non-Gaussian Properties Research Articles

Homogeneous states in driven granular mixtures: Enskog kinetic theory versus molecular dynamics simulations.

The homogeneous state of a binary mixture of smooth inelastic hard disks or spheres is analyzed. The mixture is driven by a thermostat composed by two terms: a stochastic force and a drag force proportional to the particle velocity. The combined action of both forces attempts to model the interaction of the mixture with a bath or surrounding fluid. The problem is studied by means of two independent and complementary routes. First, the Enskog kinetic equation with a Fokker-Planck term describing interactions of particles with thermostat is derived. Then, a scaling solution to the Enskog kinetic equation is proposed where the dependence of the scaled distributions φi of each species on the granular temperature occurs not only through the dimensionless velocity c = v/v0 (v0 being the thermal velocity) but also through the dimensionless driving force parameters. Approximate forms for φi are constructed by considering the leading order in a Sonine polynomial expansion. The ratio of kinetic temperatures T1/T2 and the fourth-degree velocity moments λ1 and λ2 (which measure non-Gaussian properties of φ1 and φ2, respectively) are explicitly determined as a function of the mass ratio, size ratio, composition, density, and coefficients of restitution. Second, to assess the reliability of the theoretical results, molecular dynamics simulations of a binary granular mixture of spheres are performed for two values of the coefficient of restitution (α = 0.9 and 0.8) and three different solid volume fractions (ϕ = 0.00785, 0.1, and 0.2). Comparison between kinetic theory and computer simulations for the temperature ratio shows excellent agreement, even for moderate densities and strong dissipation. In the case of the cumulants λ1 and λ2, good agreement is found for the lower densities although significant discrepancies between theory and simulation are observed with increasing density.

Filtering of the ARMAX Process with Generalized t-Distribution Noise: The Influence Function Approach

The commonly made assumption of Gaussian noise is an approximation to reality. In this paper, influence function, an analysis tool in robust statistics, is used to formulate a recursive solution for the filtering of the ARMAX process with generalized t-distribution noise. By being a superset encompassing Gaussian, uniform, t, and double exponential distributions, generalized t-distribution has the flexibility of characterizing noise with Gaussian or non-Gaussian statistical properties. The filter is formulated as a maximum likelihood problem, but instead of solving the optimization problem numerically, influence function approximation is used to obtain a recursive solution to reduce the computational load and facilitate real-time implementation. The influence function equations derived are also useful in determining the variance of the filter and the impact of outliers.

A Study on the Non-Gaussian Property by the Method of Limiting Streamline

Under wind action, the property of non-Gaussian wind pressure distribution is very important for determination of peak factor, simulation of fluctuating wind pressure and design of cladding and components. Since the distribution of non-Gaussian zone is highly irregular based on the statistical method, any effort on giving regular non-Gaussian zone regions cant reflect the real non-Gaussian property. Considering the fact that non-Gaussian property was induced by flow separation and generally the movement of vortex showed the character of time averaged steady, taking a typical low-rise building as the example, the distribution of limiting streamlines and the theory of flow separation was applied in the research. Results show that the distribution of limiting streamlines has a close relationship with the intensity of non-Gaussian property and can be used as an intuitionistic tool in the research of identification and mechanism for non-Gaussian properties.

Loss of non-Gaussianity for damped photon-subtracted thermal states

We investigate non-Gaussianity properties for a set of classical one-mode states obtained by subtracting photons from a thermal state. Three distance-type degrees of non-Gaussianity used for these states are shown to have monotonic behaviour with respect to their mean photon number. Decaying of their non-Gaussianity under damping is found to be consistently described by the distance-type measures considered here. We also compare the dissipative evolution of non-Gaussianity when starting from M-photon-subtracted and M-photon-added thermal states.

Sparse Randomized Maximum Likelihood (SpRML) for subsurface flow model calibration and uncertainty quantification

