non-Gaussian Processes Research Articles

The spectral representation method: A framework for simulation of stochastic processes, fields, and waves

The Spectral Representation Method (SRM) was developed in the 1970s to simulate Gaussian stochastic processes and fields from a Fourier series expansion according to the Spectral Representation Theorem. Since those early developments, the SRM has continuously evolved into a comprehensive framework for the simulation of stochastic processes, fields, and waves with a rigorous theoretical foundation. Its major advantages are conceptual simplicity and computational efficiency. In the 1990s, much of the theory for simulation of Gaussian stochastic processes, fields, and waves was firmly established and early methods for simulation of non-Gaussian processes, fields, and waves were introduced. In the 2000s and 2010s, methods that coupled the SRM with Translation Process Theory were improved to enable efficient and accurate simulations of stochastic processes, fields, and waves with strongly non-Gaussian marginal probability distributions. More recently, the SRM was extended for higher-order non-Gaussian processes, fields, and waves by extending the Fourier stochastic expansion to include non-linear wave interactions derived from higher-order spectra. This paper reviews the key theoretical developments related with the SRM and provides the relevant algorithms necessary for its practical implementation for the simulation of stochastic processes, fields, and waves that can be either stationary or non-stationary, homogeneous or non-homogeneous, one-dimensional or multi-dimensional, scalar or multi-variate, Gaussian or non-Gaussian, or any combination thereof. The paper concludes with some brief remarks addressing the open research challenges in SRM-based theory and simulations.

A novel class of non-Gaussian system performance assessment and controller parameter tuning methods

Traditional variance-based control performance assessment (CPA) and controller parameter tuning (CPT) methods tend to ignore non-Gaussian external disturbances. To address this limitation, this study proposes a novel class of CPA and CPT methods for non-Gaussian single-input single-output systems, denoted as data Gaussianization (inverse) transformation methods. The idea of quantile transformation is used to transform the non-Gaussian data with the goal of maximizing mutual information into virtual Gaussian data. In addition, optimal system data for the virtual loop are mapped back to the actual non-Gaussian system using quantile inverse transformation. Furthermore, a CARMA model-based recursive extended least square algorithm and a CARMA model-based least absolute deviation iterative algorithm are used to identify virtual Gaussian and non-Gaussian system process models, respectively, while implementing the CPT. Finally, a unified framework is proposed for the CPA and CPT of a non-Gaussian control system. The simulation results demonstrate that the proposed strategy can provide a consistent benchmark judgment criterion (threshold) for different non-Gaussian noises, and the tuned controller parameters have good performance.

Limit behavior in high-dimensional regime for Wishart tensors with Rosenblatt entries

In this paper, we consider an initial [Formula: see text] random matrix with non-Gaussian correlated entries on each row and independent entries from one row to another. The correlation on the rows is given by the correlation of the increments of the Rosenblatt process, which is a non-Gaussian self-similar process with stationary increments, living in the second Wiener chaos. To this initial matrix, we associate a Wishart tensor of length [Formula: see text]. We study the limit behavior in distribution of this Wishart tensor in the high-dimensional regime, i.e. when [Formula: see text] are large enough. We prove that the vector corresponding to the [Formula: see text]-Wishart tensor converges to an [Formula: see text]-dimensional non-Gaussian vector, with Rosenblatt random variables on its hyper-diagonals and zeros outside the hyper-diagonals.

The intervals between zero-crossings of non-Gaussian stable random processes

Properties of the intervals between zero-crossings of non-Gaussian stable random processes are investigated. The classes of process considered are divided into those having short- and long-term memories, characterized by a coherence function that imbues dependence between a random variable measured at different times, which replaces the auto-correlation function, which exists only for the Gaussian case. For exponential coherence functions, all moments of probability densities for the intervals exist and a persistence parameter is determined that characterizes the rate of decay of the exponential tail of the probability densities, together with the full form of the density functions for selected values of the stability index. Results are verified with numerical simulation of a Cauchy process using a method independent of the theoretical development. For power-law coherence functions, the existence of moments of the distribution is conditional upon the indices characterizing the stable process and coherence function. The probability densities of the intervals have power-law tails depending on the product of these indices, with a logarithmic correction for specific combinations of these. An outer-scale or cut-off to power-law coherence functions regains a persistence parameter to the densities and the existence of all moments of the intervals.

