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Principal Component Analysis for Multivariate Extremes
Authors:
Dan Cooley,
Anne Sabourin,
Troy Wixson
Abstract:
This chapter explores ways to reduce the dimensionality of the data while preserving key information relevant to the analysis of multivariate extreme values.
This chapter explores ways to reduce the dimensionality of the data while preserving key information relevant to the analysis of multivariate extreme values.
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Submitted 5 June, 2026;
originally announced June 2026.
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Polar Depth for Potentially Heavy-Tailed Data
Authors:
Stephan Clemençon,
Carlos Fernándes,
Pavlo Mozharovskyi,
Anne Sabourin
Abstract:
Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriat…
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Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X which norm is larger than t>0, converges to the polar depth of the limiting distribution as t converges to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a halfspace than the alternatives proposed in the literature, the halfspace depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.
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Submitted 29 May, 2026;
originally announced June 2026.
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Out-of-Distribution generalization of quantile regression with heavy tailed inputs: an SVM approach
Authors:
Baptiste Leroux,
Clément Dombry,
Anne Sabourin
Abstract:
We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional…
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We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional risk that localizes learning in the tail of the covariate distribution.
We propose a novel Support Vector Machine (SVM) framework for extreme quantile regression, leveraging reproducing kernel Hilbert spaces to handle high-dimensional and nonlinear settings. Our method also accommodates unbounded response variables and avoids restrictive transformations. We establish finite-sample learning guarantees under mild regularity assumptions.
The proposed framework unifies ideas from statistical learning and multivariate extremes, providing a tractable and theoretically grounded approach to extrapolation. We complement our theoretical findings with an empirical study on river flow data from the Danube, demonstrating the practical relevance of our methods.
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Submitted 29 May, 2026;
originally announced June 2026.
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Extrapolation in Statistical Learning with Extreme Value Theory
Authors:
Sebastian Engelke,
Nicola Gnecco,
Anne Sabourin
Abstract:
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, ge…
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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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Submitted 3 May, 2026;
originally announced May 2026.
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Adaptive tail index estimation: minimal assumptions and non-asymptotic guarantees
Authors:
Johannes Lederer,
Anne Sabourin,
Mahsa Taheri
Abstract:
A notoriously difficult challenge in extreme value theory is the choice of the number $k\ll n$, where $n$ is the total sample size, of extreme data points to consider for inference of tail quantities. Existing theoretical guarantees for adaptive methods typically require second-order assumptions or von Mises assumptions that are difficult to verify and often come with tuning parameters that are ch…
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A notoriously difficult challenge in extreme value theory is the choice of the number $k\ll n$, where $n$ is the total sample size, of extreme data points to consider for inference of tail quantities. Existing theoretical guarantees for adaptive methods typically require second-order assumptions or von Mises assumptions that are difficult to verify and often come with tuning parameters that are challenging to calibrate. This paper revisits the problem of adaptive selection of $k$ for the Hill estimator. Our goal is not an `optimal' $k$ but one that is `good enough', in the sense that we strive for non-asymptotic guarantees that might be sub-optimal but are explicit and require minimal conditions. We propose a transparent adaptive rule that does not require preliminary calibration of constants, inspired by `adaptive validation' developed in high-dimensional statistics. A key feature of our approach is the consideration of a grid for $k$ of size $ \ll n $, which aligns with common practice among practitioners but has remained unexplored in theoretical analysis. Our rule only involves an explicit expression of a variance-type term; in particular, it does not require controlling or estimating a biasterm. Our theoretical analysis is valid for all heavy-tailed distributions, specifically for all regularly varying survival functions. Furthermore, when von Mises conditions hold, our method achieves `almost' minimax optimality with a rate of $\sqrt{\log \log n}~ n^{-|ρ|/(1+2|ρ|)}$ when the grid size is of order $\log n$, in contrast to the $ (\log \log (n)/n)^{|ρ|/(1+2|ρ|)} $ rate in existing work. Our simulations show that our approach performs particularly well for ill-behaved distributions.
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Submitted 29 May, 2025; v1 submitted 28 May, 2025;
originally announced May 2025.
