The Center will propose statistical models of the mechanism underlying severe events of complex systems and provide accurate quantitative assessment of their risks. Such models will include extreme value theory, which characterizes the behavior of extremes or rare events in terms of their frequency, magnitude, and co-occurrence. The modeling of low-frequency, high-impact, and dependent shocks is particularly suitable to analyze the current situation, as it applies to both climate extremes and economic and financial crises.

Distinguishing Cause from Effect Using Quantiles: Bivariate Quantile Causal Discovery

Natasa Tagasovska, Valérie Chavez-Demoulin, Thibault Vatter

Abstract: Causal inference using observational data is challenging, especially in the bivariate case. Through the minimum description length principle, we link the postulate of independence between the generating mechanisms of the cause and of the effect given the cause to quantile regression. Based on this theory, we develop Quantile Causal Discovery (QCD), a new method to uncover causal relationships. Because it uses multiple quantile levels instead of the conditional mean only, QCD is adaptive not only to additive, but also to multiplicative or even location-scale generating mechanisms. To illustrate the effectiveness of our approach, we perform an extensive empirical comparison on both synthetic and real datasets. This study show  that QCD is robust across different implementations of the method (i.e., the quantile regression), computationally efficient, and compares favorably to state-of-the-art methods.

Causal mechanism of extreme river discharges in the upper Danube basin network

Linda Mhalla, Valérie Chavez-Demoulin, Debbie J. Dupuis

Abstract: Extreme hydrological events in the Danube river basin may severely impact human populations, aquatic organisms, and economic activity. One often characterizes the joint structure of the extreme events using the theory of multivariate and spatial extremes and its asymptotically justified models. There is interest however in cascading extreme events and whether one event causes another. In this paper, we argue that an improved understanding of the mechanism underlying severe events is achieved by combining extreme value modelling and causal discovery. We construct a causal inference method relying on the notion of the Kolmogorov complexity of extreme conditional quantiles. Tail quantities are derived using multivariate extreme value models and causal-induced asymmetries in the data are explored through the minimum description length principle. Our CausEV, for Causality for Extreme Values, approach uncovers causal relations between summer extreme river discharges in the upper Danube basin and finds significant causal links between the Danube and its Alpine tributary Lech.

Exceedance-based nonlinear regression of tail dependence
Linda Mhalla, Thomas Opitz, Valérie Chavez-Demoulin

Abstract: The probability and structure of co-occurrences of extreme values in multivariate data may critically depend on auxiliary information provided by covariates. In this contribution, we develop a flexible generalized additive modeling framework based on high threshold exceedances for estimating covariate-dependent joint tail characteristics for regimes of asymptotic dependence and asymptotic independence. The framework is based on suitably defined marginal pretransformations and projections of the random vector along the directions of the unit simplex, which lead to convenient univariate representations of multivariate exceedances based on the exponential distribution. Good performance of our estimators of a nonparametrically designed influence of covariates on extremal coefficients and tail dependence coefficients are shown through a simulation study. We illustrate the usefulness of our modeling framework on a large dataset of nitrogen dioxide measurements recorded in France between 1999 and 2012, where we use the generalized additive framework for modeling marginal distributions and tail dependence in monthly maxima. Our results imply asymptotic independence of data observed at different stations, and we find that the estimated coefficients of tail dependence decrease as a function of spatial distance and show distinct patterns for different years and for different types of stations (traffic vs. background).

Regression‐type models for extremal dependence
Linda Mhalla, Miguel de Carvalho, and Valérie Chavez‐Demoulin
Scandinavian Journal of Statistics (2019) 46(4), 1141-1167

Abstract: We propose a vector generalized additive modeling framework for taking into account the effect of covariates on angular density functions in a multivariate extreme value context. The proposed methods are tailored for settings where the dependence between extreme values may change according to covariates. We devise a maximum penalized log‐likelihood estimator, discuss details of the estimation procedure, and derive its consistency and asymptotic normality. The simulation study suggests that the proposed methods perform well in a wealth of simulation scenarios by accurately recovering the true covariate‐adjusted angular density. Our empirical analysis reveals relevant dynamics of the dependence between extreme air temperatures in two alpine resorts during the winter season.

Non-Linear Models for Extremal Dependence
Linda Mhalla, Valérie Chavez-Demoulin, and Philippe Naveau
Journal of Multivariate Analysis (2017) 159, 49-66

Abstract: The dependence structure of max-stable random vectors can be characterized by their Pickands dependence function. In many applications, the extremal dependence measure varies with covariates. We develop a flexible, semi-parametric method for the estimation of non-stationary multivariate Pickands dependence functions. The proposed construction is based on an accurate max-projection allowing to pass from the multivariate to the univariate setting and to rely on the generalized additive modeling framework. In the bivariate case, the resulting estimator of the Pickands function is regularized using constrained median smoothing B-splines, and bootstrap variability bands are constructed. In higher dimensions, we tailor our approach to the estimation of the extremal coefficient. An extended simulation study suggests that our estimator performs well and is competitive with the standard estimators in the absence of covariates. We apply the new methodology to a temperature dataset in the U.S. where the extremal dependence is linked to time and altitude.