Moloney Nicholas R, Faranda Davide, Sato Yuzuru
Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom.
Laboratoire de Sciences du Climat et de l'Environnement, UMR 8212 CEA-CNRS-UVSQ, IPSL, Universite Paris-Saclay, 91191 Gif-sur-Yvette, France.
Chaos. 2019 Feb;29(2):022101. doi: 10.1063/1.5079656.
For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of independently and identically distributed degrees of freedom, (ii) the multiplicity of a compound Poisson point process, and (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems.
对于一大类平稳时间序列，极值理论为罕见事件提供了极限分布。该理论不仅描述了极值的大小，还描述了它们出现的频率。在实践中，经常观察到极值在时间上聚集。通过所谓的极值指数，极值理论也考虑了这种短程聚集。本综述通过对极值指数的一些直观解释来介绍它。因此，根据上下文，极值指数可能表示：（i）独立同分布自由度的损失；（ii）复合泊松点过程的多重性；（iii）极端簇的平均持续时间的倒数。最近，极值指数还被用于量化（iv）动力系统中不稳定不动点周围的复发情况。