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Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast
Authors:
Nathan Huet,
Philippe Naveau,
Anne Sabourin
Abstract:
Appropriate modelling of extreme skew surges is crucial, particularly for coastal risk management. Our study focuses on modelling extreme skew surges along the French Atlantic coast, with a particular emphasis on investigating the extremal dependence structure between stations. We employ the peak-over-threshold framework, where a multivariate extreme event is defined whenever at least one location…
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Appropriate modelling of extreme skew surges is crucial, particularly for coastal risk management. Our study focuses on modelling extreme skew surges along the French Atlantic coast, with a particular emphasis on investigating the extremal dependence structure between stations. We employ the peak-over-threshold framework, where a multivariate extreme event is defined whenever at least one location records a large value, though not necessarily all stations simultaneously. A novel method for determining an appropriate level (threshold) above which observations can be classified as extreme is proposed. Two complementary approaches are explored. First, the multivariate generalized Pareto distribution is employed to model extremes, leveraging its properties to derive a generative model that predicts extreme skew surges at one station based on observed extremes at nearby stations. Second, a novel extreme regression framework is assessed for point predictions. This specific regression framework enables accurate point predictions using only the 'angle' of input variables, i.e., input variables divided by their norms. The ultimate objective is to reconstruct historical skew surge time series at stations with limited data. This is achieved by integrating extreme skew surge data from stations with longer records, such as Brest and Saint-Nazaire, which provide over 150 years of observations.
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Submitted 6 May, 2026; v1 submitted 1 May, 2025;
originally announced May 2025.
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Sharp error bounds for imbalanced classification: how many examples in the minority class?
Authors:
Anass Aghbalou,
François Portier,
Anne Sabourin
Abstract:
When dealing with imbalanced classification data, reweighting the loss function is a standard procedure allowing to equilibrate between the true positive and true negative rates within the risk measure. Despite significant theoretical work in this area, existing results do not adequately address a main challenge within the imbalanced classification framework, which is the negligible size of one cl…
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When dealing with imbalanced classification data, reweighting the loss function is a standard procedure allowing to equilibrate between the true positive and true negative rates within the risk measure. Despite significant theoretical work in this area, existing results do not adequately address a main challenge within the imbalanced classification framework, which is the negligible size of one class in relation to the full sample size and the need to rescale the risk function by a probability tending to zero. To address this gap, we present two novel contributions in the setting where the rare class probability approaches zero: (1) a non asymptotic fast rate probability bound for constrained balanced empirical risk minimization, and (2) a consistent upper bound for balanced nearest neighbors estimates. Our findings provide a clearer understanding of the benefits of class-weighting in realistic settings, opening new avenues for further research in this field.
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Submitted 16 April, 2024; v1 submitted 23 October, 2023;
originally announced October 2023.
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Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes
Authors:
Stephan Clémençon,
Nathan Huet,
Anne Sabourin
Abstract:
Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension…
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Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.
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Submitted 2 August, 2023;
originally announced August 2023.
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On Regression in Extreme Regions
Authors:
Stephan Clémençon,
Nathan Huet,
Anne Sabourin
Abstract:
We establish a statistical learning theoretical framework aimed at extrapolation, or out-of-domain generalization, on the unobserved tails of covariates in continuous regression problems. Our strategy involves performing statistical regression on a subsample of observations with continuous labels that are the furthest away from the origin, focusing specifically on their angular components. The und…
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We establish a statistical learning theoretical framework aimed at extrapolation, or out-of-domain generalization, on the unobserved tails of covariates in continuous regression problems. Our strategy involves performing statistical regression on a subsample of observations with continuous labels that are the furthest away from the origin, focusing specifically on their angular components. The underlying assumptions of our approach are grounded in the theory of multivariate regular variation, a cornerstone of extreme value theory. We address the stylized problem of nonparametric least squares regression with predictors chosen from a Vapnik-Chervonenkis class.
