scholarly journals One-component regular variation and graphical modeling of extremes

2016 ◽  
Vol 53 (3) ◽  
pp. 733-746 ◽  
Author(s):  
Adrien Hitz ◽  
Robin Evans
Keyword(s):  
Graphical Models ◽  
Regular Variation ◽  
Conditional Independence ◽  
Probability Distributions ◽  
Value Theory ◽  
High Dimensional ◽  
Regularly Varying Functions ◽  
Graphical Modeling ◽  
Multivariate Extreme Value Theory ◽  
Regularly Varying

AbstractThe 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.

scholarly journals One- versus multi-component regular variation and extremes of Markov trees

2020 ◽  
Vol 52 (3) ◽  
pp. 855-878
Author(s):  
Johan Segers
Keyword(s):  
Time Series ◽  
Random Vector ◽  
Regular Variation ◽  
Conditional Independence ◽  
Conditional Distribution ◽  
The Self ◽  
Stationary Time Series ◽  
Markov Tree ◽  
Regularly Varying ◽  
Coupled Random Walks

AbstractA Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, 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 a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

scholarly journals Gaussian Covariance Faithful Markov Trees

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Dhafer Malouche ◽  
Bala Rajaratnam
Keyword(s):  
Probability Distribution ◽  
Graphical Models ◽  
Conditional Independence ◽  
High Dimensional ◽  
Important Class ◽  
Graph Models ◽  
Selection Procedures ◽  
Gaussian Distributions ◽  
Discrete Object ◽  
Covariance Graph

Graphical models are useful for characterizing conditional and marginal independence structures in high-dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an edge between any pair of variables implies marginal independence. Covariance graph models also represent more complex conditional independence relationships between subsets of variables. When the covariance graph captures or reflects all the conditional independence statements present in the probability distribution, the latter is said to befaithfulto its covariance graph—though in general this is not guaranteed. Faithfulness however is crucial, for instance, in model selection procedures that proceed by testing conditional independences. Hence, an analysis of the faithfulness assumption is important in understanding the ability of the graph, a discrete object, to fully capture the salient features of the probability distribution it aims to describe. In this paper, we demonstrate that multivariate Gaussian distributions that have trees as covariance graphs are necessarily faithful.

scholarly journals Decompositions of dependence for high-dimensional extremes

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 587-604 ◽  
Author(s):  
D Cooley ◽  
E Thibaud
Keyword(s):  
Vector Space ◽  
Regular Variation ◽  
Tail Dependence ◽  
Positive Semidefinite ◽  
High Dimensional ◽  
Financial Returns ◽  
Random Vectors ◽  
Completely Positive ◽  
Positive Orthant ◽  
Regularly Varying

Summary We propose two decompositions that help to summarize and describe high-dimensional tail dependence within the framework of regular variation. We use a transformation to define a vector space on the positive orthant and show that transformed-linear operations applied to regularly-varying random vectors preserve regular variation. We summarize tail dependence via a matrix of pairwise tail dependence metrics that is positive semidefinite; eigendecomposition allows one to interpret tail dependence in terms of the resulting eigenbasis. This matrix is completely positive, and one can easily construct regularly-varying random vectors that share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall and financial returns data.

Probabilistic Graphical Modeling of Use Stage Energy Consumption: A Lightweight Vehicle Example1

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Cassandra Telenko ◽  
Carolyn C. Seepersad
Keyword(s):  
Energy Consumption ◽  
Graphical Models ◽  
Environmental Performance ◽  
Energy Use ◽  
Energy Savings ◽  
Probability Distributions ◽  
Design Decisions ◽  
Graphical Modeling ◽  
Use Stage ◽  
Usage Context

