模拟非高斯过程的Karhunen-Loeve 扩展式中正交变量 的求解

Evaluating Orthogonal Random Variablesin Non-Gaussian Karhunen-LoeveExpansionShuping Huang*Department of Civil engineeringshanghai Jiaotong UniversityShanghai, P. R. China 200240sphuang@AbstractA simulation algorithm is developed for generating non-Gaussian processes with a specified marginal distribution function and covariance function, especially for highly skewed non-Gaussian process. It is based on the Karhunen-Loeve (K-L) expansion in terms of deterministic orthogonal functions and uncorrelated K-L random variables. The deterministic functions are obtained from the spectral decomposition of the covariance function. For non-Gaussian processes, the distributions of the K-L variables are however unknown. The strategy for determining these unknown K-L distributions is given below. The validity of the proposed algorithm for simulating highly skewed non-Gaussian process is illustrated using numerical examples.Keywords: Monte Carlo simulation; Karhunen-Loeve expansion ；non-Gaussian processes; Johnson distributions; spectral decomposition1IntroductionOne of the most important aspects of Monte Carlo simulation technique as applied to systems with random properties or systems subjected to random excitations is the generation of sample functions representative of these random fluctuations. In many cases, these uncertain properties and excitations are best modeled using non-Gaussian processes. Examples in practical engineering problems that exhibit non-Gaussian characteristics include material and geometric properties of structural and mechanical systems, soil properties in geotechnical engineering applications, wind wave and earthquake excitations, and the response of nonlinear systems to Gaussian excitation. In particular, material and geometric properties are physical quantities that can only assume positive values (e.g. cross-section dimensions, elastic modulus, density, cone tip resistance). A Gaussian assumption leads to a non-zero probability of obtaining negative values for such quantities. The non-zero probability is usually very small for small coefficients of variation of these quantities, but it can be significant forlarger values of their coefficients of variation. If these quantities become negative, even over a small part of the domain of the system, serious numerical problems can arise during the process of analyzing the system and erroneous result can be obtained. It is therefore obvious that the development of algorithm to generate sample functions of non-Gaussian processes is of paramount importance in Monte Carlo simulation techniques.Simulation methods for Gaussian processes are quite well established. However, research on the simulation of non-Gaussian processes is comparatively recent and limited. There are a few simulation techniques [1]-[2]available for non-Gaussian processes. They make use of a spectral representation for the generation of Gaussian sample fields and a memoryless nonlinear transformation of these fields to obtain the non-Gaussian fields. Sakamoto and Ghanem[3] simulated the non-Gaussian process by specifying the non-Gaussian process as a Hermite polynomial of a suitable underlying Gaussian process. The cumulative distribution function map-based spectral correction algorithm was proposed by [4].In this paper, the Karhunen-Loeve (K-L) expansion-based simulation method is of interest due to its optimality in dimension reduction. A simulation algorithm based on K-L expansion has been developed earlier [5] for generating a non-Gaussian random process with a specified marginal distribution function and covariance function. Simulation of a non-Gaussian random process using the K-L expansion basically requires: (a) solving the homogeneous Fredholm integral equation of the second kind to obtain the eigenvalues and eigenfunctions of the covariance function (spectral decomposition) and (b) selecting uncorrelated standardized K-L random variables such that the expansion produces the desired non-Gaussian marginal distribution. The Fredholm integral equation can be solved analytically only under special circumstances. In most cases, numerical methods such as Galerkin, collocation or Rayleigh-Ritz are required. The probability distributions of the K-L random variables can be evaluated through an integral that depends on the desired random process. However, the integrand is obviously an unknown. Hence, an iterative procedure was proposed to match the target marginal distribution function. Nevertheless, the iterative procedure is complicated and works well only for weakly non-Gaussian process. It is for sure that K-L expansion is in terms of non-Gaussian variables for non-Gaussian process, but how the distributions of the non-Gaussian K-L variables related to the distribution of the non-Gaussian process that the expansion represents, remains unclear.A strategy for determining these unknown K-L distributions is proposed and investigated with a numerical example. The availability of a non-Gaussian K-L expansion has application beyond simulation. Another example is stochastic FEM.2 Simulation method2.1 Karhunen-Loeve expansionConsider a random process ),(θϖx defined on a probability space (Ω, A , P ) and indexed on abounded domain D . Assume that the process has a mean )x ϖ and a finite variance 2)](),([x x E ϖθϖ− that is bounded