non-Gaussian Random Variables Research Articles

Аналіз коефіцієнта ексцесу двокомпонентних сумішей зсунутих негаусових розподілів

The dependence of the extremes and zeros of the excess kurtosis on the weight coefficient is researched. Formulas for finding the extrema points, the values of the minimums and maximums of the excess kurtosis are obtained. Conditions on the shift parameter under which the extrema points belong to the interval are determined. Formulas for finding the zeros of the excess kurtosis are obtained and conditions on shift parameter under which the roots of the equation are real and belong to the interval are determined. Examples of calculating extremes and zeros of the excess kurtosis of two-component mixtures of shifted non-Gaussian distributions are considered. The results of the research justify the possibility of practical application of two-component mixtures of shifted distributions for mathematical and computer modeling of an infinite number of non-Gaussian random variables with negative, positive and zero excess kurtosis.

Channel Tracking and Prediction for IRS-Aided Wireless Communications

For intelligent reflecting surface (IRS)-aided wireless communications, channel estimation is essential and usually requires excessive channel training overhead when the number of IRS reflecting elements is large. The acquisition of accurate channel state information (CSI) becomes more challenging when the channel is not quasi-static due to the mobility of the transmitter and/or receiver. In this work, we study an IRS-aided wireless communication system with a practical channel model that characterizes the time-varying propagation property and propose an innovative two-stage transmission protocol. In the first stage, we send pilot symbols and track the direct/reflected channels based on the received signal, and then data signals are transmitted. In the second stage, instead of sending pilot symbols first, we directly predict the direct/reflected channels and all the time slots are used for data transmission. Based on the proposed transmission protocol, we propose a two-stage channel tracking and prediction (2SCTP) scheme to obtain the direct and reflected channels with low channel training overhead, which is achieved by exploiting the temporal correlation of the time-varying channels. Specifically, we first consider a special case where the IRS-access point (AP) channel is assumed to be static, for which a Kalman filter (KF)-based algorithm and a long short-term memory (LSTM)-based neural network are proposed for channel tracking and prediction, respectively. Then, for the more general case where the IRS-AP, user-IRS and user-AP channels are all assumed to be time-varying, we present a generalized KF (GKF)-based channel tracking algorithm, where proper approximations are employed to handle the underlying non-Gaussian random variables. Numerical simulations are provided to verify the effectiveness of our proposed transmission protocol and channel tracking/prediction algorithms as compared to existing ones.

Effect of non-Gaussian turbulence on extreme buffeting response of a high-speed railway sea-crossing bridge

The non-Gaussian property of turbulence has been emphasized in some measured wind data. However, the classic Scanlan buffeting theory with the Gaussian distribution assumption of turbulence may cause errors in predicting the structural extreme response. In this study, a hybrid approach for predicting bridge's extreme buffeting response is proposed and validated, which considers multi-dimensional non-Gaussian random variables. Then the extreme buffeting response of a cable-stayed bridge under the attack of non-Gaussian wind is investigated, where the in-situ measured wind data of Typhoon ‘Bailu’ is adopted. The sensitivity of extreme buffeting response to turbulence skewness and kurtosis is also analyzed. The study results show that the lateral and vertical buffeting displacements are in skewed distributions under the attack of Typhoon ‘Bailu’. Compared to the Gaussian wind assumption, the non-Gaussian typhoon decreases the lateral extreme response, while increases the vertical extreme response. The results from the sensitivity analysis reveal a strong correlation between the non-Gaussian intensity of component u and that of the lateral buffeting response. In addition, the lateral extreme buffeting response increases significantly with the skewness and kurtosis of component u at a high probability level.

Unconditional and conditional simulation of nonstationary and non-Gaussian vector and field with prescribed marginal and correlation by using iteratively matched correlation

In many probabilistic analysis problems, the homogeneous/nonhomogeneous non-Gaussian field is represented as a mapped Gaussian field based on the Nataf translation system. We propose a new sample-based iterative procedure to estimate the underlying Gaussian correlation for homogeneous/nonhomogeneous non-Gaussian vector or field. The numerical procedure takes advantage that the range of feasible correlation coefficients for non-Gaussian random variables is bounded if the translation system is adopted. The estimated underlying Gaussian correlation is then employed for unconditional as well as conditional simulation of the non-Gaussian vector or field according to the theory of the translation process. We then present the steps for augmenting the simulated non-Gaussian field through the Karhunen-Loeve expansion for a refined discretized grid of the field. In addition, the steps to extend the procedure described in the previous section to the multi-dimensional field are highlighted. The application of the proposed algorithms is presented through numerical examples.

