International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Simulation of narrowband non-Gaussian processes using envelope distribution Tsuchida, Takahiro; Kimura, Koji Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Simulation of Narrowband Non-Gaussian Processes Using Envelope Distribution  Takahiro Tsuchida Assistant Professor, Dept. of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo, Japan Koji Kimura Professor, Dept. of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo, Japan  ABSTRACT: An envelope distribution approach is developed for generation of a class of narrowband non-Gaussian processes. The process is prescribed by the probability density and the specified power spectrum with the non-zero dominant frequency. The proposed approach is based on the method using stochastic differential equations. In order to construct the stochastic differential equations, it is required to find a joint probability distribution, which takes the form derived from the corresponding Fokker-Planck equation and has the target non-Gaussian distribution as the marginal distribution. In the present method, the joint distribution is obtained from the envelope distribution corresponding to the target non-Gaussian probability density. The proposed approach is applicable to generation of narrowband processes with a variety of non-Gaussian probability densities. Two numerical examples are presented to illustrate the usefulness of the approach.  Monte Carlo simulation is widely used to investigate the response and the reliability of mechanical and structural systems subjected to random excitation. For the purpose of obtaining the accurate results by the simulation, the excitation should be properly modeled as a stochastic process according to the information available. Therefore, a numerical algorithm for generation of stochastic processes with given statistical characteristics is widely studied (Shinozuka and Deodatis, 1991; Bocchini and Deodatis, 2008). One of the most well-known method for Gaussian processes is the spectral representation (Shinozuka and Jan, 1972; Shinozuka and Deodatis, 1991). In practical engineering problems, the random excitation often exhibits non-Gaussianity. The examples include road roughness (Grigoriu, 1995), shallow water waves (Bitner, 1980) and wind pressure acting on low-rise buildings (Kwon and Kareem, 2011). The probability distributions of these stochastic processes are far from Gaussian distribution. It is therefore required to generate non-Gaussian processes with the proper probability distributions to accurately estimate the reliability of the non-Gaussian randomly excited systems. 	 Another important characteristic of stochastic processes is the power spectral density. Especially, the narrowband power spectrum is observed in many types of random excitation. Furthermore, the narrowband spectral density may be particularly important in view of the resonance phenomenon of the system. 	 The model of a non-Gaussian process is often constructed to match the prescribed first-order probability distribution and power spectrum. Grigoriu (1998) proposed the approach using the translation process defined by the memoryless nonlinear transform of the underlying Gaussian process. Yamazaki and Shinozuka (1988) developed the method combined with the spectral representation and the translation process to generate the prescribed non-Gaussian 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 process. Many studies for the extensions of these methods also have been conducted (Masters and Gurley, 2003; Bocchini and Deodatis, 2008; Shields and Deodatis, 2011). These methods require iteration in order to match the target distribution and/or spectral density, and therefore are sometimes computationally expensive. In this paper, a simulation method using an envelope distribution is proposed to generate a class of narrowband non-Gaussian stochastic processes with the prescribed probability distri-bution and spectral density. This method is based on the scheme developed by Cai and Lin (1996). Since no iteration is required in the procedure, the approach can be easily implemented for fast and efficient simulation. To generate the non-Gaussian process, a two-dimensional Ito stocha-stic differential equation is used. Then, in order to determine the diffusion coefficients, it is needed to find an appropriate joint probability density, which has the target probability density as the marginal distribution and satisfies a condition derived from the corresponding Fokker -Planck equation. However, it is difficult to obtain the joint probability density. 	 In the proposed procedure, the joint proba-bility density is obtained from the envelope distribution corresponding to the target non-Gaussian probability density. To demonstrate the validity of the method, this method is applied to generate non-Gaussian processes with two different types of probability distributions. The proposed approach is applicable to generation of processes with various non-Gaussian probability densities observed in engineering and scientific fields. This model may also be useful for analytical studies since the process is expressed by stochastic differential equations.   