Levy Stable Distribution Research Articles

Complex system properties in the spreading of COVID-19 pandemic

The objective of the present study was to show that the spread of the COVID-19 pandemic around the world shows complex system properties such as lognormal laws, temporal fluctuation scaling, and time correlation. First, the daily cumulative number of confirmed cases and deaths is distributed among countries as lognormals such that the time series exhibit a temporal fluctuation scaling. Second, the daily return time series of cases and deaths per day have associated Levy stable distributions and they have time correlation. The idea was to draw attention to the fact that the spread of the COVID-19 pandemic can be seen as a complex system, and, thus, contribute to the identification of the structural properties of this system, which is relevant as it is expected that future stochastic models describing the spread of the COVID-19 pandemic from a microscopic dynamics perspective should be able to explain the emergence of the structural properties identified here.

Phase-Sensitive Intracellular Doppler Fluctuation Spectroscopy

Dynamic light scattering from three-dimensional intracellular motions in living tissues or biopsies creates dynamic speckle due to broadband Doppler shifts that are recorded as broadband spectra obtained from speckle fluctuation spectroscopy. Doppler frequency shifts from active intracellular motion extend across three orders of magnitude from 10 mHz to 10 Hz. Biodynamic imaging is a full-field optical coherence tomography technique performed with low-coherence digital holography. The holographic reconstruction allows phase recovery, but mechanical stability limitations favor homodyne detection over heterodyne detection because homodyne is relatively insensitive to phase drift. However, by obtaining phase-displacement distributions from consecutive frames, the phase excursions form a robust probability distribution. The phase-probability density function of the phase-displacement distribution captures average phase displacements over time and is used to reconstruct stable heterodyne spectra. Phase-sensitive biodynamic imaging is applied to detect Levy-stable distributions from living tissue that may reflect heavy-tailed anomalous transport within intracellular media.

Fast Mixing in Heterogeneous Media Characterized by Fractional Derivative Model

This study aims at investigating non-Fickian temporal scaling of fast mixing processes using fractional advection dispersion equation (FADE) model in Indiana carbonate, multi-lognormal hydraulic conductivity field, self-affine fractures and cemented porous media, in which the fundamental solution of the FADE model is a standard symmetric Levy stable distribution. The temporal scaling of the scalar dissipation rate (SDR) induced by the FADE model is a function of fractional derivative order $$\alpha$$ , $$\chi (t) \sim t^{{ - \frac{\alpha + 1}{\alpha }}}$$ ( $$1 \le \alpha \le 2$$ ), and it reduces to the Fickian scaling $$t^{{ - \frac{3}{2}}}$$ when $$\alpha = 2$$ . Smaller values of $$\alpha$$ reflect more efficient and a better mixing state at early time. The fitted results show that the FADE model is much more accurate than the traditional model, which can also well interpret the fast mixing scaling from clearer physical mechanism than the empirical power law fitting line. The fitted values of $$\alpha$$ capture the complexity of the heterogeneous media, which are consistent with the existing empirical results. Thus, the SDR of the FADE model is feasible to describe the temporal scaling of the fast mixing for the tracer transport in heterogeneous media.

Characterizing Cryptocurrency Market with Lévy’s Stable Distributions

The recent emergence of cryptocurrencies such as Bitcoin and Ethereum has posed possible alternatives to global payments as well as financial assets around the globe, making investors and financial regulators aware of the importance of modeling them correctly. The Levy's stable distribution is one of the attractive distributions that well describes the fat tails and scaling phenomena in economic systems. In this paper, we show that the behaviors of price fluctuations in emerging cryptocurrency markets can be characterized by a non-Gaussian Levy's stable distribution with $\alpha \simeq 1.4$ under certain conditions on time intervals ranging roughly from 30 minutes to 4 hours. Our arguments are developed under quantitative valuation defined as a distance function using the Parseval's relation in addition to the theoretical background of the General Central Limit Theorem (GCLT). We also discuss the fit with the Levy's stable distribution compared to the fit with other distributions by employing the method based on likelihood ratios. Our approach can be extended for further analysis of statistical properties and contribute to developing proper applications for financial modeling.

