Graphical models for extremes

An Intrinsic Characterization of Shannon's and Rényi's Entropy

All characterizations of the Shannon entropy include the so-called chain rule, a formula on a hierarchically structured probability distribution, which is based on at least two elementary distributions. We show that the chain rule can be split into two natural components, the well-known additivity of the entropy in case of cross-products and a variant of the chain rule that involves only a single elementary distribution. The latter is given as a proportionality relation and, hence, allows a vague interpretation as self-similarity, hence intrinsic property of the Shannon entropy. Analogous characterizations are given for the Rényi entropy and its limits, the min-entropy and the Hartley entropy.

Causes of extreme events revealed by Rényi information transfer

Information-theoretic generalization of Granger causality principle, based on evaluation of conditional mutual information, also known as transfer entropy (CMI/TE), is redefined in the framework of Rényi entropy (RCMI/RTE). Using numerically generated data with a defined causal structure and examples of real data from the climate system, it is demonstrated that RCMI/RTE is able to identify the cause variable responsible for the occurrence of extreme values in an effect variable. In the presented example, the Siberian High was identified as the cause responsible for the increased probability of cold extremes in the winter and spring surface air temperature in Europe, while the North Atlantic Oscillation and blocking events can induce shifts of the whole temperature probability distribution.

Estimating changepoints in extremal dependence, applied to aviation stock prices during COVID-19 pandemic

The dependence in the tails of the joint distribution of two random variables is generally assessed using χ-measure, the limiting conditional probability of one variable being extremely high given the other variable is also extremely high. This work is motivated by the structural changes in χ-measure between the daily rate of return (RoR) of the two Indian airlines, IndiGo and SpiceJet, during the COVID-19 pandemic. We model the daily maximum and minimum RoR vectors (potentially transformed) using the bivariate Hüsler-Reiss (BHR) distribution. To estimate the changepoint in the χ-measure of the BHR distribution, we explore two changepoint detection procedures based on the Likelihood Ratio Test (LRT) and Modified Information Criterion (MIC). We obtain critical values and power curves of the LRT and MIC test statistics for low through high values of χ-measure. We also explore the consistency of the estimators of the changepoint based on LRT and MIC numerically. In our data application, for RoR maxima and minima, the most prominent changepoints detected by LRT and MIC are close to the announcement of the first phases of lockdown and unlock, respectively, which are realistic; thus, our study would be beneficial for portfolio optimization in the case of future pandemic situations.

Estimation of multiple networks with common structures in heterogeneous subgroups

Network estimation has been a critical component of high-dimensional data analysis and can provide an understanding of the underlying complex dependence structures. Among the existing studies, Gaussian graphical models have been highly popular. However, they still have limitations due to the homogeneous distribution assumption and the fact that they are only applicable to small-scale data. For example, cancers have various levels of unknown heterogeneity, and biological networks, which include thousands of molecular components, often differ across subgroups while also sharing some commonalities. In this article, we propose a new joint estimation approach for multiple networks with unknown sample heterogeneity, by decomposing the Gaussian graphical model (GGM) into a collection of sparse regression problems. A reparameterization technique and a composite minimax concave penalty are introduced to effectively accommodate the specific and common information across the networks of multiple subgroups, making the proposed estimator significantly advancing from the existing heterogeneity network analysis based on the regularized likelihood of GGM directly and enjoying scale-invariant, tuning-insensitive, and optimization convexity properties. The proposed analysis can be effectively realized using parallel computing. The estimation and selection consistency properties are rigorously established. The proposed approach allows the theoretical studies to focus on independent network estimation only and has the significant advantage of being both theoretically and computationally applicable to large-scale data. Extensive numerical experiments with simulated data and the TCGA breast cancer data demonstrate the prominent performance of the proposed approach in both subgroup and network identifications.

Linking representations for multivariate extremes via a limit set

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.

Spherical clustering in detection of groups of concomitant extremes

There is growing empirical evidence that spherical k-means clustering performs well at identifying groups of concomitant extremes in high dimensions, thereby leading to sparse models. We provide one of the first theoretical results supporting this approach, but also demonstrate some pitfalls. Furthermore, we show that an alternative cost function may be more appropriate for identifying concomitant extremes, and it results in a novel spherical k-principal-components clustering algorithm. Our main result establishes a broadly satisfied sufficient condition guaranteeing the success of this method, albeit in a rather basic setting. Finally, we illustrate in simulations that k-principal-components outperforms k-means in the difficult case of weak asymptotic dependence within the groups.