One-component regular variation and graphical modeling of extremes
Adrien S. Hitz,Robin J. Evans +1 more
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In this article, the problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density, and an approach based on graphical models which is suitable for high-dimensional vectors is proposed.Abstract:
The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.read more
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Sparse Structures for Multivariate Extremes
TL;DR: The different forms of extremal dependence that can arise between the largest observations of a multivariate random vector are described and identification of groups of variables which can be concomitantly extreme is addressed.
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Sebastian Engelke,Adrien S. Hitz +1 more
TL;DR: In this article, the authors introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes, and show that the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices.
Journal ArticleDOI
Graphical models for extremes
Sebastian Engelke,Adrien S. Hitz +1 more
TL;DR: In this paper, the authors introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes, and show that the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices.
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kth-order Markov extremal models for assessing heatwave risks
TL;DR: In this article, a kth-order Markov model framework was proposed to estimate extremal quantities, such as the probability of a heatwave event lasting as long as the devastating European 2003 heatwave.
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One- versus multi-component regular variation and extremes of Markov trees
TL;DR: In this article, the authors show that the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called tail tree.
References
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Book
Probability and Measure
TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.
Journal ArticleDOI
Sparse inverse covariance estimation with the graphical lasso
TL;DR: Using a coordinate descent procedure for the lasso, a simple algorithm is developed that solves a 1000-node problem in at most a minute and is 30-4000 times faster than competing methods.
Book
Probability: Theory and Examples
TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Book
Graphical Models, Exponential Families, and Variational Inference
TL;DR: The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.
Book
Extreme Values, Regular Variation, and Point Processes
TL;DR: In this paper, the authors present a survey of the main domains of attraction and norming constants in point processes and point processes, and their relationship with multivariate extremity processes.