1. NARROWBAND NON-GAUSSIAN PRO-CESS Let U(t) be a zero-mean stationary non-Gaussian stochastic process having the non-Gaussian prob-  Figure 1: Power spectra of process U(t) ( ][E 2U  = 1)  ability distribution pU(u) and the power spectrum )(ωUS  given by 22222222224)()(][E)( ωαωραωρααπω +++++= USU     (1) where α  is the bandwidth, ρ  is the dominant frequency and ][E 2U  is the mean square of U(t). The power spectra )(ωUS  for some combina-tions of α and ρ are shown in Fig. 1. Hereafter, the stochastic process U(t) with the non-Gaussian distribution pU(u) and the power spectrum )(ωUS  described by Eq. (1) is called narrowband non-Gaussian process. 2. GENERATION METHOD In order to generate the stochastic process U(t) with the specified distribution pU(u) and the spectral density )(ωUS  expressed by Eq. (1), the method developed by Cai and Lin (1996) is applied. In this method, the prescribed non-Gaussian process is expressed by a two-dimensional stochastic differential equation determined according to pU(u) and )(ωUS . When this method is used, it is required to find a bivariate probability distribution, which has the target non-Gaussian distribution pU(u) as the marginal distribution and satisfies a condition described later. In this paper, an approach utilizing the envelope distribution is proposed. The joint probability density can be obtained by the envelope distribution corresponding to the target probability density pU(u) of the narrowband non-Gaussian process. Then, the sample functions 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 with the desired non-Gaussian distribution pU(u) and the spectrum )(ωUS  can be generated by numerically solving the stochastic differential equations determined through the present app-roach. 2.1. Generation of narrowband non-Gaussian process In this section, the method for generation of a stationary narrowband non-Gaussian process U(t) proposed by Cai and Lin (1996) is described.  (i) Stochastic differential equation U(t) is expressed by the following pair of Ito stochastic differential equations: )(),()( 111211 tdBVUDdtVaUadU ++=     (2)  )(),()( 222221 tdBVUDdtVaUadV ++=   (3) where B1(t) and B2(t) are two independent Wiener processes and V(t) is the stochastic process orthogonal to U(t). aij is constants adjusted to match the target spectral density. The diffusion coefficients ),(1 VUD  and ),(2 VUD  are determined from the joint probability distribution pUV (u,v) of U(t) and V(t).  (ii) Constants ai j Multiplying Eqs. (2) and (3) by )( τ−tU  and taking the ensemble average, we have )()()( 1211 ττττ UVUU RaRaRdd +=              (4) )()()( 2221 ττττ UVUUV RaRaRdd +=            (5) where )(τUR  is the autocorrelation function of U(t) and )(τUVR  is the cross-correlation function of U(t) and V(t). The constants ai j in Eqs. (4) and (5) are chosen according to the target power spectrum. In the case of Eq. (1), the constants ai j are given by αρρα −=−==−= 22121211 ,,, aaaa          (6) Then, solving Eqs. (4) and (5) subjected to the following conditions 0)]()([E)0(],[E)0( 2 === tVtURUR UVU   (7) we obtain ρττατ cos)exp(][E)( 2 −= URU           (8) By using Wiener-Khintchine’s theorem, it can be confirmed that the spectral density of U(t) is given by Eq. (1).  (iii) Diffusion coefficients ),(1 VUD and ),(2 VUD  The stationary Fokker-Planck equation gover-ning the joint probability distribution pUV (u,v)  of U(t) and V(t) is obtained from Eqs. (2) and (3) as follows: 0)],(),([21)],(),([21)],()[()],()[(2222212222211211=∂∂−∂∂−+∂∂++∂∂vupvuDvvupvuDuvupvauavvupvauauUVUVUVUV                 (9) Eq. (9) is satisfied if the following three cond-itions are met (Cai and Wu, 2004). 0),(),(2112 =∂∂+∂∂vvupuauvupva UVUV        (10) 0)],(),([21),( 2111 =∂∂− vupvuDuvuupa UVUV  (11) 0)],(),([21),( 2222 =∂∂− vupvuDvvuvpa UVUV  (12) The general solution for Eq. (10) is obtained as 0),(),( 2121212221 =++= akakvkukpvup UVUV   (13) where pUV is expressed as a function of2221 vkuk + , and k1 and k2 are positive constants. Since a12 and a21 are given by Eq. (6) in this paper, k1 = k2 holds. Thus, when we set k1 and k2 to 1, Eq. (13) reduces 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 )(),( 22 vupvup UVUV +=                         (14) Substituting Eq. (14) into Eqs. (11) and (12) and taking account of Eq. (6) lead to the diffusion coefficients D1(u,v) and D2(u,v) ∫ ∞−−=uUVUVdsvsspvupvuD ),(),(2),(22α  (15) ∫ ∞−−=vUVUVdssuspvupvuD ),(),(2),(22α  (16) By numerically solving Eqs. (2) and (3) obtained through the above procedure, one can generate the stochastic process U(t) with the spectral density Eq. (1) and the probability density pU(u) which is the marginal distribution of pUV (u,v). 