Fractional diffusion with two time scales

Moving particles that rest along their trajectory lead to time-fractional diffusion equations for the scaling limit distributions. For power law waiting times with infinite mean, the equation contains a fractional time derivative of order between 0 and 1. For finite mean waiting times, the most revealing approach is to employ two time scales, one for the mean and another for deviations from the mean. For finite mean power law waiting times, the resulting equation contains a first derivative as well as a derivative of order between 1 and 2. Finite variance waiting times lead to a second-order partial differential equation in time. In this article we investigate the various solutions with regard to moment growth and scaling properties, and show that even infinite mean waiting times do not necessarily induce subdiffusion, but can lead to super-diffusion if the jump distribution has non-zero mean.     

Self-similar branching of aftershock sequences

In this paper we propose a branching aftershock sequence (BASS) model for seismicity. We suggest that the BASS model is a preferred alternative to the widely studied epidemic type aftershock sequence (ETAS) model. In the BASS model an initial, or seed, earthquake is specified. The subsequent earthquakes are obtained from the statistical distributions of magnitude, time, and location. The magnitude scaling is based on a combination of the Gutenberg-Richter scaling relation and the modified Båth’s law for the scaling relation of aftershocks relative to the magnitude of the seed earthquake. Omori’s law specifies the distribution of earthquake times, and a modified form of Omori’s law specifies the distribution of earthquake locations. Since the BASS model is specified by the four scaling relations, it is fully self-similar. This is not the case for ETAS. We also give a deterministic version of BASS and show that it satisfies Tokunaga side-branching statistics in a similar way to diffusion-limited aggregation (DLA).     

Scale-Invariant Structure of the Seismic Kinetics in Strongly Nonequilibrium Medium

Results of the scaling analysis of the waiting time series for seismic events of the 2019 Ridgecrest earthquake sequence in East California have been presented. It has been shown that the waiting time series before a strong earthquake with a magnitude of M = 7.1 acquires a scale-invariant structure. A reason for the appearance of the statistical self-similarity in the waiting time series is the transition of the medium in the focal region to a strongly nonequilibrium state. The result can be interpreted as the appearance of a temporal dissipative structure in the process of seismic kinetics. It has been proposed to use the scaling analysis of the seismic kinetic data to monitor the current state of the seismogenerating medium.

     

Acoustic emission monitoring of the Syracuse Athena temple: scale invariance in the timing of ruptures

We perform a comparative statistical analysis between the acoustic-emission time series from the ancient Greek Athena temple in Syracuse and the sequence of nearby earthquakes. We find an apparent association between acoustic-emission bursts and the earthquake occurrence. The waiting-time distributions for acoustic-emission and earthquake time series are described by a unique scaling law indicating self-similarity over a wide range of magnitude scales. This evidence suggests a correlation between the aging process of the temple and the local seismic activity.     

"Universal" distribution of interearthquake times explained

We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks, and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of interevent times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforce the view that triggering earthquakes by other earthquakes is a key physical mechanism to understand seismicity.     

Unified scaling law in the coherent noise model

The waiting time distribution between successive events and the unified scaling law is studied using the coherent noise model. It is shown that, although this model generates uncorrelated event sizes and does not exhibit criticality, it still provides the unified scaling law. We argue the role of characteristic kink observed in the unified scaling law and the meaning of the parameter C$C$ used to fix the peak of the kink to unity. Our results indicate that the parameter C$C$ is indeed a physical quantity localizing the end of the linear tendency in the scaling law, which corresponds to the completion of the dominance of correlated events in time.     

Intensity thresholds and the statistics of the temporal occurrence of solar flares

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well-described by a two-parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.     

Statistical properties and universality in earthquake and solar flare occurrence

Earthquakes are phenomena of great complexity, however some simple general laws govern the statistics of their occurrence. Some of these most important laws exhibit scale invariance, as the Gutenberg-Richter law and the Omori law. The origin of these scaling behaviours is not yet fully understood and a natural fondamental question concerns the existence of these features also in other complex phenomena. A direct inspection of experimental catalogues has shown that the stochastic processes underlying solar flare and earthquake occurrence have universal properties. Another intensively debated question is the existence of correlations between magnitudes of subsequent earthquakes. Our recent analysis of the Southern California Catalogue has shown that non-zero magnitude correlations exist. A branching model based on a dynamical scaling hypothesis, relating magnitude to time, reproduces the hierarchical organization in time and magnitude of events and the observed magnitude correlations.     

