non-Gaussian Fields Research Articles

Large-scale density and velocity field reconstructions with neural networks

ABSTRACT We assess a neural network (NN) method for reconstructing 3D cosmological density and velocity fields (target) from discrete and incomplete galaxy distributions (input). We employ second-order Lagrangian Perturbation Theory to generate a large ensemble of mock data to train an auto-encoder (AE) architecture with a Mean Squared Error (MSE) loss function. The AE successfully captures non-linear features arising from gravitational dynamics and the discreteness of the galaxy distribution. It preserves the positivity of the reconstructed density field and exhibits a weaker suppression of the power on small scales than the traditional linear Wiener filter (WF), which we use as a benchmark. In the density reconstruction, the reduction of the AE MSE relative to the WF is $\sim 15~{{\ \rm per\ cent}}$ , whereas for the velocity reconstruction a relative reduction of up to a factor of two can be achieved. The AE is advantageous to the WF at recovering the distribution of the target fields, especially at the tails. In fact, trained with an MSE loss, any NN estimate approaches the unbiased mean of the underlying target given the input. This implies a slope of unity in the linear regression of the true on the NN-reconstructed field. Only for the special case of Gaussian fields, the NN and WF estimates are equivalent. Nonetheless, we also recover a linear regression slope of unity for the WF with non-Gaussian fields.

Scalable Bayesian Transport Maps for High-Dimensional Non-Gaussian Spatial Fields

A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and uncertainty quantification of the map estimation, while resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional distributions due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for flexible marginals. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output. Supplementary materials for this article are available online.

Quantification of High-dimensional Non-Gaussianities and Its Implication to Fisher Analysis in Cosmology

It is well known that the power spectrum is not able to fully characterize the statistical properties of non-Gaussian density fields. Recently, many different statistics have been proposed to extract information from non-Gaussian cosmological fields that perform better than the power spectrum. The Fisher matrix formalism is commonly used to quantify the accuracy with which a given statistic can constrain the value of the cosmological parameters. However, these calculations typically rely on the assumption that the sampling distribution of the considered statistic follows a multivariate Gaussian distribution. In this work, we follow Sellentin & Heavens and use two different statistical tests to identify non-Gaussianities in different statistics such as the power spectrum, bispectrum, marked power spectrum, and wavelet scattering transform (WST). We remove the non-Gaussian components of the different statistics and perform Fisher matrix calculations with the Gaussianized statistics using Quijote simulations. We show that constraints on the parameters can change by a factor of ∼2 in some cases. We show with simple examples how statistics that do not follow a multivariate Gaussian distribution can achieve artificially tight bounds on the cosmological parameters when using the Fisher matrix formalism. We think that the non-Gaussian tests used in this work represent a powerful tool to quantify the robustness of Fisher matrix calculations and their underlying assumptions. We release the code used to compute the power spectra, bispectra, and WST that can be run on both CPUs and GPUs.

Hierarchical clustering with spatial adjacency constraints in heavy-tailed areal data

Some natural phenomena with areal/lattice data structures include extreme values or outliers. In such situations, the assumption of Gaussianity for the random field may not be reasonable, and no Gaussian transformation can be found because they exhibit heavy tails. A non-Gaussian stable random field, which is heavy-tailed, could be a more appropriate choice in these cases. This article introduces a sub-Gaussian α-stable (SGαS) random field for spatial analysis of multivariate areal data using a multivariate conditional autoregressive model. We, specifically, focus on the spatial clustering problem of such areal data. To address it, we develop methods that work based on adjacency constraints. To group the data, we offer an adjacency-constrained hierarchical agglomerative clustering (HAC) technique that considers both spatial and non-spatial attributes existing in the multivariate areal data. The proposed clustering algorithm is developed based on spatial adjacency constraint criteria incorporated into the HAC technique. We employed the developed algorithm to group Luxembourg communes based on simulated areal data from the SGαS distributions and French cities along the Gironde estuary based on the estuary areal dataset. We compare the results of the proposed method with the mentioned adjacency criteria, various dissimilarity, and linkages measures in clustering these two datasets.

Anomalous scaling in kinematic magnetohydrodynamic turbulence: Two-loop anomalous dimensions of leading composite operators.