Despite their apparent high dimensionality, spatially distributed hydraulic properties of geologic formations can often be compactly (sparsely) described in a properly designed basis. Hence, the estimation of high-dimensional subsurface flow properties from dynamic performance and monitoring data can be formulated and solved as a sparse reconstruction inverse problem. Recent advances in statistical signal processing, formalized under the compressed sensing paradigm, provide important guidelines on formulating and solving sparse inverse problems, primarily for linear models and using a deterministic framework. Given the uncertainty in describing subsurface physical properties, even after integration of the dynamic data, it is important to develop a practical sparse Bayesian inversion approach to enable uncertainty quantification. In this paper, we use sparse geologic dictionaries to compactly represent uncertain subsurface flow properties and develop a practical sparse Bayesian method for effective data integration and uncertainty quantification. The multi-Gaussian assumption that is widely used in classical probabilistic inverse theory is not appropriate for representing sparse prior models. Following the results presented by the compressed sensing paradigm, the Laplace (or double exponential) probability distribution is found to be more suitable for representing sparse parameters. However, combining Laplace priors with the frequently used Gaussian likelihood functions leads to neither a Laplace nor a Gaussian posterior distribution, which complicates the analytical characterization of the posterior. Here, we first express the form of the Maximum A-Posteriori (MAP) estimate for Laplace priors and then use the Monte-Carlo-based Randomize Maximum Likelihood (RML) method to generate approximate samples from the posterior distribution. The proposed Sparse RML (SpRML) approximate sampling approach can be used to assess the uncertainty in the calibrated model with a relatively modest computational complexity. We demonstrate the suitability and effectiveness of the SpRML formulation using a series of numerical experiments of two-phase flow systems in both Gaussian and non-Gaussian property distributions in petroleum reservoirs and successfully apply the method to an adapted version of the PUNQ-S3 benchmark reservoir model.

Non-Gaussian water diffusion kurtosis imaging of prostate cancer

PurposeTo evaluate the non-Gaussian water diffusion properties of prostate cancer (PCa) and determine the diagnostic performance of diffusion kurtosis (DK) imaging for distinguishing PCa from benign tissues within the peripheral zone (PZ), and assessing tumor lesions with different Gleason scores. Materials and MethodsNineteen patients who underwent diffusion weighted (DW) magnetic resonance imaging using multiple b-values and were pathologically confirmed with PCa were enrolled in this study. Apparent diffusion coefficient (ADC) was derived using a monoexponential model, while diffusion coefficient (D) and kurtosis (K) were determined using a DK model. Differences between the ADC, D and K values of benign PZ and PCa, as well as those of tumor lesions with Gleason scores of 6, 7 and ≥8 were assessed. Correlations between parameters D and K in PCa were analyzed using Pearson’s correlation coefficient. ADC, D and K values were correlated with Gleason scores of 6, 7 and ≥8, respectively. ResultsADC and D values were significantly (p<0.001) lower in PCa (0.79±0.14μm2/ms and 1.56±0.23μm2/ms, respectively) compared to benign PZ (1.23±0.19μm2/ms and 2.54±0.24μm2/ms, respectively). K values were significantly (p<0.001) greater in PCa (0.96±0.20) compared to benign PZ (0.59±0.08). D and K showed fewer overlapping values between benign PZ and PCa compared to ADC. There was a strong negative correlation between D and K values in PCa (Pearson correlation coefficient r=−0.729; p<0.001). ADC and K values differed significantly in tumor lesions with Gleason scores of 6, 7 and ≥8 (p<0.001 and p=0.001, respectively), although no significant difference was detected for D values (p=0.325). Significant correlations were found between the ADC value and Gleason score (r=−0.828; p<0.001), as well as the K value and Gleason score (r=0.729; p<0.001). ConclusionDK model may add value in PCa detection and diagnosis. K potentially offers a new metric for assessment of PCa.

TRANSPORT OF COSMIC-RAY PROTONS IN INTERMITTENT HELIOSPHERIC TURBULENCE: MODEL AND SIMULATIONS

The transport of charged energetic particles in the presence of strong intermittent heliospheric turbulence is computationally analyzed based on known properties of the interplanetary magnetic field and solar wind plasma at 1 Astronomical Unit (AU). The turbulence is assumed to be static, composite, and quasi-three-dimensional with a varying energy distribution between a one-dimensional Alfv\'enic (slab) and a structured two-dimensional component. The spatial fluctuations of the turbulent magnetic field are modeled either as homogeneous with a Gaussian probability distribution function (PDF), or as intermittent on large and small scales with a q-Gaussian PDF. Simulations showed that energetic particle diffusion coefficients both parallel and perpendicular to the background magnetic field are significantly affected by intermittency in the turbulence. This effect is especially strong for parallel transport where for large-scale intermittency results show an extended phase of subdiffusive parallel transport during which cross-field transport diffusion dominates. The effects of intermittency are found to depend on particle rigidity and the fraction of slab energy in the turbulence, yielding a perpendicular to parallel mean free path ratio close to 1 for large-scale intermittency. Investigation of higher order transport moments (kurtosis) indicates that non-Gaussian statistical properties of the intermittent turbulent magnetic field are present in the parallel transport, especially for low rigidity particles at all times.

