Théorie des valeurs extrêmes

Résumé

La théorie des valeurs extrêmes est une branche des statistiques qui s'intéresse aux valeurs extrêmes des distributions de probabilité. Elle a été développée par Émil Julius Gumbel. Méthodes La théorie des valeurs extrêmes permet de connaître le comportement asymptotique des maxima de valeurs prises par les valeurs de variables aléatoires identiquement distribuées et indépendantes. Cette loi comporte des paramètres que l'on peut estimer soit en se basant sur les valeurs extrêmes prises dans des blocs de taille fixe des données à disposition (méthode des maxima), soit en s'intéressant à la distribution des données supérieures à un certain seuil (méthode des excès). Pour pouvoir être appliquée, la théorie des valeurs extrêmes doit donc disposer de beaucoup de données. Applications La théorie des valeurs extrêmes est appliquée en hydrologie pour prévoir les crues, en océanographie dans l'étude des vagues

Source officielle

https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_valeurs_extr%C3%AAmes

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Influence de la Transjurane sur le réseau hydrologique de l’Ajoie

Les premiers travaux de construction de l’A16 en Ajoie ont débuté en 1995. La région traversée étant karstique, l’Office des Ponts et Chaussées du Canton du Jura s’est interrogé dès l’origine du projet sur l’influence de l’autoroute sur le régime hydrologique. Depuis 1989, un réseau systématique de mesures des écoulements superficiels et souterrains a donc été installé sur les zones concernées par le tronçon autoroutier. L’objectif de ce projet de master est de déterminer si un impact de la Transjurane en l’Ajoie est visible et quantifiable à partir de ces données récoltées sur plus de vingt ans. Ainsi, les périodes précédant et suivant le début des travaux de gros oeuvre sont comparés grâce notamment à l’analyse des valeurs extrêmes, de l’autocorrélation et des paramètres statistiques. Parallèlement, un modèle hydrologique (HydroRoute) semi-distribué déjà existant est calé afin de mettre en évidence un potentiel changement du régime hydrologique. Pour les sous-bassins fortement karstiques, HydroRoute étant insuffisant, un autre modèle plus adapté aux caractéristiques locales est élaboré.

2013

Chargement

Contributions to Likelihood-Based Modelling of Extreme Values

Léo Raymond-Belzile

xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis. The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed. In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package. The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.

EPFL2019

Chargement

This thesis is a contribution to financial statistics. One of the principal concerns of investors is the evaluation of portfolio risk. The notion of risk is vague, but in finance it is always linked to possible losses. In this thesis, we present some measures allowing the valuation of risk with the help of Bayesian methods. An exploratory analysis of data is presented to describe the sampling properties of financial time series. This analysis allows us to understand the origins of the daily returns studied in this thesis. Moreover, a discussion of different models is presented. These models make strong assumptions on investor behaviour, which are not always satisfied. This exploratory analysis shows some differences between the behaviour anticipated under equilibrium models, and that of real data. The Bayesian approach has been chosen because it allows one to incorporate all the variability, in particular that associated with model choice. The models studied in this thesis allow one to take heteroskedasticity into account, as well as particular shapes of the tails of returns. ARCH type models and models based on extreme value theory are studied. One original aspect of this thesis is its use of Bayesian analysis to detect change points in financial time series. We suppose that a market has two phases, and that it switches from a state to the other at random. Another new contribution is a model integrating heteroskedasticity and time dependence of extreme values, by superposition of the model proposed by Bortot and Coles (2003) and a GARCH process. This thesis uses simulation intensively for the estimation of risk measures. The drawback of simulation is the amount of time needed to obtain accurate estimates. However, simulation allows one to produce results when direct calculation is not feasible. For example, simulation allows one to compute risk estimates for time horizons greater than one day. The methods presented in this thesis are illustrated on simulated data, and on real data from European and American markets. This thesis involved the construction of a library containing C and S code to perform risk analysis using GARCH and extreme value theory models. The results show that model uncertainty can be incorporated, and that risk measures for time horizons greater than one can be obtained by simulation. The methods presented in this thesis have a natural representation involving conditioning. Thus, they permit the computation of both conditional and unconditional risk estimates. Three methods are described: the GARCH method; the two-state GARCH method; and the HBC method. Unconditional risk estimation using the GARCH method is satisfactory on data which seem stationary, but not reliable on data which are non-stationary, such as data with change points. The two-state GARCH model does a little better, but gives very satisfactory results when the risk is estimated conditionally on time. The HBC method does not give satisfactory results.

EPFL2006

Chargement

EPFL2006

Contributions to Likelihood-Based Modelling of Extreme Values

Léo Raymond-Belzile

EPFL2019