Using the field theoretic formulation of the kinematic magnetohydrodynamic turbulence, the explicit expressions for the anomalous dimensions of leading composite operators, which govern the inertial-range scaling properties of correlation functions of the weak magnetic field passively advected by the electrically conductive turbulent environment driven by the Navier-Stokes velocity field, are derived and analyzed in the second order of the corresponding perturbation expansion (in the two-loop approximation). Their properties are compared to the properties of the same anomalous dimensions obtained in the framework of the Kazantsev-Kraichnan model of the kinematic magnetohydrodynamics with the Gaussian statistics of the turbulent velocity field as well as to the analogous anomalous dimensions of the leading composite operators in the problem of the passive scalar advection by the Gaussian (the Kraichnan model) and non-Gaussian (driven by the Navier-Stokes equation) turbulent velocity field. It is shown that, regardless of the Gaussian or non-Gaussian statistics of the turbulent velocity field, the two-loop corrections to the leading anomalous dimensions are much more important in the case of the problem of the passive advection of the vector (magnetic) field than in the case of the problem of the passive advection of scalar fields. At the same time, it is also shown that, in phenomenologically the most interesting case with three spatial dimensions, higher velocity correlations of the turbulent environment given by the Navier-Stokes velocity field play a rather limited role in the anomalous scaling of passive scalar as well as passive vector quantities, i.e., that the two-loop corrections to the corresponding leading anomalous dimensions are rather close to those obtained in the framework of the Gaussian models, especially as for the problem of scalar field advection. On the other hand, the role of the non-Gaussian statistics of the turbulent velocity field becomes dominant for higher spatial dimensions in the case of the kinematic magnetohydrodynamic turbulence but remains negligible in the problem of the passive scalar advection.

Growth rate of the initial magnetic energy in isotropic velocity field with non-Gaussian distribution

A method is proposed for including the effects of non-Gaussian velocity field distribution in the estimation of the growth rate of magnetic energy in a random flow with finite memory time. The method allows a reduction to the Gaussian case that was investigates earlier. For illustration, the multivariate Laplace distribution is considered and compared with the Gaussian one. Figs 1, Refs 8.

Study of Neptune dayside magnetosheath fluctuations during Voyager-2 flyby

Turbulence in the outer planetary magnetospheres has been relatively less studied than in the Earth’s magnetosphere, especially for the ice giants which have been visited only by the Voyager-2 spacecraft so far. The aim of this work is to study the turbulence in Neptune’s dayside magnetosheath. Voyager-2 magnetometer 3-s averaged magnetic field data were analyzed with several data analysis techniques. It was found that most of the magnetosheath magnetic field is aligned in the BT direction. The BR-component spectrum showed the predominance of short-period (5minutes) oscillations, while the BT- and BN- components showed the occurrence of longer-period (20–50minutes) oscillations. The Fourier spectrum of the magnetic field showed a breakpoint near 0.00165Hz. The lower frequency portion showed index (absolute values) smaller than 1.0. The higher frequency portion showed a much steeper spectrum, with power law values varying from about 2.0 to 2.4. The statistical kurtosis parameters for BR and BN are higher than 3 in almost all scales, while the BT component (main field direction) has a more Gaussian or sub-Gaussian behavior, specially in scales higher than 20minutes. The kurtosis is very high at the low frequency spectral power of 40minutes for the BN-component, indicating asymmetric or non-Gaussian magnetic field distribution in the magnetosheath. The multifractal spectrum is well spread in the magnetosheath revealing a rich and dynamic activity in this region. Overall, the Neptune’s magnetosheath is found to have a strong turbulence activity that is comparable to that of other planetary magnetosheaths.