Approximation of the road segments travel time using Levy distributions in the reliable shortest path problem

The current trend towards an increase in the number of vehicles, especially in large cities, as well as the unavailability of the existing road infrastructure to distribute modern traffic flows leads to a higher congestion level in transportation networks. This problem emphasized the relevance of navigation problems. Despite the popularity of these problems, many existing commercial systems consider only deterministic networks, not taking into account the time-dependent and stochastic properties of traffic flows. In this paper, we consider the reliable shortest path problem in a time-dependent stochastic transportation network. The considered criterion is maximizing the probability of arriving at the destination point on time. We consider the base algorithm for the stochastic on-time arrival problem, which has a computationally complex convolution operation for calculating the arrival probability. We propose to use parametrically defined Levy stable probability distributions to describe the travel time of road segments. We show, that the use of stable distributions allows us to replace the convolution operation with the distribution value, and significantly reduces the execution time of the algorithm. Experimental analysis has shown that the use of stable distributions allows approximating the exact value of the arrival probability at a destination with a low approximation error.

Measuring Productivity Dispersion: A Parametric Approach Using the Lévy Alpha-Stable Distribution

Productivity levels and growth are extremely heterogeneous among firms. A vast literature has developed to explain the origins of productivity shocks, their dispersion, evolution and their relationship to the business cycle. We examine in detail the distribution of labor productivity levels and growth, and observe that they exhibit heavy tails. We propose to model these distributions using the four parameter Levy stable distribution, a natural candidate deriving from the generalised Central Limit Theorem. We show that it is a better fit than several standard alternatives, and is remarkably consistent over time, countries and sectors. In all samples considered, the tail parameter is such that the theoretical variance of the distribution is infinite, so that the sample standard deviation increases with sample size. We find a consistent positive skewness, a markedly different behaviour between the left and right tails, and a positive relationship between productivity and size. The distributional approach allows us to test different measures of dispersion and find that productivity dispersion has slightly decreased over the past decade.

Effect of Lévy noise on the networks of Izhikevich neurons

Levy noise plays a significant role in jump-noise processes found in neurons. We perform numerical simulations of synaptically coupled Izhikevich networks under the effect of general non-Gaussian Levy noise. Firing dynamics of an all-to-all coupled Izhikevich network and two excitatory coupled Izhikevich networks with differing adaptation properties are studied in response to applied Levy noise. Whole parameter space of Levy noise is investigated by changing its characteristic exponent, skewness, scale and mean parameters. The simulation results show that the noise generated from $$\alpha $$ -stable Levy distribution causes significant dynamical changes in the firing pattern of neuronal network. Different firing patterns exhibited by Izhikevich neurons such as tonic spiking, chattering, intrinsic bursting, low threshold spiking and fast spiking are studied in the network level with and without Levy noise. In all cases, neurons are found to be irregularly spiking when Levy noise with characteristic exponent $$\alpha $$ equal to 0.5 is applied. Levy noise with this particular value of $$\alpha $$ as 0.5 causes the Izhikevich network to have a dynamical behaviour independent of topology and heterogeneity of the network. At this value of characteristic exponent, density of the stable Levy distribution in standard parameterisation is maximum. Other parameters of Levy noise do not influence the way in which a particular neuron is firing. The effects of Levy noise on the correlation between individual neurons and on the networks are investigated using statistical measures.

Better-Fitted Probability of Hydraulic Conductivity for a Silty Clay Site and Its Effects on Solute Transport

The heterogeneous hydraulic conductivity of a subsurface medium is vital to the groundwater flow and solute transport. Probability is efficient for characterizing and quantifying the field characterization of hydraulic conductivity. Compared with sandy mediums, silty clay is paid less attention to due to its low hydraulic conductivity. For long-term solute transport and seawater intrusion, the low-permeable medium is considered as a remarkably permeable medium. This study reports on a comprehensive investigation on the hydraulic conductivity field of the Ningchegu site, located east of Tianjin City of China. Four layers recognized by 52 boreholes, plain fill, continental silty clay, mud–silt clay and marine silty clay, were deposited from the top to the bottom. The hydraulic conductivities measured via permeameter tests ranged from 2 × 10−6 m/d to 1.6 × 10−1 m/d, which corresponded to the lithology of silty clay. The magnitude and the range of the hydraulic conductivity increased with the depth. Five probability distribution models were tested with the experimental probability, indicating that a Levy stable distribution was more matched than the log-normal, normal, Weibull or gamma distributions. A simple analytical model and a Monte Carlo technique were used to inspect the effect of the silty clay hydraulic conductivity field on the statistical behavior of the solute transport. The Levy stable distribution likely generates higher peak concentrations and lower peak times compared with the widely-used log-normal distribution. This consequently guides us in describing the transport of contaminations in subsurface mediums.