for all D x ∈. The process can be expressed using the Karhunen-Loeve (K-L) expansion as:∑∞=+=1)()()(),(i i i i x f x x θξλϖθϖ(1)where i λ and are the eigenvalues and eigenfunctions of the covariance function . It has the following spectral decomposition)(x f i ),(21x x C∑∞==12121)()(),(i i i i x f x f x x C λ(2)and its eigenvalues and eigenfunctions are solutions of the homogenous Fredholm integral equation of the second kind given by(3) ∫=D i i i x f dx x f x x C )()(),(21121λEquation (3) arises from the fact that the eigenfunctions form a complete orthogonal set satisfying the equation(4) ij D j i dx x f x f δ=∫)()(where ij δ is the Kronecker-delta function. The Fredholm integral equation can be solved analytically only under special circumstances. In most cases, numerical methods such as the Galerkin, collocation or Rayleigh-Ritz method are required. The K-L random variables )(i θξ in Eq. (1) is a set of uncorrelated random variables with zero mean and unit variance, i.e. satisfying0)]([=θξi E (5a) ij j i E δθξθξ=)]()([ (5b) The distribution of the K-L random variables are standard Gaussian for Gaussian processes. However, the distributions are unknown for processes with non-Gaussian marginals. It may be noted here that Eq. (5) only requires K-L random variables to be uncorrelated. Although uncorrelated Gaussian variables are independent, this is not the case for uncorrelated non-Gaussian variables. In fact, it is typically not possible to define non-Gaussian random vectors uniquely using the the mean vector and covariance matrix alone. In other words, it is theoretically possible to generate different non-Gaussian processes from the K-L expansion that satisfy the same target covariance and marginal distribution function.2.2 Ev a lua t io n of th e o r th og on a l K -L r a nd o m v a r ia b le sIt is useful to have a non-Gaussian K-L expansion for simulation of non-stationary and multi-dimensional fields in a fairly unified way and for dimension reduction in the stochastic finite element method [6]. [5] proposed an iterative updating technique to determine the unknown cumulative distribution functions of the non-Gaussian K-L random variables. Three aspects of this technique could be improved. First, the K-L distributions were defined empirically. It would be convenient to have analytical expressions for these distributions. Second, the K-L distributions were determined iteratively. It is potentially more efficient to develop a direct solution approach. Third, the present technique does not match the prescribed marginal well at the tail for highly skewed marginals (e.g., exponential distribution).There are two facts that motivate the strategy for determining these unknown K-L distributions given below: firstly, previous K-L expansion based simulation techniques only utilized second-order statistics which is inadequate for non-Gaussian process. Independence of K-L variable need a full use for non-Gaussian process simulation. Secondly, most of the information about the PDF is contained in the first few moments, such that discover the information from the first several moments for PDF would be useful. This can be extended to multidimensional case where joint moments could be helpful for joint distribution.In this study, the original concept of matching the target covariance function by ensuring that the K-L random variables are uncorrelated is retained. The key modification is to solve for the unknown K-L distributions directly by assuming that they are independent and they follow one of the Johnson distributions [7]. The proposed procedure could be outlined as follows:1. Solve Fredholm integral equation [Eq. (3)] for i λ and using a numerical technique such as the wavelet-Galerkin method (Phoon et al. 2002b). For computation, the expansion given by Eq. (1) has to be approximated using M terms. The length of the summation M is selected such that the theoretical covariance in Eq. (2) is in reasonable agreement with the target covariance. The K-L random variables )(x f i )(θξi in the expansion must be uncorrelated as shown in Eq. (5). However, this is not sufficient to define the non-Gaussian random vector containing )(θξi uniquely. One possible solution is to assume that the K-L random variables )(θξi are independent.2. Discretize x into L nodes so that the random process becomes a random variable at each node. These random variables ),(θϖj x should follow the prescribed marginal distribution, say F . Equation(1) reduces to ∑=+=Mi j i i i j j x f x x 1)()()(),(θξλϖθϖ j = 1, …, L (6)3. Denote )(),(j j x x ϖθϖ−as random variable Y, which is a linear combination of independent random variables )(θξi as following(7) )(1θξi Mi ij a Y ∑==where the coefficients )()(j i i i ij x f a θξλ=.4. Assume that the unknown K-L distributions follow one of the Johnson distributions. The family of Johnson distributions is perhaps the most versatile choice for describing statistical distributions. It is based on translating any continuous distribution into a normal distribution defined in the following four forms, where z is the standard normal variable, x is the target random variable. The Johnson’s distribution families S b (for bounded), S u (for unbounded) and S L (for Lognomal type) are given below: a. Unbounded (Su): the set of distributions that go to infinity in both the upper or lower tail. )(sinh