Probability distribution functions for intermittent scrape-off layer plasma fluctuations

Large-amplitude, intermittent fluctuations are ubiquitous in the boundary region of magnetically confined plasmas and lead to detrimental plasma-wall interactions in next-generation, high duty cycle fusion power experiments. Using gas puff imaging data time series from the scrape-off layer in the Alcator C-Mod device, it is here demonstrated that the large-amplitude fluctuations can be described as a super-position of pulses with a fixed shape and a constant duration. By applying a new deconvolution algorithm on the data time series with a two-sided exponential pulse function, the arrival times and amplitudes of the pulses can be estimated, and the measurement time series can be reconstructed with high accuracy. The pulse amplitudes are shown to follow an exponential distribution. The waiting times between pulses are uncorrelated, their distribution has an exponential tail, and the number of arrivals is a linear function of time. This demonstrates that pulse arrivals follow a homogeneous Poisson process. Identical statistical properties apply to both ohmic and high confinement mode plasmas, clearly demonstrating the universality of the fluctuation statistics in the boundary region of Alcator C-Mod.

It is increasingly apparent that sediment property distributions on sufficiently small scales are probably irregular. This has led to the development of stochastic theory in subsurface hydrology, including statistically heterogeneous concepts based mainly on the Gaussian and Lévy-stable probability density functions (PDFs), the mathematical basis for stochastic fractals. Gaussian and Levy-stable stochastic fractals have been applied both in the field of turbulence and subsurface hydrology. However, measurements have shown that the increment frequency dis-tributions do not always follow Gaussian or Lévy-stable PDFs. Provided herein is an overview of the origin and development of a new non-stationary stochastic process, called fractional Laplace motion (flam) with stationary, correlated, incre-ments called fractional Laplace noise (fLan). It is based on the Laplace PDF and known generalizations, and does not display self-similarity. Uncorrelated versions are equivalent to a Brownian motion subordinated to the gamma process. In anal-ogy to the development of fractional Brownian motion (fBm) from Brownian motion, fLam is equivalent to fBm subordinated to a gamma process. The new

In this paper, we propose a new approach for signal level estimation in binary phase shift keying (BPSK) modulation based on the empirical characteristic function (ECF), when the probability density function (PDF) of the noise is unknown. Then, we compare our proposed method with two other estimators that are suggested for systems with known noises. Numerical results show that, in the presence of Laplace noise, the ECF estimator has a better performance in low signal-to-noise ratios (SNR) in comparison with previously proposed methods. Moreover, the proposed method does not require the knowledge of noise PDF and works without any training sequence.

We compute the Lie symmetries of characteristic function (CF) hierarchy of compressible turbulence, ignoring the effects of viscosity and heat conductivity. In the probability density function (PDF) hierarchy, a typical non-local nature is observed, which is naturally eliminated in the CF hierarchy. We observe that the CF hierarchy retains all the symmetries satisfied by compressible Euler equations. Broadly speaking, four types of symmetries can be discerned in the CF hierarchy: (i) symmetries corresponding to coordinate system invariance, (ii) scaling/dilation groups, (iii) projective groups and (iv) statistical symmetries, where the latter define measures of intermittency and non-gaussianity. As the multi-point CFs need to satisfy additional constraints such as the reduction condition, the projective symmetries are only valid for monatomic gases, that is, the specific heat ratio, $\gamma = 5/3$ . The linearity of the CF hierarchy results in the statistical symmetries due to the superposition principle. For all of the symmetries, the global transformations of the CF and various key compressible statistics are also presented.