2.2. Envelope distribution approach for genera-tion of narrowband non-Gaussian process When the method in section 2.1 is used, the joint probability distribution pUV (u,v), which takes the form of Eq. (14) and has the target non-Gaussian distribution pU(u) as the marginal distribution, is needed. In order to obtain pUV (u,v), an approach using the envelope distribution corresponding to the target probability distribution pU(u) of the non-Gaussian process is proposed. A narrowband stochastic process U(t) with the dominant frequency ρ  can be modeled as follows: )(cos)()](cos[)()( ttAtttAtU Ψ=Φ+= ρ  (17) where )()( ttt Φ+=Ψ ρ , and A(t) and )(tΦ  are slowly varying random processes with respect totρcos . The random process A(t) is called the en-velope and the process )(tΦ  is called the random phase. Similarly, the quadrature component V(t) is expressed by )(sin)()](sin[)()( ttAtttAtV Ψ=Φ+= ρ    (18) where )()()( 22 tVtUtA +=                             (19) It is assumed that )(tΦ  is independent of A(t) and the probability distribution of )(tΦ  is the uniform distribution on the interval ]2,0[ π . The assumption enables us to use Eqs. (25) and (26) and obtain the joint probability distribution pUV from the envelope distribution. Under the assumption, the distribution )(ψΨp  of )(tΨ  is also the uniform distribution on the interval ]2,0[ π  πψ 21)( =Ψp                                            (20) Using Eq. (20) and the probability density pA(a) of the envelope A(t), we obtain the joint distribution ),( ψapAΨ of A(t) and )(tΨ  )(21),( apap AA πψ =Ψ                             (21) The relationship between ),( ψapAΨ  and ),( vupUVis given by Japvup AUV ),(),( ψΨ=                          (22) where J is the Jacobian 221vuJ+=                                          (23) which is derived from Eq. (19) and )(tΨ  =     tan-1[V(t)/U(t)]. Thus, by taking into account Eqs. (19), (21), (22) and (23), pUV (u, v) can be rewri-tten as 2222 )(21),(vuvupvup AUV ++= π                  (24) The transform-pair between pU(u) and pA(a), which is known as the Blanc-Lapierre transform, are defined as follows (Rytov et al., 1988) ∫ ∫∞∞−∞ −=0 0)()(21)( dadxeapxaJup iuxAU π    (25) ∫ ∫∞ ∞∞−= 0 0 )()()( dudxeupxaxJaap iuxUA     (26) where J0(_) is the Bessel function of the first kind of order 0. Therefore, the transform-pair given by Eqs. (25) and (26) yields the envelope 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 distribution pA(a) corresponding to the target probability density pU(u). By substituting the envelope distribution pA(a) into Eq. (24), the joint probability density pUV (u,v) needed for generation of U(t) can be obtained. 3. NUMERICAL EXAMPLES 3.1 Uniformly distributed narrowband process Consider a narrowband process U(t) with the power spectral density expressed by Eq. (1) and an uniform distribution given by Δ≤≤Δ−Δ= uupU ,21)(                       (27) where Δ  is the parameter for the distribution. 3.1.1. Envelope distribution and diffusion   coefficient By using Blanc-Lapierre transform given by Eq. (26), the envelope distribution pA(a) correspond-ing to the uniform distribution can be obtained as follows: Δ≤≤−ΔΔ= aaaapA 0,)( 22            (28) Substitution of Eq. (28) into Eq. (24) leads to the joint distribution pUV (u,v) 222121),(vuvupUV −−ΔΔ= π               (29) Using Eqs. (15), (16) and (29), the diffusion coefficients used for simulation are derived as )(2),(),( 22221 vuvuDvuD −−Δ== α  (30) Then, the sample functions of uniformly distributed narrowband process can be generated by numerically solving Eqs. (2) and (3) comb-ined with Eq. (30). 3.1.2. Results The sample functions of the process U(t) are generated through the proposed procedure, and the probability distribution and power spectrum of the generated sample functions are calculated. Euler-Maruyama method (Kloeden and Platen, 1992) is used as the numerical solution for stochastic differential equations, and the compu-tational conditions are as follows:       (a)                                                (b)                                                   (c)  Figure 2: Simulation of uniformly distributed narro-wband process with the bandwidth 01.0=α and the dominant frequency 2.0=ρ . (a) Sample function. (b) Probability density function of the sample functions (solid line) and the target (dashed line). (c) Power 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 spectrum of the sample functions (solid line) and the target (dashed line) • time step: 01.0=Δt  • sample size: T = 1310.72  The parameters for the spectral density and the uniform distribution are shown below. • bandwidth: 01.0=α  • dominant frequency: 2.0=ρ  • parameter for distribution: 3=Δ   The results of a sample function of U(t), the probability distribution and the power spectrum calculated from the 200 generated sample functions are shown in Fig. 2. The solid line and the dashed line in Figs. 2(b) and 2(c) indicate the simulation results and the targets, respectively. From the comparison between the simulation results and the targets, it is seen that the present method is able to accurately match the desired characteristics.  