Non-Gaussian probability distributions for a vortex fluid

1977 ◽  
Vol 20 (3) ◽  
pp. 356 ◽  
Author(s):  
T. S. Lundgren ◽  
Y. B. Pointin
Keyword(s):  
Probability Distributions ◽  
Non Gaussian

Non-Gaussian probability distributions of solar wind fluctuations

1994 ◽  
Vol 12 (12) ◽  
pp. 1127-1138 ◽  
Author(s):  
E. Marsch ◽  
C. Y. Tu
Keyword(s):  
Solar Wind ◽  
Probability Distributions ◽  
Time Lag ◽  
Small Scale ◽  
Time Lags ◽  
Mhd Turbulence ◽  
Theoretical Concepts ◽  
Fluid Turbulence ◽  
Long Time ◽  
Non Gaussian

Abstract. The probability distributions of field differences ∆x(τ)=x(t+τ)-x(t), where the variable x(t) may denote any solar wind scalar field or vector field component at time t, have been calculated from time series of Helios data obtained in 1976 at heliocentric distances near 0.3 AU. It is found that for comparatively long time lag τ, ranging from a few hours to 1 day, the differences are normally distributed according to a Gaussian. For shorter time lags, of less than ten minutes, significant changes in shape are observed. The distributions are often spikier and narrower than the equivalent Gaussian distribution with the same standard deviation, and they are enhanced for large, reduced for intermediate and enhanced for very small values of ∆x. This result is in accordance with fluid observations and numerical simulations. Hence statistical properties are dominated at small scale τ by large fluctuation amplitudes that are sparsely distributed, which is direct evidence for spatial intermittency of the fluctuations. This is in agreement with results from earlier analyses of the structure functions of ∆x. The non-Gaussian features are differently developed for the various types of fluctuations. The relevance of these observations to the interpretation and understanding of the nature of solar wind magnetohydrodynamic (MHD) turbulence is pointed out, and contact is made with existing theoretical concepts of intermittency in fluid turbulence.

scholarly journals Heavy-tailed random matrices

2018 ◽  
Author(s):  
Vladimir Kravtsov
Keyword(s):  
Random Matrices ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Basin Of Attraction ◽  
Random Variables ◽  
Universal Properties ◽  
Random Matrix Ensembles ◽  
Free Random Variables ◽  
Heavy Tailed ◽  
Non Gaussian

This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

scholarly journals Need for Caution in Interpreting Extreme Weather Statistics

2015 ◽  
Vol 28 (23) ◽  
pp. 9166-9187 ◽  
Author(s):  
Prashant D. Sardeshmukh ◽  
Gilbert P. Compo ◽  
Cécile Penland
Keyword(s):  
Process Model ◽  
Large Scale ◽  
Probability Distributions ◽  
Extreme Weather ◽  
Recent Change ◽  
Atmospheric Variability ◽  
Tail Probabilities ◽  
The North ◽  
Special Cases ◽  
Non Gaussian

Abstract Given the reality of anthropogenic global warming, it is tempting to seek an anthropogenic component in any recent change in the statistics of extreme weather. This paper cautions that such efforts may, however, lead to wrong conclusions if the distinctively skewed and heavy-tailed aspects of the probability distributions of daily weather anomalies are ignored or misrepresented. Departures of several standard deviations from the mean, although rare, are far more common in such a distinctively non-Gaussian world than they are in a Gaussian world. This further complicates the problem of detecting changes in tail probabilities from historical records of limited length and accuracy. A possible solution is to exploit the fact that the salient non-Gaussian features of the observed distributions are captured by so-called stochastically generated skewed (SGS) distributions that include Gaussian distributions as special cases. SGS distributions are associated with damped linear Markov processes perturbed by asymmetric stochastic noise and as such represent the simplest physically based prototypes of the observed distributions. The tails of SGS distributions can also be directly linked to generalized extreme value (GEV) and generalized Pareto (GP) distributions. The Markov process model can be used to provide rigorous confidence intervals and to investigate temporal persistence statistics. The procedure is illustrated for assessing changes in the observed distributions of daily wintertime indices of large-scale atmospheric variability in the North Atlantic and North Pacific sectors over the period 1872–2011. No significant changes in these indices are found from the first to the second half of the period.

