simulation of multi-dimensional, multi- variate ...

Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

SIMULATION OF MULTI-DIMENSIONAL, MULTIVARIATE, STATIONARY AND NON-STATIONARY, STRONGLY NON-GAUSSIAN STOCHASTIC PROCESSES: THE ITERATIVE TRANSLATION APPROXIMATION METHOD (ITAM) MICHAEL D. SHIELDS1 and GEORGE DEODATIS2 1

Dept. of Civil Engineering, Johns Hopkins University, Baltimore, MD, United States. E-mail: [email protected] 2 Dept. of Civil Eng. & Eng. Mechanics, Columbia University, New York, NY, United States. E-mail: [email protected] A new class of methodologies has been recently proposed for the simulation of strongly nonGaussian stochastic processes that utilizes the Spectral Representation Method along with Translation Process Theory. These new methodologies rely on a rapidly converging iterative algorithm to identify the power spectrum (stationary or evolutionary) associated with an underlying Gaussian stochastic process that is mapped to a prescribed non-Gaussian marginal probabilistic distribution. Where such an underlying Gaussian power spectrum does not exist – due to incompatibility conditions in the Translation Process Theory – the methodologies provide a robust, accurate, and cost-effective approximation. The class of methods is presented herein in a unified framework referred to as the Iterative Translation Approximation Method (ITAM). The ITAM framework enables the simulation of stochastic processes that are multi-dimensional, multi-variate, stationary and non-stationary, and strongly non-Gaussian. The advantages of the method are discussed along with its theoretical and practical limitations. Keywords: Non-Gaussian processes, multi-variate processes, non-stationary processes, Iterative Translation Approximation Method, Translation process theory, simulation.

1

Introductions

For solving problems in stochastic mechanics – particularly those with strong nonlinearities – Monte Carlo simulation is the most robust and method and, in some cases, is the only available method. Many such applications of Monte Carlo simulation require the generation of artificial realizations of stochastic processes and fields. The generation of sample functions of stationary and Gaussian processes and fields has been deeply investigated and there are well-established methods to do so such as the Spectral Representation Method (Shinozuka and Jan 1972, Shinozuka and

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Deodatis 1991) and the Karhunen-Loeve expansion (Huang et al 2001). However, realistic processes are often complex with non-Gaussian, non-stationary, multidimensional, or multi-variate properties. In recent years, there has been an increasing interest in the generation of such processes, and there has been gradual progress in this regard [see e.g. (Sakamoto and Ghanem 2002a and 2002b, Phoon et al. 2002, Bocchini and Deodatis 2008)]. The present paper reviews the Iterative Translation Approximation Method (ITAM), which is a newly developed method for

Simulation of multi-dimensional, multi-variate, stationary and non-stationary, strongly non-Gaussian stochastic processes: The Iterative Translation Approximation Method (ITAM) Michael D. Shields and George Deodatis

simulation of general second-order nonGaussian stochastic processes and fields. The ITAM, in the presented form, utilizes the Spectral Representation Method (SRM) along with translation process theory (Grigoriu, 1995) to iteratively identify an underlying Gaussian stochastic process that, when mapped to the desired non-Gaussian distribution, matches the prescribed nonGaussian correlation with high accuracy. The various algorithms, specific to different types of stochastic processes, that together comprise the ITAM are presented and numerical examples provided that demonstrate their accuracy and computational efficiency. Furthermore, a discussion of the future ITAM developments is provided. In following sections, Translation process theory and its compatibility conditions are introduced for various classes of processes followed by detailed algorithms for the ITAM. 2

∞

∞

−∞

−∞

∫ ∫

FN−1 {FG [x1 ]}

⋅FN−1 {FG [x2 ]}φ (x1, x2 ; ρ (τ ))dx1dx2 (2) where µ N and σ N are the mean and standard deviation of the non-Gaussian process, ζ (τ ) is the non-Gaussian correlation coefficient, Φ{⋅,⋅; ρ (τ )} is the joint Gaussian PDF defined by:

φ {x1, x2 ; ρ (τ )} =

" x 2 + x22 − 2 ρ (τ )x1 x2 % exp $ − 1 ' 2(1− ρ (τ )2 ) # & 2π (1− ρ (τ )2 (3) 1

and ρ (τ ) is coefficient.

