Podobnik Boris, Jiang Zhi-Qiang, Zhou Wei-Xing, Stanley H Eugene
Faculty of Civil Engineering, University of Rijeka, 51000 Rijeka, Croatia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 2):066118. doi: 10.1103/PhysRevE.84.066118. Epub 2011 Dec 22.
For stationary time series, the cross-covariance and the cross-correlation as functions of time lag n serve to quantify the similarity of two time series. The latter measure is also used to assess whether the cross-correlations are statistically significant. For nonstationary time series, the analogous measures are detrended cross-correlations analysis (DCCA) and the recently proposed detrended cross-correlation coefficient, ρ(DCCA)(T,n), where T is the total length of the time series and n the window size. For ρ(DCCA)(T,n), we numerically calculated the Cauchy inequality -1 ≤ ρ(DCCA)(T,n) ≤ 1. Here we derive -1 ≤ ρ DCCA)(T,n) ≤ 1 for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of ρ(DCCA) within which the cross-correlations become statistically significant. For overlapping windows we numerically determine-and for nonoverlapping windows we derive--that the standard deviation of ρ(DCCA)(T,n) tends with increasing T to 1/T. Using ρ(DCCA)(T,n) we show that the Chinese financial market's tendency to follow the U.S. market is extremely weak. We also propose an additional statistical test that can be used to quantify the existence of cross-correlations between two power-law correlated time series.
对于平稳时间序列，互协方差和作为时间滞后n的函数的互相关用于量化两个时间序列的相似性。后一种度量也用于评估互相关是否具有统计显著性。对于非平稳时间序列，类似的度量是去趋势互相关分析（DCCA）和最近提出的去趋势互相关系数ρ(DCCA)(T,n)，其中T是时间序列的总长度，n是窗口大小。对于ρ(DCCA)(T,n)，我们通过数值计算了柯西不等式-1≤ρ(DCCA)(T,n)≤1。这里我们针对标准方差-协方差方法和去趋势方法推导了-1≤ρ DCCA)(T,n)≤1。对于重叠窗口，我们找到了互相关具有统计显著性时ρ(DCCA)的范围。对于重叠窗口，我们通过数值确定——对于非重叠窗口我们进行推导——ρ(DCCA)(T,n)的标准差随着T的增加趋于1/T。使用ρ(DCCA)(T,n)，我们表明中国金融市场跟随美国市场的趋势极其微弱。我们还提出了一种额外的统计检验，可用于量化两个幂律相关时间序列之间互相关的存在情况。