A multi-scale framework for probabilistic structural analysis of PSC girders considering pit corrosion of prestressing wires

Many standards and studies have emphasized the importance of considering various uncertainty factors of material properties and corrosion in assessing the structural performance of bridges. In such an analysis, however, high computational cost is a major issue because a probabilistic structural analysis requires multiple structural analyses. In this study, an efficient framework for constructing a probabilistic structural analysis model is proposed. First, a surrogate model is constructed using Gaussian process regression for the given structural analysis model. Next, random samples of model inputs and structural responses, which are assumed to be correlated non-Gaussian random variables, are generated based on the Cholesky decomposition and Morgenstern model. Finally, the statistical properties of structural responses are derived by a goodness-of-fit test; hence, they can be considered as probabilistic inputs for an upper-scale model in the form of random variables. This three-step process can be introduced accumulatively from the lower-scale to upper-scale models about a structure; accordingly, it is termed as a multi-scale framework for probabilistic structural analysis. The proposed framework was applied to construct probabilistic models for prestressed concrete (PSC) girders with corroded prestressing wires, and the probabilistic prediction results for the ultimate load and displacement showed a good agreement with the experimental data.

Generalized Correntropy Criterion-Based Performance Assessment for Non-Gaussian Stochastic Systems.

Control loop performance assessment (CPA) is essential in the operation of industrial systems. In this paper, the shortcomings of existing performance assessment methods and indicators are summarized firstly, and a novel evaluation method based on generalized correntropy criterion (GCC) is proposed to evaluate the performance of non-Gaussian stochastic systems. This criterion could characterize the statistical properties of non-Gaussian random variables more fully, so it can be directly used as the assessment index. When the expected output of the given system is unknown, generalized correntropy is used to describe the similarity of two random variables in the joint space neighborhood controlled and take it as the criterion function of the identification algorithms. To estimate the performance benchmark more quickly and accurately, a hybrid-EDA (H-EDA) combined with the idea of “wading across the stream algorithm” is proposed to obtain the system parameters and disturbance noise PDF. Through the simulation of a single loop feedback control system under different noise disturbances, the effectiveness of the improved algorithm and new indexes are verified.

НЕЛИНЕЙНЫЕ МЕТОДЫ ОЦЕНИВАНИЯ ПАРАМЕТРОВ СИГНАЛА НА ФОНЕ АСИММЕТРИЧНО-ЭКСЦЕССНЫХ НЕГАУССОВСКИХ КОРРЕЛИРОВАННЫХ ПОМЕХ

In the theory of statistical analysis of multidimensional random variables, the tasks of correlation analysis are quite important in the construction and implementation of many technical systems of control, monitoring and diagnostics. In the process of solving these tasks, the determination of the presence and nature of statistical relationship of the studied random variables is a priority. Based on the results of correlation analysis, conclusions are drawn about the presence and nature of functional dependence of random variables, the preference of the research methods used and the proposed models for describing random multidimensional processes. The application of classical mathematical apparatus of correlation analysis is widely used in the assumption that the observed random process belongs to multidimensional normal distribution law. In practice, such prerequisites for correlation analysis are not always fulfilled and most likely are a convenient mathematical ideali-zation of the processes under study. Studies show that in describing random processes, including non-Gaussian ones, an approach based on the use of moment and cumulative functions of higher orders is promising. Such a representation of random processes makes it possible to increase the accuracy of their processing under given restrictions on their algorithmic complexity and to take into account correlation relationships of the studied non-Gaussian random variables. In the proposed paper, we consider the construction of methods for constant signal parameter estimation in asymmetric-excess non-Gaussian correlated noise using the method of polynomial maximization (Kunchenko method) and its adaptation to implement non-linear algorithms and computer-aided means of functioning of signal processing systems. It is shown that polynomial processing of random variables, taking into account the parameters of a non-Gaussian distribution in the form of cumulants of one-dimensional and multidimensional distributions, makes it possible to reduce the variance of the parameter estimation in comparison with the well-known results.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads Using a Sensitivity-Based Propagation of Moments

Abstract The paper proposes a new methodology for time-dependent reliability analysis of vibratory systems using a combination of a first-order, four-moment (FOFM) method and a non-Gaussian Karhunen–Loeve (NG-KL) expansion. The approach can also be used for random vibrations studies. The vibratory system is nonlinear and is excited by stationary non-Gaussian input random processes which are characterized by their first four marginal moments and autocorrelation function. The NG-KL expansion expresses each input non-Gaussian process as a linear combination of uncorrelated, non-Gaussian random variables and computes their first four moments. The FOFM method then uses the moments of the NG-KL variables to calculate the moments and autocorrelation function of the output processes based on a first-order Taylor expansion (linearization) of the system equations of motion. Using the output moments and autocorrelation function, another NG-KL expansion expresses the output processes in terms of uncorrelated non-Gaussian variables in the time domain, allowing the generation of output trajectories. The latter are used to estimate the time-dependent probability of failure using Monte Carlo simulation (MCS). The computational cost of the proposed approach is proportional to the number of NG-KL random variables and is significantly lower than that of other recently developed methodologies which are based on sampling. The accuracy and efficiency of the proposed methodology is demonstrated using a two-degree-of-freedom nonlinear vibratory system with random coefficients excited by a stationary non-Gaussian random process.

Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality, gene flow (through pollen drift), and plant disease transmission (spore dispersion). Although Lagrangian stochastic (LS) models describe stochastic wind behaviors, such models assume that wind velocities follow Gaussian distributions. However, measured surface-layer wind velocities show a strong skewness and kurtosis. This paper presents an improved model, a non-Gaussian LS model, which incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis. Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study, October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model. Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model, such as the accuracy in the mean and variance of simulated velocities. This improvement also leads to better accuracy in friction velocity (i.e., a coupling of three-dimensional velocities). The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations. Moreover, improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations. Thus, the non-Gaussian model is worth applying to wind field simulation in the surface layer.

Methods of mathematical modeling of non-Gaussian random variables quantities and processes

Mathematical methods allowing to model non-Gaussian random variables and processes are considered. The models and description of non-Gaussian correlated processes in the form of generated Gaussian noise are analyzed, as well as the methods of formation of stationary random processes defined by the one-dimensional density distribution of Vero -abilities and the autocorrelation function. Examples of formation of non-Gaussian random variables and processes are given.

Reliability Analysis Using Second-Order Saddlepoint Approximation and Mixture Distributions

This paper proposes a new second-order saddlepoint approximation (SOSA) method for reliability analysis of nonlinear systems with correlated non-Gaussian and multimodal random variables. The proposed method overcomes the limitation of current available SOSA methods, which are applicable to problems with only Gaussian random variables, by employing a Gaussian mixture model (GMM). The latter is first constructed using the expectation maximization (EM) method to approximate the joint probability density function (PDF) of the input variables. Expressions of the statistical moments of the response variables are then derived using a second-order Taylor expansion of the limit-state function and the GMM. The standard SOSA method is finally integrated with the GMM to effectively analyze the reliability of systems with correlated non-Gaussian random variables. The accuracy of the proposed method is compared with existing methods including a SOSA based on Nataf transformation. Numerical examples demonstrate the effectiveness of the proposed approach.

