Related papers: Extremal clustering and cluster counting for spati…
Clustering, or grouping, dataset elements based on similarity can be used not only to classify a dataset into a few categories, but also to approximate it by a relatively large number of representative elements. In the latter scenario,…
We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological…
Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In…
We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator…
We use point processes theory to describe the asymptotic distribution of all upper order statistics for observations collected at renewal times. As a corollary, we obtain limiting theorems for corresponding extremal processes.
This paper presents and analyzes an approach to cluster-based inference for dependent data. The primary setting considered here is with spatially indexed data in which the dependence structure of observed random variables is characterized…
We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson…
The cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of…
We introduce the cluster index of a multivariate regularly varying stationary sequence and characterize the index in terms of the spectral tail process. This index plays a major role in limit theory for partial sums of regularly varying…
Clustering is a well-known unsupervised machine learning approach capable of automatically grouping discrete sets of instances with similar characteristics. Constrained clustering is a semi-supervised extension to this process that can be…
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable…
Motivated by the fundamental problem of measuring species diversity, this paper introduces the concept of a cluster structure to define an exchangeable cluster probability function that governs the joint distribution of a random count and…
Clustering provides a common means of identifying structure in complex data, and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields. A natural question is how many clusters are appropriate…
We establish sharp tail asymptotics for component-wise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the…
It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate…
We investigate how large deviations events cluster in the framework of an infinite moving average process with light-tailed noise and long memory. The long memory makes clusters larger, and the asymptotic behaviour of the size of the…
Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails, including both power-like tails and certain lighter tails, e.g.…
As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles…
The $k$-means clustering algorithm and its variant, the spherical $k$-means clustering, are among the most important and popular methods in unsupervised learning and pattern detection. In this paper, we explore how the spherical $k$-means…
A central issue in the theory of extreme values focuses on suitable conditions such that the well-known results for the limiting distributions of the maximum of i.i.d. sequences can be applied to stationary ones. In this context, the…