Stable Distribution Function Research Articles

Multivariate Subgaussian Stable Distributions in R

We introduce and showcase [mvpd](https://CRAN.R-project.org/package=mvpd) (an acronym for *m*ulti*v*ariate *p*roduct *d*istributions), a package that uses a product distribution approach to calculating multivariate subgaussian stable distribution functions. The family of multivariate subgaussian stable distributions are elliptically contoured multivariate stable distributions that contain the multivariate Cauchy and the multivariate normal distribution. These distributions can be useful in modeling data and phenomena that have heavier tails than the normal distribution (more frequent occurrence of extreme values). Application areas include log returns for stocks, signal processing for radar and sonar data, astronomy, and hunting patterns of sharks.

A generative model for age and income distribution

Each individual in society experiences an evolution of their income during their lifetime. Macroscopically, this dynamic creates a statistical relationship between age and income for each society. In this study, we investigate income distribution and its relationship with age and identify a stable joint distribution function for age and income within the United Kingdom and the United States. We demonstrate a flexible calibration methodology using panel and population surveys and capture the characteristic differences between the UK and the US populations. The model here presented can be utilised for forecasting income and planning pensions.

Exact and approximate analytical time-domain Green's functions for space-fractional wave equations.

The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α0|ω|y. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.

Approximate analytical time-domain Green’s functions for space-fractional wave equations

Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the power law exponent and attenuation constant for breast and for liver. The results for both breast and liver converge very close to the source, and similar performance is observed in time-domain Green’s functions computed for linear with frequency attenuation. Despite minor differences in the arguments, the approximate analytical time-domain Green’s functions derived for dispersive time-fractional and space-fractional wave equations are also quite similar. [Work supported in part by NIH Grants EB023051 and EB012079.] Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the po...

Weak and one-sided strong laws for random variables with infinite mean

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.

Data distribution and transformation in population based sampling survey of viral load in HIV positive men who have sex with men in China

Objective: To understand the distribution of population viral load (PVL) data in HIV infected men who have sex with men (MSM), fit distribution function and explore the appropriate estimating parameter of PVL. Methods: The detection limit of viral load (VL) was ≤ 50 copies/ml. Box-Cox transformation and normal distribution tests were used to describe the general distribution characteristics of the original and transformed data of PVL, then the stable distribution function was fitted with test of goodness of fit. Results: The original PVL data fitted a skewed distribution with the variation coefficient of 622.24%, and had a multimodal distribution after Box-Cox transformation with optimal parameter (λ) of-0.11. The distribution of PVL data over the detection limit was skewed and heavy tailed when transformed by Box-Cox with optimal λ=0. By fitting the distribution function of the transformed data over the detection limit, it matched the stable distribution (SD) function (α=1.70, β=-1.00, γ=0.78, δ=4.03). Conclusions: The original PVL data had some censored data below the detection limit, and the data over the detection limit had abnormal distribution with large degree of variation. When proportion of the censored data was large, it was inappropriate to use half-value of detection limit to replace the censored ones. The log-transformed data over the detection limit fitted the SD. The median (M) and inter-quartile ranger (IQR) of log-transformed data can be used to describe the centralized tendency and dispersion tendency of the data over the detection limit.

Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

Stable distributions are an important class of infinitely-divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form, and is expressible only through the variate's characteristic function or other integral forms. In this paper we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no pre-existing efficient method for their computation. The novel evaluation schemes consist of optimized generalized Gaussian quadrature rules for integral representations of the density function, complemented by various asymptotic expansions near various values of the shape and argument parameters. We report several numerical examples illustrating the efficiency of our methods. The resulting code has been made available online.

On the properties of Laplace transform originating from one-sided Lévy stable laws

We consider the conventional Laplace transform of f(x), denoted by with . For we furnish the closed form expressions for the inverse Laplace transforms and . In both cases they involve definite integration with kernels which are appropriately rescaled one-sided Lévy stable probability distribution functions , , . Since are exactly and explicitly known for rational α, i.e. for with , , our results extend the known and tabulated case of to any rational . We examine the integral kernels of this procedure as well as the resulting two kinds of Lévy integral transformations.

On limit distributions of the time of first passage over a high level

We describe a simple method of finding the limit distribution of the time of first passage over a high level for a random sequence (or random process). The method reduces the considered problem to a well-studied problem of finding the distribution of this random sequence (or process) at a certain fixed time. This allows to consider not only traditional sums of independent random variables but also sums of dependent random variables. As examples, sums of independent random variables with stable limit distribution functions, sequential testing of hypotheses, and problems of the first passage over a high level for Gaussian processes with strong dependencies are considered.