Diffusional kurtosis imaging: a promising technique for detecting microstructural changes in neural development and regeneration

Diffusional kurtosis imaging: a promising technique for detecting microstructural changes in neural development and regeneration

Complex network approach to characterize the statistical features of the sunspot series

Complex network approaches have been recently developed as an alternative framework to study the statistical features of time-series data. We perform a visibility-graph analysis on both the daily and monthly sunspot series. Based on the data, we propose two ways to construct the network: one is from the original observable measurements and the other is from a negative-inverse-transformed series. The degree distribution of the derived networks for the strong maxima has clear non-Gaussian properties, while the degree distribution for minima is bimodal. The long-term variation of the cycles is reflected by hubs in the network that span relatively large time intervals. Based on standard network structural measures, we propose to characterize the long-term correlations by waiting times between two subsequent events. The persistence range of the solar cycles has been identified over 15–1000 days by a power-law regime with scaling exponent γ = 2.04 of the occurrence time of two subsequent strong minima. In contrast, a persistent trend is not present in the maximal numbers, although maxima do have significant deviations from an exponential form. Our results suggest some new insights for evaluating existing models.

A domain decomposition method of stochastic PDEs: An iterative solution techniques using a two-level scalable preconditioner

Recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions. A two-level domain decomposition method is reported in this investigation to devise a linear solver for the large-scale system in the Galerkin spectral stochastic finite element method (SSFEM). In particular, a two-level scalable preconditioner is introduced in order to iteratively solve the large-scale linear system in the intrusive SSFEM using an iterative substructuring based domain decomposition solver. The implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains. The numerical and parallel scalabilities of the two-level preconditioner are contrasted with the previously developed one-level preconditioner for two-dimensional flow through porous media and elasticity problems with spatially varying non-Gaussian material properties. A distributed implementation of the parallel algorithm is carried out using MPI and PETSc parallel libraries. The scalabilities of the algorithm are investigated in a Linux cluster.

Biased cosmological parameter estimation with galaxy cluster counts in the presence of primordial non-Gaussianities

The redshift dependence of the abundance of galaxy clusters is very sensitive to the statistical properties of primordial density perturbations. It can thus be used to probe small deviations from Gaussian initial conditions. Such deviations constitute a very important signature of many inflationary scenarios, and are thus expected to provide crucial information on physical processes which took place in the very early Universe. We have determined the biases which may be introduced in the estimation of cosmological parameters by wrongly assuming the absence of primordial non-Gaussianities. Although we find that the estimation of the present-day dark energy density using cluster counts is not very sensitive to the non-Gaussian properties of the density field, we show that the biases can be considerably larger in the estimation of the dark energy equation of state parameter w and of the amplitude of the primordial density perturbations. Our results suggest that a significant level of non-Gaussianity at cluster scales may be able to reconcile the constraint on the amplitude of the primordial perturbations obtained using galaxy cluster number counts from the Planck Sunyaev–Zeldovich Catalog with that obtained from the primary cosmic microwave background anisotropies measured by the Planck satellite.

AVO inversion based on generalized extreme value distribution with adaptive parameter estimation

Amplitude Variation with Offset (AVO) inversion is very sensitive to noise and other uncertainties which are often far from Gaussian in the actual pre-stack seismic data, sometimes causing AVO inversion to give poor results. Since the generalized extreme value (GEV) distribution can efficiently fit any distribution with different parameters, to obtain steady and rational AVO inversion results, we use GEV distribution to model the distribution of the iterative residual error in the AVO inversion. The maximum likelihood method is used to re-evaluate the GEV distribution parameters of each iterative residual error to efficiently fit its non-Gaussian property and reduce the sensitivity of AVO inversion to non-Gaussian residual error. Quasi-Newton based Conjugate Gradient (QCG) AVO algorithm is derived from adaptive adjusted parameters GEV distribution to make sure the direction is always a descent direction for the objective function. Finally, a new choice of adaptive variable step length obtained by the Taylor expression of the residual vector can adaptively adjust the step length to accelerate the convergent speed. Both the synthetic and real seismic data tests illustrate the effectiveness of the proposed method.

