

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arXiv2025

Intrinsic Whittle--Mat\'ern fields and sparse spatial extremes

Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Mat\'ern Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

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arXiv2025

Numerical models outperform AI weather forecasts of record-breaking extremes

Artificial intelligence (AI)-based models are revolutionizing weather forecasting and have surpassed leading numerical weather prediction systems on various benchmark tasks. However, their ability to extrapolate and reliably forecast unprecedented extreme events remains unclear. Here, we show that for record-breaking weather extremes, the numerical model High RESolution forecast (HRES) from the European Centre for Medium-Range Weather Forecasts still consistently outperforms state-of-the-art AI models GraphCast, GraphCast operational, Pangu-Weather, Pangu-Weather operational, and Fuxi. We demonstrate that forecast errors in AI models are consistently larger for record-breaking heat, cold, and wind than in HRES across nearly all lead times. We further find that the examined AI models tend to underestimate both the frequency and intensity of record-breaking events, and they underpredict hot records and overestimate cold records with growing errors for larger record exceedance. Our findings underscore the current limitations of AI weather models in extrapolating beyond their training domain and in forecasting the potentially most impactful record-breaking weather events that are particularly frequent in a rapidly warming climate. Further rigorous verification and model development is needed before these models can be solely relied upon for high-stakes applications such as early warning systems and disaster management.

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arXiv2025

Theoretical guarantees for neural estimators in parametric statistics

Neural estimators are simulation-based estimators for the parameters of a family of statistical models, which build a direct mapping from the sample to the parameter vector. They benefit from the versatility of available network architectures and efficient training methods developed in the field of deep learning. Neural estimators are amortized in the sense that, once trained, they can be applied to any new data set with almost no computational cost. While many papers have shown very good performance of these methods in simulation studies and real-world applications, so far no statistical guarantees are available to support these observations theoretically. In this work, we study the risk of neural estimators by decomposing it into several terms that can be analyzed separately. We formulate easy-to-check assumptions ensuring that each term converges to zero, and we verify them for popular applications of neural estimators. Our results provide a general recipe to derive theoretical guarantees also for broader classes of architectures and estimation problems.

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arXiv2025

Extreme Conformal Prediction: Reliable Intervals for High-Impact Events

Conformal prediction is a popular method to construct prediction intervals with marginal coverage guarantees from black-box machine learning models. In applications with potentially high-impact events, such as flooding or financial crises, regulators often require very high confidence for such intervals. However, if the desired level of confidence is too large relative to the amount of data used for calibration, then classical conformal methods provide infinitely wide, thus, uninformative prediction intervals. In this paper, we propose a new method to overcome this limitation. We bridge extreme value statistics and conformal prediction to provide reliable and informative prediction intervals with high-confidence coverage, which can be constructed using any black-box extreme quantile regression method. A weighted version of our approach can account for nonstationary data. The advantages of our extreme conformal prediction method are illustrated in a simulation study and in an application to flood risk forecasting.

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arXiv2025

Extremes of structural causal models

The behavior of extreme observations is well-understood for time series or spatial data, but little is known if the data generating process is a structural causal model (SCM). We study the behavior of extremes in this model class, both for the observational distribution and under extremal interventions. We show that under suitable regularity conditions on the structure functions, the extremal behavior is described by a multivariate Pareto distribution, which can be represented as a new SCM on an extremal graph. Importantly, the latter is a sub-graph of the graph in the original SCM, which means that causal links can disappear in the tails. We further introduce a directed version of extremal graphical models and show that an extremal SCM satisfies the corresponding Markov properties. Based on a new test of extremal conditional independence, we propose two algorithms for learning the extremal causal structure from data. The first is an extremal version of the PC-algorithm, and the second is a pruning algorithm that removes edges from the original graph to consistently recover the extremal graph. The methods are illustrated on river data with known causal ground truth.

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arXiv2024

Progression: an extrapolation principle for regression

The problem of regression extrapolation, or out-of-distribution generalization, arises when predictions are required at test points outside the range of the training data. In such cases, the non-parametric guarantees for regression methods from both statistics and machine learning typically fail. Based on the theory of tail dependence, we propose a novel statistical extrapolation principle. After a suitable, data-adaptive marginal transformation, it assumes a simple relationship between predictors and the response at the boundary of the training predictor samples. This assumption holds for a wide range of models, including non-parametric regression functions with additive noise. Our semi-parametric method, progression, leverages this extrapolation principle and offers guarantees on the approximation error beyond the training data range. We demonstrate how this principle can be effectively integrated with existing approaches, such as random forests and additive models, to improve extrapolation performance on out-of-distribution samples.

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arXiv2024

L\'evy graphical models

Conditional independence and graphical models are crucial concepts for sparsity and statistical modeling in higher dimensions. For L\'evy processes, a widely applied class of stochastic processes, these notions have not been studied. By the L\'evy-It\^o decomposition, a multivariate L\'evy process can be decomposed into the sum of a Brownian motion part and an independent jump process. We show that conditional independence statements between the marginal processes can be studied separately for these two parts. While the Brownian part is well-understood, we derive a novel characterization of conditional independence between the sample paths of the jump process in terms of the L\'evy measure. We define L\'evy graphical models as L\'evy processes that satisfy undirected or directed Markov properties. We prove that the graph structure is invariant under changes of the univariate marginal processes. L\'evy graphical models allow the construction of flexible, sparse dependence models for L\'evy processes in large dimensions, which are interpretable thanks to the underlying graph. For trees, we develop statistical methodology to learn the underlying structure from low- or high-frequency observations of the L\'evy process and show consistent graph recovery. We apply our method to model stock returns from U.S. companies to illustrate the advantages of our approach.

