Degree Of non-Gaussianity Research Articles

On the accuracy and precision of correlation functions and field-level inference in cosmology

ABSTRACT We present a comparative study of the accuracy and precision of correlation function methods and full-field inference in cosmological data analysis. To do so, we examine a Bayesian hierarchical model that predicts lognormal (LN) fields and their two-point correlation function. Although a simplified analytic model, the LN model produces fields that share many of the essential characteristics of the present-day non-Gaussian cosmological density fields. We use three different statistical techniques: (i) a standard likelihood-based analysis of the two-point correlation function; (ii) a likelihood-free (simulation-based) analysis of the two-point correlation function; (iii) a field-level analysis, made possible by the more sophisticated data assimilation technique. We find that (a) standard assumptions made to write down a likelihood for correlation functions can cause significant biases, a problem that is alleviated with simulation-based inference; and (b) analysing the entire field offers considerable advantages over correlation functions, through higher accuracy, higher precision, or both. The gains depend on the degree of non-Gaussianity, but in all cases, including for weak non-Gaussianity, the advantage of analysing the full field is substantial.

Non-Gaussian subdiffusion of single-molecule tracers in a hydrated polymer network.

Single molecule tracking experiments inside a hydrated polymer network have shown that the tracer motion is subdiffusive due to the viscoelastic environment inside the gel-like network. This property can be related to the negative autocorrelation of the instantaneous displacements at short times. Although the displacements of the individual tracers exhibit Gaussian statistics, the displacement distribution of all the trajectories combined from different spatial locations of the polymer network exhibits a non-Gaussian distribution. Here, we analyze many individual tracer trajectories to show that the central portion of the non-Gaussian distribution can be well approximated by an exponential distribution that spreads sublinearly with time. We explain all these features seen in the experiment by a generalized Langevin model for an overdamped particle with algebraically decaying correlations. We show that the degree of non-Gaussianity can change with the extent of heterogeneity, which is controlled in our model by the experimentally observed distributions of the motion parameters.

Multi-spectral optical imaging of the spatiotemporal dynamics of ionospheric intermittent turbulence

Equatorial plasma depletions have significant impact on radio wave propagation in the upper atmosphere, causing rapid fluctuations in the power of radio signals used in telecommunication and GPS navigation, thus playing a crucial role in space weather impacts. Complex structuring and self-organization of equatorial plasma depletions involving bifurcation, connection, disconnection and reconnection are the signatures of nonlinear evolution of interchange instability and secondary instabilities, responsible for the generation of coherent structures and turbulence in the ionosphere. The aims of this paper are three-fold: (1) to report the first optical imaging of reconnection of equatorial plasma depletions in the South Atlantic Magnetic Anomaly, (2) to investigate the optical imaging of equatorial ionospheric intermittent turbulence, and (3) to compare nonlinear characteristics of optical imaging of equatorial plasma depletions for two different altitudes at same times. We show that the degree of spatiotemporal complexity of ionospheric intermittent turbulence can be quantified by nonlinear studies of optical images, confirming the duality of amplitude-phase synchronization in multiscale interactions. By decomposing the analyses into North-South and East-West directions we show that the degree of non-Gaussianity, intermittency and multifractality is stronger in the North-South direction, confirming the anisotropic nature of the interchange instability. In particular, by using simultaneous observation of multi-spectral all-sky emissions from two different heights we show that the degree of non-Gaussianity and intermittency in the bottomside F-region ionosphere is stronger than the peak F-region ionosphere. Our results are confirmed by two sets of observations on the nights of 28 September 2002 and 9 November 2002.

