粗粒化对长程相关和反相关信号标度行为的影响
Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals
摘要：
我们研究了各种粗粒化方法如何影响长程幂律相关和反相关信号的标度特性，通过去趋势波动分析进行量化。具体而言，对于信号幅度的粗粒化，我们考虑（i）地板，（ii）对称和（iii）中心对称粗粒化方法。我们发现，反相关信号的幅度粗粒化导致在大尺度上的随机行为的交叉，并且随着粗粒化分区间隔$Δ$的宽度的增加，这种交叉移动到中间和小尺度。相比之下，正相关信号的缩放受粗粒度化的影响较小，当$Δ<1$时没有可观察到的变化，而对于$Δ>1$，交叉出现在小尺度处，并且随着$Δ$的增加而移动到中等尺度和大尺度。对于基于地板和对称方法的非常粗糙的粗粒化（$Δ>3$），交叉的位置稳定，与中心对称方法相反，中心对称方法中交叉连续跨尺度移动并导致所有尺度上的随机行为，从而表明与地板和对称方法相比，中心对称的效果要强得多。对于粗粒度的时间，其中数据点在非重叠的时间窗口中平均，我们发现，反相关和正相关信号的缩放实际上是保留的。我们的模拟结果是有用的符号序列的相关性和标度性质的正确解释。
英文摘要：
We investigate how various coarse-graining methods affect the scaling properties of long-range power-law correlated and anti-correlated signals, quantified by the detrended fluctuation analysis. Specifically, for coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii) the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find, that for anti-correlated signals coarse-graining in the magnitude leads to a crossover to random behavior at large scales, and that with increasing the width of the coarse-graining partition interval $Δ$ this crossover moves to intermediate and small scales. In contrast, the scaling of positively correlated signals is less affected by the coarse-graining, with no observable changes when $Δ<1$, while for $Δ>1$ a crossover appears at small scales and moves to intermediate and large scales with increasing $Δ$. For very rough coarse-graining ($Δ>3$) based on the Floor and Symmetry methods, the position of the crossover stabilizes, in contrast to the Centro-Symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales, thus indicating a much stronger effect of the Centro-Symmetry compared to the Floor and the Symmetry methods. For coarse-graining in time, where data points are averaged in non-overlapping time windows, we find that the scaling for both anti-correlated and positively correlated signals is practically preserved. The results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences.
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