non-Gaussian Probability Distribution Function Research Articles

Enhance primordial black hole abundance through the non-linear processes around bounce point

The non-singular bouncing cosmology is an alternative paradigm to inflation, wherein the background energy density vanishes at the bounce point, in the context of Einstein gravity. Therefore, the non-linear effects in the evolution of density fluctuations (δρ) may be strong in the bounce phase, which potentially provides a mechanism to enhance the abundance of primordial black holes (PBHs). This article presents a comprehensive illustration for PBH enhancement due to the bounce phase. To calculate the non-linear evolution of δρ, the Raychaudhuri equation is numerically solved here. Since the non-linear processes may lead to a non-Gaussian probability distribution function for δρ after the bounce point, the PBH abundance is calculated in a modified Press-Schechter formalism. In this case, the criterion of PBH formation is complicated, due to complicated non-linear evolutionary behavior of δρ during the bounce phase. Our results indicate that the bounce phase indeed has potential to enhance the PBH abundance sufficiently. Furthermore, the PBH abundance is applied to constrain the parameters of bounce phase, providing a complementary to the surveys of cosmic microwave background and large scale structure.

An amended approximation of the non-Gaussian probability distribution function

Network models of rubber elasticity are based on the conformational entropy of an idealized chain and mostly motivated by the non-Gaussian statistical theory by Kuhn and Grün. However, the non-Gaussian probability distribution function cannot be expressed in a closed form and requires an approximation. All such approximations applied in the literature demonstrate pronounced inaccuracies in comparison to the analytical solution. The ideal choice of the approximation function depends on a variety of factors, such as the chain parameters or the desired application of the approximation (the probability distribution function itself, the corresponding entropic energy, or the force developed by the chain). In addition, when making a choice regarding the best approximation for a given application, the applied error measure plays a significant role since the approximation that grants, for example, the best maximal relative error is not necessarily the same that provides the best mean absolute error. In the literature, this application-specific evaluation of available approximations is commonly disregarded. In this paper, we evaluate previously proposed approximations on the application-specific basis and develop an approach to derive a family of approximations for the free energy of a polymer chain in a broader range of the number of its chain segments. The analytical method based on the Padé technique delivers an approximation of the non-Gaussian probability distribution function that can be easily tailored depending on the desired application. The proposed approach is capable to provide much stronger predictions in comparison to the Kuhn and Grün model in a wide range of chain segment numbers.

Automated glaucoma detection from fundus images using wavelet-based denoising and machine learning

Glaucoma is a domineering and irretrievable neurodegenerative eye disease produced by the optical nerve head owed to extended intra-ocular stress inside the eye. Recognition of glaucoma is an essential job for ophthalmologists. In this paper, we propose a methodology to classify fundus images into normal and glaucoma categories. The proposed approach makes use of image denoising of digital fundus images by utilizing a non-Gaussian bivariate probability distribution function to model the statistics of wavelet coefficients of glaucoma images. The traditional image features were extracted followed by the popular feature selection algorithm. The selected features are then fed to the least square support vector machine classifier employing various kernel functions. The comparison result shows that the proposed approach offers maximum classification accuracy of nearly 91.22% over the existing best approaches.

Probabilistic observation model correction using non-Gaussian belief fusion

This paper presents a framework for state estimation which tolerates uncertainty in observation model parameters by (1) incorporating this uncertainty in state observation, and (2) correcting model parameters to improve future state observations. The first objective is met by an uncertainty propagation approach, while the second is achieved by gradient-descent optimization. The novel framework allows state estimates to be represented by non-Gaussian probability distribution functions. By correcting observation model parameters, estimation performance is enhanced since the accuracy of observations is increased. Monte Carlo simulation experiments validate the efficacy of the proposed approach in comparison with conventional estimation techniques, showing that as model parameters converge to ground-truth over time, state estimation correspondingly improves when compared to a static model estimate. Because observation models cannot be known with perfect accuracy and existing approaches do not address parametric uncertainties in non-Gaussian estimation, this work has both novelty and usefulness in most state estimation contexts.