the

Gaussian

correlation

2.2 Stationary vector translation process theory

Translation process theory

2.1 Stationary scalar translation process theory Translation process theory, introduced by Grigoriu (1995), is one of the most commonly used transformation models for presenting stationary and non-Gaussian processes. Let X(t) be a stationary Gaussian stochastic process with zero mean and standard deviation σ . The non-Gaussian translation processes with marginal cumulative distribution function (CDF) FN (⋅) is defined by the following nonlinear transformation:

Z(t) = FN−1 {FG [X(t)]}

RN (τ ) = µ N2 + σ N2 ζ (t) =

(1)

The autocorrelation function (ACF) RN (τ ) of the non-Gaussian process can be computed from the corresponding Gaussian ACF RG (τ ) by:

2

Gioffrè et al. (2000) extended translation vector process for one dimensional, multivariate, non-Gaussian stochastic process. Let X(t) = {X1 (t), X 2 (t),!, X m (t)}T be a zero mean, unit standard deviation, m-variate, stationary, and Gaussian stochastic process with cross-correlation matrix (CCM) R G (τ ) defined as:

RGjk (τ ) = E[X j (t)X k (t + τ )]

(4)

where j, k = 1, 2,!, m . The zero-mean, stationary, non-Gaussian vector translation process Z(t) = {Z1 (t), Z 2 (t),!, Z m (t)}T is defined by:

Z j (t) = g j [X j (t)] = FN−1 {FG [X j (t)]} (5)

Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

The components of the non-Gaussian CCM R N (τ ) can be determined from the underlying Gaussian CCM as follows:

RNjk (τ ) = E[Z j (t)Z k (t + τ )] =

∫ ∫

g j (x j1 )gk (yk 2 )

(6)

⋅ φ X j1Xk 2 {x j1, xk 2 ; ρ jk (τ )}dx j1dxk 2 where j, k = 1, 2,!, m , and the joint Gaussian PDF is defined by Eq. (3). 2.3 Non-stationary scalar translation process theory Lastly, Ferrante et al. (2005) generalized translation theory for non-stationary processes. Let Y (t) be uni-variate, nonstationary, and non-Gaussian stochastic process with non-Gaussian CDF FN (⋅, t) . The prescribed process can be computed as:

Y (t) = FN−1 {FG [X(t)], t}

(7)

where the underlying Gaussian process is, in general non-stationary, and the non-Gaussian CDF 𝑭𝑵 ⋅, 𝒕 is now time dependent. The ACF of the non-stationary and non-Gaussian Y (t) can be expressed as:

RN (s, t) = µ N (s)µ N (t) + σ N (s)σ N (t)ζ (s, t) =

∞

∞

−∞

−∞

∫ ∫

g(x1 , s)g(y2 , t)

(8)

⋅ φ X1X2 {x1, x2 ; ρ (s, t)}dx j1dxk 2 where µ N (t) , σ N (t) , and 𝜻(𝒔, 𝒕) are the mean, standard deviation and correlation coefficient of the non-Gaussian and nonstationary process Y (t) .

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2.4

Translation process incompatibility

In all cases described above, (uni-variate, multi-variate, stationary or non-stationary processes) it is always possible to map a known Gaussian ACF to a prescribed nonGaussian ACF. However, the inverse trasformation, where the non-Gaussian ACF is known and the underlying Gaussian ACF is unknown is not always well defined because of incompatibility between the arbitrarily prescribed non-Gaussian ACF and marginal non-Gaussian probability distribution. The conditions for this incompatibility are explored in various works (Grigoriu 1995, Gioffre et al. 2000, Ferrante et al. 2005) and, for brevity, are not discussed explicitly here. Nonetheless, it is often desirable to use the translation process model for simulation of non-Gaussian fields even when this incompatibility exists. The ITAM provides such a capability by iteratively identifying an underlying Gaussian process that is compatible with translation process theory and yields a non-Gaussian process possessing correlation that is very close to the prescribed yet incompatible one. 3

Iterative Translation Approximation Method (ITAM)