Application of wavelet transforms to the simulation of corrosion fields on buried pipelines

The present study presents a framework to analyze and simulate nonhomogeneous non-Gaussian corrosion fields on the external surface of buried in-service pipelines by using continuous and discrete wavelet transforms. The considered transforms are the two-dimensional continuous wavelet transform (CWT) using the complex Morlet wavelets, dual-tree complex discrete wavelet transform (DT-CDWT), and dual-tree complex discrete wavelet with hyperbolic wavelet transform scheme (DT-CHWT); the natural corrosion field is measured using a high-resolution laser scan. Scalograms and marginal distribution of the measured corrosion field are incorporated into the iterative power and amplitude correction (IPAC) algorithm to generate synthetic corrosion fields. The framework is explained and illustrated using a numerical example. The results indicate that the proposed framework can generate synthetic corrosion fields that effectively capture probabilistic characteristics of the measured corrosion field in terms of the scalogram, textural features, and burst capacity of the pipe segment containing the corrosion field. The results suggest that DT-CDWT and DT-CHWT combined with IPAC are efficient and viable options to simulate synthetic corrosion fields to facilitate the pipeline corrosion management practice, whereas the use of CWT in conjunction with IPAC is associated with high computational cost.

Probabilistic Analysis of Strip Footings Using the Subset Simulation Approach and the Influence of Spatial Variability of Two Clay Layers

Abstract This article studies the probabilistic analysis at the ultimate limit state (ULS) of a strip footing resting on a purely cohesive soil composed of two layers with spatial variability of the upper or lower layer and treats the simultaneous effect of the variability of the two layers on the probability of failure Pf value using the subset simulation approach. The results are examined by comparing the Pf values obtained with the Pf value corresponding to the reference case of two homogenous clay layers. In general, the subset simulation approach is used in cases of uncertain parameters modeled by random variables. But in this paper, it is employed with uncertain parameters modeled by random fields because the spatial variability of soil properties strongly affects the behavior of geotechnical structures and causes a significant change in the variability of their responses. The soil cohesion parameter is modelled as a non-Gaussian (log-normal) anisotropic random field using a square exponential autocorrelation function by the optimal linear expansion estimation (EOLE) and the deterministic model is based on numerical simulations using the finite difference software FLAC3D.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

Dynamic response of monopile-supported wind turbines considering non-Gaussian characteristics of wind inflow

Available wind turbine design standards usually assume that the inflow wind field is a smooth Gaussian process, but the in-situ measured data show that the inflow wind presents a certain degree of non-Gaussian characteristics. This study investigates the dynamic response of the supporting structure of the wind turbine under both Gaussian and non-Gaussian wind fields. The Mises stress at the tower base and the mudline of the monopile are considered. Through the global maxima method and the average conditional exceedance rate method, the extreme value response of the turbine under a given mean recurrence interval is obtained. The results indicate that by taking the non-Gaussian characteristics of wind fields into account, the short-term extreme values are generally larger than those under Gaussian wind fields. For the long-term extreme values of the stress at the two critical points, the influence of kurtosis is significantly greater than the skewness. Meanwhile, the long-term stress extreme value also increases with the increase of kurtosis. The non-Gaussian characteristic of the wind field has a particular influence on the safe operation of the wind turbine and should be considered when designing wind turbines.

Minkowski Functionals in Joint Galaxy Clustering & Weak Lensing Analyses

We investigate the inclusion of clustering maps in a weak lensing Minkowski functional (MF) analysis of DES-like and LSST-like simulations to constrain cosmological parameters. The standard 3x2pt approach to lensing and clustering data uses two-point correlations as its primary statistic; MFs, morphological statistics describing the shape of matter fields, provide additional information for non-Gaussian fields. Previous analyses have studied MFs of lensing convergence maps; in this project we explore their simultaneous application to clustering maps. We employ a simplified linear galaxy bias model, and using a lognormal curved sky measurement and Monte Carlo Markov Chain (MCMC) sampling process for parameter inference, we find that MFs do not yield any information in the $\Omega_{\rm m}$ -- $\sigma_8$ plane not already generated by a 3x2pt analysis. However, we expect that MFs should improve constraining power when nonlinear baryonic and other small-scale effects are taken into account. As with a 3x2pt analysis, we find a significant improvement to constraints when adding clustering data to MF-only and MF$+C_\ell$ shear measurements, and strongly recommend future higher order statistics be measured from both convergence and clustering maps.