<p><span>Measurement uncertainty analysis based on combining the state-of-knowledge distributions requires evaluation of the probability density function (PDF), the cumulative distribution function (CDF), and/or the quantile function (QF) of a random variable reasonably associated with the measurand. This can be derived from the characteristic function (CF), which is defined as a Fourier transform of its probability distribution function. Working with CFs provides an alternative and frequently much simpler route than working directly with PDFs and/or CDFs. In particular, derivation of the CF of a weighted sum of independent random variables is a simple and trivial task. However, the analytical derivation of the PDF and/or CDF by using the inverse Fourier transform is available only in special cases. Thus, in most practical situations, a numerical derivation of the PDF/CDF from the CF is an indispensable tool. In metrological applications, such approach can be used to form the probability distribution for the output quantity of a measurement model of additive, linear or generalized linear form. In this paper we propose new original algorithmic implementations of methods for numerical inversion of the characteristic function which are especially suitable for typical metrological applications. The suggested numerical approaches are based on the Gil-Pelaez inverse formulae and on using the approximation by discrete Fourier transform and the fast Fourier transform (FFT) algorithm for computing PDF/CDF of the univariate continuous random variables. As illustrated here, for typical metrological applications based on linear measurement models the suggested methods are an efficient alternative to the standard Monte Carlo methods.</span></p>

What Jump Process to Use to Model S&P 500 Returns?

Within the general framework of multiplicative log‐normal models, we propose methods to characterise non‐Gaussian and intermittent fluctuations, and study basic characteristics of non‐Gaussian stochastic processes displaying slow convergence to a Gaussian with an increasing coarse‐grained level of the time series. Here the multiplicative log‐normal model stands for a stochastic process described by the multiplication of Gaussian and log‐normally distributed variables. In other words, using two Gaussian variables, ξ and ω, the time series {xi} of this process can be described as xi = ξi expωi. Depending on the variance of ω, λ2, the probability density function (PDF) of x exhibits a non‐Gaussian shape. As the non‐Gaussianity parameter λ2 increases, the non‐Gaussian tails become fatter. On the other hand, when λ2 → 0, the PDF converges to a Gaussian distribution. For the purpose of estimating the non‐Gaussianity parameter λ2 from the observed time series, we evaluate a novel method based on analytical expre...

Postural constraints recruit shorter-timescale processes into the non-Gaussian cascade processes

Issues related to effect of multiplicative (modulating) (otherwise amplitude distortions) and additive noises on processed signal are considered. Analysis of statistical characteristics of probability density function (PDF) of instantaneous signal values against background of modulating interference is carried out. Expressions of the joint PDF envelope, phase and instantaneous signal values are obtained, for the case of only phase distortions, as well as the case when fluctuations in amplitude and phase are independent of each other. It is noted that in the second case, with the independence of phase and amplitude distortions, the PDF of the signal, first of all, is determined by the PDF of its instantaneous values and practically does not depend on the PDF of the signal phase. Expressions are obtained for the PDF useful signal and the most common PDF envelope when affecting the signal with modulating noise for deep phase distortions and for its uniform distribution over the interval (0, 2π). In addition, for the case of some functional relationship of phase and amplitude fluctuations, the expression for PDF instantaneous signal values is defined. The presence of such a functional connection allows calculating the PDF of the signal through the statistical characteristics of its envelope. It is obtained that PDF of mixture of signal and additive noise contains arbitrary distribution of phase and amplitude, as well as arbitrary law of distribution of envelope of processed signal. It is obtained that weight coefficients are determined by derivative of characteristic function of amplitude distortion in case when PDFs of said mixture are distributed according to normal law and functional connection exists between amplitude and phase distortions.

For the first time, fluctuation characteristics at the high-field side (HFS) and low-field side (LFS) scrape-off-layers (SOLs) in L-mode plasmas of the JT-60U tokamak are compared in detail; the plasma transport is studied using reciprocating Langmuir probes. A statistical analysis based on probability density functions (PDFs) is employed to describe intermittent (nondiffusive) transport in SOL plasma fluctuations. It is found that positive bursty events associated with blobby plasma transport appear frequently at the LFS midplane. Further, the PDF at the LFS midplane is strongly positively skewed, while the PDF at the HFS SOL is close to a Gaussian distribution. The conventional conditional averaging method is improved by using the variable-interval time-averaging method along with it, which enables the precise reconstruction of the burst profile exhibiting a rapid increase and a slow decay. The radial evolution of plasma blobs is analysed by using an autocorrelation function, the short-time Fourier transform and wavelet analysis.