Physics-Based Anomaly Detection Defined on Manifold Space

Current popular anomaly detection algorithms are capable of detecting global anomalies but often fail to distinguish local anomalies from normal instances. Inspired by contemporary physics theory (i.e., heat diffusion and quantum mechanics), we propose two unsupervised anomaly detection algorithms. Building on the embedding manifold derived from heat diffusion, we devise Local Anomaly Descriptor (LAD), which faithfully reveals the intrinsic neighborhood density. It uses a scale-dependent umbrella operator to bridge global and local properties, which makes LAD more informative within an adaptive scope of neighborhood. To offer more stability of local density measurement on scaling parameter tuning, we formulate Fermi Density Descriptor (FDD), which measures the probability of a fermion particle being at a specific location. By choosing the stable energy distribution function, FDD steadily distinguishes anomalies from normal instances with any scaling parameter setting. To further enhance the efficacy of our proposed algorithms, we explore the utility of anisotropic Gaussian kernel (AGK), which offers better manifold-aware affinity information. We also quantify and examine the effect of different Laplacian normalizations for anomaly detection. Comprehensive experiments on both synthetic and benchmark datasets verify that our proposed algorithms outperform the existing anomaly detection algorithms.

Induced Emission of Alfvén Waves in Inhomogeneous Streaming Plasma: Implications for Solar Corona Heating and Solar Wind Acceleration

The results of a self-consistent kinetic model of heating the solar corona and accelerating the fast solar wind are presented for plasma flowing in a nonuniform magnetic field configuration of near-Sun conditions. The model is based on a scale separation between the large transit or inhomogeneity scales and the small dissipation scales. The macroscale instability of the marginally stable particle distribution function compliments the resonant frequency sweeping dissipation of transient Alfvén waves by their induced emission in inhomogeneous streaming plasma that provides enough energy for keeping the plasma temperature decaying not faster than r(-1) in close agreement with in situ heliospheric observations.

An empirical model for trip distribution of commuters in The Netherlands: transferability in time and space reconsidered

In this paper, we evaluate the distribution of commute trips in The Netherlands, to assess its transferability in space and time. We used Dutch Travel Surveys from 1995 and 2004–2008 to estimate the empirical distribution from a spatial interaction model as function of travel time and distance. We find that the distribution function is not transferable in time and space. Instead, the propensity to travel longer to work has grown over time. As a result, the average commute length has increased. Contrary, land use changes appeared to have a much smaller effect on commuting distances in this period. The propensity to make longer commute trips also increases with city-size. However, this does not lead to an increase in average commuting distance with city-size, because it is offset by smaller distances between residences and jobs in large cities. The results are quite relevant for spatial interaction modeling. They indicate that the concept of one stable distribution function is insufficient to accurately describe aggregated travel behavior.

Quasilinear evolution of plasma distribution functions and consequences on wave spectrum and perpendicular ion heating in the turbulent solar wind

The measured spectrum of kinetic Alfven wave fluctuations in the turbulent solar wind plasma is used to calculate the quasi-linear evolution of the initially stable electron and ion distribution functions. The resulting ion distribution function is found to be unstable to electromagnetic left hand polarized ion cyclotron-Alfven waves as well as right hand polarized magnetosonic-whistler waves. These waves can pitch angle scatter the ion super-thermal velocity component to provide perpendicular ion heating. Additionally, right hand polarized waves transfer some part of kinetic Alfven wave flux to whistler waves.

German Real Estate Return Distributions: Is There Anything Normal?

Executive Summary. This paper uses a sample ofGerman commercial and residential property returns toestimate parameters for stable distribution functions. Aquantile-based estimation methodology is used to examinedistributions of income, capital growth, and totalreturns. There are controls for the effects of propertycharacteristics and for possible differences betweenappraisal-based and transaction-based return distributionparameters. The findings reveal that the assumptionof normality in return distributions can be rejected forvirtually all subsamples of all property types, and for allyears from 2000 to 2009. However, income return distributionsare found to be less leptokurtic than those ofcapital growth. Building characteristics do not havea strong influence on distribution parameters, whiletransaction-based returns do differ from appraisal-basedreturns for retail and residential properties.

Electrostatic fluctuations in nonequilibrium plasmas with particle collisions

We present a method for calculation of plasma fluctuations in a nonequilibrium plasma with stable particle distribution functions. The method takes into account particle collisions and collective particle interactions which affect the collision process. In contrast to the known approaches, our method makes no reference to a specific form of the collision integrals and distribution functions. Using the developed method, we calculate the high-frequency spectrum of the electric field fluctuations in a collisional plasma with arbitrary particle distribution functions.