Some observations on Levy stability and intermittency in nucleus-nucleus interactions at SPS energies

The power law relation between higher order and second order scaled factorial moments is studied in one dimensional pseudo-rapidity phase (η) space in the interactions of 32S beam with CNO, AgBr and Emulsion at incident energy of 200 AGeV. Observation for such a power law may indicate a self similar cascade mechanism in multiparticle production process. The values of slope, βq are found to be independent of target size. The value of the scaling exponent υ = 1.412 obtained is higher than the critical value υ = 1.304, indicating that no second order phase transition exists in our data. The ratio of anomalous fractal dimensions, dq/d2 is found to increase with increase in the order of moments, q. The dependence of dq/d2 on q indicates a multifractal structure and the presence of self-similar cascading mechanism in our data. The dq/d2 values are well described by the Levy-stable distribution with Levy index μ = 1.562 which is consistent with and lies within the Levy stable region (0 ≤ μ ≤ 2). The multifractal spectrum is concave downward with a maximum at q = 0. The decrease in Dq with increasing q shows that there is a self affine multifractal behaviour in multiparticle production in our data.

SIMPLE ESTIMATOR AND CONSISTENT STRONGLY OF STABLE DISTRIBUTIONS

Stable distributions are extensively used to analyze earnings of financial assets, such as exchange rates and stock prices assets. In this paper we propose a simple and strongly consistent estimator for the scale parameter of a symmetric stable Levy distribution. The advantage of this estimator is that your computational time is minimum thus it can be used to initialize intensive computational procedure such as maximum likelihood. With random samples of sized n we tested the efficacy of these estimators by Monte Carlo method. We also included applications for three data sets.

On the Coherence of VAR Risk Measures for Levy Stable Distributions

The Value-at-Risk (VaR) risk measure has been widely used in finance and insurance for capital and risk management. However, in recent years it has fallen somewhat out of favour due to a seminal paper by Artzner et al. (1999) who showed that VaR does not in general have all the four coherence properties which are desirable for a risk measure. In particular, the violation of the sub-additive property discourages diversification and is counter-intuitive to risk finance. In this paper, it is proved (Theorem 3.1) that VaR for independent Levy-stable random variates is a coherent risk measure being translational invariant, monotonic, positively homogeneous and sub-additive. That is, Levy-stable distributions are VaR coherent. As Levy-stable distributions are a rich class of probability distributions, the VaR risk measure may still have widespread applications. A brief comparative discussion is also given for L-stable variates for the expected shortfall risk measure.

Anomalous Subdiffusive Measurements by Fluorescence Correlations Spectroscopy and Simulations of Translational Diffusive Behavior in Live Cells

Using a combination of optical experiments and computer simulations, we found that live cells act as traps to produce anomalous subdiffusive translational motion of molecules exemplified by the glucocorticoid receptor α. The glucocorticoid receptor α was expressed as a fusion protein with the green fluorescent protein in living mammalian U2OS cells. The measurements were carried out by fluorescence correlation spectroscopy. The glucocorticoid receptor α was either a homodimer or a monomer in the nucleoplasm depending on the presence or absence of the stimulus dexamethasone. Our simulations showed that the experimentally measured rates of translational diffusion could be attributed to spatial and temporal traps. Thus, traps acted by spatial and temporal heterogeneity and randomness, respectively. As we prove here for the first time, anomalous translational diffusion caused by spatial randomness was different from anomalous translational diffusion caused by heterogeneous temporal randomness. For this purpose, we explored two classes of transformations in the time domain for which the assumption that they are stable and have probability density functions was satisfied. The first one was the Inverse Gamma distribution and the second class was a stable Levy distribution. The spatial exponent α that was extracted from time series expressed scale invariance in the space domain. The time exponent γ measured limited time scaling of the embedding complex dynamics. Hence, both exponents must not be mixed up by a single exponent. Fluorescence fluctuation spectroscopy (FCS) is the method of choice for measuring the exponents of anomalous subdiffusive translational motion of molecules in live cells.