An Adaptive Bayesian Parameter Estimation of a Synchronous Generator Under Gross Errors

Polynomial-chaos-expansion-based surrogate models have recently been advocated in the literature for power system dynamic parameter estimation. Regarding the estimation of the uncertain generator parameters, a Bayesian inference framework has been proposed based on a polynomial-based reduced-order representation of the synchronous machines using assumed parameter values. Then, the non-Gaussian posterior probability distribution functions (pdfs) of these parameters are recovered through the stochastic sampling approach efficiently. However, facing very large parameter errors, the reliability of the surrogate model decreases, yielding biased estimation results. To overcome this problem, this article develops a hierarchical Bayesian inference framework that processes the measurements provided by phasor measurement units, while making use of multifidelity surrogates together with the importance sampling method. The latter allows us to estimate in an efficient manner the posterior pdfs of the uncertain parameters through the normalized weights of the prior samples. To improve the accuracy of the posterior pdfs, an adaptive procedure is further adopted in the importance sampling for the gradual evolution of its proposal functions. The new proposals assist in fine-tuning the sample space and thereby help to construct surrogates with higher fidelity. Through an iterative process, this approach is able to estimate accurately and efficiently non-Gaussian posterior pdfs of the uncertain generator parameters subject to gross errors.

Statistical description of turbulent transport for flux driven toroidal plasmas

A novel methodology to analyze non-Gaussian probability distribution functions (PDFs) of intermittent turbulent transport in global full-f gyrokinetic simulations is presented. In this work, the auto-regressive integrated moving average (ARIMA) model is applied to time series data of intermittent turbulent heat transport to separate noise and oscillatory trends, allowing for the extraction of non-Gaussian features of the PDFs. It was shown that non-Gaussian tails of the PDFs from first principles based gyrokinetic simulations agree with an analytical estimation based on a two fluid model.

Turbulent fluctuations during pellet injection into a dipole confined plasma torus

We report measurements of the turbulent evolution of the plasma density profile following the fast injection of lithium pellets into the Levitated Dipole Experiment (LDX) [Boxer et al., Nat. Phys. 6, 207 (2010)]. As the pellet passes through the plasma, it provides a significant internal particle source and allows investigation of density profile evolution, turbulent relaxation, and turbulent fluctuations. The total electron number within the dipole plasma torus increases by more than a factor of three, and the central density increases by more than a factor of five. During these large changes in density, the shape of the density profile is nearly “stationary” such that the gradient of the particle number within tubes of equal magnetic flux vanishes. In comparison to the usual case, when the particle source is neutral gas at the plasma edge, the internal source from the pellet causes the toroidal phase velocity of the fluctuations to reverse and changes the average particle flux at the plasma edge. An edge particle source creates an inward turbulent pinch, but an internal particle source increases the outward turbulent particle flux. Statistical properties of the turbulence are measured by multiple microwave interferometers and by an array of probes at the edge. The spatial structures of the largest amplitude modes have long radial and toroidal wavelengths. Estimates of the local and toroidally averaged turbulent particle flux show intermittency and a non-Gaussian probability distribution function. The measured fluctuations, both before and during pellet injection, have frequency and wavenumber dispersion consistent with theoretical expectations for interchange and entropy modes excited within a dipole plasma torus having warm electrons and cool ions.

Non-Gaussian normal diffusion induced by delocalization.

Non-Gaussian normal diffusion, i.e., the probability density function (PDF) is non-Gaussian but the mean squared displacement (MSD) depends on time linearly, has been observed in particle motions. Here we show by numerical simulations that this phenomenon may manifest itself in energy diffusion along a lattice at a nonzero, finite temperature. The studied model is a one-dimensional disordered lattice with on-site potential. We find that the energy density fluctuations are spatially localized if the nonlinear interaction is suppressed, but may spread with a non-Gaussian PDF and a linear time-dependent MSD when the nonlinear interaction is turned on. Our analysis suggests that the mechanism lies in the delocalization properties of the localized modes.

Dynamic properties of small-scale solar wind plasma fluctuations.

The paper presents the latest results of the studies of small-scale fluctuations in a turbulent flow of solar wind (SW) using measurements with extremely high temporal resolution (up to 0.03 s) of the bright monitor of SW (BMSW) plasma spectrometer operating on astrophysical SPECTR-R spacecraft at distances up to 350,000 km from the Earth. The spectra of SW ion flux fluctuations in the range of scales between 0.03 and 100 s are systematically analysed. The difference of slopes in low- and high-frequency parts of spectra and the frequency of the break point between these two characteristic slopes was analysed for different conditions in the SW. The statistical properties of the SW ion flux fluctuations were thoroughly analysed on scales less than 10 s. A high level of intermittency is demonstrated. The extended self-similarity of SW ion flux turbulent flow is constantly observed. The approximation of non-Gaussian probability distribution function of ion flux fluctuations by the Tsallis statistics shows the non-extensive character of SW fluctuations. Statistical characteristics of ion flux fluctuations are compared with the predictions of a log-Poisson model. The log-Poisson parametrization of the structure function scaling has shown that well-defined filament-like plasma structures are, as a rule, observed in the turbulent SW flows.