The ITAM was initially proposed by Shields et al. (2011) for simulation of one dimensional (1D), uni-variate (1V), stationary, and nonGaussian stochastic processes. The methodology has since been extended to multi-variate, stationary processes by Shields and Deodatis (2013a) to non-stationary processes as well (Shields and Deodatis 2013b). The ITAM possesses a series of advantages when compared to other existing methods: (1) it converges rapidly (generally within 10 iterations; (2) it approximates the computed non-Gaussian correlation very accurately even for strongly non-Gaussian

Simulation of multi-dimensional, multi-variate, stationary and non-stationary, strongly non-Gaussian stochastic processes: The Iterative Translation Approximation Method (ITAM) Michael D. Shields and George Deodatis

processes; (3) because it is built upon translation process theory, the non-Gaussian process possesses the marginal non-Gaussian distribution exactly and other properties such as crossing-rates are preserved; and (4) it identifies the underlying Gaussian process without the need for sample function generation as required with previous methods (Bocchini and Deodatis 2008). In the following sections, the specific ITAM algorithms are presented for different classes of second-order non-Gaussian stochastic processes.

correlation function and using the mapping in Eq. (2). (iii) Upgrade the underlying Gaussian PSDF by: β

(i+1) G

S

! S T (ω ) $ (i) N (ω ) = # (i) & SG (ω ) " SN (ω ) %

(9)

where β is an exponent to optimize the convergence rate. (iv) Iterate back to step 2. The method generally converges quite rapidly requiring only a small number of iterations using Eq. (9).

3.1 ITAM: 1D, 1V Stationary and NonGaussian processes For 1D, 1V, stationary stochastic processes, the ITAM identifies a non-Gaussian power spectral density function (PSDF) that: (a) is compatible with the prescribed marginal distribution, (b) approximates the target incompatible non-Gaussian PSDF very accurately. It does so by iteratively identifying an underlying Gaussian PSDF that, when mapped to the non-Gaussian distribution using translation process theory yields an accurate estimate of the incompatible non-Gaussian PSDF. This procedure yields also the underlying Gaussian PSDF, which is useful for simulation purposes. The methodology proceeds as follows: (i) Initialize the underlying Gaussian PSDF SG(0) (ω ) . This initialization can be any arbitrary well-defined PSDF although it is recommended to set the initial underlying Gaussian PSDF equal to the prescribed target non-Gaussian PSDF 𝑆!! (𝜔). (ii) Map the Gaussian PSDF SG(i) (ω ) to the PSDF possessing the SN(i) (ω ) prescribed marginal non-Gaussian distribution FX (⋅) using translation process theory. This is done by computing the corresponding Gaussian

4

3.2 ITAM: 1D, mV Stationary and NonGaussian processes For simulation of multi-variate (mV) processes, it is necessary to identify a nonGaussian cross-spectral density matrix (CSDM) that approaches the incompatible target non-Gaussian CSDM as closely as possible. The methodology differs from the 1V stationary ITAM because the identification of the full CSDM imposes additional constraints on the problem. Specifically, the Gaussian CSDM SG (ω ) must be both Hermitian and positive definite at every frequency. Using a Cholesky decomposition and upgrading the decomposed lower triangular matrix of the underlying Gaussian CSDM SG(i) (ω ) , the methodology is able to preserve these properties for the new CSDM SG(i+1) (ω ) at each iteration. The procedure for the m-V stationary ITAM is as follows: (i) Initialize the underlying Gaussian CSDM SG(0) (ω ) . Again, this can be any arbitrary valid CSDM but is generally set equal to the target non-Gaussian CSDM 𝑺!! (𝜔).

Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

(ii) Using SG(i) (ω ) and the prescribed marginal non-Gaussian CDF FX j (⋅), j = 1, 2,!, m , compute the corresponding marginal CSDM S(i) N (ω ) by translation vector process theory. (iii) Decompose the target non-Gaussian CSDM STN (ω ) , and the computed CSDMs SG(i) (ω ) , S(i) N (ω ) Cholesky’s method such that: 𝐒 = 𝐇𝐇 !

using

(10)