МОДЕЛЮВАННЯ ПОСЛІДОВНОГО ОЦІНЮВАННЯ ПАРАМЕТРА ЗСУВУ АСИМЕТРИЧНО-РОЗПОДІЛЕНИХ ВИПАДКОВИХ ВЕЛИЧИН МЕТОДОМ МАКСИМІЗАЦІЇ ПОЛІНОМА

An original approach to finding sequential estimates of the parameter of bias of non-Gaussian asymmetric-distributed random variables is investigated in the paper. The polynomial maximization method (PlMM), which is based on the mathematical apparatus of stochastic Kunchchenko polynomials and a partial description of random variables by higher order statistics (moments or cumulants) is the basis of this approach. The classic approach to solving a posed problem, which is based on simple linear recurrent statistics, that does not take into account the peculiarities of probabilistic data distribution and is optimal only for Gaussian model, is analyzed. Analytical expressions for finding the estimates by polynomial maximization method at the second degree polynomial are obtained. A comparative analysis of the efficiency on the basis of the criterion of the magnitude of asymptotic dispersion of the estimates of various methods parameters is performed. It is shown that theoretical value of the coefficient of the reduction of PlMM-estimates dispersion (in comparison with linear estimates) depends on the magnitude of cumulative coefficients of asymmetry and excess of statistical data. On the basis of the received results, in MATLAB software environment a set of m-functions that realize statistical modeling by Monte-Carlo method of linear and polynomial sequential grading algorithms for the parameter of bias of non-Gaussian random variables with different types of distributions (ex-ponential, gamma, lognormal, Weibull, double-Gaussian ones) is developed. The combination of the obtained results shows that the application of the proposed approach can provide a significant reduc-tion in the time to make decisions when diagnosing the state of technical systems and technological processes.

Time domain stochastic finite element simulation towards probabilistic seismic soil-structure interaction analysis

This paper exploits the recent advances in the field of stochastic finite elements in time-domain simulation of the seismic behavior of soil-foundation-structure systems in three-dimension considering the soil parameters to be heterogeneous, anisotropic, non-Gaussian random fields, structural material parameters to be non-Gaussian random variables, and earthquake motion to be a non-stationary random process. Both the formulation and its implementation are discussed. Scalable numerical schemes for distributed memory parallel computing of hundreds of millions stochastic degrees of freedom, that may arise for any typical uncertain soil-structure interaction (SSI) problem, are presented. Although the presented approach allows for treatment of material nonlinearity, behaviors of only linear systems are illustrated in this study. Through comparisons of the stochastic SSI analysis results with that of the deterministic SSI, deterministic rigid base structural response history, and 1-D deterministic geotechnical site response analyses, and through an extensive parametric study by varying the input uncertainty parameters, the shortcomings of the conventional deterministic approaches are highlighted.

Calculating the Malliavin derivative of some stochastic mechanics problems.

The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how to calculate the Malliavin derivative, the derivative of the expectation of a quantity of interest of a model with respect to its underlying stochastic parameters, for four problems found in mechanics. The non-intrusive approach uses the Malliavin Weight Sampling (MWS) method in conjunction with a standard Monte Carlo method. The models are expressed as ODEs or PDEs and discretised using the finite difference or finite element methods. Specifically, we consider stochastic extensions of; a 1D Kelvin-Voigt viscoelastic model discretised with finite differences, a 1D linear elastic bar, a hyperelastic bar undergoing buckling, and incompressible Navier-Stokes flow around a cylinder, all discretised with finite elements. A further contribution of this paper is an extension of the MWS method to the more difficult case of non-Gaussian random variables and the calculation of second-order derivatives. We provide open-source code for the numerical examples in this paper.

Chance-Constrained Economic Dispatch With Non-Gaussian Correlated Wind Power Uncertainty

Extending traditional deterministic economic dispatch to incorporate significant stochastic wind power is an important but challenging task in today's power system decision making. In this paper, this issue is formulated as a chance-constrained economic dispatch (CCED) problem. Usually, in the presence of non-Gaussian correlated random variables, both the objective function and constraints are difficult to handle. To address this issue, this paper provides a novel method dealing with non-Gaussian random variables. First, the Gaussian mixture model is adopted to represent the joint probability density function of power output for multiple wind farms. Then, analytical formulae are derived that can be used for fast computation of partial derivatives of the objective function and transformation of chance constraints into linear ones. Thereafter, the CCED can be solved as a deterministic linear convex optimization with a global optimal solution. The effectiveness and efficiency of the proposed methodology are validated via a case study with a modified IEEE 39-bus system.

