Minimizing the effect of sinusoidal trends in detrended fluctuation analysis

The detrended fluctuation analysis (DFA) [Peng et al., 1994] and its extensions (MF-DFA) [Kantelhardt et al., 2002] have been used extensively to determine possible long-range correlations in self-affine signals. While the DFA has been claimed to be a superior technique, recent reports have indicated its susceptibility to trends in the data. In this report, a smoothing filter is proposed to minimize the effect of sinusoidal trends and distortion in the log-log plots obtained by DFA and MF-DFA techniques

Scale invariant correlations and the distribution of prime numbers

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.Comment: 13 pages, 8 figures, version to appear in J. Phys.

Cross-correlation of long-range correlated series

A method for estimating the cross-correlation $C_{xy}(\tau)$ of long-range correlated series $x(t)$ and $y(t)$, at varying lags $\tau$ and scales $n$, is proposed. For fractional Brownian motions with Hurst exponents $H_1$ and $H_2$, the asymptotic expression of $C_{xy}(\tau)$ depends only on the lag $\tau$ (wide-sense stationarity) and scales as a power of $n$ with exponent ${H_1+H_2}$ for $\tau\to 0$. The method is illustrated on (i) financial series, to show the leverage effect; (ii) genomic sequences, to estimate the correlations between structural parameters along the chromosomes.Comment: 14 pages, 8 figure

Continuously Sheared Granular Matter Reproduces in Detail Seismicity Laws

International audienceWe introduce a shear experiment that quantitatively reproduces the main laws of seismicity. By continuously and slowly shearing a compressed monolayer of disks in a ringlike geometry, our system delivers events of frictional failures with energies following a Gutenberg-Richter law. Moreover, foreshocks and aftershocks are described by Omori laws and interevent times also follow exactly the same distribution as real earthquakes, showing the existence of memory of past events. Other features of real earthquakes qualitatively reproduced in our system are both the existence of a quiescence preceding some main shocks, as well as magnitude correlations linked to large quakes. The key ingredient of the dynamics is the nature of the force network, governing the distribution of frictional thresholds

The covariance structure of multifractional Brownian motion with application to long range dependence

SIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 22522, issue : a.2000 n.9 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc