Altmann Eduardo G, Kantz Holger
Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71(5 Pt 2):056106. doi: 10.1103/PhysRevE.71.056106. Epub 2005 May 12.
The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent gamma, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different gamma, which shows a lack of invariance under the change of observables.
极端事件之间的重现时间一直是许多不同科学领域统计分析的核心要点。同时，庞加莱重现时间已被广泛用于表征非线性动力系统。我们比较了这些统计方法的主要特性，指出它们对时间序列中进行的重现分析的影响。特别是，我们分析了平均重现时间和重现时间统计量对概率密度函数、观察到重现的区间以及时间序列的时间相关性的依赖性。在长期相关性的情况下，我们验证了拉伸指数分布的有效性，它由指数γ唯一确定，同时表明它仅限于线性长期相关过程的类别。简单的变换能够改变时间序列的相关性，导致具有不同γ的拉伸指数重现时间统计量，这表明在可观测量变化下缺乏不变性。