λεηγ−+=x z (8)b. Bounded (Sb): the set of distributions that have a fixed boundary on either the upper or lower tail, or both. )(In xx z −+−+=ελεηγ (9) c. Log Normal (S l ): a border between the Unbounded and Bounded distribution forms. (In λεηγ−+=x z (10)The flexibility provided by the choice of form and fitting parameters allows for great flexibility is very flexible in terms of distribution shape in representing random variables.These distributions can model any distribution function. The unknown K-L distributions can be expressed using one of the analytical Johnson probability density functions with unknown parameters.5. Taking Laplace transform on both sides of Eq. (7), the right hand side becomes product of the Laplace transform of each independent random variable (i ij a ξ):(8) ∏==Mi a Y s L s L i ij 1)()(ξwhere)(1)(ijij a a s L a s L i i ij ξξ= (9) Taking logarithm of Eq. (9) gives (10) )())((1∑==Mi a Y i ij L Log s L Log ξ6. Evaluate successive derivatives of each side of Eq. (10) with respect to s at s = 0, a set of nonlinear Equations can be obtained in which the unknowns to be solved are the parameters in a Johnson distribution. Additional equations are needed to ensure that the K-L random variables have zero-mean and unit-variance.7. Generate N sample functions of the non-Gaussian process using the truncated K-L expansion as Eq.(1) with K-L random variables generated according to the non-Gaussian Johnson distributions.3 Numerical exampleNumerical simulation of a non-Gaussian stationary stochastic process is performed to investigate the performance of the method. The following Gamma distribution with three parameters is selected to illustrate the proposed simulation algorithm: 0 , ; x ; exp )(1),,,(1>≥⎟⎠⎞⎜⎝⎛−−⎟⎠⎞⎜⎝⎛−Γ=−λσμσμσμλσλσμλx x x f (13) where location parameter μ is 2−, scale parameter σ = 2/1and shape parameter 2=λsuch that the process is zero-mean and unit-variance.The target covariance function is:(13) 221||21),(x x e x x C −−=The K-L variables are solved using the procedure described above. They are Gamma variables with location parameter vector μ=[-1.0345 -1.0778 -1.2329 -1.4400 -1.7308]; scale parameter σ =[0.966667 0.927778 0.811111 0.694444 0.577778] and shape parameter vector λ = [1.07021.1617 1.52002.0736 2.9956]. 5000 Samples are generated using these K-L variables by Eq. (1). Figure 1 shows the target Gamma marginal distribution and Standard Gaussian distribution. Figure 2 compares the target marginal distribution function with the simulated distribution function computedfrom sample functions. It can be seen that the two curves in Figure 2 agree very well. Note that the proposed works well for this example as the target distribution deviate significantly from the Gaussian case as shown in Figure 1. The target covariance function, the theoretical covariance and the simulated covariance function computed from sample functions are compared in Figure 3. The comparison is good as to be expected. In fact, the comparison will be good if Eq. (5) is satisfied. The merit of the proposed method is the full use of independence of the K-L variables and no iterative procedure is involved. Johnson distributions provide parametric models, which make evaluation of probability distributions of unknown variables reduce to parameter identification and thus feasible. Nevertheless, parametric models are not as flexible as non-parametric models in fitting arbitrary distributions and leave less space for numerical approximation.4ConclusionsKarhunen-Loeve expansion has been developed for generating non-Gaussian processes with a specified marginal distribution and covariance function. The unknown K-L variables are evaluated by general parametric representation and Laplace transform of product of independent variables. A stationary random process with Gamma marginal distribution was used to illustrate the proposed simulation algorithm.(a) （b）Figure 1: Comparing distributions:（a）Target Gamma marginal distribution and standard Gaussian distribution（b）Target Gamma distribution function and the simulated distribution functionFigure 2: Comparing covariance functionsReferences[1]Grigoriu, M. (1998), “Simulation of stationary non-Gaussian translation processes”, ASCEJournal of Engineering Mechanics,124(2): 121-126.[2]Deodatis G. and Micaletti R.C. (2001), “Simulation of Highly-Skewed Non-Gaussian StochasticProcesses”, ASCE Journal of Engineering Mechanics, 127(12), 1284-1295.[3]Gavriliadis, P. N. and Athanassoulis, G. A. (2003), “Moment data can be analytically completed”,Probabilistic Engineering Mechanics, 17,329-338.[4]Sakamoto, S. and Ghanem, R. (2002), “Simulation of multi-dimensional non-gaussiannon-stationary random fields”, Probabilistic Engineering Mechanics, 17,167-176.[5]Masters, F. and Gurley, K.R.(2003), “Non-Gaussian Simulation: Cumulative Distribution FunctionMap-Based Spectral Correction”, ASCE Journal of Engineering Mechanics, 129(12): 1418-1428.[6]Phoon K.K, Huang S.P. and Quek S.T. (2002a), “Simulation of non-Gaussian processes usingKarhunen-Loeve Expansion”, Computers & Structures, 80,1049-1060.[7]Schüeller G.I., Pradlwarter H.J. and Schenk C.A. (2003), “Non-stationary response of large linearFE models under stochastic loading”, Computers & Structures, 81 (8-11), 937-947.[8]Johnson, M. E. (1987), “Multivariate Statistical Simulation”, New York: John Wiley.。