Figure 1

(a) rms of detrended variance $F_{\mathrm{DFA}}(n)$ and detrended covariance, $F_{\mathrm{DCCA}}(n)$, where $n$ is a scale. For two time series generated by two ARFIMA processes: $\{y_i\}$ with $\rho =0.1$ and $\{y_i^{\prime }\}$ with ${\rho }^{\prime }=0.4$ we show the DFA curves $F_{\mathrm{DFA}}(n)$ for both $\{y_i\}$ and $\{y_i^{\prime }\}$, which can be fitted by power laws $F_{\mathrm{DFA}}\sim n^H$. Cross correlations are generated since we choose the error term to be equal for both time series: ${\eta }_i={\eta }_i^{\prime }$, where ${\eta }_i$ corresponds to $\{y_i\}$ and ${\eta }_i^{\prime }$ corresponds to $\{y_i^{\prime }\}$. When cross correlations are present, the same weights are responsible for power-law cross correlations between $\{y_i\}$ and $\{y_i^{\prime }\}$. For $n\gg 1$ we find $F_{\mathrm{DCCA}}(n)\approx n^{\lambda }$ [see Eq. (4)], where $\lambda =0.73$. This example illustrates the relation: $\lambda \approx (H+H^{\prime })/2$. If we choose the error terms ${\eta }_i^{\prime }=-{\eta }_i$, then $F_{\mathrm{DCCA}}^2(n)$ becomes negative for every $n$. For that case the cross-correlation function $X(n)$ becomes also negative. (b) Detrended covariance $F_{\mathrm{DCCA}}^2(n)$ of Eq. (4). We generate two pairs of two ARFIMA processes, where for each pair the time series are power-law autocorrelated, but not cross correlated, since each ARFIMA is generated by its own error term. The fluctuations, both positive and negative, indicate that two time series are not power-law cross correlated with an unique exponent, but either short-range cross correlated or not at all cross correlated.Reuse & Permissions

Figure 2

Power-law autocorrelations and cross correlations in successive differences of air humidity $\{y_i\}$ and air temperature $\{y_i^{\prime }\}$, recorded each 10 minutes. For time series of their absolute values, $\{|y_i|\}$ and $\{|y_i^{\prime }|\}$, we find that both time series show sudden bursts of large changes. We show the rms of detrended variance $F_{\mathrm{DFA}}(n)$ together with detrended covariance $F_{\mathrm{DCCA}}(n)$. We find that DFA curves of $\{|y_i|\}$ and $\{|y_i^{\prime }|\}$ and DCCA curve are very similar, and can be approximated with power laws $F_{\mathrm{DFA}}(n)\sim n^H$ with scaling exponents $H=0.72$ and $H^{\prime }=0.76$, and $F_{\mathrm{DCCA}}(n)\sim n^{\lambda }$ with $\lambda =0.75$.Reuse & Permissions

Figure 3

Power-law autocorrelations and cross correlations between two different EEG time series: $C4/A1$ ($\{y_i\}$) and $C3/A2$) ($\{y_i^{\prime }\}$) with 32 000 data points, recorded every second. For time series of their absolute values, $\{|y_i|\}$ and $\{|y_i^{\prime }|\}$, we show rms of detrended variance $F_{\mathrm{DFA}}(n)$ together with detrended covariance $F_{\mathrm{DCCA}}(n)$. We find that $F_{\mathrm{DFA}}(n)$ of $\{|y_i|\}$ and $\{|y_i^{\prime }|\}$ and DCCA curve, $F_{\mathrm{DCCA}}(n)$ are very similar, and can be approximated with power laws $n^{\alpha }(n^{\lambda })$ with scaling exponents $\alpha =0.83$ and $\alpha =0.84$, $\lambda =0.84$, respectively.Reuse & Permissions

Figure 4

Long-range autocorrelations and cross correlations in absolute values of price changes and trading volume for both Dow Jones index and Nasdaq index, recorded daily, in the period from July 1993 to November 2003. (a) Integrated profiles $I(n)\equiv \sum _{i=1}^n(|y_i|-|{\overline{y}}_i|)$ of the time series of absolute values $\{|y_i|\}$ and $\{|y_i^{\prime }|\}$ of logarithmic changes in price for Dow Jones and Nasdaq, and (c) the corresponding absolute values $\{|z_i|\}$ and $\{|z_i^{\prime }|\}$ for trading volume changes. (b) For price, rms of the detrended variance $F_{\mathrm{DFA}}(n)$ curves for each $\{|z_i^{\prime }|\}$ and $\{|z_i^{\prime }|\}$, and also the rms of the detrended covariance, $F_{\mathrm{DCCA}}(n)$. (d) Three curves represents the same measures but for “volume” (absolute of trading volume changes). For all DFA and DCCA curves we find both power-law autocorrelations and power-law cross correlations. Power-law cross correlations between Nasdaq and Dow Jones indices imply that current price changes of Nasdaq depend on its previous changes but also on previous price changes of Dow Jones. For trading volumes we also analyze time series $\{z_i\}$ and $\{z_i^{\prime }\}$, and we find that the DFA and DCCA analyses show the anticorrelated behavior with DFA exponents $\alpha =0.07$, $\alpha =0.11$, and DCCA exponent $\lambda =0.04$.Reuse & Permissions