EVOLUTION OF THE MAGNETIC FIELD LINE DIFFUSION COEFFICIENT AND NON-GAUSSIAN STATISTICS

ABSTRACT The magnetic field line random walk (FLRW) plays an important role in the transport of energy and particles in turbulent plasmas. For magnetic fluctuations that are transverse or almost transverse to a large-scale mean magnetic field, theories describing the FLRW usually predict asymptotic diffusion of magnetic field lines perpendicular to the mean field. Such theories often depend on the assumption that one can relate the Lagrangian and Eulerian statistics of the magnetic field via Corrsin’s hypothesis, and additionally take the distribution of magnetic field line displacements to be Gaussian. Here we take an ordinary differential equation (ODE) model with these underlying assumptions and test how well it describes the evolution of the magnetic field line diffusion coefficient in 2D+slab magnetic turbulence, by comparisons to computer simulations that do not involve such assumptions. In addition, we directly test the accuracy of the Corrsin approximation to the Lagrangian correlation. Over much of the studied parameter space we find that the ODE model is in fairly good agreement with computer simulations, in terms of both the evolution and asymptotic values of the diffusion coefficient. When there is poor agreement, we show that this can be largely attributed to the failure of Corrsin’s hypothesis rather than the assumption of Gaussian statistics of field line displacements. The degree of non-Gaussianity, which we measure in terms of the kurtosis, appears to be an indicator of how well Corrsin’s approximation works.

Non-Gaussianity of petrophysical parameters using [formula omitted] entropy and a multifractal random walk

Geological systems such as petroleum reservoirs can be investigated using Tsallis entropy and multiplicative hierarchical cascade models. The occurrence of non-Gaussianity is a sign of uncertainty and a phase transition, which could indicate the existence of a petroleum reservoir. Two important parameters that describe a system at any scale are determined: the degree of non-Gaussianity, q, for the entropy and the intermittency, λ2, which explains critical behavior in a system. Some petrophysical indicators can be used to characterize a reservoir, but there is a lack of methods for measuring scaling information. This study compares non-Gaussianity for three selected indicators at various scales: gamma radiation (GR), sonic transient time (DT) and neutron porosity (NPHI). The results show that GR has a fat-tailed probability distribution function (PDF) at all scales, which is a sign of phase transition in the system and indicates high q and λ2. This provides valuable information about GR. NPHI shows scale dependence and the PDF converges to a Gaussian distribution at large scales. This is indicative of separated and uncorrelated porosity at large scales. For the DT series, small λ2 and q at all scales are a hallmark of local DT correlations.

Constrained maximum likelihood estimation of the diffusion kurtosis tensor using a Rician noise model

A computational framework to obtain an accurate quantification of the Gaussian and non-Gaussian component of water molecules' diffusion through brain tissues with diffusion kurtosis imaging, is presented. The diffusion kurtosis imaging model quantifies the kurtosis, the degree of non-Gaussianity, on a direction dependent basis, constituting a higher order diffusion kurtosis tensor, which is estimated in addition to the well-known diffusion tensor. To reconcile with the physical phenomenon of molecular diffusion, both tensor estimates should lie within a physically acceptable range. Otherwise, clinically and artificially significant changes in diffusion (kurtosis) parameters might be confounded. To guarantee physical relevance, we here suggest to estimate both diffusional tensors by maximizing the joint likelihood function of all Rician distributed diffusion weighted images given the diffusion kurtosis imaging model while imposing a set of nonlinear constraints. As shown in this study, correctly accounting for the Rician noise structure is necessary to avoid significant overestimation of the kurtosis values. The performance of the constrained estimator was evaluated and compared to more commonly used strategies during simulations. Human brain data were used to emphasize the need for constrained estimators as not imposing the constraints give rise to constraint violations in about 70% of the brain voxels.

Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states

Non Gaussian states and processes are useful resources in quantum information with continuous variables. An experimentally accessible criterion has been proposed to measure the degree of non Gaussianity of quantum states, based on the conditional entropy of the state with a Gaussian reference. Here we adopt such criterion to characterise an important class of non classical states, single-photon added coherent states. Our studies demonstrate the reliability and sensitivity of this measure, and use it to quantify how detrimental is the role of experimental imperfections in our realisation.

OPTIMAL DIRECTED CURRENT OF A BROWNIAN MOTOR UNDER A NON-GAUSSIAN NOISE GENERATED BY A MULTIPLICATIVE PROCESS

Unidirectional motion is achieved when a particle, moving under the influence of an underlying noise source, is subjected to a ratchet asymmetric periodic potential. Here, we investigate how deviations from the Gaussian nature of the noise distribution function impacts the average particle's current. The input noise is considered to be produced by a Langevin process including both multiplicative and additive random noise sources. The resulting input random signal has a power-law amplitude distribution and a finite correlation time. These features are controlled by the average of the multiplicative noise. We show that the average particle's velocity depends non-monotonically on the degree of non-Gaussianity of the input noise. It exhibits a maximum at an intermediate value of the effective power-law exponent that characterizes the asymptotic decay of the noise probability distribution function.

