Lennartz Sabine, Bunde Armin
Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jun;79(6 Pt 2):066101. doi: 10.1103/PhysRevE.79.066101. Epub 2009 Jun 5.
Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records x_{i} of length N that consist of a long-term correlated component (1-a)y_{i} characterized by a correlation exponent gamma , 0<gamma<1 , and a white-noise component aeta_{i} , 0< or =a< or =1 . We show that the autocorrelation function C_{N}(s) has the general form C_{N}(s)=[C_{infinity}(s)-E_{a}]/(1-E_{a}) , where C_{infinity}(0)=1 , C_{infinity}(s>0)=B_{a}s;{-gamma} , and E_{a}={2B_{a}/[(2-gamma)(1-gamma)]}N;{-gamma}+O(N;{-1}) . The finite-size parameter E_{a} also occurs in related quantities, for example, in the variance Delta_{N};{2}(s) of the local mean in time windows of length s : Delta_{N};{2}(s)=[Delta_{infinity};{2}(s)-E_{a}]/(1-E_{a}) . For purely long-term correlated data B_{0} congruent with(2-gamma)(1-gamma)/2 yielding E_{0} congruent withN;{-gamma} , and thus C_{N}(s)=[(2-gamma)(1-gamma)/2s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] and Delta_{N};{2}(s)=[s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] . We show how to estimate E_{a} and C_{infinity}(s) from a given data set and thus how to obtain accurately the exponent gamma and the amount of white noise a .
长期记忆在自然界中普遍存在，并且对自然灾害的发生有着重要影响，但其检测常常因所考虑记录的长度较短以及数据中的加性白噪声而变得复杂。在此，我们研究长度为(N)的合成高斯分布记录(x_{i})，它由一个具有相关指数(\gamma)（(0\lt\gamma\lt1)）的长期相关分量((1 - a)y_{i})和一个白噪声分量(a\eta_{i})（(0\leq a\leq1)）组成。我们表明自相关函数(C_{N}(s))具有一般形式(C_{N}(s)=[C_{\infty}(s)-E_{a}]/(1 - E_{a}))，其中(C_{\infty}(0)=1)，(C_{\infty}(s\gt0)=B_{a}s^{-\gamma})，并且(E_{a}={2B_{a}/[(2 - \gamma)(1 - \gamma)]}N^{-\gamma}+O(N^{-1}))。有限尺寸参数(E_{a})也出现在相关量中，例如，在长度为(s)的时间窗口内局部均值的方差(\Delta_{N}^{2}(s))中：(\Delta_{N}^{2}(s)=[\Delta_{\infty}^{2}(s)-E_{a}]/(1 - E_{a}))。对于纯长期相关数据，(B_{0})等于((2 - \gamma)(1 - \gamma)/2)，从而(E_{0})等于(N^{-\gamma})，因此(C_{N}(s)=[((2 - \gamma)(1 - \gamma)/2)s^{-\gamma}-N^{-\gamma}]/[1 - N^{-\gamma}])且(\Delta_{N}^{2}(s)=[s^{-\gamma}-N^{-\gamma}]/[1 - N^{-\gamma}])。我们展示了如何从给定数据集中估计(E_{a})和(C_{\infty}(s))，进而如何准确获得指数(\gamma)和白噪声量(a)。
