non-Gaussian Stochastic Processes Research Articles

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.

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.

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.

Neural network-aided simulation of non-Gaussian stochastic processes

Simulation of non-Gaussian stochastic processes is of paramount importance to dynamic reliability assessment of structures driven by non-Gaussian loads. In this paper, a novel translation model based on neural network is proposed to convert the target non-Gaussian power spectrum to the underlying Gaussian power spectrum for simulating non-Gaussian stochastic processes. First, a valid neural network model, whose structure is similar to the transformation from a non-Gaussian power spectrum to its underlying Gaussian power spectrum, is established. Then, starting with an initial underlying Gaussian power spectrum, the non-Gaussian power spectrum corresponding to the target non-Gaussian distribution is obtained through the Wiener-Khintchine transform and translation process. After that, the discrete data of non-Gaussian power spectrum is employed as the input layer in the neural network above, while the discrete data of Gaussian power spectrum serves as the output layer. With the trained neural network, the required underlying Gaussian power spectrum can be directly obtained by inputting the target non-Gaussian power spectrum. Finally, Gaussian samples can be generated by using the spectral representation method, which are then converted into non-Gaussian samples via the translation process. The effectiveness, accuracy, and applicability of the proposed method are demonstrated through two numerical examples.

Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.

Modeling and simulation of financial returns under non-Gaussian distributions

It is well known that the probability distribution of high-frequency financial returns is characterized by a leptokurtic, heavy-tailed shape. This behavior undermines the typical assumption of Gaussian log-returns behind the standard approach to risk management and option pricing. Yet, there is no consensus on what class of probability distributions should be adopted to describe financial returns and different models used in the literature have demonstrated, to varying extent, an ability to reproduce empirically observed stylized facts. In order to provide some clarity, in this paper we perform a thorough study of the most popular models of return distributions as obtained in the empirical analyses of high-frequency financial data. We compare the statistical properties and simulate the dynamics of non-Gaussian financial fluctuations by means of Monte Carlo sampling from the different models in terms of realistic tail exponents. Our findings show a noticeable consistency between the considered return distributions in the modeling of the scaling properties of large price changes. We also discuss the convergence rate to the asymptotic distributions of the non-Gaussian stochastic processes and we study, as a first example of possible applications, the impact of our results on option pricing in comparison with the standard Black and Scholes approach.

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.

Simulation of non-stationary and non-Gaussian random processes by 3rd-order Spectral Representation Method: Theory and POD implementation

This paper introduces the 3rd-order Spectral Representation Method for simulation of non-stationary and non-Gaussian stochastic processes. The proposed method extends the classical 2nd-order Spectral Representation Method to expand the stochastic process from an evolutionary bispectrum and an evolutionary power spectrum, thus matching the process completely up to third-order. A Proper Orthogonal Decomposition (POD) approach is further proposed to enable an efficient FFT-based implementation that reduces computational cost significantly. Two examples are presented, including the simulation of a fully non-stationary seismic ground motion process, highlighting the accuracy and efficacy of the proposed method.

Simulation of stationary non-Gaussian stochastic vector processes using an eigenvalue-based iterative translation approximation method

The simulation of stationary non-Gaussian stochastic vector processes is of great importance in the time-domain response analysis of structures under non-Gaussian excitations. When incompatibility between the target cross power spectral density matrix and the marginal probabilistic density function of the stochastic vector process occurs in the sense of translation processes, an iterative simulation scheme must be required. Currently, the Cholesky decomposition-based iterative translation approximation method is widely utilized. However, the use of this method still encounters some difficulties. First, it requires performing Cholesky decompositions three times at each iteration and is therefore cumbersome. Second, a complicated random shuffling procedure is necessary, which renders the final iteration result non-unique. Third, it is very difficult for this method to handle cases with complex-valued spectral matrices. Lastly, it can generally be used to simulate stochastic vector processes with very limited components. To overcome these difficulties, an eigenvalue-based iterative translation approximation method is proposed in this study. This method can not only avoid the cumbersome Cholesky decomposition and complex random shuffling, but can also handle cases with complex-valued and large spectral matrices more easily. Four numerical examples with no incompatibility, significant incompatibility, complex-valued coherence and a large number of component processes are respectively employed to demonstrate the effectiveness of the proposed method. The results show that the proposed method has similar accuracy to the Cholesky decomposition-based iterative method in the first two examples. For the last two examples, the simulation accuracy is also satisfactory. The above results verify that the proposed method is superior to the Cholesky decomposition-based iterative translation approximation method overall.

