Multivariate probability distribution of ordered peaks of vector Gaussian random processes

The problem of determining the joint probability distribution of ordered peaks of jointly stationary Gaussian random processes is considered. The solution is obtained by modeling the number of times a specified threshold is crossed by the component processes as a multivariate Poisson process. Based on this, the joint probability distribution of the time required for the nth crossing of a specified level with positive slope is derived. This formulation is further extended to derive the joint distribution of ordered peaks in a given time interval. An illustrative example on a bivariate Gaussian random process is presented and the analytical predictions are shown to compare reasonably well with corresponding results from Monte Carlo simulations. Also presented is an analysis of response of a randomly driven multi-degree of freedom system with emphasis on the sensitivity of ordered peak characteristics with respect to changes in system parameters. It is demonstrated that higher order statistics are generally more sensitive to changes in system characteristics—a property that has potential for application in structural model updating and damage detection.     

Non-Gaussian random processes in ocean engineering

This paper presents a state-of-the-art review on stochastic analysis and probabilistic prediction of non-Gaussian random processes in ocean engineering. The derivation of probability density functions which constitute the basis for stochastic analysis of non-Gaussian processes is discussed in detail, and then the probability distributions of peaks and troughs of non-Gaussian random process is discussed to provide information necessary for engineering design. As an example of application of these probability distribution functions, the procedure for predicting responses of an offshore structure which has substantial non-linear characteristics in random seas is presented.     

Dynamic reliability analysis for non-stationary non-Gaussian response based on the bivariate vector translation process

Calculation of probability of exceedance for nonstationary non-Gaussian responses remains a great challenge to researchers in the field of structural reliability. In this paper, an analytical solution is proposed for calculating the mean upcrossing rate (MCR) of the non-stationary non-Gaussian responses by approximating the displacement and velocity responses with the bivariate vector translation process, in which the unified Hermite polynomial model (UHPM) is selected as the mapping function. The first four moments (i.e., mean value, standard deviation, skewness, and kurtosis) and cross-correlation function of the displacement and velocity responses needed in UHPM are estimated from some representative samples generated by random function-spectral representation method (RFSRM) and time-domain analysis. Under the Poisson assumption of the upcrossing events, the calculation of extreme value distribution or probability of exceedance for structural response can be determined with the proposed method. The proposed method is applicable to a wide range of structural responses, including asymmetric and hardening or softening responses. Three numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method. It can be concluded that the proposed method provides an accurate and useful tool for dynamic reliability assessment in engineering applications.     

A translation model for non-stationary, non-Gaussian random processes

A model for simulation of non-stationary, non-Gaussian processes based on non-linear translation of Gaussian random vectors is presented. This method is a generalization of traditional translation processes that includes the capability of simulating samples with spatially or temporally varying marginal probability density functions. A formal development of the properties of the resulting process includes joint probability density function, correlation distortion and lower and upper bounds that depend on the target marginal distributions. Examples indicate the possibility of exactly matching a wide range of marginal pdfs and second order moments through a simple interpolating algorithm. Furthermore, the application of the method in simulating statistically inhomogeneous random media is investigated, using the specific case of binary translation with stationary and non-stationary target correlations.     

Equations for probability density of response of dynamic systems to a class of non-Poisson random impulse process excitations

The excitation considered in the present paper is a random train of impulses driven by two classes of non-Poisson counting processes. The impulse processes are obtained by selecting impulses from a Poisson and from an Erlang-driven trains of impulses with the aid of an additional, purely jump stochastic process, assumed as an auxiliary state variable. The variable introduced for the first class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Poisson processes, with different parameters, and is tantamount to a two-state Markov chain. The variable introduced for the second class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Erlang processes, with different parameters. As each Erlang process is tantamount to a number of Markov states, the Markov chain for the whole problem is constructed. The equations governing the joint probability density-distribution function of the state vector of the dynamic system and of the Markov states are derived from the general integro-differential forward Chapman–Kolmogorov equation. The necessary jump probability intensity functions are evaluated for both classes of impulse processes and for purely external as well as parametric excitations. Parametric excitation multiplicative to the displacement and to the velocity state variable is considered. The resulting set of coupled integro-partial differential equations is obtained.     

