Scaling theory for the $1/f$ noise

Abstract

We show that in a broad class of processes that show a 1/f α spectrum, the power also explicitly depends on the characteristic time scale. Despite an enormous amount of work, this generic behavior remains so far overlooked and poorly understood. An intriguing example is how the power spectrum of a simple random walk on a ring with L sites shows 1/f 3/2 not 1/f 2 behavior in the frequency range 1/L 2 f 1/2. We address the fundamental issue by a scaling method and discuss a class of solvable processes covering physically relevant applications.

Linking space-time correlations for a class of self-organized critical systems

“…Clearly, the power spectrum is a homogeneous function of the arguments, and one can apply the scaling methods [30] to re-express Eq. ( 4) as…”

“…In particular, we show intensive analysis for a SOC model, earlier studied by Davidsen and Paczuski [11]. Our main tool of analysis is the scaling method [27][28][29][30] that allows us to understand the spatial correlations by establishing a relationship for temporal correlations between microscopic and macroscopic fluctuations. The paper structure is as follows.…”

Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes

“…Recently it was shown that a certain class of nonlinear Hawkes processes can reproduce power-law distributions [14], which may appeal to practitioners dealing with complex systems exhibiting power-law distributions, such as financial markets. Long-range memory and 1/f β noise (with 0.5 β 1.5), as its characteristic feature [15][16][17][18][19][20], are of particular interest as they are observed across different physical [21][22][23], biological [9] and social systems [24]. Typically long-range memory is modeled using various non-Markovian models: fractional Brownian motion [25][26][27][28][29], ARCH models [30][31][32][33][34][35] or non-Markovian mechanisms such as trapping [36,37].…”