A framework for conditional simulation of nonstationary non-Gaussian random field and multivariate processes

We propose a conditional simulation framework for the nonstationary/nonhomogeneous non-Gaussian/Gaussian field and multivariate processes. There are three distinct stages within the proposed framework. The first one is to sample directly the unconditional nonstationary non-Gaussian field or multivariate processes without resorting to the underlying Gaussian characteristics. The second stage maps the unconditional samples and the conditions (i.e., observations) to the Gaussian space and uses them to find the conditional mean and covariance. In the final stage, the conditional samples by considering the effect of the conditioning are generated in Gaussian space and mapped back to the original non-Gaussian field. The efficiency of the framework relies on the efficient unconditional simulation algorithms (i.e., the iterative rank-dependent shuffling algorithm, and the iterative power and amplitude correction algorithm), and the reuse of the unconditional samples for the conditional simulation. The conditional samples match the conditional mean and conditional variance–covariance. The framework can directly apply to the random field and multivariate processes defined by the marginal probability distribution function and the covariance-variance information, or the time–frequency dependent PSD function. The framework is illustrated using several examples.

Bridge vibration under complex wind field and corresponding measurements: A review

Due to increased demands in transportation development, bridge construction has been extended to areas with complex wind fields. In addition to common flow features such as high wind speeds and strong turbulence intensity, in coastal areas, bridges are likely to be exposed to non-stationary and non-Gaussian high-speed wind fields during hurricane events. Bridges built in mountainous regions will experience varying wind directions, incident angles of attack (AOA), and wind speed variation due to undulating terrain. The complex wind fields are challenging to the bridge engineering community to which accurate calculation methods and suppression measures for bridge vibration are urgently required. In this paper, state-of-the-art research results are introduced through three categories: (a) the distinctive characteristics of complex wind fields in offshore and mountainous areas, (b) the categories of wind-induced vibrations for bridges, and (c) the suppression measures employed in engineering practices. Until now, the studies regarding the coupled bridge vibration in complex wind fields have made considerable progress, but a richer database of actual wind measurement results and more efficient simulation methods still need to be further established.

Simulating Nonhomogeneous Non-Gaussian Field by Using Iterative Rank-Dependent Reordering versus Translation Process-Based Procedure

We compare two commonly used procedures, namely, the iterative rank-dependent reordering (IRDR) procedure and the translation process based procedure, for simulating homogeneous/nonhomogeneous non-Gaussian fields. We identify the limitations and the implicit assumptions of the procedures. We provide a new interpretation of the IRDR procedure, point out that there is no guarantee that the algorithm converges, and suggest modifications in terms of the initial samples, iteration involving decomposition, and convergence requirement to the IRDR procedure for it to become more efficient and robust. The numerical results show that, depending on the prescribed marginal probability distribution, the use of the IRDR procedure may not achieve a prescribed correlation function, a feature that is well-known if the translation process (i.e., Nataf translation system) based procedure is employed. It is shown that the performance of the modified IRDR procedure is comparable to that of the translation process based procedures in terms of limitations and matching the prescribed correlation function. The numerical results also show that the suggested modifications to IRDR in the present study make the algorithm more efficient and robust.

Characterization of the non-Gaussian hydraulic conductivity field via deep learning-based inversion of hydraulic-head and self-potential data

Accurate characterization of the spatial heterogeneity of hydraulic properties such as hydraulic conductivity (K) is essential for understanding groundwater flow and contaminant transport processes. Deep learning-based ensemble smoother methods have proven to be effective in estimating non-Gaussian K fields by assimilating hydrodynamic measurements. However, traditional hydrogeological investigations (e.g., hydraulic tomography) often suffer from the limited number of invasive borehole data (e.g., hydraulic head), making the non-Gaussian K inversion problem strongly underdetermined. As a non-invasive and high-sampling-density geophysical approach, the self-potential (SP) method can measure fluctuations of the electrical field caused by subsurface fluid movement. Thus, the recorded SP signals are electro-kinetic responses of the groundwater flow and can provide complementary information for delineating non-Gaussian heterogeneity. In this paper, we performed a joint hydrogeophysical inversion of hydraulic-head and SP data to characterize the non-Gaussian K field within a deep learning-based inversion framework. We conducted three synthetic cases in a 3-D non-Gaussian aquifer with 16,000 unknown K, to assess the ability of the proposed joint inversion framework for K imaging by assimilating different types of data. The results show that using limited hydraulic-head data alone may capture the large-scale features of the non-Gaussian reference field but might lose fine features. By integrating the hydraulic-head and SP data, the non-Gaussian K field can be reconstructed with an improved accuracy and reduced estimation uncertainty. The low-cost SP method opens up new opportunities for non-Gaussian K characterization and fluid-dynamic monitoring.