Simulation of non-Gaussian stochastic processes with prescribed rainflow cycle count using short-time Fourier transform

Random phenomena described by non-Gaussian processes are ubiquitous in many scientific fields. Properly understanding their characteristics is crucial in safety analysis, where the random occurrence of extreme values is often critical. In practice, the information available for fully characterizing a stochastic process is often limited by experimental constraints. Thus, prediction based on simulations comes as a valid alternative. Motivated by applications in structural fatigue assessment, this research presents a simulation method for non-Gaussian stochastic processes that is able to jointly reproduce a prescribed power spectral density and a rainflow cycle count. The main idea is to model non-Gaussian behavior by a frequency-dependent non-linearity in the short-time Fourier transform. As such, the power spectral density is directly imposed, while the non-linear model is optimized to match the rainflow cycle count. It is shown that the model is flexible enough to be interpolated between different operating points, so as to allow the simulation of stochastic processes at non-measured operating conditions. The full methodology is illustrated for assessing the fatigue assessment of a hydroelectric turbine runner.

Generalized white noise analysis and topological algebras

In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-Itô expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of non-Gaussian stochastic processes.

Variance of the fatigue damage in non-Gaussian stochastic processes with narrow-band power spectrum

This paper presents two theoretical models to assess the variance of the fatigue damage in stationary narrow-band and non-Gaussian stochastic processes. The models extend two solutions existing in the literature and restricted to Gaussian processes. The new models here developed exploit a non-linear transformation that links Gaussian and non-Gaussian domains based on skewness and kurtosis coefficients, which are used to quantify the deviation from the Gaussian distribution. Monte Carlo numerical simulations in time-domain are performed to confirm the correctness of the proposed non-Gaussian models, and to investigate the sensitivity of the variance of the damage to the skewness, kurtosis, and inverse slope of the stress versus life (S-N) curve. An example is finally presented to demonstrate the increase of the failure probability due to non-Gaussian effects in the stochastic loading.

An algorithm to simulate nonstationary and non-Gaussian stochastic processes

We proposed a new iterative power and amplitude correction (IPAC) algorithm to simulate nonstationary and non-Gaussian processes. The proposed algorithm is rooted in the concept of defining the stochastic processes in the transform domain, which is elaborated and extend. The algorithm extends the iterative amplitude adjusted Fourier transform algorithm for generating surrogate and the spectral correction algorithm for simulating stationary non-Gaussian process. The IPAC algorithm can be used with different popular transforms, such as the Fourier transform, S-transform, and continuous wavelet transforms. The targets for the simulation are the marginal probability distribution function of the process and the power spectral density function of the process that is defined based on the variables in the transform domain for the adopted transform. The algorithm is versatile and efficient. Its application is illustrated using several numerical examples.

Long-term temperature field of steel-box girder of a long-span bridge: Measurement and simulation

The temperature field of a long-span bridge is variant with an underlying statistical property due to the periodically time-varying solar radiation and the unceasing structural heat exchange with surrounding environment. A comprehensive understanding of the long-term temperature field of a steel-box girder is essential for the design and maintenance of cable-supported bridges. However, it is challenging to obtain the long-term temperature variations because of the complex geometric configuration of the main girder and the complicated heat transfer process. In this context, this paper presents a strategy to overcome this problem by simulating the long-term temperature field of a steel-box girder based on limited monitored data. In the simulation, the long-term temperature field is treated as a stochastic process with prescribed spectral and probabilistic properties. By analyzing the monitored temperature data on Sutong Bridge, the models of power spectral density (PSD), coherence function, and high-order moments are developed. A noteworthy feature captured in measurements is that the temperature is non-Gaussian with considerable kurtosis and skewness. Hence, the temperature field simulation is converted to the simulation of multivariate non-Gaussian stochastic processes. Based on the spectral representation method accompanied with Hermite transformation, the temperature field of the steel-box girder of Sutong Bridge is simulated with a two-year duration considering the non-Gaussianity. Verifications of the simulated field are made through comparisons with the developed spectral and probabilistic models. It is shown that the simulations match well with the measured temperature characteristics, specifically including the time histories and their PSD, PDF, and coherence functions. The good agreement indicates the effectiveness of the simulated long-term temperature field.

