M. Alaburda

Point process model of 1/f noise vs a sum of Lorentzians

We present a simple point process model of 1/f(beta) noise, covering different values of the exponent beta . The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence, or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields 1/f(beta) spectra in a wide range of frequencies. We obtain explicit expressions for the power spectra and present numerical illustrations of the model. Further we analyze the relation of the point process model of 1/f noise with the Bernamont-Surdin-McWhorter model, representing the signals as a sum of the uncorrelated components. We show that the point process model is complementary to the model based on the sum of signals with a wide-range distribution of the relaxation times. In contrast to the Gaussian distribution of the signal intensity of the sum of the uncorrelated components, the point process exhibits asymptotically a power-law distribution of the signal intensity. The developed multiplicative point process model of 1/f(beta)noise may be used for modeling and analysis of stochastic processes in different systems with the power-law distribution of the intensity of pulsing signals.

Nonlinear stochastic models of 1/f noise and power-law distributions

Starting from the developed generalized point process model of 1/f noise [B. Kaulakys et al., Phys. Rev. E 71 (2005) 051105] we derive the nonlinear stochastic differential equations for the signal exhibiting 1/fβ noise and 1/xλ distribution density of the signal intensity with different values of β and λ. The processes with 1/fβ are demonstrated by the numerical solution of the derived equations with the appropriate restriction of the diffusion of the signal in some finite interval. The proposed consideration may be used for modeling and analysis of stochastic processes in different systems with the power-law distributions, long-range memory or with the elements of self-organization.

Modeling long-memory processes by stochastic difference equations and superstatistical approach

It is shown that the Poissonian-like process with slowly diffusing-like time-dependent average interevent time may be represented as the superstatistical one and exhibits 1= f noise. The distribution of the Poissonian-like interevent time may be expressed as q-exponential distribution of the Nonextensive Statistical Mechanics.

Interplay between positive feedbacks in the generalized CEV process

The dynamics of the generalized CEV process dXt=aXtndt+bXtmdWt(gCEV) is due to an interplay of two feedback mechanisms: State-to-Drift and State-to-Diffusion, whose degrees are n and m respectively. We particularly show that the gCEV, in which both feedback mechanisms are positive, i.e. n,m>1, admits a stationary probability distribution P provided that n 2. Furthermore the power spectral density obeys S(f)∼1fβ, where β=2−1+ϵ2(m−1), ϵ>0. The tail behavior of the stationary pdf as well as of the power-spectral density thus are both independent of the drift feedback degree n but governed by the diffusion feedback degree m. Bursting behavior of the gCEV is investigated numerically. Burst intensity S and burst duration T are shown to be related by S∼T2.

Evolution of Complex Systems and 1/f Noise: from Physics to Financial Markets

We introduce the stochastic multiplicative model of time intervals between the events, defining a multiplicative point process and analyze the statistical properties of the signal. Such a model system exhibits power-law spectral density S(f)~1/fβ, scaled as power of frequency for various values of β between 0.5 and 2. We derive explicit expressions for the power spectrum and other statistics and analyze the model system numerically. The specific interest of our analysis is related with the theoretical modeling of the nonlinear complex systems exhibiting fractal behavior and self-organization.

Modeling scaled processes and clustering of events by the nonlinear stochastic differential equations

We present and analyze the nonlinear stochastic differential equations generating scaled signals with the power‐law statistics, including 1/fβ noise and q‐Gaussian distribution. Numerical analysis reveals that the process exhibits some peaks, bursts or extreme events, characterized by power‐law distributions of the burst statistics and, therefore, the model may simulate self‐organized critical and other systems exhibiting avalanches, bursts or clustering of events.

Modeling non-Gaussian 1/f Noise by the Stochastic Differential Equations

We consider stochastic model based on the linear stochastic differential equation with the linear relaxation and with the diffusion‐like fluctuations of the relaxation rate. The model generates monofractal signals with the non‐Gaussian power‐law distributions and 1/fβ noise.

Modeling of Flows with Power-Law Spectral Densities and Power-Law Distributions of Flow Intensities

We present analytical and numerical results of modeling of flows represented as correlated non-Poissonian point process and as Poissonian sequence of pulses of different size. Both models may generate signals with power-law distributions of the intensity of the flow and power-law spectral density. Furthermore, different distributions of the interevent time of the point process and different statistics of the size of pulses may result in 1/f β noise with 0.5 ≲ β ≲ 2. A combination of the models is applied for modeling Internet traffic.

1/f noise from the nonlinear transformations of the variables

The origin of the low-frequency noise with power spectrum

Modeling the inverse cubic distributions by nonlinear stochastic differential equations

1/f^\beta