Application of Two Bayesian Filters to Estimate Unknown Heat Fluxes in a Natural Convection Problem

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Marcelo J. Colaço ◽  
Helcio R. B. Orlande ◽  
Wellington B. da Silva ◽  
George S. Dulikravich
Keyword(s):  
Natural Convection ◽  
Sequential Monte Carlo ◽  
Nonlinear Models ◽  
Probability Distributions ◽  
Heat Fluxes ◽  
Large Set ◽  
Sampling Importance Resampling ◽  
Filter Methods ◽  
Importance Resampling ◽  
Non Gaussian

Sequential Monte Carlo (SMC) or particle filter methods, which have been originally introduced in the beginning of the 1950s, became very popular in the last few years in the statistical and engineering communities. Such methods have been widely used to deal with sequential Bayesian inference problems in the fields like economics, signal processing, and robotics, among others. SMC methods are an approximation of sequences of probability distributions of interest, using a large set of random samples, named particles. These particles are propagated along time with a simple Sampling Importance distribution. Two advantages of this method are: they do not require the restrictive hypotheses of the Kalman filter, and they can be applied to nonlinear models with non-Gaussian errors. This paper uses two SMC filters, namely the SIR (sampling importance resampling filter) and the ASIR (auxiliary sampling importance resampling filter) to estimate a heat flux on the wall of a square cavity encasing a liquid undergoing natural convection. Measurements, which contain errors, taken at the boundaries of the cavity were used in the estimation process. The mathematical model as well as the initial condition are supposed to have some errors, which were taken into account in the probabilistic evolution model used for the filter. Also, the results using different grid sizes and patterns for the direct and inverse problems were used to avoid the so-called inverse crime. In these results, additional errors were considered due to the different location of the grid points used. The final results were remarkably good when using the ASIR filter.

Response of a Tethered Aerostat to Simulated Turbulence

2004 ◽  
Author(s):  
Keith A. Stanney ◽  
Christopher D. Rahn
Keyword(s):  
Atmospheric Turbulence ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Decomposition Theorem ◽  
Worst Case ◽  
Gaussian Density ◽  
Time Histories ◽  
Wind Input ◽  
Non Gaussian ◽  
Proper Orthogonal

Aerostats are lighter-than-air vehicles tethered to the ground by a cable and used for broadcasting, communications, surveillance, and drug interdiction. The dynamic response of tethered aerostats subject to extreme atmospheric turbulence often dictates survivability. This paper develops a theoretical model that predicts the planar response of a tethered aerostat subject to atmospheric turbulence and simulates the response to 1000 simulated hurricane scale turbulent time histories. The aerostat dynamic model assumes the aerostat hull to be a rigid body with nonlinear fluid loading, instantaneous weathervaning for planar response, and a continuous tether. Galerkin’s method discretizes the coupled aerostat and tether partial differential equations to produce a nonlinear initial value problem that is integrated numerically given initial conditions and wind inputs. The proper orthogonal decomposition theorem generates, based on Hurricane Georges wind data, turbulent time histories that possess the sequential behavior of actual turbulence, are spectrally accurate, and have non-Gaussian density functions. The generated turbulent time histories are simulated to predict the aerostat response to severe turbulence. The resulting probability distributions for the aerostat position, pitch angle, and confluence point tension predict the aerostat behavior in high gust environments. The results uncover a worst case wind input consisting of a two-pulse vertical gust.

Multiplicative Noise and Non-Gaussianity: A Paradigm for Atmospheric Regimes?

2005 ◽  
Vol 62 (5) ◽  
pp. 1391-1409 ◽  
Author(s):  
Philip Sura ◽  
Matthew Newman ◽  
Cécile Penland ◽  
Prashant Sardeshmukh
Keyword(s):  
Large Scale ◽  
Multiplicative Noise ◽  
Probability Distributions ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Stochastic Perturbations ◽  
The Past ◽  
State Dependent ◽  
Non Gaussian ◽  
Noise Statistics