This work contributes to a broader initiative to develop statistical learning theoretical foundations for supervised learning strategies that enhance performance on the supposedly heavy tails of covariates. Previous efforts in this area have focused exclusively on binary classification on extreme covariates. Although the continuous target setting necessitates different techniques and regularity assumptions, our main results echo findings from earlier studies. We quantify the predictive performance on tail regions in terms of excess risk, presenting it as a finite sample risk bound with a clear bias-variance decomposition. Numerical experiments with simulated and real data illustrate our theoretical findings.
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Submitted 12 September, 2025; v1 submitted 6 March, 2023;
originally announced March 2023.
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Tail inverse regression for dimension reduction with extreme response
Authors:
Anass Aghbalou,
François Portier,
Anne Sabourin,
Chen Zhou
Abstract:
We consider the problem of supervised dimension reduction with a particular focus on extreme values of the target $Y\in\mathbb{R}$ to be explained by a covariate vector $X \in \mathbb{R}^p$. The general purpose is to define and estimate a projection on a lower dimensional subspace of the covariate space which is sufficient for predicting exceedances of the target above high thresholds. We propose…
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We consider the problem of supervised dimension reduction with a particular focus on extreme values of the target $Y\in\mathbb{R}$ to be explained by a covariate vector $X \in \mathbb{R}^p$. The general purpose is to define and estimate a projection on a lower dimensional subspace of the covariate space which is sufficient for predicting exceedances of the target above high thresholds. We propose an original definition of Tail Conditional Independence which matches this purpose. Inspired by Sliced Inverse Regression (SIR) methods, we develop a novel framework (TIREX, Tail Inverse Regression for EXtreme response) in order to estimate an extreme sufficient dimension reduction (SDR) space of potentially smaller dimension than that of a classical SDR space. We prove the weak convergence of tail empirical processes involved in the estimation procedure and we illustrate the relevance of the proposed approach on simulated and real world data.
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Submitted 24 February, 2023; v1 submitted 30 July, 2021;
originally announced August 2021.
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Concentration bounds for the empirical angular measure with statistical learning applications
Authors:
Stéphan Clémençon,
Hamid Jalalzai,
Stéphane Lhaut,
Anne Sabourin,
Johan Segers
Abstract:
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, t…
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The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.
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Submitted 17 October, 2022; v1 submitted 7 April, 2021;
originally announced April 2021.
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Heavy-tailed Representations, Text Polarity Classification & Data Augmentation
Authors:
Hamid Jalalzai,
Pierre Colombo,
Chloé Clavel,
Eric Gaussier,
Giovanna Varni,
Emmanuel Vignon,
Anne Sabourin
Abstract:
The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation. In this paper, we develop a novel method to learn a heavy-tailed embedding with desirable regularity properties regarding the distributional tails, which allows to analyze the points far away from the…
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The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation. In this paper, we develop a novel method to learn a heavy-tailed embedding with desirable regularity properties regarding the distributional tails, which allows to analyze the points far away from the distribution bulk using the framework of multivariate extreme value theory. In particular, a classifier dedicated to the tails of the proposed embedding is obtained which performance outperforms the baseline. This classifier exhibits a scale invariance property which we leverage by introducing a novel text generation method for label preserving dataset augmentation. Numerical experiments on synthetic and real text data demonstrate the relevance of the proposed framework and confirm that this method generates meaningful sentences with controllable attribute, e.g. positive or negative sentiment.
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Submitted 25 March, 2021; v1 submitted 25 March, 2020;
originally announced March 2020.
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A Multivariate Extreme Value Theory Approach to Anomaly Clustering and Visualization
Authors:
Maël Chiapino,
Stéphan Clémençon,
Vincent Feuillard,
Anne Sabourin
Abstract:
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 pheno…
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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 by means of a novel mixture model that permits to describe the distribution of extremal observations and where the anomaly type $α$ is viewed as a latent variable. One may then take advantage of the model by assigning to any extreme point a posterior probability for each anomaly type $α$, defining implicitly a similarity measure between anomalies. It is explained at length how the latter permits to cluster extreme observations and obtain an informative planar representation of anomalies using standard graph-mining tools. The relevance and usefulness of the clustering and 2-d visual display thus designed is illustrated on simulated datasets and on real observations as well, in the aeronautics application domain.
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Submitted 17 July, 2019;
originally announced July 2019.
