Generalized extreme value statistics and sum of correlated variables

Abstract

Abstract. We show that generalised extreme value statistics -the statistics of the k th largest value among a large set of random variables-can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Fréchet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalisations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

The Weibull manifold in low-level image processing: An application to automatic image focusing

“…Quite recently [18,16] have established a link between the Weibull distribution and difference filters on images. In [18] the authors suggested the connection to extreme value theory [21], via the properties of sums of correlated variables [2], whereas [16] follows the alternative path of fragmentation theory [7]. We have also subsequently exploited these ideas and presented a connection between the dihedral filters from Section 2.1 and extreme value theory in [49].…”

“…The responses of difference-based filter functions on image data, are known to be Weibull distributed [16,2,49]. As such, every filtered image may be represented as a single, unique point on a statistical manifold.…”

Spatio-chromatic Image Content Descriptors and Their Analysis Using Extreme Value Theory

“…If we consider u(t), s(n) as stochastic processes and select a finite number N of random samples u 1 ,...u N , then their joint distribution J(u 1 ,...,u N ) and the distribution Y (s N ) of s N , depend on the underlying original distribution F (X N ). At this point we may pose two questions: In [7] the authors have demonstrated that under certain conditions on Y (s) the possible limiting forms of Φ(s) are the familiar forms in (1) and depend on the tail behaviour of F (X) at large X. In our particular case, we use as units U the black-box that computes the absolute value of the filter result vectors from the irreducible representations of the dihedral groups.…”

“…In addition, these sums are calculated over a small, finite neighbourhood, and for this reason, the random variables are highly correlated. In short, the output for each filter has a form similar to the sums described in [7], and so it should be possible to use the EVT to model their distribution. As we will show experimentally later, the EVT models in (1) provide a good fit to our filtered data, which is a strong indication that the requirements for EVT equivalence from [7] generally hold.…”

“…In short, the output for each filter has a form similar to the sums described in [7], and so it should be possible to use the EVT to model their distribution. As we will show experimentally later, the EVT models in (1) provide a good fit to our filtered data, which is a strong indication that the requirements for EVT equivalence from [7] generally hold. We also note, that since we are always dealing with positive quantities (norms of sums) that have a strictly positive support, we do not use the Gumbel model, which is unbounded, but only the Weibull and Fréchet models.…”