Influence of Ground Motion Non-Gaussianity on Seismic Performance of Buildings

The non-Gaussian feature of seismic ground motion has been reported in some works. However, there remains a lack of research on the influence of the ground motion non-Gaussianity on the seismic performance of buildings, which motivates this study. By employing a non-Gaussian non-stationary random process simulation method previously proposed by the authors, 40,000 ground motion acceleration signals are efficiently generated, including 20,000 Gaussian and 20,000 non-Gaussian records. As computational examples, a four-story frame building and a three-tower super-tall building are selected. The generated acceleration signals serve as external excitations for the two buildings, allowing for a comparison of the differences in seismic structural responses caused by the Gaussian and non-Gaussian earthquake groups. Probability analysis is performed using top-layer displacement and maximum inter-story drift ratio as damage indicators. The results show that the structural responses induced by both Gaussian and non-Gaussian earthquake groups have identical first- and second-order moments but different higher-order moments. The responses from non-Gaussian earthquakes display distinct non-Gaussian traits, with their distribution of extreme values exhibiting a longer tail compared to the Gaussian counterparts. This leads to a notably larger value of non-Gaussian responses under high crossing probabilities, with an amplification that can surpass 18%.

Investigation on the sampling distributions of non-Gaussian wind pressure skewness and kurtosis

Accurate estimation of skewness and kurtosis is crucial for addressing issues related to non-Gaussian wind pressure distribution fitting, signal simulation, and extreme estimation. The skewness and kurtosis determined from finite-length signals are inherent random variables characterized by obvious value fluctuations. Very limited research has been conducted on the sampling distributions of skewness and kurtosis for non-Gaussian wind pressure, motivating the present study. Firstly, the investigation focuses on the non-Gaussian white noise. The expressions of main statistical indicators such as the variance, covariance, correlation coefficient, skewness, and kurtosis of the two sampling distributions are derived and then verified using the Hermite polynomial model. The primary factors influencing the sampling distribution variance (SDV) of white noise are analyzed, including the signal length and marginal moments of the parent distribution. Secondly, the concentration shifts to colored non-Gaussian processes. The composition of SDV is analyzed through theoretical derivation. On this basis, a method for estimating SDV values using the Gaussian process regression model is proposed, with accuracy and feasibility verified based on the long-duration wind pressure data measured in wind tunnel. Furthermore, the relationship between the SDVs of wind pressure processes, signal length, higher-order correlation functions, and marginal moments of the parent process is discussed.

A novel method for the non-Gaussian wind pressure simulation based on autoregressive model and maximum entropy method

The simulation of wind pressures with non-Gaussian characteristics is a prerequisite to apply the Monte Carlo method in the structural wind-resistant reliability analysis. Due to its simplicity and accuracy, the autoregressive (AR) model has been widely used for the non-Gaussian process simulation. However, AR-based simulation methods perform poorly for the wind pressures with strong non-Gaussian characteristics. Moreover, they cannot reproduce the bimodal characteristics of some hardening wind pressures. To improve the simulation accuracy, a novel non-Gaussian simulation approach is proposed in this study based on AR model and maximum entropy method. In this approach, a set of formulas to determine the first ten moments of the input process as a function of the corresponding moments of the output process in the AR model have been firstly derived. Then, the probability distribution of the input process is reconstructed by the enhanced maximum entropy method. Finally, the translation process theory is introduced to simulate the non-Gaussian wind pressure process combined with AR model. Numerical results show the proposed method has higher simulation accuracy for the wind pressures with strong non-Gaussianity compared with the conventional AR-based method. Additionally, it can reproduce the bimodal characteristic for the wind pressures with bimodal distribution.

Green Measures for a Class of Non-Markov Processes

In this paper, we investigate the Green measure for a class of non-Gaussian processes in Rd. These measures are associated with the family of generalized grey Brownian motions Bβ,α, 0<β≤1, 0<α≤2. This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability 1 for dα>2 and 1<α≤2. The Green measure then generalizes those measures of all these classes.

An integrated approach combining randomized kernel PCA, Gaussian mixture modeling and ICA for fault detection in non-linear processes

Principal component analysis (PCA) and independent component analysis (ICA), as well as their kernel extensions, have been widely applied in the past for industrial fault detection with Gaussian or non-Gaussian process data with linear or non-linear characteristics. Kernel-based techniques lead to computational complexity due to the high dimensionality of the dataset in the feature space. In this work, a randomization approach is used to obtain a low-rank approximation of the high-dimensional kernel matrix. A hybrid machine learning technique is proposed that integrates randomized kernel PCA (RKPCA) with ICA and Gaussian mixture modeling (GMM). The proposed approach, ICA-RKPCA-GMM, addresses the Gaussian and non-Gaussian characteristics of non-linear process data. Another hybrid algorithm combining three basic techniques of ICA, PCA and GMM is also developed (ICA-PCA-GMM). The fault detection performances of the proposed techniques (ICA-RKPCA-GMM and ICA-PCA-GMM) are compared with PCA, ICA, KPCA and combined ICA-PCA techniques by applying the techniques to two benchmark systems. Monitoring performances were evaluated by determining the false alarm rate and fault detection rate for different types of process and sensor faults. The simulation results show that the proposed ICA-RKPCA-GMM approach yields better results than individual ICA, PCA and KPCA techniques, the combined ICA-PCA and the proposed ICA-PCA-GMM technique.

