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.     

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.     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

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.     

Generation of non-Gaussian stochastic processes using nonlinear filters

Non-Gaussian stochastic processes are generated using nonlinear filters in terms of Itô differential equations. In generating the stochastic processes, two most important characteristics, the spectral density and the probability density, are taken into consideration. The drift coefficients in the Itô differential equations can be adjusted to match the spectral density, while the diffusion coefficients are chosen according to the probability density. The method is capable to generate a stochastic process with a spectral density of one peak or multiple peaks. The locations of the peaks and the band widths can be tuned by adjusting model parameters. For a low-pass process with the spectrum peak at zero frequency, the nonlinear filter can match any probability distribution, defined either in an infinite interval, a semi-infinite interval, or a finite interval. For a process with a spectrum peak at a non-zero frequency or with multiple peaks, the nonlinear filter model also offers a variety of profiles for probability distributions. The non-Gaussian stochastic processes generated by the nonlinear filters can be used for analysis, as well as Monte Carlo simulation.     

A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process

Some widely used methodologies for simulation of non-Gaussian processes rely on translation process theory which imposes certain compatibility conditions between the non-Gaussian power spectral density function (PSDF) and the non-Gaussian probability density function (PDF) of the process. In many practical applications, the non-Gaussian PSDF and PDF are assigned arbitrarily; therefore, in general they can be incompatible. Several techniques to approximate such incompatible non-Gaussian PSDF/PDF pairs with a compatible pair have been proposed that involve either some iterative scheme on simulated sample functions or some general optimization approach. Although some of these techniques produce satisfactory results, they can be time consuming because of their nature. In this paper, a new iterative methodology is developed that estimates a non-Gaussian PSDF that: (a) is compatible with the prescribed non-Gaussian PDF, and (b) closely approximates the prescribed incompatible non-Gaussian PSDF. The corresponding underlying Gaussian PSDF is also determined. The basic idea is to iteratively upgrade the underlying Gaussian PSDF using the directly computed (through translation process theory) non-Gaussian PSDF at each iteration, rather than through expensive ensemble averaging of PSDFs computed from generated non-Gaussian sample functions. The proposed iterative scheme possesses two major advantages: it is conceptually very simple and it converges extremely fast with minimal computational effort. Once the underlying Gaussian PSDF is determined, generation of non-Gaussian sample functions is straightforward without any need for iterations. Numerical examples are provided demonstrating the capabilities of the methodology.     

Spherical harmonics in quadratic forms for testing multivariate normality

We study two statistics for testing goodness of fit to the null hypothesis of multivariate normality, based on averages over the standardized sample of multivariate spherical harmonics, radial functions and their products. These statistics (of which one was studied in the two-dimensional case in Quiroz and Dudley, 1991) have, as limiting distributions, linear combinations of chi-squares. In arbitrary dimension, we obtain closed form expressions for the coefficients that describe the limiting distributions, which allow us to produce Monte Carlo approximate limiting quantiles. We also obtain Monte Carlo approximate finite sample size quantiles and evaluate the power of the statistics presented against several alternatives of interest. A power comparison with other relevant statistics is included. The statistics proposed are easy to compute (with Fortran code available from the authors) and their finite sample quantiles converge relatively rapidly, with increasing sample size, to their limiting values, a behaviour that could be explained by the large number of orthogonal functions used in the quadratic forms involved.     

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.     

Efficient stationary multivariate non-Gaussian simulation based on a Hermite PDF model

An efficient stationary multivariate non-Gaussian simulation method is developed using spectral representation and third order Hermite polynomial translation. An approximate closed form relationship is employed to identify the Hermite translation parameters based on target skewness and kurtosis. This preserves a high degree of accuracy over the entire admissible range of the Hermite translation, and eliminates the need for iterative solution of the translation parameters. The Hermite PDF model is suitable for a wide range of strongly non-Gaussian stochastic process. In addition, an explicit bidirectional relationship between the target non-Gaussian and Gaussian correlation is developed to eliminate the need for iteration or numerical integration to identify the underlying Gaussian correlation. Examples apply the simulation method to both theoretical targets and experimental wind pressure data.     

Translation models revisited

Translation models have been defined as memoryless mappings of Gaussian elements which match exactly/approximately target marginal distributions/correlations. We extend this class of translation models to include memoryless mappings of non-Gaussian elements. It is shown that quantities of interest inferred from equivalent translation models, i.e., models which share the same marginal distributions and have similar second moments, can differ significantly. It is suggested to construct families of equivalent translation models and select members of these families which are optimal for given quantities of interest.     

A method for the efficient construction and sampling of vector-valued translation random fields

A method is developed for the efficient construction and sampling of vector-valued translation random processes and fields. Given a target marginal CDF and target covariance function, the approach is to approximate the spectral densities of the Gaussian image by a linear sum of shape functions, where each is scaled by a constant. An efficient optimization algorithm is developed to solve for the unknown constants. The objective function to be minimized is equal to the mean-square difference between the target covariance function, and the translated version of the approximate covariance function; a complex set of constraint equations is enforced during the optimization routine to ensure that the resulting covariance function of the Gaussian image is positive definite. It is shown that classical Monte Carlo simulation techniques can be used to generate samples of the Gaussian images of these models and map them into desired non-Gaussian samples. Several examples are considered to illustrate the application of the proposed method and to assess its accuracy.     