Phase transition in d-dimensional long-range interacting systems

We present new iterative method for the derivation one-point distribution function in the equilibrium state. For derivation of the distribution function, we must solve Lane-Emden equation. In general case, we solve the equation with iterative method. However the traditional method is not ensured convergence of the algorithm, we cannot often obtain solutions. In order to obtain the stable stationary distribution function, we apply an iterative method, inspired by a previous one used in 2D turbulence. Our method ensures entropy increase and convergence of the algorithm. Furthermore, our method can obtain the distribution function quickly [1]. Here we present the phase transition in longrange interacting systems. The Hamiltonian Mean-Field (HMF) model describes the motion of globally coupled particles on a 1D circle. Nevertheless the interaction is described only one cosine function, both the dynamical and the thermodynamical properties of this mode are quite various and complicated. For the HMF model, the extension for 2D model had been proposed [3]. We have extended the HMF model for 3D and 4D models. Then we analyze the phase-transition for d-dimensional models. The Hamiltonian of the HMF models is written as

Diffusion in a half-space: From Lord Kelvin to path integrals

Many important transport phenomena are described by simple mathematical models rooted in the diffusion equation. Geometrical constraints present in such phenomena often have a global influence and manifest themselves in scaling relations and stable distribution functions. I treat a random walk confined to a half-space using several approaches: diffusion equations, lattice walks, and path integrals. Potential generalizations are discussed.

Physical and geometrical optics for 2-D rough surfaces with power-law height spectra

Random rough surfaces with slowly decaying power spectral density can have infinite slope variance. Such surfaces do not satisfy the classical curvature criterion for validity of the physical optics (PO) approximation, and the infinite frequency geometrical optics limit or specular point scattering model breaks down. We show for two-dimensional surfaces with infinite slope variance that the Gaussian form of the classical geometrical limit generalizes to a stable distribution function. We also show that the PO integral is insensitive to surface components with spatial frequency above a cutoff wavenumber, which explains past observations that PO can be accurate for surfaces with power law spectra. This result leads to a general validity condition for the PO approximation in the backscattering direction for power-law surfaces, which in the case of a k/sup -4/ spectrum requires that the significant slope of the surface be less than 0.03.

The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: finite shear rates

The purpose of this paper is to extend the rheological predictions of the Doi-Hess kinetic theory for sheared nematic polymers from the anomalous weak shear regime (Forest et al. 2004a) to arbitrary shear rates, and to associate salient rheological and optical properties with the solution space of kinetic theory. Using numerical bifurcation software (AUTO), we provide the phase diagram of all stable monodomain orientational probability distribution functions (PDFs) and their phase transitions (bifurcations) vs nematic concentration (N) and normalized shear rate (Peclet number, Pe) for Pe≥1. Shear stresses, normal stress differences, the peak direction of the orientational distribution, and birefringence order parameters are calculated and illustrated for each type of PDF attractor: steady flow-aligning, both in and out of the flow deformation plane and along the vorticity axis; unsteady limit cycles, where the peak orientation direction rotates in-plane or around the vorticity axis or in bi-stable orbits tilted between them; and chaotic attractors first observed in kinetic simulations by Grosso et al. (2001). We pay particular attention to correlations between rheological features and the variety of monodomain phase transitions. Together with the weak flow regime, these results provide a nearly complete picture of the rheological consequences of the Doi-Hess kinetic theory for sheared monodomains of rigid, extreme aspect ratio, nematic rods or platelets.

Quantitative Measurement of Two-Dimensional Distribution Functions of Diffusion and Relaxation in Grossly Inhomogeneous Fields

We demonstrate the quantitative extraction of multidimensional distribution functions in the presence of grossly inhomogeneous fields. Examples are shown for diffusion— T 2 distribution functions and T 1− T 2 distribution functions. The pulse sequences consist of an initial editing sequence followed by a long series of nominal 180° pulses. They are designed such that the kernels describing the relationships between the distribution functions and the measured data are separable. The required phase cycling is discussed. We analyze in detail the extra spin dynamics effects due to the strong field inhomogeneities including the effects on diffusion and relaxation. A recently developed algorithm is used to invert the data and extract stable multidimensional distribution functions in an efficient manner. We present examples for several applications of this new technique. Diffusion–relaxation distribution functions can be used for fluid identification and for the characterization of pore geometry of porous media based on the effects of restricted diffusion. We have also determined T 1− T 2 distribution functions of water saturated sedimentary rock and find excellent agreement with previous measurements performed in homogeneous fields.