1,269 research outputs found
Spectral tail processes and max-stable approximations of multivariate regularly varying time series
A regularly varying time series as introduced in Basrak and Segers (2009) is
a (multivariate) time series such that all finite dimensional distributions are
multivariate regularly varying. The extremal behavior of such a process can
then be described by the index of regular variation and the so-called spectral
tail process, which is the limiting distribution of the rescaled process, given
an extreme event at time 0. As shown in Basrak and Segers (2009), the
stationarity of the underlying time series implies a certain structure of the
spectral tail process, informally known as the "time change formula". In this
article, we show that on the other hand, every process which satisfies this
property is in fact the spectral tail process of an underlying stationary
max-stable process. The spectral tail process and the corresponding max-stable
process then provide two complementary views on the extremal behavior of a
multivariate regularly varying stationary time series
-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
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