The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus

This paper deals with the study of the Green’s functions of a layered soil with random characteristics. The dynamic shear modulus of the soil is modelled as a non-Gaussian random process that varies in the vertical direction and is characterized by a marginal probability density function and a correlation function. The stochastic finite element method is applied to a hybrid thin layer — direct stiffness formulation in order to obtain the stochastic system equations, which are solved by means of a Monte Carlo simulation. The influence of the variations of the dynamic shear modulus on the Green’s functions is illustrated for different excitation frequencies and receiver positions.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

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.     

Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion

The non-Gaussian Karhunen–Loeve (K–L) expansion is very attractive because it can be extended readily to non-stationary and multi-dimensional fields in a unified way. However, for strongly non-Gaussian processes, the original procedure is unable to match the distribution tails well. This paper proposes an effective solution to this tail mismatch problem using a modified orthogonalization technique that reduces the degree of shuffling within columns containing empirical realizations of the K–L random variables. Numerical examples demonstrate that the present algorithm is capable of matching highly non-Gaussian marginal distributions and stationary/non-stationary covariance functions simultaneously to a very accurate degree. The ability to converge correctly to an abrupt lower bound in the target marginal distributions studied is noteworthy.     

A Ground State Monte Carlo Approach for Studies of Dipolar Systems with Rotational Degrees of Freedom

We have developed a path integral ground state Monte Carlo (PIGSMC) algorithm for quantum simulations of rotating dipolar molecules, using a highly accurate sixth-order algorithm. The method allows us to calculate unbiased estimates of ground state properties of dipolar molecules in a variety of geometries, with or without an external electric field. To demonstrate the capability of the approach, we calculate the orientational phase diagram of a one dimensional lattice system of rotating point dipoles in the absence of any external electric fields. We find that for finite lattice size, this system exhibits an order?Cdisorder transition at finite dipolar interaction strength in contrast to the well-known orientational disorder of the corresponding one dimensional O(3) quantum rotor models. Comparison of the quantum Monte Carlo results with a self-consistent field estimate of the phase transition shows the emergence of an ordered phase at non-zero dipolar strength, confirming the symmetry breaking role of the anisotropic dipole?Cdipole interaction.     

Simulation of non-Gaussian field applied to wind pressure fluctuations

A simulation algorithm to generate non-Gaussian wind pressure fields is proposed. This algorithm uses the correlation–distortion method based on translation vector processes. Conditions on the matrix of cross-covariance functions are given to assure the applicability of the model. The proposed method does not require iterative procedures and it is well suited when experimental data are available. In particular it requires cross-covariance functions and marginal distribution that can be directly estimated from data. To illustrate the procedure, the model is calibrated on experimental results obtained from wind tunnel tests on a tall building. The efficiency of the proposed methodology for reproducing the non-Gaussian nature of pressure fluctuations on separated flow regions is demonstrated.     

联合分布函数蒙特卡罗模拟及结构可靠度分析

针对复杂极限状态方程可靠度计算问题,提出了基于理论联合分布函数以及2 种近似联合分布函数的结构失效概率蒙特卡罗模拟方法,并给出了计算流程图.采用2 个算例证明了所提方法的有效性.结果表明:所提的失效概率模拟方法的计算精度很高,尤其适用于复杂极限状态方程的可靠度计算问题.2 种联合分布函数近似构造方法得到的失效概率精度相当,近似方法与精确方法结果的差异随失效概率的减小而增大,而且随着变量间相关性的增加而增加.当失效概率小于10-3时,近似方法的失效概率误差较大.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Assessing multivariate process/product yield via discrete point approximation

The assessment of multivariate yield is central to the robust design of products/processes. Currently, yield is evaluated via Monte Carlo simulation. However, it requires thousands of replications per simulation to achieve an acceptably precise estimate of yield, this is often tedious and time consuming, thereby rendering it unattractive as an evaluation tool. We propose a discrete point approximation on each design variable, using general Beta distributions, for assessing reasonably precise multivariate yield estimates, which require only a minute fraction of the Monte Carlo replications/simulations required to estimate yield (e.g., 3 and 5 design variables would require only 3³ = 27 and 3⁵ = 243 replications, respectively). The Beta distribution has the desirable property of being able to characterize a wide variety of processes that may or may not be symmetric and which may or may not have a finite operating range. Using an approach that computes the roots of a polynomial, whose degree is determined by the number of discrete points, discrete three point approximations are obtained and tabulated for twenty-five different Beta distributions. Based on several test examples, where design parameters are modeled as independent Beta random variates, our approach appears to be highly accurate, achieving virtually the same multivariate yield estimate as that obtained via Monte Carlo simulation. The substantial reduction in the number of replications and associated computational time required to assess yield makes the iterative adjustment of design parameters a more practical design strategy.