Soil Probabilistic Slope Stability Analysis Using Stochastic Finite Difference Method

The paper contrasts results obtained by the partially factored limit state design method and a more advanced Random Finite Difference Method (RFDM) in a benchmark problem of slope stability analysis with variable undrained shear strength. Local Average Subdivision method was used to simulate the non-Gaussian random variables. The key difference between the methods is that RFDM takes into account spatial variability of soil parameters allowing slope failure to occur naturally along the path of least resistance. The probabilistic method leads to predictions of the "probability of slope failure" as opposed to the more traditional "factor of safety" measure of slope safety in the limit state design method; however, they give significant different results depending on the level of the variability. Analyses conducted using Monte Carlo Simulation show that the same partial factor can have very different levels of risk depending on the degree of uncertainty of the mean value of the soil shear strength. Calibration studies show the partial factor necessary to achieve target probability values.

The Analysis of Stochastic Properties of the SVD Decomposition at Approximation of the Experimental Data

The stochastic method is offered for the analysis of the SVD decomposition of experimental data matrices for complex non-Gaussian random variables generating independent variables of regression models (RM). To analyse the stability of RM desired solutions, the distortion of data matrix elements is simulated. There are considered therein both common-type classical matrices and binary normal matrices suggested by the author. Within the stochastic analysis of the distribution of singular values σi for matrices an algorithm is applied to identify the probable density functions σi on limited-scope samples. The research results are described and recommendations are given for the correct application of singular decomposition.

The extended Davenport peak factor as an extreme-value estimation method for linear combinations of correlated non-Gaussian random variables

This article develops a method for obtaining the distribution function of the extreme values that result from the linear combination of a finite number of non-Gaussian random variables that come from a common agent within a period of time. The approach incorporates the variability of both the components and their interactions, identifies the conditions for the arrival of extreme values and bounds the uncertainty in the results. The outcome of the method is an “extended Davenport peak factor” (EDPF) that relates the descriptors of the resultant, the parameters of the distribution function of the peaks, and the probability of not exceeding a given threshold.The method was applied to the analysis of the extreme wind actions on a parabolic trough. The model parameters were expressed in terms of 3 magnitudes which characterize the spatial (correlation), temporal (frequency of peaks) and intrinsic (variance) variability of the actions. The results are consistent with methods that consider the variability of the components and the variability that is introduced by their combination. Nonetheless, the methods that do not incorporate this information result in peak factors that are an average of 13% lower. Also, the methods for Gaussian variables do not capture the variability between the case studies.

Surrogate-enhanced stochastic search algorithms to identify implicitly defined functions for reliability analysis

The paper proposes two adaptive stochastic search algorithms to locate and trace an implicitly defined function with samples that are used to construct a surrogate model for reliability analysis. Both methods begin by propagating a series of surrogate-informed stochastic processes toward the implicit performance function. Having located the function, the first method conducts a “global” tracing of the function while the second traces the function by propagating samples locally. A key feature of both proposed algorithms is that the surrogate model evolves continuously with the sample selection and is used to inform the selection of new samples such that it converges rapidly to an accurate representation of the limit surface. In the present implementation, an artificial neural network surrogate model is employed but the method can, in principle, be applied with any surrogate model form. Performance of the algorithms is illustrated through problems with highly nonlinear limit state functions and high dimensional non-Gaussian random variables.

Stochastic Creep Damage Estimation in Pipings with Spatial Non-gaussian Uncertainties Using Spectral Stochastic Finite Element Method

This study focuses on the development of stochastic finite element basedmethodology for damage assessment of piping components that carry high temperature fluids. The primary damagingmechanism is assumed to be due to thermal creep. The material properties are assumed to be spatially inhomogeneous and are modeled as 2-D non-Gaussian random fields. A spectral stochastic finite element based methodology is developed where the non-Gaussian random fields are determined by using optimal linear expansion scheme (OLE). The corresponding finite element models for the structure are shown to be functions of a vector of correlated non-Gaussian random variables. Monte Carlo simulations (MCS) are carried out to estimate the 2-D non-Gaussian stress fields, which inturn lead to estimates of the creep damage field at different time instants. The developments are illustrated through a numerical example involving an axisymmetric section of a circular pipe.