Figure 1

Log${}_{10}$-log${}_{10}$ plots of the fluctuation $F\left(n\right)$ vs the subset size $n$ for a correlated signal, superimposed additively by various trends. The correlated signals are generated by using the method proposed in Ref. [15]. For DFA, $F\left(n\right)$ vs $n$ is plotted for the quadratic (open squares), periodic (open circles), and power-law (open triangles) trends together with the arrows indicating the crossover points. For ADFA, $F\left(n\right)$ vs $n$ is plotted for the same superimposed trends: quadratic (solid squares), periodic (solid circles), and power-law (solid triangles) trends. By using ADFA with a significance level $\alpha =0.1$, we obtained the scaling exponents, estimated by the least-squares fit (dotted lines), $\beta =0.68,0.67,0.68$ for the quadratic, periodic, and power-law trends, respectively. The plots for periodic and power-law trends are vertically shifted for clarity. The inset shows a plot of the average degree of the regression polynomial over the $N/n$ subsets vs $n$, where ADFA is applied to the correlated signal superimposed by the quadratic trend. The error bars are twice the estimated sample standard deviations from the $N/n$ subsets for each $n$.Reuse & Permissions

Figure 2

Plots of the scaling exponent $\beta$ vs the correlation exponent $\gamma$ [15] obtained by DFA (open squares) and ADFA (solid squares) with the correlated data of various $\gamma$'s. The solid line is a least-squares fit of the results of ADFA being $\beta \approx 0.81-0.47\gamma$, and the dotted line represents the theoretical relation between $\beta$ and $\gamma$ for DFA, $\beta =1-\gamma /2$ [17]. The correlated signals are generated by using the method proposed in Ref. [15], and the error bars are the standard deviation of 10 independent trials. The inset shows the log${}_{10}$-log${}_{10}$ plots of $F\left(n\right)$ vs $n$ obtained by DFA (open squares) and ADFA (solid squares) when $\gamma =0.65$.Reuse & Permissions