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arXiv2024

Validating Deep Learning Weather Forecast Models on Recent High-Impact Extreme Events

The forecast accuracy of machine learning (ML) weather prediction models is improving rapidly, leading many to speak of a "second revolution in weather forecasting". With numerous methods being developed and limited physical guarantees offered by ML models, there is a critical need for a comprehensive evaluation of these emerging techniques. While this need has been partly fulfilled by benchmark datasets, they provide little information on rare and impactful extreme events or on compound impact metrics, for which model accuracy might degrade due to misrepresented dependencies between variables. To address these issues, we compare ML weather prediction models (GraphCast, PanguWeather, and FourCastNet) and ECMWF's high-resolution forecast system (HRES) in three case studies: the 2021 Pacific Northwest heatwave, the 2023 South Asian humid heatwave, and the North American winter storm in 2021. We find that ML weather prediction models locally achieve similar accuracy to HRES on the record-shattering Pacific Northwest heatwave but underperform when aggregated over space and time. However, they forecast the compound winter storm substantially better. We also highlight structural differences in how the errors of HRES and the ML models build up to that event. The ML forecasts lack important variables for a detailed assessment of the health risks of the 2023 humid heatwave. Using a possible substitute variable, prediction errors show spatial patterns with the highest danger levels over Bangladesh being underestimated by the ML models. Generally, case-study-driven, impact-centric evaluation can complement existing research, increase public trust, and aid in developing reliable ML weather prediction models.

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arXiv2024

Extremal graphical modeling with latent variables via convex optimization

Extremal graphical models encode the conditional independence structure of multivariate extremes and provide a powerful tool for quantifying the risk of rare events. Prior work on learning these graphs from data has focused on the setting where all relevant variables are observed. For the popular class of H\"usler-Reiss models, we propose the \texttt{eglatent} method, a tractable convex program for learning extremal graphical models in the presence of latent variables. Our approach decomposes the H\"usler-Reiss precision matrix into a sparse component encoding the graphical structure among the observed variables after conditioning on the latent variables, and a low-rank component encoding the effect of a few latent variables on the observed variables. We provide finite-sample guarantees of \texttt{eglatent} and show that it consistently recovers the conditional graph as well as the number of latent variables. We highlight the improved performances of our approach on synthetic and real data.

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arXiv2024

Graphical models for multivariate extremes

Graphical models in extremes have emerged as a diverse and quickly expanding research area in extremal dependence modeling. They allow for parsimonious statistical methodology and are particularly suited for enforcing sparsity in high-dimensional problems. In this work, we provide the fundamental concepts of extremal graphical models and discuss recent advances in the field. Different existing perspectives on graphical extremes are presented in a unified way through graphical models for exponent measures. We discuss the important cases of nonparametric extremal graphical models on simple graph structures, and the parametric class of H\"usler--Reiss models on arbitrary undirected graphs. In both cases, we describe model properties, methods for statistical inference on known graph structures, and structure learning algorithms when the graph is unknown. We illustrate different methods in an application to flight delay data at US airports.

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arXiv2023

Boosted Control Functions: Distribution generalization and invariance in confounded models

Modern machine learning methods and the availability of large-scale data have significantly advanced our ability to predict target quantities from large sets of covariates. However, these methods often struggle under distributional shifts, particularly in the presence of hidden confounding. While the impact of hidden confounding is well-studied in causal effect estimation, e.g., instrumental variables, its implications for prediction tasks under shifting distributions remain underexplored. This work addresses this gap by introducing a strong notion of invariance that, unlike existing weaker notions, allows for distribution generalization even in the presence of nonlinear, non-identifiable structural functions. Central to this framework is the Boosted Control Function (BCF), a novel, identifiable target of inference that satisfies the proposed strong invariance notion and is provably worst-case optimal under distributional shifts. The theoretical foundation of our work lies in Simultaneous Equation Models for Distribution Generalization (SIMDGs), which bridge machine learning with econometrics by describing data-generating processes under distributional shifts. To put these insights into practice, we propose the ControlTwicing algorithm to estimate the BCF using nonparametric machine-learning techniques and study its generalization performance on synthetic and real-world datasets compared to robust and empirical risk minimization approaches.

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arXiv2023

Extremal Dependence of Moving Average Processes Driven by Exponential-Tailed L\'evy Noise

Moving average processes driven by exponential-tailed L\'evy noise are important extensions of their Gaussian counterparts in order to capture deviations from Gaussianity, more flexible dependence structures, and sample paths with jumps. Popular examples include non-Gaussian Ornstein--Uhlenbeck processes and type G Mat\'ern stochastic partial differential equation random fields. This paper is concerned with the open problem of determining their extremal dependence structure. We leverage the fact that such processes admit approximations on grids or triangulations that are used in practice for efficient simulations and inference. These approximations can be expressed as special cases of a class of linear transformations of independent, exponential-tailed random variables, that bridge asymptotic dependence and independence in a novel, tractable way. This result is of independent interest since models that can capture both extremal dependence regimes are scarce and the construction of such flexible models is an active area of research. This new fundamental result allows us to show that the integral approximation of general moving average processes with exponential-tailed L\'evy noise is asymptotically independent when the mesh is fine enough. Under mild assumptions on the kernel function we also derive the limiting residual tail dependence function. For the popular exponential-tailed Ornstein--Uhlenbeck process we prove that it is asymptotically independent, but with a different residual tail dependence function than its Gaussian counterpart. Our results are illustrated through simulation studies.

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