Simulating multivariate stationary non-Gaussian process based on wavenumber–frequency spectrum and unified Hermite polynomial model

It is of importance to model the input excitations considering the spatial variation and non-Gaussianity for the reliability evaluation of long span structures. The existing multivariate stationary non-Gaussian simulation methods are mainly based on the target marginal distribution and cross-spectrum matrix. Due to the Cholesky decomposition of cross-spectrum matrix, the computational cost of these methods increases rapidly with the increase of spatial points. Therefore, this paper develops a novel simulation method based on the wavenumber–frequency spectrum and unified Hermite polynomial model (UHPM). Firstly, UHPM is extended to the homogenous and stationary non-Gaussian field. Then, a complete transformation model from homogenous and stationary non-Gaussian auto-correlation function (ACF) into underlying Gaussian ACF with its applicable range is proposed. Furthermore, two types of incompatibility between the first four marginal moments and wavenumber–frequency spectrum are discussed, and the corresponding remedies are provided. To facilitate the calculation, the 2D-Fast Fourier Transform (FFT) technique is embedded in the Wiener–Khintchine transformation and spectral representation method (SRM). Finally, a unified simulation framework for multivariate stationary non-Gaussian process based on its relationship to the homogenous and stationary non-Gaussian field is presented. Two numerical examples, involving the simulations of non-Gaussian wind velocities and ground motion accelerations, are investigated to verify the capabilities of the proposed method.

A new perspective on the simulation of cross-correlated random fields

Cross-correlated random fields are widely used to model multiple uncertain parameters and/or phenomena with inherent spatial/temporal variability in numerous engineering systems. The effective representation of such fields is therefore the key element in the stochastic simulation, reliability analysis and safety assessment of engineering problems with mutual correlations. However, the simulation of such fields is generally not straightforward given the complexity of correlation structure. In this paper, we develop a unified framework for simulating non-Gaussian and non-stationary cross-correlated random fields that have been specified by their correlation structure and marginal cumulative distribution functions. Our method firstly represents the cross-correlated random fields by means of a new general stochastic expansion, in which the fields are expanded in terms of a set of deterministic functions with corresponding random variables. A finite element discretization scheme is then developed to further approximate the fields, so that the sets of deterministic functions reflecting the cross-covariance structure can be straightforwardly determined from the spectral decomposition of the resulting discretized fields. For non-Gaussian random fields, an iterative mapping procedure is developed to generate random variables to fit non-Gaussian marginal distribution of the fields. By virtue of the remarkable property of the presented stochastic expansion, i.e., various random fields share an identical set of random variables, the framework we develop is conceptually simple for simulating non-Gaussian cross-correlated fields with arbitrary covariance functions, which need not be stationary. In particular, the developed method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. Five illustrative examples, including a spatially varying non-Gaussian and nonstationary seismic ground motions, are used to demonstrate the application of the developed method.

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space–time fields on arbitrary manifolds, and practical computational considerations.

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: Second-order effects

The second-order formula of Minkowski functionals in weakly non-Gaussian fields is compared with the numerical $N$-body simulations. Recently, the weakly non-Gaussian formula of Minkowski functionals is extended to include the second-order effects of non-Gaussianity in general dimensions. We apply this formula to the three-dimensional density field in the large-scale structure of the Universe. The parameters of the second-order formula include several kinds of skewness and kurtosis parameters. We apply tree level of nonlinear perturbation theory to estimate these parameters, including novel calculations of quartic cumulants. First we compare the theoretical values with those of numerical simulations on the basis of parameter values, and next we test the performance of the analytic formula combined with the perturbation theory. The second-order formula outperforms the first-order formula in general. The performance of the perturbation theory depends on the smoothing radius applied in defining the Minkowski functionals. The quantitative comparisons are presented in detail.