MRI quantification of non‐Gaussian water diffusion by kurtosis analysis

Quantification of non-Gaussianity for water diffusion in brain by means of diffusional kurtosis imaging (DKI) is reviewed. Diffusional non-Gaussianity is a consequence of tissue structure that creates diffusion barriers and compartments. The degree of non-Gaussianity is conveniently quantified by the diffusional kurtosis and derivative metrics, such as the mean, axial, and radial kurtoses. DKI is a diffusion-weighted MRI technique that allows the diffusional kurtosis to be estimated with clinical scanners using standard diffusion-weighted pulse sequences and relatively modest acquisition times. DKI is an extension of the widely used diffusion tensor imaging method, but requires the use of at least 3 b-values and 15 diffusion directions. This review discusses the underlying theory of DKI as well as practical considerations related to data acquisition and post-processing. It is argued that the diffusional kurtosis is sensitive to diffusional heterogeneity and suggested that DKI may be useful for investigating ischemic stroke and neuropathologies, such as Alzheimer's disease and schizophrenia.

Imprints of primordial non-Gaussianities in X-ray and SZ signals from galaxy clusters

Several inflationary models predict the possibility that the primordial perturbations of the density field may contain a degree of non-Gaussianity which would influence the subsequent evolution of cosmic structures at large scales. In order to study their impact, we use a set of three cosmological DM-only simulations starting from initial conditions with different levels of non-Gaussianity: f_NL=0,+/-100. More specifically, we focus on the distribution of galaxy clusters at different redshifts and, using suitable scaling relations, we determine their X-ray and SZ signals. Our analysis allows us to estimate the differences in the logN-logS and logN-logY due to the different initial conditions and to predict the cluster counts at different redshifts expected for future surveys (eROSITA and SPT). We also use a second set of simulations assuming a different cosmological scenario to estimate how the dependence on f_NL is degenerate with respect to other parameters. Our results indicate that the effects introduced by a realistic amount of primordial non-Gaussianity are small when compared to the ones connected with current uncertainties in cosmological parameters, particularly with sigma_8. However, if future surveys will be associated with optical follow-up campaigns to determine the cluster redshift, an analysis of the samples at z>1 can provide significant constraints on f_NL. In particular we predict that the SPT cluster survey will be able to detect ~1000 clusters at z>1 for the Gaussian case, with a difference of 15-20 per cent associated to f_NL=+/-100.

Kullback relative entropy and characterization of partially polarized optical waves

Different properties of partially polarized light are discussed using the Kullback relative entropy, which provides a physically meaningful measure of proximity between probability density functions (PDFs). For optical waves with a Gaussian PDF, the standard degree of polarization is a simple function of the Kullback relative entropy between the considered optical light and a totally depolarized light of the same intensity. It is shown that the Kullback relative entropies between different PDFs allow one to define other properties such as a degree of anisotropy and a degree of non-Gaussianity. It is also demonstrated that, in dimension three, the Kullback relative entropy between a partially polarized light and a totally depolarized light can lead to natural definitions of two degrees of polarization needed to characterize the polarization state. These analyses enlighten the physical meaning of partial polarization of light waves in terms of a measure of disorder provided by the Shannon entropy.