Extreme Design Events of Non-Gaussian Stochastic Processes

Extreme Design Events of Non-Gaussian Stochastic Processes

A sample-based iterative scheme for simulating non-stationary non-Gaussian stochastic processes

This paper presents a new numerical scheme for simulating stochastic processes specified by their marginal distribution functions and covariance functions. Stochastic samples are first generated to satisfy target marginal distribution functions. An iterative algorithm is proposed to match the simulated covariance function of stochastic samples to the target covariance function, and only a few iterations can converge to a required accuracy. Several explicit representations, based on Karhunen-Loève expansion and Polynomial Chaos expansion, are further developed to represent the obtained stochastic samples in series forms. Proposed methods can be applied to non-stationary non-Gaussian stochastic processes, and three examples illustrate their accuracies and efficiencies.

A time-variant uncertainty propagation analysis method based on a new technique for simulating non-Gaussian stochastic processes

In this paper, the time-variant uncertainty propagation analysis is defined to solve the output stochastic process of a time-variant function with uncertainty. And a time-variant uncertainty propagation analysis method is constructed with the combination of an extended orthogonal series expansion method (extended OSE) and sparse grid numerical integration (SGNI). The SGNI serving as a classical uncertainty propagation method is utilized here to solve the moments and autocorrelation function at discrete time points of the time-variant performance function. And the extended OSE is proposed to simulate the output stochastic process based on the results from SGNI. By extended OSE, a non-Gaussian stochastic process is represented as the sumof orthogonal time functions with random coefficients, and these coefficients can be directly obtained by discretization of the target process. For these coefficients are correlated and non-Gaussian, the correlated polynomial chaos expansion method (c-PCE) is presented to represent them in terms of correlated standard Gaussian variables, and then the principal component analysis (PCA) is adopted to transform them into independent ones with dimension reduction. Finally we can obtain an explicit expression to represent the non-Gaussian process whatever it is stationary or non-stationary. Three illustrative examples are used to verify the performance of the extended OSE. In addition, two engineering problems are investigated to demonstrate the effectiveness of the time-variant uncertainty propagation method.

Generation of strongly non-Gaussian stochastic processes by iterative scheme upgrading phase and amplitude contents

Random excitations, such as wind velocity, always exhibit non-Gaussian features. Sample realisations of stochastic processes satisfying given features should be generated, in order to perform the dynamical analysis of structures under stochastic loads based on the Monte Carlo simulation. In this paper, an efficient method is proposed to generate stationary non-Gaussian stochastic processes. It involves an iterative scheme that produces a class of sample processes satisfying the following conditions. (1) The marginal cumulative distribution function of each sample process is perfectly identical to the prescribed one. (2) The ensemble-averaged power spectral density function of these non-Gaussian sample processes is as close to the prescribed target as possible. In this iterative scheme, the underlying processes are generated by means of the spectral representation method that recombines the upgraded power spectral density function with the phase contents of the new non-Gaussian processes in the latest iteration. Numerical examples are provided to demonstrate the capabilities of the proposed approach for four typical non-Gaussian distributions, some of which deviate significantly from the Gaussian distribution. It is found that the estimated power spectral density functions of non-Gaussian processes are close to the target ones, even for the extremely non-Gaussian case. Furthermore, the capability of the proposed method is compared to two other methods. The results show that the proposed method performs well with convergence speed, accuracy, and random errors of power spectral density functions.

The Wright Functions of the Second Kind in Mathematical Physics

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.