Figure 1

Analysis of the multifractal cascade model: (a) MFDFA fluctuation function for $q=1$ (○), 2 (□), and 5 (⋄). The autocorrelation function $C_x(s)$ of the data is shown in the inset. (b) Scaled PDFs $P_Q(r)$ of the return intervals for return periods $R_Q=10$ (○), 70 (□), and 500 (⋄). To avoid overlapping, symbols were shifted downwards by a factor of 10 (□) and 100 (⋄). The relevant filled symbols show the corresponding PDFs for the shuffled data, shifted downwards by a factor of ${10}^4$. (c) Exponents $\delta (Q)$ vs $R_Q$, for different systems sizes $N=2^{16}$ (○), $2^{21}$ (◃), and $2^{26}$ (*). (d) Return intervals autocorrelation function $C_Q(s)$ for the same $R_Q$ values as in (b). (e) Conditional return periods $R_Q(r_0)$ in units of $R_Q$ vs $r_0/R_Q$ for the same $R_Q$ values as in (b) (○), The filled symbols are for the shuffled data. The curves for $R_Q=70$ and 500 were raised by a factor of 10 and 100, respectively, to avoid overlapping symbols. (f) Exponents $\nu (Q)$, for the same systems sizes as in (c). The results in (a), (b), (d), and (e) are based on data sets of length $N=2^{21}$ and averaged over 150 configurations.Reuse & Permissions

Figure 2

Analysis of the arithmetic returns of IBM daily stock closing prices (open symbols), and simulated multifractal data of similar length $N=2^{14}$ (dashed lines), averaged over 500 configurations: (a) MFDFA fluctuation function for three moments $q=1$ (○), 2 (□), and 5 (⋄). The autocorrelation function $C_x(s)$ of the IBM returns is shown in the inset. (b) Scaled PDFs $P_Q(r)$ of the return intervals versus $r/R_Q$ for $R_Q=10$ (○), 30 (▵), and 70 (□). To avoid overlapping, symbols were shifted downwards by a factor of 10 (▵) and 100 (□). (c) Autocorrelation function $C_Q(s)$ of return intervals $(r_j)$ for $R_Q=10$ and 30 [symbols correspond to those in (b)]. (d) Conditional return periods $R_Q(r_0)$ in units of $R_Q$ vs $r_0/R_Q$ for the same $R_Q$ values as in (c). The curve for $R_Q=30$ was raised by a factor of 10 to avoid overlapping symbols.Reuse & Permissions