arXiv subjects

Clément Dombry

Publications and source records attributed to Clément Dombry.

At least 37 records · Page 2Linked to original sources

arXiv2018

Tail measure and tail spectral process of regularly varying time series

The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main result is to prove in an abstract framework that there is a one to one correspondance between these two objets, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For non negative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.

math.PR↗

arXiv2018

Full likelihood inference for max-stable data

We show how to perform full likelihood inference for max-stable multivariate distributions or processes based on a stochastic Expectation-Maximisation algorithm, which combines statistical and computational efficiency in high-dimensions. The good performance of this methodology is demonstrated by simulation based on the popular logistic and Brown--Resnick models, and it is shown to provide dramatic computational time improvements with respect to a direct computation of the likelihood. Strategies to further reduce the computational burden are also discussed.

stat.ME↗

arXiv2017

Total variation regularized non-negative matrix factorization for smooth hyperspectral unmixing

Hyperspectral analysis has gained popularity over recent years as a way to infer what materials are displayed on a picture whose pixels consist of a mixture of spectral signatures. Computing both signatures and mixture coefficients is known as unsupervised unmixing, a set of techniques usually based on non-negative matrix factorization. Unmixing is a difficult non-convex problem, and algorithms may converge to one out of many local minima, which may be far removed from the true global minimum. Computing this true minimum is NP-hard and seems therefore out of reach. Aiming for interesting local minima, we investigate the addition of total variation regularization terms. Advantages of these regularizers are two-fold. Their computation is typically rather light, and they are deemed to preserve sharp edges in pictures. This paper describes an algorithm for regularized hyperspectral unmixing based on the Alternating Direction Method of Multipliers.

math.OC↗

arXiv2017

Maximum likelihood estimators based on the block maxima method

The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observed data. Among other semi-parametric methods (such as the popular Hill's estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of most common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.

math.ST↗

arXiv2016

High order expansions for renewal functions and applications to ruin theory

A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function g on a neighborhood of 0. This expansion relies on complex analysis and is expressed in terms of the residues of the function 1/(1 -- g). Under the assumption that g can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well known Crámer-Lundberg approximation of the ruin probability when the initial reserve is large.

math.PR↗

arXiv2016

Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Cesàro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property.

math.PR↗

arXiv2016

A Semi-Definite Programming approach to low dimensional embedding for unsupervised clustering

This paper proposes a variant of the method of Guédon and Verhynin for estimating the cluster matrix in the Mixture of Gaussians framework via Semi-Definite Programming. A clustering oriented embedding is deduced from this estimate. The procedure is suitable for very high dimensional data because it is based on pairwise distances only. Theoretical garantees are provided and an eigenvalue optimisation approach is proposed for computing the embedding. The performance of the method is illustrated via Monte Carlo experiements and comparisons with other embeddings from the literature.

stat.ML↗

arXiv2016

Random tessellations associated with max-stable random fields

With any max-stable random process $η$ on $\mathcal{X}=\mathbb{Z}^d$ or $\mathbb{R}^d$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the max-stable process $η$ which is seen as the pointwise maximum of a random collection of functions $Φ=\{ϕ\_i, i\geq 1\}$. The tessellation is constructed as follows: two points $x,y\in \mathcal{X}$ are in the same cell if and only if there exists a function $ϕ\inΦ$ that realizes the maximum $η$ at both points $x$ and $y$, i.e. $ϕ(x)=η(x)$ and $ϕ(y)=η(y)$. We characterize the distribution of cells in terms of coverage and inclusion probabilities. Most interesting is the stationary case where the asymptotic properties of the cells are strongly related to the ergodic properties of the non-singular flow generating the max-stable process. For example, we show that: i) the cells are bounded almost surely if and only if $η$ is generated by a dissipative flow, ii) the cells have positive asymptotic density almost surely if and only if $η$ is generated by a positive flow.

math.PR↗

arXiv2015

On functional records and champions

Records among a sequence of iid random variables $X_1,X_2,\dotsc$ on the real line have been investigated extensively over the past decades. A record is defined as a random variable $X_n$ such that $X_n>\max(X_1,\dotsc,X_{n-1})$. Trying to generalize this concept to the case of random vectors, or even stochastic processes with continuous sample paths, the question arises how to define records in higher dimensions. We introduce two different concepts: A simple record is meant to be a stochastic process (or a random vector) $X_n$ that is larger than $ X_1,\dotsc, X_{n-1}$ in at least one component, whereas a complete record has to be larger than its predecessors in all components. The behavior of records is investigated. In particular, the probability that a stochastic process $ X_n$ is a record as $n$ tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the distribution of $ X_n$, given that $ X_n$ is a record is derived.

