Eichner Jan F, Kantelhardt Jan W, Bunde Armin, Havlin Shlomo
Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016130. doi: 10.1103/PhysRevE.73.016130. Epub 2006 Jan 25.
Many natural records exhibit long-term correlations characterized by a power-law decay of the autocorrelation function, C(s) approximately s-gamma, with time lag s and correlation exponent 0<gamma<1. We study how the presence of such correlations affects the statistics of the extreme events, i.e., the maximum values of the signal within time segments of the fixed duration R. We find numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the deviations for finite R depend on the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. The last item implies that conditional mean maxima and conditional maxima distributions (with the value of the previous maximum as condition) should be considered for an improved extreme event prediction. We provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long-term correlated records.
许多自然记录呈现出长期相关性，其特征是自相关函数C(s)随时间滞后s呈幂律衰减，即C(s)≈s^(-γ)，相关指数0<γ<1。我们研究了这种相关性的存在如何影响极端事件的统计特性，即固定持续时间R的时间段内信号的最大值。我们通过数值计算发现：(i) 对于大的R，最大值的积分分布函数收敛到与不相关信号类似的耿贝尔分布；(ii) 有限R时的偏差取决于记录的初始分布及其相关特性；(iii) 最大值序列呈现出与原始数据类似的长期相关性，最显著的是(iv) 最大值分布以及平均最大值显著依赖于历史，特别是前一个最大值。最后一点意味着为了改进极端事件预测，应考虑条件平均最大值和条件最大值分布（以前一个最大值的值为条件）。我们给出的迹象表明，平均最大值对前一个最大值的这种依赖性在观测到的长期相关记录中也会出现。