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Principal Component Analysis for Multivariate Extremes
Authors:
Holger Drees,
Anne Sabourin
Abstract:
The first order behavior of multivariate heavy-tailed random vectors above large radial thresholds is ruled by a limit measure in a regular variation framework. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than other…
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The first order behavior of multivariate heavy-tailed random vectors above large radial thresholds is ruled by a limit measure in a regular variation framework. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than others. Identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we apply Principal Component Analysis (PCA) to a re-scaled version of radially thresholded observations. Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radial thresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are re-scaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection sub-space. Numerical experiments illustrate the relevance of the proposed framework for practical purposes.
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Submitted 26 June, 2019;
originally announced June 2019.
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Identifying groups of variables with the potential of being large simultaneously
Authors:
Maël Chiapino,
Anne Sabourin,
Johan Segers
Abstract:
Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collection of all possible groups of variables in a systema…
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Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collection of all possible groups of variables in a systematic way, from smaller groups to larger ones. The issue that the tolerance level in the stopping criterion should depend on the size of the groups is circumvented by the use of a conditional tail dependence coefficient. Alternatively, such stopping criteria can be based on limit distributions of rank-based estimators of the coefficient of tail dependence, quantifying the speed of decay of joint survival functions. Numerical experiments indicate that the algorithm's effectiveness for detecting tail-dependent groups of variables is highest when paired with a criterion based on a Hill-type estimator of the coefficient of tail dependence.
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Submitted 27 February, 2018;
originally announced February 2018.
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Max K-armed bandit: On the ExtremeHunter algorithm and beyond
Authors:
Mastane Achab,
Stephan Clémençon,
Aurélien Garivier,
Anne Sabourin,
Claire Vernade
Abstract:
This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduc…
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This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.
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Submitted 27 July, 2017;
originally announced July 2017.
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Sparse Representation of Multivariate Extremes with Applications to Anomaly Ranking
Authors:
Nicolas Goix,
Anne Sabourin,
Stéphan Clémençon
Abstract:
Extremes play a special role in Anomaly Detection. Beyond inference and simulation purposes, probabilistic tools borrowed from Extreme Value Theory (EVT), such as the angular measure, can also be used to design novel statistical learning methods for Anomaly Detection/ranking. This paper proposes a new algorithm based on multivariate EVT to learn how to rank observations in a high dimensional space…
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Extremes play a special role in Anomaly Detection. Beyond inference and simulation purposes, probabilistic tools borrowed from Extreme Value Theory (EVT), such as the angular measure, can also be used to design novel statistical learning methods for Anomaly Detection/ranking. This paper proposes a new algorithm based on multivariate EVT to learn how to rank observations in a high dimensional space with respect to their degree of 'abnormality'. The procedure relies on an original dimension-reduction technique in the extreme domain that possibly produces a sparse representation of multivariate extremes and allows to gain insight into the dependence structure thereof, escaping the curse of dimensionality. The representation output by the unsupervised methodology we propose here can be combined with any Anomaly Detection technique tailored to non-extreme data. As it performs linearly with the dimension and almost linearly in the data (in O(dn log n)), it fits to large scale problems. The approach in this paper is novel in that EVT has never been used in its multivariate version in the field of Anomaly Detection. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.
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Submitted 31 March, 2016;
originally announced March 2016.
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Sparsity in Multivariate Extremes with Applications to Anomaly Detection
Authors:
Nicolas Goix,
Anne Sabourin,
Stéphan Clémençon
Abstract:
Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g. portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (non-parametric) characterization of extremal dependence in the framework of multivariate Extreme Value Theory (EVT) is the angu…
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Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g. portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (non-parametric) characterization of extremal dependence in the framework of multivariate Extreme Value Theory (EVT) is the angular measure, which provides direct information about the probable 'directions' of extremes, that is, the relative contribution of each feature/coordinate of the 'largest' observations. Modeling the angular measure in high dimensional problems is a major challenge for the multivariate analysis of rare events. The present paper proposes a novel methodology aiming at exhibiting a sparsity pattern within the dependence structure of extremes. This is done by estimating the amount of mass spread by the angular measure on representative sets of directions, corresponding to specific sub-cones of $R^d\_+$. This dimension reduction technique paves the way towards scaling up existing multivariate EVT methods. Beyond a non-asymptotic study providing a theoretical validity framework for our method, we propose as a direct application a --first-- anomaly detection algorithm based on multivariate EVT. This algorithm builds a sparse 'normal profile' of extreme behaviours, to be confronted with new (possibly abnormal) extreme observations. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.