math.PR↗

arXiv2015

Exact simulation of max-stable processes

Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by \citet{DM-2014} for Brown--Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.

stat.ME↗

arXiv2015

Probabilities of concurrent extremes

The statistical modelling of spatial extremes has recently made major advances. Much of its focus so far has been on the modelling of the magnitudes of extreme events but little attention has been paid on the timing of extremes. To address this gap, this paper introduces the notion of extremal concurrence. Suppose that one measures precipitation at several synoptic stations over multiple days. We say that extremes are concurrent if the maximum precipitation over time at each station is achieved simultaneously, e.g., on a single day. Under general conditions, we show that the finite sample concurrence probability converges to an asymptotic quantity, deemed extremal concurrence probability. Using Palm calculus, we establish general expressions for the extremal concurrence probability through the max-stable process emerging in the limit of the componentwise maxima of the sample. Explicit forms of the extremal concurrence probabilities are obtained for various max-stable models and several estimators are introduced. In particular, we prove that the pairwise extremal concurrence probability for max-stable vectors is precisely equal to the Kendall's $τ$. The estimators are evaluated by using simulations and applied to study the concurrence patterns of temperature extremes in the United States. The results demonstrate that concurrence probability can provide a powerful new perspective and tools for the analysis of the spatial structure and impact of extremes.

math.ST↗

arXiv2013

Stationary max-stable processes with the Markov property

We prove that the class of discrete time stationary max-stable process satisfying the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order $1$. A similar statement is also proved for continuous time processes.

math.PR↗

arXiv2013

Approximation of stable random measures and applications to linear fractional stable integrals

Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the approximations weakly converge as the mesh-size goes to zero. As an application, we improve upon previous approximation schemes for integrals with respect to linear fractional stable motions.

math.PR↗

arXiv2013

Maximum likelihood estimators for the extreme value index based on the block maxima method

The maximum likelihood method offers a standard way to estimate the three parameters of a generalized extreme value (GEV) distribution. Combined with the block maxima method, it is often used in practice to assess the extreme value index and normalization constants of a distribution satisfying a first order extreme value condition, assuming implicitely that the block maxima are exactly GEV distributed. This is unsatisfactory since the GEV distribution is a good approximation of the block maxima distribution only for blocks of large size. The purpose of this paper is to provide a theoretical basis for this methodology. Under a first order extreme value condition only, we prove the existence and consistency of the maximum likelihood estimators for the extreme value index and normalization constants within the framework of the block maxima method.

math.PR↗

arXiv2012

Conditional simulations of Brown-Resnick processes

Since many environmental processes such as heat waves or precipitation are spatial in extent, it is likely that a single extreme event affects several locations and the areal modeling of extremes is therefore essential if the spatial dependence of extremes has to be appropriately taken into account. Although some progress has been made to develop a geostatistic of extremes, conditional simulation of max-stable processes is still in its early stage. This paper proposes a framework to get conditional simulations of Brown-Resnick processes. Although closed forms for the regular conditional distribution of Brown-Resnick processes were recently found, sampling from this conditional distribution is a considerable challenge as it leads quickly to a combinatorial explosion. To bypass this computational burden, a Markov chain Monte-Carlo algorithm is presented. We test the method on simulated data and give an application to extreme rainfall around Zurich. Results show that the proposed framework provides accurate conditional simulations of Brown-Resnick processes and can handle real-sized problems.

stat.ME↗

arXiv2012

Conditional simulation of extremal Gaussian processes

Recently the regular conditional distributions of max-infinitely divisible processes were derived by \citet{Dombry2011} and although these conditional distributions have complicated closed forms, \citet{Dombry2011b} introduce an algorithm to get conditional realizations of Brown-Resnick processes. In this paper we derive the regular conditional distributions of the max-stable process introduced by \citet{Schlather2002} and adapt the framework of \citet{Dombry2011b} to this specific process. We test the methods on simulated data and give an application to extreme temperatures in Switzerland. Results show that the proposed sampling scheme provide accurate conditional simulations and can handle real-sized problems.

math.ST↗

arXiv2012

Conditional simulation of max-stable processes

stat.ME↗

arXiv2012

Regular conditional distributions of max infinitely divisible processes

This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{η(t)\}_{t\in T}$ given observations $\{η(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is the point process representation of max-infinitely divisible processes by Giné, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent related result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. We believe this work offers new tools and perspective for prediction in extreme value theory together with numerous potential applications.

math.PR↗