Scaling of Lévy-Student processes

Student’s t-distributions are widely used in financial studies as heavy-tailed alternatives to normal distributions. As these distributions are not closed under convolution, there exist no Lévy processes with Student’s t-marginals at all points in time. In this article we show that a Student’s t-approximation of these marginals is still suitable, while not exact. Using this approximation, we are able to describe the scaling behavior of such Lévy-Student processes and the parameters of its marginal distributions by a simple analytical scaling law. This scaling law drastically simplifies the use of Lévy-Student processes as a general diffusion process in various interdisciplinary applications. We explicitly provide an application in the context of modelling high-frequency price returns.     

Inverse statistics in the foreign exchange market

We investigate intra-day foreign exchange (FX) time series using the inverse statistic analysis developed by Simonsen et al. (Eur. Phys. J. 27 (2002) 583) and Jensen et al. (Physica A 324 (2003) 338). Specifically, we study the time-averaged distributions of waiting times needed to obtain a certain increase (decrease) ρ in the price of an investment. The analysis is performed for the Deutsch Mark (DM) against the US$ for the full year of 1998, but similar results are obtained for the Japanese Yen against the US$. With high statistical significance, the presence of “resonance peaks” in the waiting time distributions is established. Such peaks are a consequence of the trading habits of the market participants as they are not present in the corresponding tick (business) waiting time distributions. Furthermore, a new stylized fact, is observed for the (normalized) waiting time distribution in the form of a power law Pdf. This result is achieved by rescaling of the physical waiting time by the corresponding tick time thereby partially removing scale-dependent features of the market activity.     

Critical behavior in earthquake energy dissipation

We explore bursty multiscale energy dissipation from earthquakes flanked by latitudes 29° S and 35.5° S, and longitudes 69.501° W and 73.944° W (in the Chilean central zone). Our work compares the predictions of a theory of nonequilibrium phase transitions with nonstandard statistical signatures of earthquake complex scaling behaviors. For temporal scales less than 84 hours, time development of earthquake radiated energy activity follows an algebraic arrangement consistent with estimates from the theory of nonequilibrium phase transitions. There are no characteristic scales for probability distributions of sizes and lifetimes of the activity bursts in the scaling region. The power-law exponents describing the probability distributions suggest that the main energy dissipation takes place due to largest bursts of activity, such as major earthquakes, as opposed to smaller activations which contribute less significantly though they have greater relative occurrence. The results obtained provide statistical evidence that earthquake energy dissipation mechanisms are essentially “scale-free”, displaying statistical and dynamical self-similarity. Our results provide some evidence that earthquake radiated energy and directed percolation belong to a similar universality class.     

Waiting time distributions in financial markets

We study waiting time distributions for data representing two completely different financial markets that have dramatically different characteristics. The first are data for the Irish market during the 19th century over the period 1850 to 1854. A total of 10 stocks out of a database of 60 are examined. The second database is for Japanese yen currency fluctuations during the latter part of the 20th century (1989-1992). The Irish stock activity was recorded on a daily basis and activity was characterised by waiting times that varied from one day to a few months. The Japanese yen data was recorded every minute over 24 hour periods and the waiting times varied from a minute to a an hour or so. For both data sets, the waiting time distributions exhibit power law tails. The results for Irish daily data can be easily interpreted using the model of a continuous time random walk first proposed by Montroll and applied recently to some financial data by Mainardi, Scalas and colleagues. Yen data show a quite different behaviour. For large waiting times, the Irish data exhibit a cut off; the Yen data exhibit two humps that could arise as result of major trading centres in the World. Received 31 December 2001     

Unified scaling law for earthquakes

We show that the distribution of waiting times between earthquakes occurring in California obeys a simple unified scaling law valid from tens of seconds to tens of years. The short time clustering, commonly referred to as aftershocks, is nothing but the short time limit of the general hierarchical properties of earthquakes. There is no unique operational way of distinguishing between main shocks and aftershocks. In the unified law, the Gutenberg-Richter b value, the exponent -1 of the Omori law for aftershocks, and the fractal dimension d(f) of earthquakes appear as critical indices.     

