The p-sphere and the geometric substratum of power-law probability distributions

Abstract. The theory of scale similarity and breakdown coefficients is applied here to intermittent rainfall data consisting of time series and spatial rain fields. The probability distributions (pdf) of the logarithm of the breakdown coefficients are the principal descriptor used. Rain fields are distinguished as being either multiscaling or multiaffine depending on whether the pdfs of breakdown coefficients are scale similar or scale dependent, respectively. Parameter estimation techniques are developed which are applicable to both multiscaling and multiaffine fields. The scale parameter (width), σ, of the pdfs of the log-breakdown coefficients is a measure of the intermittency of a field. For multiaffine fields, this scale parameter is found to increase with scale in a power-law fashion consistent with a bounded-cascade picture of rainfall modelling. The resulting power-law exponent, H, is indicative of the smoothness of the field. Some details of breakdown coefficient analysis are addressed and a theoretical link between this analysis and moment scaling analysis is also presented. Breakdown coefficient properties of cascades are also investigated in the context of parameter estimation for modelling purposes.

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Probability of ruin in discrete insurance risk model with dependent Pareto claims

Dependence Modeling ◽

10.1515/demo-2019-0011 ◽

2019 ◽

Vol 7 (1) ◽

pp. 215-233

Author(s):

Corina D. Constantinescu ◽

Tomasz J. Kozubowski ◽

Haoyu H. Qian

Keyword(s):

Power Law ◽

Pareto Distribution ◽

Risk Model ◽

Probability Distributions ◽

Insurance Risk ◽

Probability Of Ruin ◽

New Class ◽

Compound Binomial ◽

Compound Binomial Risk Model ◽

Positive Integers

AbstractWe present basic properties and discuss potential insurance applications of a new class of probability distributions on positive integers with power law tails. The distributions in this class are zero-inflated discrete counterparts of the Pareto distribution. In particular, we obtain the probability of ruin in the compound binomial risk model where the claims are zero-inflated discrete Pareto distributed and correlated by mixture.

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Generalized Lotka-Volterra (GLV) Models of Stock Markets

Advances in Complex Systems ◽

10.1142/s0219525900000224 ◽

2000 ◽

Vol 03 (01n04) ◽

pp. 301-322 ◽

Cited By ~ 10

Author(s):

Sorin Solomon

Keyword(s):

Power Law ◽

Wide Class ◽

Stock Markets ◽

Probability Distributions ◽

Economic Systems ◽

Lévy Flights ◽

Levy Flights ◽

General Method

The Generalized Lotka-Volterra (GLV) model: [Formula: see text] provides a general method to simulate, analyze and understand a wide class of phenomena that are characterized by power-law probability distributions: [Formula: see text] and truncated Levy flights fluctuations [Formula: see text]. We show how the model applies to economic systems.

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Investigations on the power-law burst lifetime distribution characteristics of the magnetospheric system

10.5194/egusphere-egu2020-20011 ◽

2020 ◽

Author(s):

Prince Prasad ◽

Santhosh Kumar G ◽

Sumesh Gopinath

Keyword(s):

Power Law ◽

Waiting Time ◽

Probability Distributions ◽

Sunspot Cycle ◽

Lifetime Distribution ◽

Characteristic Time Scale ◽

General Strategy ◽

Scaling Exponent ◽

Time Duration ◽

Scaling Properties

<p>The waiting time distributions and associated statistical relationships can be considered as a general strategy for analyzing space weather and inner magnetospheric processes to a large extent. It measures the distribution of delay times between subsequent hopping events in such processes. In a physical system the time duration between two events is called a waiting-time, like the time between avalanches. The burst lifetime can be considered as the time duration when magnitude of fluctuations are above a given threshold intensity.&#160; If a characteristic time scale is absent then the probability densities vary with power-law relations having a scaling exponent. The burst lifetime distribution of the substorm index called as the Wp index (Wave and planetary), which reflects Pi2 wave power at low-latitude is considered for the present analysis. Our analysis shows that the lifetime probability distributions of Wp index yield power-law exponents. Even though power-law exponents are observed in magnetospheric proxies for different solar activity periods, not many studies were made to analyze whether these features will repeat or differ depending on sunspot cycle. We compare the variations of power-law exponents of Wp index and other magnetospheric proxies, such as AE index, during solar maxima and solar minima. Thus the study classifies the activity bursts in Wp and other magnetospheric proxies that may have different dynamical critical scaling features. We also expect that the study sheds light into certain stochastic aspects of scaling properties of the magnetosphere which are not developed as global phenomena, but in turn generated due to inherent localized properties of the magnetosphere.</p>

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FRACTAL METHODS APPLIED TO DESCRIBE CELLULAR AUTOMATON COMBAT MODELS

Fractals ◽

10.1142/s0218348x01000609 ◽

2001 ◽

Vol 09 (02) ◽

pp. 177-184 ◽

Cited By ~ 5

Author(s):

MICHAEL K. LAUREN

Keyword(s):

Cellular Automaton ◽

Complexity Theory ◽

Power Law ◽

Chaotic Systems ◽

Probability Distributions ◽

High Dimensional ◽

Compelling Evidence ◽

Military Equipment ◽

The Difference ◽

Maneuver Warfare

Analysis of warfare data provides compelling evidence that intensity of conflicts obeys a power-law (fractal) dependence on frequency. There is also evidence for the existence of other power-law dependences and traits characteristic of high-dimensional chaotic systems, such as fat-tailed probability distributions and intermittency in warfare data. In this report, it is discussed how a cellular automaton model used to describe modern maneuver warfare produces casualty distributions which exhibit these properties. This points to a possible origin of the characteristics of the larger timescale data. More interesting, the techniques of fractal analysis offer a method by which to characterize these behaviors, and to quantify the difference between models based on complexity theory (such as cellular automata models), and more traditional combat models based on the physics of military equipment.