An improved analytical solution to outcrossing rate for scalar nonstationary and non-gaussian processes

For time-dependent reliability methods, there is a lack of analytical solutions to outcrossing rates of nonstationary and non-Gaussian processes. This paper aims to propose an analytical method to determine this type of outcrossing rate with improved accuracy and efficiency. The novelty of this proposed method is that it can consider the probabilistic properties of non-Gaussian processes by combining a two-component parallel system model with higher-order moments-based reliability indexes. The main contributions of this method are: (1) compared with PHI2+ method, it is more accurate for the calculation of the outcrossing rate of nonstationary and non-Gaussian processes with nonlinear limit state functions; (2) compared with MPHI2 method, it is analytical and not only applicable for the nonstationary stochastic processes but also insensitive to time increments; and (3) it is not only more accurate than the existing methods for nonstationary and non-Gaussian processes but also more computation efficient than the Monte Carlo simulation method. The proposed method has shown its advantages for practical structures with neither Gaussian processes nor linear limit state functions, which are beneficial for both researchers and asset managers to evaluate the time-dependent reliability of structures accurately and to develop risk-informed maintenance schemes with a view to prolonging their service life.

Novel Hermite Polynomial Model Based on Probability-Weighted Moments for Simulating Non-Gaussian Stochastic Processes

Novel Hermite Polynomial Model Based on Probability-Weighted Moments for Simulating Non-Gaussian Stochastic Processes

The higher-order analysis of response for linear structure under non-Gaussian stochastic excitations with known statistical moments and power spectral density

The frequency-domain analysis method is a fundamental component of the random vibration analysis, in which the corresponding moment spectrum of excitations are the prerequisite. Nevertheless, the determination of the higher-order moment spectra for non-Gaussian stochastic excitations continues to pose a significant challenge in the existing research works. This paper introduces an accurate and efficient computational approach for determining higher-order moment spectra of non-Gaussian stochastic excitations with known statistical moments and power spectral density (PSD). Firstly, following the idea of the simulation method of non-Gaussian stochastic processes, the transformation model between the stationary non-Gaussian stochastic process and underlying Gaussian stochastic process is determined based on the known statistical moments, the PSD of the underlying Gaussian stochastic process is determined using the transformation model. Secondly, the approximate higher-order moment spectra models of stationary non-Gaussian stochastic processes are presented by the PSD of the underlying Gaussian process. Subsequently, the higher-order moment spectra models of non-Gaussian excitations are utilized to compute the higher-order moment spectra of response for the linear structure via the auxiliary harmonic excitation generalized method. Finally, three numerical examples are examined to assess the efficiency and accuracy of the proposed approximate models for the higher-order moment spectra of non-Gaussian stochastic processes.

Conditional simulation of stationary non-Gaussian processes based on unified hermite polynomial model

The conditional simulation of non-Gaussian excitations utilizing records from the monitoring system is of great significance for hazard mitigation. To this end, this paper proposes a novel conditional non-Gaussian simulation method. In this method, the Unified Hermite Polynomial Model (UHPM) is used to describe the transformation relationship between recorded and unrecorded non-Gaussian processes and their underlying Gaussian counterparts. Meanwhile, an explicit transformation model between their correlation functions is also provided. Then, the covariance matrix of Fourier coefficients of the underlying Gaussian processes is constructed. Based on this covariance matrix, the conditional samples of Fourier coefficients are generated and substituted into the Spectral Representation Method (SRM) to perform the conditional simulation of the underlying Gaussian processes. Finally, the conditionally simulated samples of the underlying Gaussian processes are transformed into the non-Gaussian samples by the UHPM. To showcase the precision and efficacy of the proposed method, two numerical examples involving the conditional simulations of non-Gaussian ground motions and non-Gaussian wind pressure coefficients are provided.

Intermittent Kac's flights and the generalized telegrapher's equation.

A generalized one-dimensional telegrapher equationassociated with an intermittent change of sign in the velocity of a Kac's flight has been proposed. To solve this random differential equation, we used the enlarged master equationapproach to obtain the exact differential equationfor the evolution of a normalized positive distribution. This distribution is associated with a generalized finite-velocity diffusionlike process. We studied the robustness of the ballistic regime, the cutoff of its domain, and the time-dependent Gaussian convergence. The second moment for the evolution of the profile has been studied as a function of non-Poisson statistics (possibly intermittent) for the time intervals Δ_{ij} in the Kac's flight. Numerical results for the evolution of sharp and wide initial profiles have also been presented. In addition, for comparison with a non-Gaussian process at all times, we have revisited the non-Markov Poisson's flight with exponential pulses. A theory for generalized random flights with intermittent stochastic velocity and in the presence of a force is also presented, and the stationary distribution for two classes of potential has been obtained.