一种非高斯随机振动过程数值模拟方法

目的 考虑真实随机振动的非高斯特性,提出一种根据已知信息生成与其相符的非高斯随机振动过程的数值模拟方法。方法 基于均值、方差、偏斜度、峭度及功率谱密度函数（或自相关函数）等约束条件,对非高斯随机振动进行模拟。根据功率谱获取非高斯过程的自相关矩阵;通过Hermite多项式的正交性质和多项式混沌展开方法推导出的公式,构造满足标准正态分布随机过程的协方差矩阵,并对其进行谱分解和主成分分析;最后,利用Karhunen-Loeve展开和多项式混沌展开来表示所模拟的非高斯振动过程。结果 随着采样点个数的增加,实测数据与模拟数据之间的误差越来越小,该方法具有较好的模拟精度。结论 应用多项式混沌展开、Karhunen-Loeve展开以及蒙特卡洛等方法,可生成非高斯随机振动过程,并得到准确有效的各项统计参数模拟值。     

Reliability of dynamic systems in random environment by extreme value theory

A practical method is developed for estimating the performance of highly reliable dynamic systems in random environment. The method uses concepts of univariate extreme value theory and a relatively small set of simulated samples of system states. Generalized extreme value distributions are fitted to state observations and used to extrapolate Monte Carlo estimates of reliability and failure probability beyond data. There is no need to postulate functional forms of extreme value distributions since they are selected by the estimation procedure. Our approach can be viewed as an alternative implementation of the method in [7] for estimating system reliability. Numerical examples involving Gaussian and non-Gaussian system states are used to illustrate the implementation of the proposed method and assess its accuracy.     

Probability distributions of peaks and troughs of non-Gaussian random processes

This paper deals with the development of probability density functions applicable for peaks, troughs and peak-to-trough excursions of a non-Gaussian random process where the response of a non-linear system is represented in the form of Volterra's second-order functional series. The density functions of peaks and troughs are derived in closed form and presented separately. It is found that the probability density function applicable to peaks (and troughs) is equivalent to the density function of the envelope of a random process consisting of the sum of a narrow-band Gaussian process and sine wave having the same frequency. Furthermore, for a non-Gaussian random process for which the skewness of the distribution is less than 1.2, the density function of peaks (and troughs) can be approximately presented in the form of a Rayleigh distribution. The parameter of the Rayleigh distribution is given as a function of parameters representing the non-Gaussian characteristics. The results of comparisons between newly derived density functions and histograms of peaks, troughs and peak-to-trough excursions constructed from data with strong non-linear characteristics show that the distributions well represent the histograms for all cases.     

Response of linear dynamic systems to non-Erlang renewal impulses: Stochastic equations approach

Linear dynamical systems under random trains of impulses driven by a class of non-Erlang renewal processes are considered. The class considered is the one where the renewal events are selected from an Erlang renewal process. The original train of impulses is recast, with the aid of an auxiliary stochastic variable, in terms of two independent Poisson processes. Thus, by augmenting the state vector of the dynamic system with the auxiliary stochastic variables, the original non-Markov problem is converted to a Markov one.

The differential equations for the response statistical moments can then be derived from the generalised Ito's differential rule.

Numerical results obtained for a few different models and various sets of parameters, show that the present approach allows to account for a variety of inter arrival time's probability distributions. Transient mean value and variance of the response of a linear oscillator have been obtained from the equations for moments.     

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the Itô stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single-time) statistics or probability distributions.     

卡特里娜飓风的启示——有关海洋和水利工程的风险分析

2005年卡特里娜（Katrina）和丽塔（Rita）飓风对美国新奥尔良市和佛罗里达东部海岸带来的灾难性破坏,验证了笔者在20世纪80年代初期提出的复合极值分布理论及其对上述海域飓风强度预测结果的正确性。以此为鉴,讨论了海岸、近海、水利和城市防灾工程中引入不确定性分析和多维联合概率理论进行风险分析的必要性。     

Probability distribution applicable to non-Gaussian random processes

This paper presents a probability density function representing a non-Gaussian random process in closed form. The probability density is based on the Kac-Siegert solution of Volterra's stochastic series expansion of a nonlinear system. A method is developed, however, to obtain the Kac-Siegert solution from knowledge of the time history only of the random process, and the result is expressed as a function of a normal distribution. Then, by applying the change of random variable technique, the asymptotic probability density function applicable to the response of a nonlinear system (which is a non-Gaussian random process) is developed in closed form. A comparison of the presently developed probability density function and the histogram constructed from a record indicating strong non-Gaussian characteristics shows excellent agreement.     

Joint distribution of peaks and valleys in a stochastic process

General expressions and numerical results are presented pertaining to the occurrence of two local extrema of a stochastic process at prescribed time values. The extrema may be either peaks or valleys and the process may be either stationary or nonstationary. General formulas are presented for the rates of occurrence, the joint and conditional probability distributions, and the moments of the extreme values. These formulas are relatively simple multiple-integral expressions, but the integrands involve the joint probability density function for six random variables. The procedures are then applied for the special case of a stationary mean-zero Gaussian process for which the calculations are greatly simplified. Numerical results for three different spectral density functions demonstrate that conditioning on either only the existence or both the existence and the value of one peak can have a very significant effect on both the rate of occurrence and the probability distribution of a second peak.     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.     

Space–time extreme value statistics of non-Gaussian random fields

This paper focuses on two new methods for predicting the extreme values of a non-Gaussian random field in both space and time. Both methods rely on the use of scalar time series expressing spatial extremes. These time series are constructed by sampling the available realizations of the random field over a suitable grid defining the domains in question and extracting the extreme values for each time point. In this way, time series of spatial extremes are produced. The realizations of the random field are obtained from either measurements or Monte Carlo simulations. The obtained time series provide the basis for estimating the extreme value distribution using recently developed techniques for time series, which results in an accurate practical procedure. The proposed prediction methods are applied to two specific cases. One is a second-order random ocean wave field, whose statistics deviate only mildly from the Gaussian, and the other is an example of a random field whose statistics is strongly non-Gaussian.     

Translation vectors with non-identically distributed components

A model for non-Gaussian random vectors is presented that relies on a modification of the standard translation transformation which has previously been used to model stationary non-Gaussian processes and non-Gaussian random vectors with identically distributed components. The translation model has the ability to exactly match target marginal distributions and a broad variety of correlation matrices. Joint distributions of the new class of translation vectors are derived, as are upper and lower bounds on the target correlation that depend on the target marginal distributions. Examples are presented that demonstrate the applicability of the approach to the modelling of heterogeneous material properties, and also illustrate the possible shortcomings of using second moment characterizations for such random vectors. Lastly, an outline is given of a method under development for extending the model to non-stationary, non-Gaussian random processes.     

Stochastic finite element analysis of layered composite beams with spatially varying non-Gaussian inhomogeneities

This study focuses on the development of a stochastic finite element-based methodology for failure assessment of composite beams with spatially varying non-Gaussian distributed inhomogeneities. The material properties in the individual laminae are modeled as non-Gaussian random fields, whose probability density functions and the correlations are estimated from the test data. The non-Gaussian random fields are discretized into a vector of correlated non-Gaussian random variables using the optimal linear expansion scheme that preserves the second-order non-Gaussian characteristics of the fields. Subsequently, the estimates of the failure probability are obtained from Monte Carlo simulations carried out on the vector of correlated random variables. Issues related to the computational efficiency of the proposed framework and the variabilities in the material properties are discussed. Numerical examples are presented, which highlight the salient features of the proposed method.     

A Monte Carlo simulation model for stationary non-Gaussian processes

A class of stationary non-Gaussian processes, referred to as the class of mixtures of translation processes, is defined by their finite dimensional distributions consisting of mixtures of finite dimensional distributions of translation processes. The class of mixtures of translation processes includes translation processes and is useful for both Monte Carlo simulation and analytical studies. As for translation processes, the mixture of translation processes can have a wide range of marginal distributions and correlation functions. Moreover, these processes can match a broader range of second order correlation functions than translation processes. The paper also develops an algorithm for generating samples of any non-Gaussian process in the class of mixtures of translation processes. The algorithm is based on the sampling representation theorem for stochastic processes and properties of the conditional distributions. Examples are presented to illustrate the proposed Monte Carlo algorithm and compare features of translation processes and mixture of translation processes.     

Numerical Evaluation of Compound Poisson Distributions

A useful interpretation is applied to probability generating functions for discrete random variables in order to provide an algorithm for calculating probability mass functions for compound Poisson distributions. This interpretation involves a theorem for geometric transforms which is developed to provide a method for calculating the inverse transform of exp (az^k). The result of this theorem is then applied to formulate an algorithm for numerically inverting the geometric transform of a compound Poisson process. The evaluation of these probabilities may be carried out to a prespecified point with an accuracy dependent only on the accuracy of the input distributions. The procedure is illustrated by the numerical calculation of the distribution of daily demand probabilities required in the analysis of an inventory system.