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Application of the fractional-stable distributions for approximation of the gene expression profiles

Statistical Applications in Genetics and Molecular Biology ◽

10.1515/sagmb-2014-0094 ◽

2015 ◽

Vol 14 (3) ◽

Cited By ~ 2

Author(s):

Viacheslav Saenko ◽

Yury Saenko

Keyword(s):

Gene Expression ◽

Power Law ◽

Limit Distribution ◽

Probability Distributions ◽

Expression Profiles ◽

Gene Expression Profiles ◽

Stable Distributions ◽

Scientific Data ◽

Microarray Experiments ◽

Individual Gene

AbstractNowadays, there are reliable scientific data highlighting that the probability density function of the gene expression demonstrates a number of universal features commonly observed in the microarray experiments. First of all, these distributions demonstrate the power-law asymptotics and, secondly, the shape of these distributions is inherent for all organisms and tissues. This fact leads to appearance of a number of works where authors investigate various probability distributions for an approximation of the empirical distributions of the gene expression. Nevertheless, all these distributions are not a limit distribution and are not a solution of any equation. These facts, in our opinion, are essential shortcoming of these probability laws. Besides, the expression of the individual gene is not an accidental phenomenon and it depends on the expression of the other genes. This suggests an existence of the genic regulatory net in a cell. The present work describes the class of the fractional-stable distributions. This class of the distributions is a limit distribution of the sums of independent identically distributed random variables. Due to the power-law asymptotics, these distributions are applicable for the approximation of the experimental densities of the gene expression for microarray experiments. The parameters of the fractional stable distributions are statistically estimated by experimental data and the functions of the empirical and theoretical densities are compared. Here we describe algorithms for simulation of the fractional-stable variables and estimation of the parameters of the the fractional stable densities. The results of such a comparison allow to conclude that the empirical densities of the gene expression can be approximated by the fractional-stable distributions.

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On the sensitivity of tidal network characterization to power law estimation

Advances in Geosciences ◽

10.5194/adgeo-39-69-2014 ◽

2014 ◽

Vol 39 ◽

pp. 69-73 ◽

Cited By ~ 3

Author(s):

M. Jiménez ◽

S. Castanedo ◽

Z. Zhou ◽

G. Coco ◽

R. Medina ◽

...

Keyword(s):

Power Law ◽

Probability Distributions ◽

Correlation Coefficients ◽

Drainage Area ◽

Cumulative Distribution ◽

Volume Distribution ◽

Drainage Volume ◽

Open Issue ◽

The Impact

Abstract. Long-term simulations (3000 yr) of an idealized basin using different tidal ranges (1, 2 and 3 m) and grain sizes (120, 480 and 960 μm) have been performed in order to cover a range of hydrodynamic and sedimentary conditions. Two different cell sizes (50 and 100 m) have been used to study the impact of cell size on tidal network development. The probability distributions of the drainage area and the drainage volume have been computed for every simulation (during an ebb and a flood phase). Power law distributions are observed in drainage area and drainage volume distribution. As an objective estimation of the exponent of a power law is an open issue, different approaches (linear binning, normalized logarithmic binning, cumulative distribution function and maximum likelihood) proposed by White et al. (2008) to estimate the exponent have been used to carry out a sensitivity analysis. Our findings indicate that although all methods results in high and significant correlation coefficients, more work is needed to develop a universal, objective estimation of the exponent.

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The luminosity–volume test for cosmological fast radio bursts

Astronomy and Astrophysics ◽

10.1051/0004-6361/201834722 ◽

2019 ◽

Vol 625 ◽

pp. A109 ◽

Cited By ~ 6

Author(s):

N. Locatelli ◽

M. Ronchi ◽

G. Ghirlanda ◽

G. Ghisellini

Keyword(s):

Star Formation ◽

Power Law ◽

Star Formation Rate ◽

Probability Distributions ◽

Formation Rate ◽

Radio Spectrum ◽

Radio Bursts

We have applied the luminosity–volume test, also known as ⟨V/Vmax⟩, to fast radio bursts (FRBs). We compare the 23 FRBs, recently discovered by ASKAP, with 20 of the FRBs found by Parkes. These samples have different flux limits and correspond to different explored volumes. We put constrains on their redshifts with probability distributions (PDFs) and applied the appropriate cosmological corrections to the spectrum and rate in order to compute the ⟨V/Vmax⟩ for the ASKAP and Parkes samples. For a radio spectrum of FRBs ℱν ∝ ν−1.6, we found ⟨V/Vmax⟩ = 0.68 ± 0.05 for the ASKAP sample, that includes FRBs up to z = 0.72+0.42−0.26, and 0.54 ± 0.04 for Parkes, that extends up to z = 2.1+0.47−0.38. The ASKAP value suggests that the population of FRB progenitors evolves faster than the star formation rate, while the Parkes value is consistent with it. Even a delayed (as a power law or Gaussian) star formation rate cannot reproduce the ⟨V/Vmax⟩ of both samples. If FRBs do not evolve in luminosity, the ⟨V/Vmax⟩ values of ASKAP and Parkes sample are consistent with a population of progenitors whose density strongly evolves with redshift as ∼z2.8 up to z ∼ 0.7.

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