20,250 research outputs found
Linear prediction and partial tail correlation for extremes
2022 Summer.Includes bibliographical references.This dissertation consists of three main studies for extreme value analyses: linear prediction for extremes, uncertainty quantification for predictions, and investigating conditional relationships between variables at their extreme levels. We employ multivariate regular variation to provide a framework for modeling dependence in the upper tail, which is assumed to be a direction of interest. Cooley and Thibaud [2019] consider transformed-linear operations to define a vector space on the nonnegative orthant and show regular variation is preserved by these transformed-linear operations. Extending the approach of Cooley and Thibaud [2019], we first consider the problem of performing prediction when observed values are at extreme levels. This linear approach is motivated by the limitation that traditional extreme value models have difficulties fitting a high dimensional extreme value model. We construct an inner product space of nonnegative random variables from transformed-linear combinations of independent regularly varying random variables. Rather than fully characterizing extremal dependence in high dimensions, we summarize tail behavior via a matrix of pairwise tail dependencies. The projection theorem yields the optimal transformed-linear predictor, which has a similar form to the best linear unbiased predictor in non-extreme prediction. We then quantify uncertainty for the prediction of extremes by using information contained in the tail pairwise dependence matrix. We create the 95% prediction interval based on the geometry of regular variation. We show that the prediction intervals have good coverage in a simulation study as well as in two applications: prediction of high NO2 air pollution levels, and prediction of large financial losses. We also compare prediction intervals with a linear approach to ones with a parametric approach. Lastly, we develop the novel notion of partial tail correlation via projection theorem in the inner product space. Partial tail correlations are the analogue of partial correlations in non-extreme statistics but focus on extremal dependence. Partial tail correlation can be represented by the inner product of prediction errors associated with the previously defined best transformed-linear prediction for extremes. We find a connection between the partial tail correlation and the inverse matrix of tail pairwise dependencies. We then develop a hypothesis test for zero elements in the inverse extremal matrix. We apply the idea of partial tail correlation to assess flood risk in application to extreme river discharges in the upper Danube River basin. We compare the extremal graph constructed from the idea of the partial tail correlation to physical flow connections on the Danube
-means clustering of extremes
The -means clustering algorithm and its variant, the spherical -means
clustering, are among the most important and popular methods in unsupervised
learning and pattern detection. In this paper, we explore how the spherical
-means algorithm can be applied in the analysis of only the extremal
observations from a data set. By making use of multivariate extreme value
analysis we show how it can be adopted to find "prototypes" of extremal
dependence and we derive a consistency result for our suggested estimator. In
the special case of max-linear models we show furthermore that our procedure
provides an alternative way of statistical inference for this class of models.
Finally, we provide data examples which show that our method is able to find
relevant patterns in extremal observations and allows us to classify extremal
events
Two-dimensional Black Holes in a Higher Derivative Gravity and Matrix Model
We construct perturbatively a class of charged black hole solutions in type
0A string theory with higher derivative terms. They have extremal limit, where
the solution interpolates smoothly between near horizon AdS_2 geometry and the
asymptotic linear dilaton geometry. We compute the free energy and the entropy
of those solution using various methods. In particular, we show that there is
no correction in the leading term of the free energy in the large charge limit.
This supports the duality of the type 0A strings on the extremal black hole and
the 0A matrix model in which the tree level free energy is exact without any
alpha' corrections.Comment: Latex, 19 page
Extremal t processes: Elliptical domain of attraction and a spectral representation
The extremal t process was proposed in the literature for modeling spatial
extremes within a copula framework based on the extreme value limit of
elliptical t distributions (Davison, Padoan and Ribatet (2012)). A major
drawback of this max-stable model was the lack of a spectral representation
such that for instance direct simulation was infeasible. The main contribution
of this note is to propose such a spectral construction for the extremal t
process. Interestingly, the extremal Gaussian process introduced by Schlather
(2002) appears as a special case. We further highlight the role of the extremal
t process as the maximum attractor for processes with finite-dimensional
elliptical distributions. All results naturally also hold within the
multivariate domain
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