Aging in coherent noise models and natural time

Predictability of the coherent-noise model and its applications

We study the threshold distribution function of the coherent-noise model for the case of infinite number of agents. This function is piecewise constant with a finite number of steps n. The latter exhibits a 1/f behavior as a function of the order of occurrence of an avalanche and hence versus natural time. An analytic expression of the expectation value E(S) for the size S of the next avalanche is obtained and used for the prediction of the next avalanche. Apart from E(S), the number of steps n can also serve for this purpose. This enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data.

Magnitude correlations in global seismicity

By employing natural time analysis, we analyze the worldwide seismicity and study the existence of correlations between earthquake magnitudes. We find that global seismicity exhibits nontrivial magnitude correlations for earthquake magnitudes greater than M(w) 6.5.

Analysis of return distributions in the coherent noise model

Return distributions of the coherent noise model are studied for the system-size-independent case. It is shown that, in this case, these distributions are in the shape of q Gaussians, which are the standard distributions obtained in nonextensive statistical mechanics. Moreover, an exact relation connecting the exponent τ of avalanche size distribution and the q value of appropriate q Gaussian has been obtained as q=(τ+2)/τ . Making use of this relation one can easily determine q parameter values of the appropriate q Gaussians a priori from one of the well-known exponents of the system. Since the coherent noise model has the advantage of producing different τ values by varying a model parameter σ , clear numerical evidences on the validity of the proposed relation have been achieved for various cases. Finally, the effect of the system size has also been analyzed and an analytical expression has been proposed, which is corroborated by the numerical results.

Nonextensivity and natural time: The case of seismicity

Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.

Origin of the usefulness of the natural-time representation of complex time series

The concept of natural time turned out to be useful in revealing dynamical features behind complex time series including electrocardiograms, ionic current fluctuations of membrane channels, seismic electric signals, and seismic event correlation. However, the origin of this empirical usefulness is yet to be clarified. Here, it is shown that this time domain is in fact optimal for enhancing the signals in time-frequency space by employing the Wigner function and measuring its localization property.