In this paper, a mixture noise model, which is a sum of symmetric Cauchy and zero-mean Gaussian random variables in time domain, is studied. The Cauchy and Gaussian distributions are characterized by the unknown median γ and variance σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively. The probability density function (PDF) and characteristic function (CF) of the mixture are also investigated which are calculated by the convolution of the two PDFs, and product of the two CFs, respectively. Due to the complication of the resultant PDF, typical approaches such as maximum likelihood estimator may not be able to estimate γ and σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> reliably. Based on the resultant CF, we propose to employ the fractional lower-order moment estimator for their computation. Simulation results show the mean square error performance of the proposed method and a comparison with the Cramér-Rao lower bound is also provided.

The problem of estimating the parameters of a non causal ARMA system, driven by an unknown input noise with unknown symmetrical probability density function (PDF) is addressed. A maximum likelihood approach is proposed in this paper. The main idea of our approach is that the assumed PDF of the input noise is the PDF minimizing the Fisher information among PDFs matching the estimated cumulants of 2nd and 4th order. This minimization problem is hard to solve, so we use an over-parameterized PDF model, which is a gaussian mixture. We obtain two different models for the classes of sub-Gaussian and super-Gaussian PDFs. For this latter class, we get the most robust estimator in Huber's sense, among these generated by this class. A new parameter estimation method is given and its robustness and optimality properties are detailed. The performances of the resulting identification scheme are compared to those of another higher order method.

In order to assess the availability of wind potential energy in a given place, a fundamental part is to investigate the behavior of the wind during a certain period. A standard way to characterize wind data is by using statistical models such as the Weibull distribution. To define the parameters of the Weibull probability density function (PDF) which best describes the input data, the estimation of two parameters is needed. Rather than testing different estimation methods for the Weibull distribution parameters and choosing the one with greater fitting level, we propose a iterative method aiming to maximize the suitability of the PDF into the raw data histogram. The method is based on the Newton-Raphson algorithm and uses as initial guess the parameters yielded by a Maximum Likelihood estimation. We also assess the performance of the hybrid Weibull distribution proposed by Tackle and Brown for different types of data and compare it to the standard Weibull PDF. The results show that, for every set of data analyzed, the proposed method outperformed the state-of-the-art Maximum Likelihood method. Also the hybrid Weibull PDF model yielded better fit than the standard Weibull PDF in all cases.

This paper examines the estimation of the Stochastic Conditional Duration model by the empirical characteristic function and the generalized method of moments when maximum likelihood is unavailable. The joint characteristic function for the durations along with general expressions for the moments are derived, leading naturally to estimation via the empirical characteristic function and generalized method of moments. In a Monte Carlo study as well as an empirical application, these alternative methods are compared with quasi maximum likelihood. These experiments reveal that the empirical characteristic function approach outperforms the quasi maximum likelihood and generalized method of moments in terms of both bias and root mean square error. Copyright The Author(s). Journal compilation Royal Economic Society 2008

Financial data such as asset returns, exchange rates, or option prices cannot be modeled effectively by classical distributions such as the Gaussian. These types of data have probability density functions that are thick-tailed and negatively skewed. To account for these features, we propose a new method of generating classes of distribution functions through convolution of smooth and non-smooth characteristic functions where the smoothing parameter is used to control the thickness of the density tails. To illustrate the advantages of using such class of distributions, we consider special cases in which the smooth characteristic functions are of those of the uniform, the normal and the compact supported cosine distributions and the non-smooth is the characteristic function of the Cauchy distribution. As a comparison criterion between distributions, we use the Stiltjes-Hamburger conditions for moments’ existence and show how the proposed distributions outperform the Student and Pearson IV distributions, which are commonly used by financial engineers to model stock returns.