Adrien S. Hitz

Bio: Adrien S. Hitz is an academic researcher from University of Oxford. The author has contributed to research in topics: Graphical model & Extreme value theory. The author has an hindex of 4, co-authored 9 publications receiving 89 citations.

Graphical Models for Extremes

Abstract: Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max‐stable and multivariate Pareto distributions. Statistical modelling in this field has been limited to moderate dimensions so far, partly owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures. We introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes. A Hammersley–Clifford theorem links this new notion to the factorization of densities of extreme value models on graphs. For the popular class of Husler–Reiss distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices. New parametric models can be built in a modular way and statistical inference can be simplified to lower dimensional marginals. We discuss learning of minimum spanning trees and model selection for extremal graph structures, and we illustrate their use with an application to flood risk assessment on the Danube river.

Graphical models for extremes

Abstract: Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max‐stable and multivariate Pareto distributions. Statistical modelling in this field has been limited to moderate dimensions so far, partly owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures. We introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes. A Hammersley–Clifford theorem links this new notion to the factorization of densities of extreme value models on graphs. For the popular class of Husler–Reiss distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices. New parametric models can be built in a modular way and statistical inference can be simplified to lower dimensional marginals. We discuss learning of minimum spanning trees and model selection for extremal graph structures, and we illustrate their use with an application to flood risk assessment on the Danube river.

One-component regular variation and graphical modeling of extremes

Abstract: The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

Graphical Models for Extremes

Abstract: Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max-stable and multivariate Pareto distributions. Statistical modelling in this field has been limited to moderate dimensions so far, partly owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures. We introduce a general theory of conditional independence for multivariate Pareto distributions that allows the definition of graphical models and sparsity for extremes. A Hammersley-Clifford theorem links this new notion to the factorization of densities of extreme value models on graphs. For the popular class of Husler-Reiss distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices. New parametric models can be built in a modular way and statistical inference can be simplified to lower-dimensional marginals. We discuss learning of minimum spanning trees and model selection for extremal graph structures, and illustrate their use with an application to flood risk assessment on the Danube river.

Discrete Extremes

Abstract: Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data Extreme values of a random variable $X$ are commonly modeled using the generalized Pareto distribution, a method that often gives good results in practice When $X$ is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births

Higher-dimensional spatial extremes via single-site conditioning

Abstract: 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 a single site, we suggest a joint inference scheme based on all observation locations, and implement an importance sampling algorithm to provide spatial realizations and estimates of quantities conditioning upon the process being extreme at any of one of an arbitrary set of locations. The modelling approach is applied to Australian summer temperature extremes, permitting assessment the spatial extent of high temperature events over the continent.

Advances in statistical modeling of spatial extremes

Abstract: We thank Thomas Opitz for helpful discussions, as well as the Editor and two anonymous referees for various comments that have improved the paper. This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. OSR-CRG2017-3434.

Sparse Structures for Multivariate Extremes

Abstract: 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 extremal dependence that can arise between the largest observations of a multivariate random vector. We then discuss the current research topics including clustering, principal component analysis and graphical modeling for extremes. Identification of groups of variables which can be concomitantly extreme is also addressed. The methods are illustrated with an application to flood risk assessment.