MODELING OF THE NON-GAUSSIAN PDFs OF FIELD VARIATIONS AND INTERMITTENCY IN THE TURBULENT SOLAR WIND

The probability distribution functions (PDFs) of magnetic field variations display strong scale-dependent non-Gaussianity in the turbulent solar wind. This is a typical signature of intermittent turbulence. Physical modeling of the turbulent field variations based on the characteristics of the observed turbulence, including the variability of its power level, produces, free of parameter adjustment and over a broad range of inertial scales, accurate fits of the non-Gaussian PDFs. The effects of phase randomization and time resolution of the Fourier power spectra are further tested to determine which of the phase correlation or the spectral variability is responsible for the strong non-Gaussianity of the observed PDFs of field variations. The periods of enhanced power level are found to be responsible for the non-Gaussian tails of the PDFs.

Statistical properties of entropy-consuming fluctuations in jammed states of laponite suspensions: Fluctuation relations and generalized Gumbel distribution

We report a universal large deviation behavior of spatially averaged global injected power just before the rejuvenation of the jammed state formed by an aging suspension of laponite clay under an applied stress. The probability distribution function (PDF) of these entropy consuming strongly non-Gaussian fluctuations follow an universal large deviation functional form described by the generalized Gumbel (GG) distribution like many other equilibrium and nonequilibrium systems with high degree of correlations but do not obey the Gallavotti-Cohen steady-state fluctuation relation (SSFR). However, far from the unjamming transition (for smaller applied stresses) SSFR is satisfied for both Gaussian as well as non-Gaussian PDF. The observed slow variation of the mean shear rate with system size supports a recent theoretical prediction for observing GG distribution.

Digital simulation of two-dimensional random fields with arbitrary power spectra and non-Gaussian probability distribution functions

Methods for simulation of two-dimensional signals with arbitrary power spectral densities and signal amplitude probability density functions are disclosed. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In most cases the method provides satisfactory results and can thus be considered an engineering approach. Several illustrative examples with relevance for optics are given.

A Multicriteria Statistical Based Comparison Methodology for Evaluating Evolutionary Algorithms

This paper presents a statistical based comparison methodology for performing evolutionary algorithm comparison under multiple merit criteria. The analysis of each criterion is based on the progressive construction of a ranking of the algorithms under analysis, with the determination of significance levels for each ranking step. The multicriteria analysis is based on the aggregation of the different criteria rankings via a non-dominance analysis which indicates the algorithms which constitute the efficient set. In order to avoid correlation effects, a principal component analysis pre-processing is performed. Bootstrapping techniques allow the evaluation of merit criteria data with arbitrary probability distribution functions. The algorithm ranking in each criterion is built progressively, using either ANOVA or first order stochastic dominance. The resulting ranking is checked using a permutation test which detects possible inconsistencies in the ranking—leading to the execution of more algorithm runs which refine the ranking confidence. As a by-product, the permutation test also delivers <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> -values for the ordering between each two algorithms which have adjacent rank positions. A comparison of the proposed method with other methodologies has been performed using reference probability distribution functions (PDFs). The proposed methodology has always reached the correct ranking with less samples and, in the case of non-Gaussian PDFs, the proposed methodology has worked well, while the other methods have not been able even to detect some PDF differences. The application of the proposed method is illustrated in benchmark problems.

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.

Plasma diffusion in self-consistent fluctuations

The problem of particle diffusion in position space, as a consequence of electromagnetic fluctuations is addressed. Numerical results obtained with a self-consistent hybrid code are presented, and a method to calculate diffusion coefficient in the direction perpendicular to the mean magnetic field is proposed. The diffusion is estimated for two different types of fluctuations. The first type (resulting from an agyrotropic initial setting) is stationary, wide band white noise, and associated to Gaussian probability distribution function for the magnetic fluctuations. The second type (resulting from a Kelvin-Helmholtz instability) is non-stationary, with a power-law spectrum, and a non-Gaussian probability distribution function. The results of the study allow revisiting the question of loading particles of solar wind origin in the Earth magnetosphere.