(iv) Upgrade the Cholesky decomposed underlying Gaussian CSDM as follows: β

H

(i+1) G

! HT (ω ) $ (i) N (ω ) = # (i) & HG (ω ) (11) " H N (ω ) %

extension of translation process by Ferrante et al. (2005). With this in mind, there are two main challenge problems in the simulation of non-stationary and non-Gaussian processes by spectral representation. The first case (referred to as “forward” problem) requires estimating the non-Gaussian ES from a known underlying Gaussian ES. This is particularly challenging because, in general, no unique ES can be identified from a given non-stationary ACF. The second case (called as “inverse” problem) is the practically more interesting case requires estimating the underlying Gaussian ES from an known non-Gaussian ES which is incompatible with the prescribe the marginal PDF according to translation process. The ITAM method is applied in this reverse case. 3.3.1

(v) Normalize the estimated Gaussian CSDM S(i+1) N (ω ) . (vi) Randomly shuffle the components of the vector process. This step is necessary to ensure adequate convergence of all components as described by Shields and Deodatis (2013a). (vii) Iterate back to step 2. 3.3 ITAM: 1D, 1V Non-Stationary and Non-Gaussian processes Lastly, Shields and Deodatis (2013b) extended the ITAM for non-stationary and non-Gaussian stochastic processes. The method, which relies upon the evolutionary spectrum (ES) defined for non-stationary processes by Priestley (1965) requires some additional approximations to address specific theoretical constraints posed by the theory of evolutionary power in addition to those posed by incompatibility of the non-stationary

5

Forward problem

In a slightly simplified form, Priestley’s theory of evolutionary power states that the non-stationary ACF 𝑅(𝑠, 𝑡) can be computed from the ES 𝑆(𝜔, 𝑡) as:

R(s, t) =

∫

∞ −∞

S(ω, s)S(ω, t)e Iω (t−s) dω (12)

where I is the imaginary unit. However, in general no unique inverse to this equation exists. This problem prohibits the direct translation from a Gaussian ES to a nonGaussian ES. Shields and Deodatis (2013b) proposed an approximate transformation by defining the “pseudo-autocorrelation” as:

R P (t, τ ) =

∫

∞ −∞

S(ω, t)e Iωτ dω

(13)

under the assumption that the ES can be approximated by series of independent stationary PSDFs at each time t. This implies that the non-Gaussian pseudo-ACF can be

Simulation of multi-dimensional, multi-variate, stationary and non-stationary, strongly non-Gaussian stochastic processes: The Iterative Translation Approximation Method (ITAM) Michael D. Shields and George Deodatis

computed by the stationary translation of the ES at each time instant. Furthermore, the nonGaussian ES at each time instant is estimated as the inverse Wiener-Khintchine transform of the translated the pseudo-ACF. The limitations of this approximation have been explored in some detail by Shields and Deodatis (2013b). The procedure to estimate the nonGaussian ES from the underlying Gaussian ES is summarized as follows: (i) From the known Gaussian ES SG (ω, t) , calculate the Gaussian pseudo-ACF RGP (t, τ ) using Eq. (13). (ii) Using stationary translation process theory, compute the non-Gaussian pseudo-ACF RNP (t, τ ) . (iii) Estimate the non-Gaussian ES SˆN (ω, t) using the inverse Wiener-Khintchine transform. 3.3.2

Inverse problem

The so-called “inverse” problem requires estimating the underlying Gaussian ES from a known non-Gaussian ES with the prescribed marginal CDF. For this case, the ITAM has been extended such that the approximation developed in the forward case is utilized at each iteration. The procedure is described in the following steps: (i) Initialize the underlying Gaussian ES SG(0) (ω, t) . (ii) Estimate the non-Gaussian ES SˆN(i) (ω, t) from the underlying Gaussian ES SG(i) (ω, t) following the forward problem procedure in Section 3.3.1. (iii) Upgrade the underlying Gaussian ES by:

6

β

(i+1) G

S

! S T (ω, t ) $ (i) N (ω, t) = # (i) & SG (ω, t) (14) " SN (ω, t) %

(iv) Iterate back to step 2. 4

Numerical Examples

Examples for one dimensional, unit-variate stationary and non-stationary ITAM are presented in the following sections to demonstrate the capabilities of the methodology. For both stationary and nonstationary cases, we consider a weakly nonGaussian distribution and a strongly nonGaussian distribution. 4.1 Example 1: 1D, 1V Stationary and NonGaussian processes In this example, two target non-Gaussian marginal CDFs are given as follows. The first is weakly non-Gaussian while the second is very strongly non-Gaussian. 4.1.1