Violation of the scaling relation and non-Markovian nature of earthquake aftershocks

The statistical properties of earthquake aftershocks are studied. The scaling relation for exponents of the Omori law and the power-law calm time distribution (i.e., the interoccurrence time distribution), which is valid if a sequence of aftershocks is a singular Markovian process, is carefully examined. Data analysis shows significant violation of the scaling relation, implying the non-Markovian nature of aftershocks.     

The distribution of first-passage times and durations in FOREX and future markets

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely a distribution derived from the Mittag-Leffler survival function and the Weibull distribution. For the Mittag-Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t_max, whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other hand, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results more efficiently than a simple Weibull distribution. It should be stressed that a Weibull distribution with a power-law tail is more flexible than the Mittag-Leffler distribution, which itself can be approximated by a Weibull distribution and a power-law. Indeed, the key point is that in the former case there is freedom of choice for the exponent of the power-law attached to the Weibull distribution, which can exceed 1 in order to reproduce decays faster than possible with a Mittag-Leffler distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull law for short times and do not have too heavy a tail.     

Performance limitations of flat-histogram methods

We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based upon the exact density of states of 2D Ising models. The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins N as the minimal N2 of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the +/-J nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Fréchet extremal value distribution that obeys exponential scaling. Furthermore, the shape parameters of these distributions indicate that statistical sample means become ill defined already for moderate system sizes within these complex energy landscapes.     

A relationship between Hurst exponents of slip and waiting time data of earthquakes

We propose a new relationship between the slip and waiting time of real earthquake series. We calculated the Hurst exponents for both time series of slip and waiting time of earthquake sequence in Taiwan CWB (Central Weather Bureau) catalogue. Our findings suggest a good correlation with a correlation coefficient of about 0.8 between the two exponents. Such a good correlation is highly similar to the ones expected from time- or slip-predictable earthquake recurrence models and suggests that the recurrence of real seismicity could be reduced to the time- or slip-predictability in certain sense. This paper, thus, initiates a new direction re-considering earthquake recurrence.     

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.     

Local waiting times in critical systems

The quiet times at a fixed point in space are investigated in a system close to or at a non-equilibrium phase transition. The statistics for such first-return times follow from the universality class of the dynamics and the ensemble: for a power-law waiting time distribution the exponent depends on the dimension and the underlying model. We study the two-dimensional Manna sandpile, with both the continously driven self-organized version and the tuned one. The latter has an absorbing state or depinning phase transition at a critical value of the control parameter. The connection to a driven interface in a random medium gives the exponent of the waiting time distribution. In the open ensemble, differences ensue due to the spatial inhomogeneity and the properties of the driving signal. For both ensembles, the waiting time distributions are found to exhibit logarithmic corrections to scaling.Received: 13 September 2004, Published online: 23 December 2004PACS: 05.70.Ln Nonequilibrium and irreversible thermodynamics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 52.25.Fi Transport properties     

电子商务中人类活动的标度行为实证研究

应用去趋势波动分析法,对电子商务中人类网上购物行为进行研究,首次探讨了人类浏览及购买行为时间序列(数量波动)标度律.首先,研究发现人类网上购物行为呈现出明显的周期性,其时间序列的概率密度函数具有显著的双模态特征.其次,利用傅里叶变换方法分析浏览以及购买行为时间序列的功率谱,发现其演化过程不同于无关联的泊松过程.最后,基于功率谱过滤周期性趋势的影响,对去除周期趋势后的浏览和购买行为时间序列进行去趋势波动分析,发现其标度行为表明其具有较强的长程关联特性,且平均标度值近似为1,表明其具有自组织临界性.实证研究结果与其他领域如因特网交通流和金融市场价格波动的标度行为相似,有助于理解人类活动如何影响电子商务系统演化和提高在线商务活动效率,对分析电子商务中人类行为活动的机制和预测其波动趋势具有重要的启示作用.