Classification of stochastic processes based on deep learning

Stochastic processes model the time evolution of fluctuation phenomena widely observed in physics, chemistry, biology, and even social science. Typical examples include the dynamics of molecular interactions, cellular signalling, animal feeding, disease transmission, financial market fluctuation, and climate change. We create three datasets based on the codes obtained from the published article; the first one is for 12 stochastic processes, the second one for the Markov and non-Markov processes, and the third one for the Gaussian and non-Gaussian processes. We do the stochastic process classification by employing a series of convolution neural networks (CNNs), i.e. VGG16, VGG19, AlexNet, and MobileNetV2, achieving the accuracy rates of ‘99%’, ‘98%’, ‘95%’, and ‘94%’ on the first dataset, respectively; in the second dataset, the test accuracy of VGG16 is ‘100%’, and for the rest of the models, it is ‘99%’; and in the third dataset, the test accuracy of all models is ‘100%’, except the VGG19, which is ‘99%’. According to the findings, CNNs have slightly higher accuracy than classic feature-based approaches in the majority of circumstances, but at the cost of much longer training periods.

A method for evaluation of the probability density function of white noise filtered non-Gaussian stochastic process

Passing non-Gaussian white noises through a pre-designed linear filter can generate non-Gaussian stochastic processes with desired probabilistic and spectral properties. This linear filtering methodology has been extensively employed in non-Gaussian signal processing and simulation. At present, the investigation of probability properties for the output non-Gaussian processes is primarily concentrated on the higher-order statistics, spectra, and correlation functions. However, evaluating the probability density function (PDF) of the output process remains a challenging task. In order to address this issue, this short communication commences with the theoretical derivation of the PDF formula for the white noise filtered non-Gaussian stochastic process. The derived PDF is the result of a convolution of multiple PDFs, each obtained by applying a scaling transformation to the input white noise PDF. Subsequently, a discretized numerical calculation method is proposed to overcome the analytical difficulties associated with the computation of multiple convolutions. Finally, the accuracy and efficiency of the proposed method are demonstrated through several numerical examples.

Process monitoring via dependence description based on variable selection and vine copula

Process monitoring is crucial to ensure the safety of industrial processes. Generally, the monitoring process involves all measured variables; however, large industrial processes contain many redundant variables. For a method based on describing the intrinsic correlation relationships among variables, the vine copula-based dependence description (VCDD) method shows significant advantages for describing nonlinear and non-Gaussian processes. However, redundant and irrelevant variables adversely affect the correlation between variables containing the most important information, reducing model performance. The lack of research in this area may substantially weaken the advantages of VCDD for fault monitoring. Therefore, this article introduces a variable selection vine copula dependence description monitoring model. It utilizes known faults as validation data to select the relevant variables for constructing the VCDD model, specifically designed for monitoring known faults. Furthermore, to prevent information loss, the remaining unselected variables are also employed to create a separate VCDD model, dedicated to monitoring unknown faults. The performance of the proposed method is verified by a numerical example, the Tennessee-Eastman process and the Penicillin fermentation process.

Inverse elastic scattering by random periodic structures

This paper investigates an inverse scattering problem of time-harmonic elastic waves incident on a rigid random periodic structure. A novel and efficient numerical method is proposed to quantify the randomness, i.e., to reconstruct key statistical properties of random structures from boundary measurements of the scattering data. This method consists of two ingredients, the Monte Carlo technique for sampling the probability space and a continuation method with respect to the wavenumber. Unlike existing methods, this approach does not require a priori knowledge of the randomness and can handle a wide range of random structures, including non-Gaussian processes with extremely high levels of complexity. To illustrate the accuracy and effectiveness of the proposed method, numerical examples involving Gaussian and non-Gaussian processes with various covariances are provided.

An efficient methodology for simulating multivariate non-Gaussian stochastic processes

Simulation of multivariate non-Gaussian stochastic processes is still a challenging task, where the efficiency and accuracy may not be balanced. In this paper, an efficient methodology is presented to address this challenge. This methodology makes use of a probability weighted moments-based Hermite polynomial model and a wavenumber–frequency spectrum density-based spectral representation approach. First, a probability weighted moments-based Hermite polynomial model is employed to establish a transformation relationship between Gaussian and non-Gaussian samples. Importantly, this model only requires the solution of a solvable system of equations. Second, the target non-Gaussian wavenumber–frequency spectrum density can be promptly transformed into the Gaussian one, where a fast root matching strategy and fast Fourier transform are utilized. Finally, the underlying Gaussian samples are generated by the spectral representation method, and the non-Gaussian samples are obtained via the proposed transformation model. The computational results from numerical examples demonstrate that the proposed methodology is considered to be an accurate, applicable, and efficient approach for simulating multivariate non-Gaussian stochastic processes.

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses