ON THE UBIQUITY OF 1/f NOISE

1989 ◽  
Vol 03 (06) ◽  
pp. 795-819 ◽  
Author(s):  
BRUCE J. WEST ◽  
MICHAEL F. SHLESINGER

A generic mechanism for the ubiquitous phenomenon of 1/f noise is reviewed. This mechanism arises in random processes expressible as a product of several random variables. Under mild conditions this product form leads to the log-normal distribution which we show straightforwardly generates 1/f noise. Thus, 1/f noise is tied directly to a probability limit distribution. A second mechanism involving scaling is introduced to provide a natural crossover from log-normal to inverse power-law behavior and generates 1/fα noise instead of pure 1/f noise. Examples of these distributions and the transitions between them are drawn from such diverse areas as economics, scientific productivity, bronchial structure and cardiac activity.

1992 ◽  
Vol 263 (1) ◽  
pp. R141-R147 ◽  
Author(s):  
H. H. Szeto ◽  
P. Y. Cheng ◽  
J. A. Decena ◽  
Y. Cheng ◽  
D. L. Wu ◽  
...  

The dynamic pattern of fetal breathing was studied in 17 fetal lambs with chronically implanted electromyographic electrodes in the diaphragm. The instantaneous breathing rate time series appeared similar on different time scales, with clusters of faster breathing rates interspersed with periods of relative quiescience, suggesting self-similarity. Distribution histograms of the interbreath intervals (IBIs) showed log-normal distribution for IBIs less than 1 s and inverse power-law distribution for IBIs greater than 1 s. The ratio of log-normal distribution to power-law distribution varied from approximately 2 at 102 days to approximately 30 by 130 days of gestation. Fast Fourier transform of the breathing rate time series revealed 1/f beta power spectra for all animals, with beta increasing linearly from 0.43 to 0.88 between 102 and 139 days. Studies in the newborn lamb showed further maturation in both the distribution characteristics of the IBIs, as well as in the 1/f power spectra, with beta approaching 1.0 at 2 days after birth. The inverse power-law relationship in the distribution of the IBIs, together with the 1/f beta power spectra, indicate scale invariance and suggest that fractal mechanisms are involved in the regulation of fetal breathing.


2007 ◽  
Vol 10 (01) ◽  
pp. 29-51 ◽  
Author(s):  
STEFANO BATTISTON ◽  
JOAO F. RODRIGUES ◽  
HAMZA ZEYTINOGLU

We present an analysis of inter-regional investment stocks within Europe from a complex networks perspective. We consider two different levels: first, we compute the inward–outward investment stocks at the level of firms, based on ownership shares and number of employees; then we estimate the inward–outward investment stock at the level of regions in Europe, by aggregating the ownership network of firms, based on their headquarter location. To our knowledge, there is no similar approach in the literature so far, and we believe that it may lead to important applications for policy making. In the present paper, we focus on the statistical distributions and the scaling laws, while in further studies we will analyze the structure of the network and its relation to geographical space. We find that while outward investment and activity of firms are power law distributed with a similar exponent, for regions these quantities are better described by a log-normal distribution. At both levels we also find scaling laws relating investment to activity and connectivity. In particular, we find that investment stock scales as a power law of the connectivity, as previously found for stock market data.


2006 ◽  
Vol 17 (10) ◽  
pp. 1429-1436 ◽  
Author(s):  
LUCIEN BENGUIGUI ◽  
EFRAT BLUMENFELD-LIEBERTHAL

We propose a new classification of the size distributions of entities based on an exponent α defined from the shape of the log–log Rank Size plot. From an inspection of a large number of cases in different fields, one finds three possibilities: α = 1 giving a power law, α > 1 (parabola like curve) and 0 < α < 1 (analogous to a log normal distribution). A fourth possibility that can be defined when α < 0 was never observed. We present a modified version of models based on a random multiplicative process and an introduction of new entities during the growth. We recover all three kinds of distributions and show that the type of a distribution is conditioned by the rate of the introduction of new entities.


Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 908
Author(s):  
Atushi Ishikawa ◽  
Shouji Fujimoto ◽  
Arturo Ramos ◽  
Takayuki Mizuno

We analytically derived and confirmed by empirical data the following three relations from the quasi-time-reversal symmetry, Gibrat’s law, and the non-Gibrat’s property observed in the urban population data of France. The first is the relation between the time variation of the power law and the quasi-time-reversal symmetry in the large-scale range of a system that changes quasi-statically. The second is the relation between the time variation of the log-normal distribution and the quasi-time-reversal symmetry in the mid-scale range. The third is the relation among the parameters of log-normal distribution, non-Gibrat’s property, and quasi-time-reversal symmetry.


2019 ◽  
Vol 6 (8) ◽  
pp. 190944 ◽  
Author(s):  
Lukas Schneider ◽  
Claudius Gros

Analysing the timeline of US, UK, German and Dutch music charts, we find that the evolution of album lifetimes and of the size of weekly rank changes provide evidence for an acceleration of cultural processes. For most of the past five decades, number one albums needed more than a month to climb to the top, nowadays an album is in contrast top ranked either from the start, or not at all. Over the last three decades, the number of top-listed albums increased as a consequence from roughly a dozen per year, to about 40. The distribution of album lifetimes evolved during the last decades from a log-normal distribution to a power law, a profound change. Presenting an information–theoretical approach to human activities, we suggest that the fading relevance of personal time horizons may be causing this phenomenon. Furthermore, we find that sales and airplay- based charts differ statistically and that the inclusion of streaming affects chart diversity adversely. We point out in addition that opinion dynamics may accelerate not only in cultural domains, as found here, but also in other settings, in particular in politics, where it could have far reaching consequences.


2021 ◽  
Vol 13 (3) ◽  
pp. 1361
Author(s):  
Rafael González-Val

This paper analyses the probability distribution of worldwide forest areas. We find moderate support for a Pareto-type distribution (power law) using FAO data from 1990 to 2015. Power laws are common features of many complex systems in nature. A power law is a plausible model for the world probability distribution of forest areas in all examined years, although the log-normal distribution is a plausible alternative model that cannot be rejected. The random growth of forest areas could generate a power law or log-normal distribution. We study the change in forest coverage using parametric and non-parametric methods. We identified a slight convergence of forest areas over the time reviewed; however, random forest area growth cannot be rejected for most of the distribution of forest areas. Therefore, our results give support to theoretical models of stochastic forest growth.


2016 ◽  
Author(s):  
M. Olin ◽  
T. Anttila ◽  
M. Dal Maso

Abstract. We present the combined power law and log-normal distribution (PL+LN) model, a computationally efficient model to be used in simulations where the particle size distribution cannot be accurately represented by log-normal distributions, such as in simulations involving the initial steps of aerosol formation, where new particle formation and growth occur simultaneously, or in the case of inverse modelling. The model was validated against highly accurate sectional models using input parameter values that reflect conditions typical to particle formation occurring in the atmosphere and in vehicle exhaust, and tested in the simulation of a particle formation event performed in a mobile aerosol chamber at Mäkelänkatu street canyon measurement site in Helsinki, Finland. The number, surface area, and mass concentrations in the chamber simulation were conserved with the relative errors lower than 2 % using the PL+LN model, whereas a moment-based log-normal model and sectional models with the same computing time as with the PL+LN model caused relative errors up to 10 % and 135 %, respectively.


2016 ◽  
Vol 16 (11) ◽  
pp. 7067-7090 ◽  
Author(s):  
Miska Olin ◽  
Tatu Anttila ◽  
Miikka Dal Maso

Abstract. We present the combined power law and log-normal distribution (PL+LN) model, a computationally efficient model to be used in simulations where the particle size distribution cannot be accurately represented by log-normal distributions, such as in simulations involving the initial steps of aerosol formation, where new particle formation and growth occur simultaneously, or in the case of inverse modeling. The model was evaluated against highly accurate sectional models using input parameter values that reflect conditions typical to particle formation occurring in the atmosphere and in vehicle exhaust. The model was tested in the simulation of a particle formation event performed in a mobile aerosol chamber at Mäkelänkatu street canyon measurement site in Helsinki, Finland. The number, surface area, and mass concentrations in the chamber simulation were conserved with the relative errors lower than 2 % using the PL+LN model, whereas a moment-based log-normal model and sectional models with the same computing time as with the PL+LN model caused relative errors up to 17 and 79 %, respectively.


2006 ◽  
Author(s):  
Gerardo Ramirez ◽  
Sonia Perez ◽  
John G. Holden

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