Symmetric beta distribution

The shifted beta distribution has probability density function (PDF) given by:

f (x) =

Γ(C + D) (x − A)C−1 (B − x)D−1 Γ(C)Γ(D)(B − A)C+D−1 (15)

where 𝑋 is defined in the range A < x < B and Γ(⋅) is the gamma function. The parameters for the weakly non-Gaussian distribution are A = −3.317 , B = 3.317 , C = 5 , and D = 5 . The distribution is quite close in shape to the standard Gaussian distribution as can be seen in Fig. 1. 4.1.1 Inverse L-shaped beta distribution We prescribe an inverted L-shaped beta distribution as a strongly non-Gaussian distribution with PDF given in Eq. (15) and

Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

Inv. L-beta Symm. beta Normal

0.4

PDF

,

0.3 0.2 0.1 0 -6

-4

-2

0

x

2

4

6

Figure 1. Weakly and strongly non-Gaussian beta marginal PDFs shown with the standard normal distribution.

The target non-Gaussian PSDF is prescribed as follows:

SNT (ω ) =

125 2 −5 ω ω e 4

(16)

having unit standard deviation. In the iterative process, the relative difference between non-Gaussian computed PSDF and the target non-Gaussian PSDF is computed as:

ε(i) = 100

∑

2

M −1

"#SN(i) (ω n ) − SNT (ω n )$% 2 M −1 ∑ "#SNT (ω n )

%$n=0

(17)

Nonetheless, this approximation can be considered quite accurate given the degree of incompatibility between the non-Gaussian marginal PDF and the PSDF. Non-Gaussian PSDF

0.5

B = 0.457

0.8

Target Computed ϵ=0.47%

0.6 0.4 0.2 0 0

0.5

1

1.5

2

ω

(a) Symmetric beta distribution non-Gaussian PSDF

parameters A = −28.429 , C = 0.1895 , and D = 11.795 .

0.8

Target Computed ϵ=18.19%

0.6 0.4 0.2 0 0

0.5

1

1.5

2

ω

(b) Inverse L-shaped beta distribution Figure 2. Comparison of computed compatible non-Gaussian PSDFs with the target non-Gaussian PSDFs for (a) a weakly non-Gaussian process and (b) a strongly non-Gaussian process.

4.2 Examples: 1D, 1V Non-Stationary and Non-Gaussian processes

n=0

where M is the number of the frequency steps in the prescribed domain. This measure of difference is used as a metric for convergence of the method. The resulting compatible non-Gaussian PSDFs and the target non-Gaussian PSDFs identified using the ITAM are shown in Fig. 2. The calculated difference between the nonGaussian PSDFs is very small for the weakly non-Gaussian distribution. On the other hand, the relative difference for the strongly nonGaussian PDF is significantly larger

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For the one dimensional, uni-variate, nonstationary and non-Gaussian process, the target incompatible non-Gaussian ES is defined as:

SNT (ω, t) =

A(t)3 2 − A(t ) ω ω e 4

(18)

with the time-dependent shape parameter give by:

Simulation of multi-dimensional, multi-variate, stationary and non-stationary, strongly non-Gaussian stochastic processes: The Iterative Translation Approximation Method (ITAM) Michael D. Shields and George Deodatis

(19)

where A0 = 10 and b = 0.075. The shape of distribution is shown in Fig. 3.

1

Non-Gaussia ACF

A(t) = A0 − bt

Target Computed ϵ=13.52%

0.5 0 -0.5 0

5

10

Lag

15

20

In this example, we consider a strongly non-Gaussian process with inverted L-shaped beta PDF given in Eq. (15) with parameter A = −28.429 , B = 0.457 , C = 0.1895 , and D = 11.795 . The relative difference is computed by comparing the non-Gaussian ACF computed from the ITAM method and the target nonGaussian ACF as:

ε(i) = 100

M −1

M −1

n=0

m=0 M −1

∑ ∑ "# Rˆ ∑ ∑ n=0

(i) N

Target Computed ϵ=14.38%

0.5 0 -0.5 0

5

10

Lag

15

20

1

Target Computed ϵ=14.85%

0.5 0 -0.5 0

5

10

Lag

15

20

(a) t = 60

2

(sn , tm ) − RNT (sn , tm )

%$M −1

1

(b) t = 30 Non-Gaussian ACF

Figure 3. Target non-Gaussian and non-stationary ES.