Constraints on Primordial Non‐Gaussianity from the High‐Redshift Cluster MS 1054−03

The implications of the massive, X-ray selected cluster of galaxies MS 1054-03 at z = 0.83 are discussed in light of the hypothesis that the primordial density fluctuations may be non-Gaussian. We generalize the Press-Schechter (PS) formalism to the non-Gaussian case and calculate the likelihood that a cluster as massive as MS 1054 would have been found in the Einstein Medium Sensitvity Survey (EMSS). A flat universe (ΩM + ΩΛ = 1) is assumed, and the mass fluctuation amplitude is normalized to the present-day cluster abundance. The probability of finding an MS 1054-like cluster then depends only on ΩM and the extent of primordial non-Gaussianity. We quantify the latter by adopting a specific functional form for the probability distribution function (PDF), denoted ψλ, which tends to Gaussianity for λ 1 but is significantly non-Gaussian for λ 10, and show how λ is related to the more familiar statistic T, the probability of ≥3 σ fluctuations for a given PDF relative to a Gaussian. Special attention is given to a careful calculation of the virial mass of MS 1054 from the available X-ray temperature, galaxy velocity, and weak-lensing data. We find that Gaussian initial density fluctuations are consistent with the data on MS 1054 only if ΩM 0.2. For ΩM ≥ 0.25 a significant degree of non-Gaussianity is required, unless the mass of MS 1054 has been substantially overestimated by X-ray and weak-lensing data. The required amount of non-Gaussianity is a rapidly increasing function of ΩM for 0.25 ≤ ΩM ≤ 0.45, with λ ≤ 1 (T 7) at the upper end of this range. For a fiducial ΩM = 0.3, ΩΛ = 0.7 universe, favored by several lines of evidence we obtain an upper limit λ ≤ 10, corresponding to a T ≥ 3. This finding is consistent with the conclusions of Koyama, Soda, & Taruya, who applied the generalized PS formalism to low (z 0.1) and intermediate (z 0.6) redshift cluster data sets.

Nonuniversality of the scaling exponents of a passive scalar convected by a random flow.

We consider a passive scalar convected by a multiscale random velocity field with short yet finite temporal correlations. Taking Kraichnan's limit of a white Gaussian velocity as a zero approximation we develop the perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus nonuniversal as was predicted by Shraiman and Siggia on a phenomenological ground.

間欠空調時の室温・熱負荷の確率性状に関する研究 : 非正規外乱の場合の解析

A new method is proposed which gives the probability density function of room air temperature and heating load taking account of non-Gaussian external climates such as outdoor temperature and solar radiation. This method is a simple extension to the second-order analysis already shown in another article. A thermal system composed of one mass is used to analyze the relationship between the eigenvalue of the system and the degree of non-Gaussianity. Next, the similar calculation using a simple room clarifies the influence of non-Gaussianity on room air temperature and heating load. The utility of the previously proposed thermal calculation method, which gives only the mean and variance of the heating load, is discussed through comparison with the present results.

Weakly decaying turbulence in an equivalent-barotropic fluid

Numerical solutions are analyzed for the evolution of turbulent flow in an equivalent-barotropic fluid (i.e., a shallow water layer in the limit of strong rotation) from random, narrow-band initial conditions with small viscosity. Particular attention is given to the regime of small deformation radius: the solutions are weakly dissipative of both energy and potential enstrophy; after a brief initial interval of broadening in wave number, the shapes of the energy and potential enstrophy spectra are nearly invariant in time but their centroids move toward larger scales, and coherent vortex structures spontaneously develop with a preferred shape of axisymmetric potential vorticity monopoles. An asymptotic model is derived for small deformation radius, and its behavior exhibits the essential features of the equivalent-barotropic model. Solutions are analyzed for increasing deformation radii, ranging from small to infinite (i.e., nondivergent, two-dimensional flow). In this sequence, both the dynamical evolution rates (e.g., dissipation) and the degree of non-Gaussianity (intermittency) increase substantially. In particular, the spatial sparseness of the coherent vortices increases with deformation radius.

Detection of non-Gaussian signals in non-Gaussian noise using the bispectrum

The problem of detecting a non-Gaussian time series in the presence of additive Gaussian or non-Gaussian noise is cast into a classical hypothesis testing framework, using the sample bispectrum as the test statistic. The power of the test is demonstrated as a function of signal-to-noise ratio, the degree of skewness of the signal, and processing parameters. The results are compared to the power of a classical energy detection test. It is concluded that the bispectrum can be used effectively to detect non-Gaussian signals in the presence of interfering noise and that it may perform better, depending on the degree of non-Gaussianity, than energy detection. >