Modeling of Compound Gaussian Sea Clutter Based on Inverse Gaussian Distribution

The Compound-Gaussian (CG) distribution is widely used for modeling non-Gaussian clutter, as its texture component describes the non-Gaussian properties of the clutter. In this paper, a CG model with an inverse-Gaussian texture distribution, called the Inverse-Gaussian Compound-Gaussian (IG-CG) distribution, is proposed, and its distributional properties are derived. IPIX radar lake-clutter measurements have been analyzed, and the results show that the two-parameter IG-CG distribution model fits real radar data better than a single parameter IG-CG distribution model or a K distribution model.

A note on institutional hierarchy and volatility in financial markets

From a statistical point of view, the prevalence of non-Gaussian distributions in financial returns and their volatilities shows that the Central Limit Theorem (CLT) often does not apply in financial markets. In this article, we take the position that the independence assumption of the CLT is violated by herding tendencies among market participants, and investigate whether a generic probabilistic herding model can reproduce non-Gaussian statistics in systems with a large number of agents. It is well known that the presence of a herding mechanism in the model is not sufficient for non-Gaussian properties, which crucially depend on the details of the communication network among agents. The main contribution of this article is to show that certain hierarchical networks, which portray the institutional structure of fund investment, warrant non-Gaussian properties for any system size and even lead to an increase in system-wide volatility. Viewed from this perspective, the mere existence of financial institutions with socially interacting managers contributes considerably to financial volatility.

Homogeneous steady states in a granular fluid driven by a stochastic bath with friction

The homogeneous state of a granular flow of smooth inelastic hard spheres or disks described by the Enskog–Boltzmann kinetic equation is analyzed. The granular gas is fluidized by the presence of a random force and a drag force. The combined action of both forces, which act homogeneously on the granular gas, tries to mimic the interaction of the set of particles with a surrounding fluid. The first stochastic force thermalizes the system, providing for the necessary energy recovery to keep the system in its gas state at all times, whereas the second force allows us to mimic the action of the surrounding fluid viscosity. After a transient regime, the gas reaches a steady state characterized by a scaled distribution function φ that depends not only on the dimensionless velocity c ≡ v/v0 (v0 being the thermal velocity) but also on the dimensionless driving force parameters. The dependence of φ and its first relevant velocity moments a2 and a3 (which measure non-Gaussian properties of φ) on both the coefficient of restitution α and the driven parameters is widely investigated by means of the direct simulation Monte Carlo method. In addition, approximate forms for a2 and a3 are also derived from an expansion of φ in Sonine polynomials. The theoretical expressions of the above Sonine coefficients agree well with simulation data, even for quite small values of α. Moreover, the third order expansion of the distribution function makes a significant improvement in accuracy for larger velocities and inelasticities. Results also show that the non-Gaussian corrections to the distribution function φ are smaller than those observed for undriven granular gases.

Detection of low observable moving target in sea clutter via fractal characteristics in fractional Fourier transform domain

Effective detection of low observable moving target at sea is important for remote sensing and radar signal processing. The non-Gaussian property of sea clutter and lack of accurate model make the detection difficult for statistics based detectors. Also the fractal techniques in time domain cannot achieve high detection probability in heavy sea clutter. To help solve the problems, fractal characteristics of IPIX datasets in fractional Fourier transform (FRFT) domain are analysed making use of the fluctuation of FRFT amplitudes and moving target detection algorithms are proposed based on the fractal characteristics in FRFT domain. Firstly, fractal model in FRFT domain is established with fractional Brownian motion model and two judgment and extraction methods are employed for calculating the fractal characteristics in FRFT domain. It is found that sea clutter of different polarisations exhibit fractal behaviours in FRFT domain, that is, self-similarity property, within its corresponding scale-invariant interval. Then, we find that four specific fractal statistics in the best FRFT domain can provide valuable information for developing simple and effective detectors. Finally, traditional amplitude detector and Hurst exponent detector in time domain are compared and the results prove the superior detection ability of low observable moving target without complex computations.

Fault detection and diagnosis for data incomplete industrial systems with new Bayesian network approach

For the fault detection and diagnosis problem in large-scale industrial systems, there are two important issues: the missing data samples and the non-Gaussian property of the data. How-ever, most of the existing data-driven methods cannot be able to handle both of them. Thus, a new Bayesian network classifier based fault detection and diagnosis method is proposed. At first, a non-imputation method is presented to handle the data incomplete samples, with the property of the proposed Bayesian network classifier, and the missing values can be marginalized in an elegant manner. Furthermore, the Gaussian mixture model is used to approximate the non-Gaussian data with a linear combination of finite Gaussian mixtures, so that the Bayesian network can process the non-Gaussian data in an effective way. Therefore, the entire fault detection and diagnosis method can deal with the high-dimensional incomplete process samples in an efficient and robust way. The diagnosis results are expressed in the manner of probability with the reliability scores. The proposed approach is evaluated with a benchmark problem called the Tennessee Eastman process. The simulation results show the effectiveness and robustness of the proposed method in fault detection and diagnosis for large-scale systems with missing measurements.