Average Sample Number Function for Pareto Heavy Tailed Distributions

The main purpose of this work is shortly to give the average sample number function after a sequential probability ratio test on the index parameter alpha of stable densities, which we give a mean of the number of data required to take decision in the case , we use the fact that the tails of Levy-stable distributions are asymptotically equivalent to a Pareto law for large data. Stable distributions are a rich class of probability distributions that allow skewness and heavy tails and have many intriguing mathematical properties. The lack of closed formulas for densities and distribution functions for all has been a major drawback to the use of stable distributions by practitioners, but few stable distributions have the analytical formula of their densities functions which are Gauss, Levy, and Cauchy.

Fat tails in marine microbial population fluctuations

The fluctuations of natural populations have deep impact on important ecological issues, such as pest outbreaks, fisheries or the formation of harmful algal blooms (HABs). However, consensus on the appropriate descriptor of such fluctuations is still lacking. Here, using 16 to 20 years of weekly data on marine microbial population abundance comprising more than 200 species, we analysed the distribution of the population fluctuations. We found that population fluctuations of all groups and in 12 out of 17 species were not Gaussian (D’Agostino test p < 0.05) but instead were adequately described by Levy‐stable distributions (LSD). Consistent with ecological theories, the LSD characteristic parameter (α) characterizes as highly volatile those groups known to form HABs, such as dinoflagellates (α= 1.48), and as lowly volatile, nanoflagellates (α= 1.92), a group which can be subjected to predatory control. Moreover, zooplankton groups composed of species with sexual reproduction and complex life cycles such as crustacean and Appendicullaria also showed departures from Gaussian population fluctuations and adequate fits to LSD. Our results suggest that heavy‐tailed population fluctuations are widespread, implying that extreme population fluctuations are more likely than previously expected, a fact that has important consequences for the predictability of population outbreaks.

Non-Gaussian nature of fracture and the survival of fat-tail exponents.

We study the fluctuations of the global velocity V(l)(t), computed at various length scales l, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity--and extreme event distributions--by measuring only at a global scale.

Image restoration of the open-loop adaptive optics retinal imaging system based on optical transfer function analysis

The residual aberrations of the adaptive optics retinal imaging system will decrease the quality of the retinal images. To overcome this obstacle, we found that the optical transfer function (OTF) of the adaptive optics retinal imaging system can be described as the Levy stable distribution. Then a new method is introduced to estimate the OTF of the open-loop adaptive optics system, based on analyzing the residual aberrations of the open-loop adaptive optics system in the residual aberrations measuring mode. At last, the estimated OTF is applied to restore the retinal images of the open-loop adaptive optics retinal imaging system. The contrast and resolution of the restored image is significantly improved with the Laplacian sum (LS) from 0.0785 to 0.1480 and gray mean grads (GMG) from 0.0165 to 0.0306.

Fluctuations in multiplicative systems with jumps

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.

Operator solutions for fractional Fokker-Planck equations

We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided Lévy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.

BEC, the $\tau$-Model, and Jets at the Z Pole

Bose-Einstein correlations of pairs of identical charged pions produced in hadronic Z decays are analyzed in terms of various parametrizations. A good description is achieved using a Levy stable distribution in conjunction with a model where a particle's momentum is highly correlated with its space-time point of production, the τ -model. However, a small but significant elongation of the particle emission region is observed in the Longitudinal Center of Mass frame, which is not accommodated in the τ -model. Further, for three-jet events the region is found to be larger in the event plane than out of the plane.

Lévy stable two-sided distributions: Exact and explicit densities for asymmetric case

We investigate functions g(α,β;x) which are heavy-tailed Lévy stable probability distributions of index 0<α≤2 and appropriate asymmetry parameter β. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g(α,β;x) for all admissible rational values of α and β. We reproduce all the previously known cases and furnish distributions for values α and β which were previously inaccessible. We point out many instances of experimental and statistical data that could be described by our results.