Although energy consumption during product use can lead to significant environmental impacts, the relationship between a product's usage context and its environmental performance is rarely considered in design evaluations. Traditional analyses rely on broad, average usage conditions and do not differentiate between contexts for which design decisions are highly beneficial and contexts for which the same decision may offer limited benefits or even penalties in terms of environmental performance. In contrast, probabilistic graphical models (PGMs) provide the capability of modeling usage contexts as variable factors. This research demonstrates a method for representing the usage context as a PGM and illustrates it with a lightweight vehicle design example. Factors such as driver behavior, alternative driving schedules, and residential density are connected by conditional probability distributions derived from publicly available data sources. Unique scenarios are then defined as sets of conditions on these factors to provide insight into sources of variability in lifetime energy use. The vehicle example demonstrates that implementation of realistic usage scenarios via a PGM can provide a much higher fidelity investigation of use stage energy savings than commonly found in the literature and that, even in the case of a universally beneficial design decisions, distinct scenarios can have significantly different implications for the effectiveness of lightweight vehicle designs.

scholarly journals Integrating additional knowledge into the estimation of graphical models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yunqi Bu ◽  
Johannes Lederer
Keyword(s):  
Graphical Models ◽  
Tuning Parameter ◽  
Functional Magnetic Resonance ◽  
Resonance Imaging ◽  
Graphical Modeling ◽  
Standard Methods ◽  
Neighborhood Selection ◽  
Selection For ◽  
The Brain

Abstract Graphical models such as brain connectomes derived from functional magnetic resonance imaging (fMRI) data are considered a prime gateway to understanding network-type processes. We show, however, that standard methods for graphical modeling can fail to provide accurate graph recovery even with optimal tuning and large sample sizes. We attempt to solve this problem by leveraging information that is often readily available in practice but neglected, such as the spatial positions of the measurements. This information is incorporated into the tuning parameter of neighborhood selection, for example, in the form of pairwise distances. Our approach is computationally convenient and efficient, carries a clear Bayesian interpretation, and improves standard methods in terms of statistical stability. Applied to data about Alzheimer’s disease, our approach allows us to highlight the central role of lobes in the connectivity structure of the brain and to identify an increased connectivity within the cerebellum for Alzheimer’s patients compared to other subjects.

scholarly journals Clustering of Maxima: Spatial Dependencies among Heavy Rainfall in France

2013 ◽  
Vol 26 (20) ◽  
pp. 7929-7937 ◽  
Author(s):  
Elsa Bernard ◽  
Philippe Naveau ◽  
Mathieu Vrac ◽  
Olivier Mestre
Keyword(s):  
Spatial Patterns ◽  
Extreme Value Theory ◽  
Heavy Rainfall ◽  
Spatial Clustering ◽  
Relevant Information ◽  
Value Theory ◽  
Research Field ◽  
Multivariate Extreme Value Theory ◽  
Spatial Dependencies ◽  
The Mean

Abstract One of the main objectives of statistical climatology is to extract relevant information hidden in complex spatial–temporal climatological datasets. To identify spatial patterns, most well-known statistical techniques are based on the concept of intra- and intercluster variances (like the k-means algorithm or EOFs). As analyzing quantitative extremes like heavy rainfall has become more and more prevalent for climatologists and hydrologists during these last decades, finding spatial patterns with methods based on deviations from the mean (i.e., variances) may not be the most appropriate strategy in this context of studying such extremes. For practitioners, simple and fast clustering tools tailored for extremes have been lacking. A possible avenue to bridging this methodological gap resides in taking advantage of multivariate extreme value theory, a well-developed research field in probability, and to adapt it to the context of spatial clustering. In this paper, a novel algorithm based on this plan is proposed and studied. The approach is compared and discussed with respect to the classical k-means algorithm throughout the analysis of weekly maxima of hourly precipitation recorded in France (fall season, 92 stations, 1993–2011).

Joint estimation of multiple high-dimensional Gaussian copula graphical models

2017 ◽  
Vol 59 (3) ◽  
pp. 289-310 ◽  
Author(s):  
Yong He ◽  
Xinsheng Zhang ◽  
Jiadong Ji ◽  
Bin Liu
Keyword(s):  
Graphical Models ◽  
Joint Estimation ◽  
Gaussian Copula ◽  
High Dimensional

Pseudo-Regularly Varying Functions and Generalized Renewal Processes

2018 ◽  
Author(s):  
Valeriĭ V. Buldygin ◽  
Karl-Heinz Indlekofer ◽  
Oleg I. Klesov ◽  
Josef G. Steinebach
Keyword(s):  
Renewal Processes ◽  
Regularly Varying Functions ◽  
Regularly Varying
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