A Preliminary Study of Epilepsy in Children Using Diffusional Kurtosis Imaging

To study brain abnormalities, in terms of non-Gaussian water diffusion properties using diffusional kurtosis imaging (DKI) in children with electroencephalography (EEG) confirmed epilepsy lateralized to both hemispheres. A total of 15 children with epileptiform waves on EEG in both hemispheres and 18 children as normal controls (NC) matched for age and sex were recruited. Data from DKI for all children were used to characterize non-Gaussian properties. Fractional anisotropy (FA), mean diffusivity (MD), and mean kurtosis (MK) maps were estimated from the DKI datasets. Voxel-based analyses (VBA) based on these measures were performed and compared between the epilepsy and NC groups. The VBA showed abnormal regions in both white matter (WM) and gray matter (GM) in those with epilepsy. Analysis of FA values revealed that the abnormal regions were significant mainly in the left frontal and temporal lobes of the WM. Analysis of MD values revealed that differences were significant mainly in the right hemisphere of the limbic lobe, uncus, parahippocampal region, both in GM and WM of frontal and temporal lobes, and GM of the rectus of the left cerebrum. Finally, analysis of MK values revealed significant differences mainly in WM of the frontal lobes of both cerebrum, and GM and WM of the parietal lobe of the right cerebrum. These preliminary results suggest that DKI is sensitive for the characterization of diffusion abnormalities in both WM and GM of children with epilepsy.

Gaussian Lagrangian stochastic models for multi-particle dispersion

We have extended the “well-mixed” two-particle stochastic models for 3D Gaussian turbulence to n particles, and have performed calculations for clusters of n ⩽ 6 particles. The particle joint motions are Gaussian and are constrained by pair-wise spatial correlations. This neglects non-Gaussian properties of the two-point velocity distribution and also neglects multi-point correlations. It also takes no account of intermittency. Although the models do not predict the growth of the separation of particles in the cluster satisfactorily, we find that they do give a good representation of the shape statistics for the cluster in comparison with direct numerical simulation results. We conclude that the pair-wise spatial structure of the turbulence accounts for most of the observed characteristics of the shape of multi-particle clusters in turbulence, and that non-Gaussian and multi-point features of the turbulence are of secondary importance.

A Better Characterization of Spinal Cord Damage in Multiple Sclerosis: A Diffusional Kurtosis Imaging Study

The spinal cord is a site of predilection for MS lesions. While diffusion tensor imaging is useful for the study of anisotropic systems such as WM tracts, it is of more limited utility in tissues with more isotropic microstructures (on the length scales studied with diffusion MR imaging) such as gray matter. In contrast, diffusional kurtosis imaging, which measures both Gaussian and non-Gaussian properties of water diffusion, provides more biomarkers of both anisotropic and isotropic structural changes. The aim of this study was to investigate the cervical spinal cord of patients with MS and to characterize lesional and normal-appearing gray matter and WM damage by using diffusional kurtosis imaging. Nineteen patients (13 women, mean age = 41.1 ± 10.7 years) and 16 controls (7 women, mean age = 35.6 ± 11.2-years) underwent MR imaging of the cervical spinal cord on a 3T scanner (T2 TSE, T1 magnetization-prepared rapid acquisition of gradient echo, diffusional kurtosis imaging, T2 fast low-angle shot). Fractional anisotropy, mean diffusivity, and mean kurtosis were measured on the whole cord and in normal-appearing gray matter and WM. Spinal cord T2-hyperintense lesions were identified in 18 patients. Whole spinal cord fractional anisotropy and mean kurtosis (P = .0009, P = .003), WM fractional anisotropy (P = .01), and gray matter mean kurtosis (P = .006) were significantly decreased, and whole spinal cord mean diffusivity (P = .009) was increased in patients compared with controls. Mean spinal cord area was significantly lower in patients (P = .04). Diffusional kurtosis imaging of the spinal cord can provide a more comprehensive characterization of lesions and normal-appearing WM and gray matter damage in patients with MS. Diffusional kurtosis imaging can provide additional and complementary information to DTI on spinal cord pathology.