Abstract Atmospheric circulation statistics are not strictly Gaussian. Small bumps and other deviations from Gaussian probability distributions are often interpreted as implying the existence of distinct and persistent nonlinear circulation regimes associated with higher-than-average levels of predictability. In this paper it is shown that such deviations from Gaussianity can, however, also result from linear stochastically perturbed dynamics with multiplicative noise statistics. Such systems can be associated with much lower levels of predictability. Multiplicative noise is often identified with state-dependent variations of stochastic feedbacks from unresolved system components, and may be treated as stochastic perturbations of system parameters. It is shown that including such perturbations in the damping of large-scale linear Rossby waves can lead to deviations from Gaussianity very similar to those observed in the joint probability distribution of the first two principal components (PCs) of weekly averaged 750-hPa streamfunction data for the past 52 winters. A closer examination of the Fokker–Planck probability budget in the plane spanned by these two PCs shows that the observed deviations from Gaussianity can be modeled with multiplicative noise, and are unlikely the results of slow nonlinear interactions of the two PCs. It is concluded that the observed non-Gaussian probability distributions do not necessarily imply the existence of persistent nonlinear circulation regimes, and are consistent with those expected for a linear system perturbed by multiplicative noise.

THEORY OF RANDOM ADVECTION IN TWO DIMENSIONS

1996 ◽  
Vol 10 (18n19) ◽  
pp. 2273-2309 ◽  
Author(s):  
M. CHERTKOV ◽  
G. FALKOVICH ◽  
I. KOLOKOLOV ◽  
V. LEBEDEV
Keyword(s):  
Velocity Field ◽  
Probability Distribution ◽  
Probability Distributions ◽  
Classical Problem ◽  
Passive Scalar ◽  
Two Dimensions ◽  
Change Of Variables ◽  
The Matrix ◽  
Two Dimensional Flow ◽  
Non Gaussian

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n < ln (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.

scholarly journals Fatigue Test of 6082 Aluminum Alloy under Random Load with Controlled Kurtosis

Materials ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 856
Author(s):  
Robert Owsiński ◽  
Adam Niesłony
Keyword(s):  
Aluminum Alloy ◽  
Fatigue Life ◽  
Probability Distributions ◽  
Experimental Tests ◽  
Random Load ◽  
Additional Weight ◽  
Random Loads ◽  
Kurtosis Parameter ◽  
6082 Aluminum Alloy ◽  
Non Gaussian

This paper presents the results of experimental tests carried out on an electromagnetic shaker where the excited element was a specimen with additional weight attached to the slip table. The load was random with a different kurtosis parameter value, i.e., it was performed for non-Gaussian loads. The experiment was accompanied by basic fatigue calculations in the frequency domain and their verification with experimental results. A significant decrease in fatigue life was found to take place with an increase in kurtosis and the maintenance of the same standard deviation of the specimen load. The fatigue effect, caused by the deviation from the normal distribution that was described by the kurtosis parameter, on the fatigue life of aluminum alloy 6082 was presented. An analysis revealed the different amplitude probability distributions for the loading signal and the recorded deformation signal. It was concluded that there was a lack of sensitivity of the numerical model to the change in the kurtosis parameter of the distribution of random loads.

scholarly journals Density estimation with Gaussian processes for gravitational wave posteriors

2021 ◽  
Vol 508 (2) ◽  
pp. 2090-2097
Author(s):  
V D’Emilio ◽  
R Green ◽  
V Raymond
Keyword(s):  
Gaussian Processes ◽  
Posterior Probability ◽  
Probability Function ◽  
Probability Distributions ◽  
Source Parameters ◽  
Kernel Density ◽  
Marginal Probability ◽  
Stochastic Sampling ◽  
Interpolation Accuracy ◽  
Non Gaussian

ABSTRACT The properties of black hole and neutron-star binaries are extracted from gravitational waves (GW) signals using Bayesian inference. This involves evaluating a multidimensional posterior probability function with stochastic sampling. The marginal probability distributions of the samples are sometimes interpolated with methods such as kernel density estimators. Since most post-processing analysis within the field is based on these parameter estimation products, interpolation accuracy of the marginals is essential. In this work, we propose a new method combining histograms and Gaussian processes (GPs) as an alternative technique to fit arbitrary combinations of samples from the source parameters. This method comes with several advantages such as flexible interpolation of non-Gaussian correlations, Bayesian estimate of uncertainty, and efficient resampling with Hamiltonian Monte Carlo.

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