Sparse Structures for Multivariate Extremes

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

Using a priori knowledge to construct copulas

Our purpose is to model the dependence between two random variables, taking into account a priori knowledge on these variables. For example, in many applications (oceanography, finance...), there exists an order relation between the two variables; when one takes high values, the other cannot take low values, but the contrary is possible. The dependence for the high values of the two variables is, therefore, not symmetric. However a minimal dependence also exists: low values of one variable are associated with low values of the other variable. The dependence can also be extreme for the maxima or the minima of the two variables. In this paper, we construct step by step asymmetric copulas with asymptotic minimal dependence, and with or without asymptotic maximal dependence, using mixture variables to get at first asymmetric dependence and then minimal dependence. We fit these models to a real dataset of sea states and compare them using Likelihood Ratio Tests when they are nested, and BIC- criterion (Bayesian Information criterion) otherwise

Exploratory data analysis for moderate extreme values using non-parametric kernel methods

In many settings it is critical to accurately model the extreme tail behaviour of a random process. Non-parametric density estimation methods are commonly implemented as exploratory data analysis techniques for this purpose as they possess excellent visualisation properties, and can naturally avoid the model specification biases implied by using parametric estimators. In particular, kernel-based estimators place minimal assumptions on the data, and provide improved visualisation over scatterplots and histograms. However kernel density estimators are known to perform poorly when estimating extreme tail behaviour, which is important when interest is in process behaviour above some large threshold, and they can over-emphasise bumps in the density for heavy tailed data. In this article we develop a transformation kernel density estimator, and demonstrate that its mean integrated squared error (MISE) efficiency is equivalent to that of standard, non-tail focused kernel density estimators. Estimator performance is illustrated in numerical studies, and in an expanded analysis of the ability of well known global climate models to reproduce observed temperature extremes in Sydney, Australia

High-dimensional peaks-over-threshold inference

Max-stable processes are increasingly widely used for modelling complex extreme events, but existing fitting methods are computationally demanding, limiting applications to a few dozen variables. $r$-Pareto processes are mathematically simpler and have the potential advantage of incorporating all relevant extreme events, by generalizing the notion of a univariate exceedance. In this paper we investigate score matching for performing high-dimensional peaks over threshold inference, focusing on extreme value processes associated to log-Gaussian random functions and discuss the behaviour of the proposed estimators for regularly-varying distributions with normalized marginals. Their performance is assessed on grids with several hundred locations, simulating from both the true model and from its domain of attraction. We illustrate the potential and flexibility of our methods by modelling extreme rainfall on a grid with $3600$ locations, based on risks for exceedances over local quantiles and for large spatially accumulated rainfall, and briefly discuss diagnostics of model fit. The differences between the two fitted models highlight the importance of the choice of risk and its impact on the dependence structure

Modeling asymptotically independent spatial extremes based on Laplace random fields

We tackle the modeling of threshold exceedances in asymptotically independent stochastic processes by constructions based on Laplace random fields. These are defined as Gaussian random fields scaled with a stochastic variable following an exponential distribution. This framework yields useful asymptotic properties while remaining statistically convenient. Univariate distribution tails are of the half exponential type and are part of the limiting generalized Pareto distributions for threshold exceedances. After normalizing marginal tail distributions in data, a standard Laplace field can be used to capture spatial dependence among extremes. Asymptotic properties of Laplace fields are explored and compared to the classical framework of asymptotic dependence. Multivariate joint tail decay rates for Laplace fields are slower than for Gaussian fields with the same covariance structure; hence they provide more conservative estimates of very extreme joint risks while maintaining asymptotic independence. Statistical inference is illustrated on extreme wind gusts in the Netherlands where a comparison to the Gaussian dependence model shows a better goodness-of-fit in terms of Akaike's criterion. In this specific application we fit the well-adapted Weibull distribution as univariate tail model, such that the normalization of univariate tail distributions can be done through a simple power transformation of data

Some variations of EM algorithms for Marshall-Olkin bivariate Pareto distribution with location and scale

Recently Asimit et. al used an EM algorithm to estimate Marshall-Olkin bivariate Pareto distribution. The distribution has seven parameters. We describe few alternative approaches of EM algorithm. A numerical simulation is performed to verify the performance of different proposed algorithms. A real-life data analysis is also shown for illustrative purposes

Factor Copula Models for Replicated Spatial Data

We propose a new copula model that can be used with replicated spatial data. Unlike the multivariate normal copula, the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process. Moreover, the proposed copula can model tail dependence and tail asymmetry. The model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature, such as the Matern model. For some choice of common factors, the joint copula density is given in closed form and therefore likelihood estimation is very fast. In the general case, one-dimensional numerical integration is needed to calculate the likelihood, but estimation is still reasonably fast even with large data sets. We use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors. We apply the proposed model to spatial temperature data and compare its performance with some popular geostatistics models.Comment: 34 pages, 3 tables and 5 figure

Robust Estimation of Bivariate Tail Dependence Coefficient

The problem of estimating the coefficient of bivariate tail dependence is considered here from the robustness point of view; it combines two apparently contradictory theories of robust statistics and extreme value statistics. The usual maximum likelihood based or the moment type estimators of tail dependence coefficient are highly sensitive to the presence of outlying observations in data. This paper proposes some alternative robust estimators obtained by minimizing the density power divergence with suitable model assumptions; their robustness properties are examined through the classical influence function analysis. The performance of the proposed estimators is illustrated through an extensive empirical study considering several important bivariate extreme value distributions.Comment: Pre-Print, 20 page

A direct verification argument for the Hamilton-Jacobi equation continuum limit of nondominated sorting

Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. The original proof relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this proof is new in the homogenization literature, and may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle

Simple models for multivariate regular variations and the H\"usler-Reiss Pareto distribution

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 H\"usler-Reiss models. In a second part of the paper, we focus on the H\"usler-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 H\"usler-Reiss Pareto model with different tail indices and a likelihood ratio test for discriminating constant tail index versus varying tail indices