Non-Gaussian ACF

(a) t = 0

2

" RT (sn , tm )$% m=0 # N

(19) where M is the number of time steps, and Rˆ N is the estimated compatible non-stationary and non-Gaussian ACF calculated from the converged non-Gaussian ES. The estimated non-stationary and nonGaussian ACF is shown in Fig. 4. In spite of the significant incompatibility, the computed non-Gaussian and non-stationary ACF converged to a reasonable approximation at each time step showing that the proposed technique enables a very accurate estimate of the ES and that it maintains the shape of prescribed ACF suitably.

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Figure 4. Comparison of computed compatible non-Gaussian ACFs with the target non-Gaussian ACF at (a) t = 0, (b) t = 30, and (c) t = 60.

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Conclusions

A series of algorithms for simulation of general second-order non-Gaussian stochastic processes using the Iterative Translation Approximation Method (ITAM) have been reviewed in this paper. The ITAM is demonstrated to be an efficient and accurate methodology to approximate a general nonGaussian process by a translation process despite existing incompatibility between the prescribed autocorrelation function and the marginal non-Gaussian PDF. The cases of

Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

uni-variate, multi-variate, stationary and nonstationary processes are considered and examples are given to demonstrate the methods’ effectiveness. References Bocchini, P. and Deodatis, G., Critical review and latest developments of a class of simulation algorithms for strongly non-Gaussian random fields. Probabilistic Engineering Mechanics. 23, 393-407, 2008. Ferrante, F., Arwade, S., Graham-Brady, L., A translation model for non-stationary, nonGaussian random processes, Probabilistic Engineering Mechanics, 20, 215–28, 2005. Grigoriu, M., Crossings of non-Gaussian translation processes, Journal of Engineering Mechanics, 110, 610–20, 1984. Grigoriu, M., Applied non-Gaussian processes. New Jersey, Prentice Hall, New Jersey, NY, 1995. Gioffrè, M., Gusella, V., Grigoriu, M., Simulation of non-Gaussian field applied to wind pressure fluctuations, Probabilistic Engineering Mechanics, 15, 339–45, 2000. Huang, S.P., Quek, S.T., and Phoon, K.K., Convergence study of the truncated KarhunenLoeve expansion for simulation of stochastic processes. International Journal for Numerical Methods in Engineering. 52, 1029-1043, 2001. Phoon, K.K., Huang, S.P., and Quek, S.T. Simulation of second-order processes using Karhunen-Loeve expansion. Computers and Structures. 80, 1049-1060, 2002.

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Priestley, M., Evolutionary spectra and nonstationary processes, Journal of the Royal Statistical Society. Section B, 27, 204-37, 1965. Sakamoto, S. and Ghanem, R. (2002a). Polynomial chaos decomposition for the simulation of nonGaussian nonstationary stochastic processes. Journal of Engineering Mechanics. 128, 190201, 2002a. Sakamoto, S. and Ghanem, R. (2002b). Simulation of multi-dimensional non-gaussian nonstationary random fields. Probabilistic Engineering Mechanics. 17, 167-176, 2002b. Shields, M., Deodatis, G., Bocchini, P., A simple and efficient methodology to approximate a general non-Gaussian stochastic process by a translation process, Probabilistic Engineering Mechanics, 26, 511–9, 2011. Shields, M., Deodatis, G., A simple and efficient methodology to approximate a general nonGaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation, Probabilistic Engineering Mechanics, 31, 19-29, 2013a. Shields, M., Deodatis, G., Estimation of evolutionary spectra for simulation of nonstationary and non-Gaussian stochastic processes, Computers and Structures, 126, 14963, 2013b. Shinozuka, M. and Jan, C.-M., Digital simulation of random processes and its applications. Journal of Sounds and Vibration. 25, 111-128, 1972. Shinozuka, M. and Deodatis, G., Simulation of stochastic processes by spectral representation. Applied Mechanics Reviews, 44, 191-204, 1991.