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Submitted 14 March, 2016; v1 submitted 21 July, 2015;
originally announced July 2015.
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On Anomaly Ranking and Excess-Mass Curves
Authors:
Nicolas Goix,
Anne Sabourin,
Stéphan Clémençon
Abstract:
Learning how to rank multivariate unlabeled observations depending on their degree of abnormality/novelty is a crucial problem in a wide range of applications. In practice, it generally consists in building a real valued "scoring" function on the feature space so as to quantify to which extent observations should be considered as abnormal. In the 1-d situation, measurements are generally considere…
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Learning how to rank multivariate unlabeled observations depending on their degree of abnormality/novelty is a crucial problem in a wide range of applications. In practice, it generally consists in building a real valued "scoring" function on the feature space so as to quantify to which extent observations should be considered as abnormal. In the 1-d situation, measurements are generally considered as "abnormal" when they are remote from central measures such as the mean or the median. Anomaly detection then relies on tail analysis of the variable of interest. Extensions to the multivariate setting are far from straightforward and it is precisely the main purpose of this paper to introduce a novel and convenient (functional) criterion for measuring the performance of a scoring function regarding the anomaly ranking task, referred to as the Excess-Mass curve (EM curve). In addition, an adaptive algorithm for building a scoring function based on unlabeled data X1 , . . . , Xn with a nearly optimal EM is proposed and is analyzed from a statistical perspective.
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Submitted 5 February, 2015;
originally announced February 2015.
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Semi-parametric modeling of excesses above high multivariate thresholds with censored data
Authors:
Anne Sabourin
Abstract:
How to include censored data in a statistical analysis is a recur-rent issue in statistics. In multivariate extremes, the dependence structure of large observations can be characterized in terms of a non parametric angular measure, while marginal excesses above asymptotically large thresholds have a parametric distribution. In this work, a flexible semi-parametric Dirichlet mix-ture model for angu…
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How to include censored data in a statistical analysis is a recur-rent issue in statistics. In multivariate extremes, the dependence structure of large observations can be characterized in terms of a non parametric angular measure, while marginal excesses above asymptotically large thresholds have a parametric distribution. In this work, a flexible semi-parametric Dirichlet mix-ture model for angular measures is adapted to the context of censored data and missing components. One major issue is to take into account censoring intervals overlapping the extremal threshold, without knowing whether the correspond-ing hidden data is actually extreme. Further, the censored likelihood needed for Bayesian inference has no analytic expression. The first issue is tackled using a Poisson process model for extremes, whereas a data augmentation scheme avoids multivariate integration of the Poisson process intensity over both the censored intervals and the failure region above threshold. The implemented MCMC algorithm allows simultaneous estimation of marginal and dependence parameters, so that all sources of uncertainty other than model bias are cap-tured by posterior credible intervals. The method is illustrated on simulated and real data.
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Submitted 2 December, 2014;
originally announced December 2014.
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Combining regional estimation and historical floods: a multivariate semi-parametric peaks-over-threshold model with censored data
Authors:
Anne Sabourin,
Benjamin Renard
Abstract:
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, t…
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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-parametric Dirichlet Mixture model. The existence of partially missing and censored observations (historical data) is accounted for within a data augmentation scheme. This model is applied to a case study involving four catchments in Southern France, for which historical data are available since 1604. The comparison of marginal estimates from four versions of the model (with or without regionalizing the shape parameter; using or ignoring historical floods) highlights significant differences in terms of return level estimates. Moreover, the availability of historical data on several nearby catchments allows investigating the asymptotic dependence properties of extreme floods. Catchments display a a significant amount of asymptotic dependence, calling for adapted multivariate statistical models.
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Submitted 28 November, 2014;
originally announced November 2014.