On the probability distribution function of the velocity field and its derivative in multi-scale turbulence

Using a combination of active and passive grids, we produce an inhomogeneous turbulent flow with strongly non-Gaussian velocity probability distribution (density) functions (pdfs). This new, multi-scale flow shares features with the turbulent jet in freestream turbulence, the near field of turbulence-generating grids, and atmospheric flows. In particular, we study unskewed velocity distributions with both super- and sub-Gaussian tails and demonstrate the linear dependence of the normalized odd and even moments of the distributions on the first and second spatial derivatives of the turbulence r.m.s. fields, respectively. We also note bimodal behavior of the pdfs. The experiments demonstrate a clear effect of the flow organization and the large-scale intermittency on the velocity derivative pdfs and the small scales of the turbulence. The work is motivated by the complex wind fields, and associated, intermittent high stresses, encountered by wind turbines. We also draw comparisons to recent studies of multi-scale turbulence produced by fractal grids.

Metastable phenomena in lutetium nitrate crystal

Abstract The ac conductivities along the c -axis of the lutetium nitrate crystal were measured successively in the frequency range from 0.5 to 10 5 Hz at temperatures from 290 to 210 K. The metastable behaviors of the conductivity were found in the temperature region above 200 K. The fluctuation was given by the successive measurements of the time series of the ac conductivities and represented by non-Gaussian probability distribution function. The effect of the metastable singularity on Raman scattering spectra was observed in the metastable temperature region.

Non-Gaussian statistics and superdiffusion in a driven-dissipative dusty plasma

Particle random motion can exhibit both anomalous diffusion and non-Gaussian statistics in some physical systems. Anomalous diffusion is quantified by a deviation from alpha=1 in a power law for a particle's mean-square displacement, MSD proportional, variant(Deltat)alpha. A deviation from Gaussian statistics for a probability distribution function (PDF) is quantified by fitting to a kappa function or Tsallis distribution, with a fit parameter q. We report an experiment and simulations to test a theory that connects anomalous diffusion and non-Gaussian statistics. In the experiment, a single-layer dusty plasma, which behaved as a two-dimensional (2D) driven-dissipative system, had a non-Gaussian PDF. By adjusting an externally applied laser heating, q was varied over a wide range. A correlation between the deviations from Gaussian statistics and normal diffusion for a 2D liquid was found in the experiment. This correlation indicates a connection between anomalous diffusion and non-Gaussian statistics. However, such a connection is lacking in equilibrium 2D Yukawa liquids, as demonstrated in numerical simulations.

Statistical analysis of the turbulent Reynolds stress and its link to the shear flow generation in a cylindrical laboratory plasma device

The statistical properties of the turbulent Reynolds stress arising from collisional drift turbulence in a magnetized plasma column are studied and a physical picture of turbulent driven shear flow generation is discussed. The Reynolds stress peaks near the maximal density gradient region, and is governed by the turbulence amplitude and cross-phase between the turbulent radial and azimuthal velocity fields. The amplitude probability distribution function (PDF) of the turbulent Reynolds stress is non-Gaussian and positively skewed at the density gradient maximum. The turbulent ion-saturation (Isat) current PDF shows that the region where the bursty Isat events are born coincides with the positively skewed non-Gaussian Reynolds stress PDF, which suggests that the bursts of particle transport appear to be associated with bursts of momentum transport as well. At the shear layer the density fluctuation radial correlation length has a strong minimum (∼4–6mm∼0.5Cs∕Ωci, where Cs is the ion acoustic speed and Ωci is the ion gyrofrequency), while the azimuthal turbulence correlation length is nearly constant across the shear layer. The results link the behavior of the Reynolds stress, its statistical properties, generation of bursty radially going azimuthal momentum transport events, and the formation of the large-scale shear layer.

Safety in Numbers: Gravitational Lensing Degradation of the Luminosity Distance–Redshift Relation

Observation of the expansion history of the Universe allows exploration of the physical properties and energy density of the Universe's various constituents. Standardizable candles such as Type Ia supernovae remain one of the most promising and robust tools in this endeavor, by allowing for a direct measure of the luminosity distance-redshift curve, and thereby producing detailed studies of the dark energy responsible for the Universe's currently accelerating expansion. As such observations are pushed to higher redshifts, the observed flux is increasingly affected by gravitational lensing magnification due to intervening structure along the line-of-sight. We simulate and analyze the non-Gaussian probability distribution function of de/amplification due to lensing of standard candles, quantify the effect of a convolution over many independent sources (which acts to restore the intrinsic average (unlensed) luminosity due to flux conservation), and compute the additional uncertainty due to lensing on derived cosmological parameters. For example, the ``degradation factor'' due to lensing is a factor of three reduction in the effective number of usable supernovae at z=1.5 (for sources with intrinsic flux dispersion of 10%). We also derive a useful expression for the effective increased dispersion in standard candles due to lensing, as a function of redshift.