3.2 Laplace-distributed narrowband process Consider a narrowband process U(t) with the power spectral density expressed by Eq. (1) and a Laplace distribution given by )exp(2)( uupU ββ −=                              (31) where β  is the parameter for the distribution.  3.2.1 Envelope distribution and diffusion coefficient Application of Blanc-Lapierre transform given by Eq. (26) yields the envelope distribution pA(a) corresponding to the Laplace distribution as follows: )()( 02 aaKapA ββ=                                 (32) where K0(• ) is the modified Bessel function of the second kind of order 0. Substituting Eq. (32) into Eq. (24), the joint distribution pUV (u,v) can be obtained, then from pUV (u,v) and Eqs. (15) and (16), the diffusion coefficients used for simulation are derived as )()(),(),(2202212221vuKvuKvuvuDvuD+++==ββαβ    (33) where K1(• ) is the modified Bessel function of the second kind of order 1. The sample functions of the Laplace-distributed narrowband process can be obtained by numerically solving Eqs. (2) and (3) combined with Eq. (33).  3.2.2 Results The sample functions of the process U(t) are generated by using the present method, and the probability distribution and power spectrum are calculated from the 200 generated sample functions. The computational conditions are the same as that shown in section 3.1.2. The parameters for the spectral density and the Laplace distribution are as follows: • bandwidth: 05.0=α  • dominant frequency: 1=ρ  • parameter for distribution: 2=β   The results are shown in Fig. 3. The solid line and the dashed line in Figs. 3(b) and 3(c) denote the simulation results and the targets for comparison, respectively. The probability distri-bution and spectral density of the sample functions generated by using the proposed procedure agree very well with the correspon-ding targets. 4. CONCLUSIONS An envelope distribution approach has been pro-posed for generation of a class of narrowband non-Gaussian process. The non-Gaussian process is prescribed by the probability density and the specified power spectrum with the non-zero dominant frequency. The proposed approach is based on the method using a two-dimensional stochastic differential equation developed by Cai and Lin (1996). In order to construct the stocha-stic differential equation, a joint probability distribution, which has the target probability density  as the marginal distribution and satisfies  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7                                         (a)                                                (b)                                                   (c)  Figure 3: Simulation of Laplace-distributed narrow-band process with the bandwidth 05.0=α and the dominant frequency 1=ρ . (a) Sample function. (b) Probability density function of the sample functions (solid line) and the target (dashed line). (c) Power spectrum of the sample functions (solid line) and the target (dashed line) a condition derived from the corresponding Fokker-Planck equation, is needed. In the present method, the joint probability density is obtained by utilizing the envelope distribution corresponding to the target probab-ility density of the non-Gaussian process. The proposed approach can be applied to generation of narrowband processes with a variety of non-Gaussian probability densities. In order to demonstrate the validity of the present method, uniformly and Laplace distri-buted processes with the different bandwidth and dominant frequency was generated. In either case, the results were in good agreement with the target probability density and power spectrum. 5. REFERENCES Bitner, E.M. (1980). "Non-linear effects of the statis-tical model of shallow-water wind waves”, Applied Ocean Research, 2 (2), 63–73. Bocchini, P. and Deodatis, G. (2008). "Non-linear ef-fects of the statistical model of shallow-water wind waves”, Probabilistic Engineering Mechanics, 23 (4), 393–407. Cai, G.Q. and Lin, Y.K. (1996). "Generation of non-Gaussian stationary stochastic processes”, Physical Review. E, 54 (1), 299–303. Cai, G.Q. and Wu., C. (2004). "Modeling of bounded stochastic processes”, Probabilistic Engineer-ing Mechanics, 19 (3), 197–203. Grigoriu, M. (1995). "Applied non-Gaussian pro-cesses”, Englewood Cliffs, NJ: Prentice Hall. Grigoriu, M. (1998). "Simulation of stationary non-Gaussian translation processes”, Journal of Engineering Mechanics, 124 (2), 121–126. Kay, S (2010). "Representation and Generation of Non-Gaussian Wide-Sense Stationary Random Processes With Arbitrary PSDs and a Class of PDFs”, IEEE Transactions on Signal Processing, 58 (7), 3448–3458. Kloeden, P.E. and Platen, E. (1992). "Numerical Solution of Stochastic Differential Equations”, Springer. Kwon, D.K. and Kareem, A. (2011). "Peak factors for non-Gaussian load effects revisited”, Journal of Structural Engineering, 137 (12), 1611–1619. Masters, F. and Gurley, K. (2003). "Non-Gaussian simulation: cumulative distribution function 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 map-based spectral correction”, Journal of Engineering Mechanics, 129 (12), 1418–1428. Rytov, S. M., Kravtsov, Y. A. and Tatarskii, V. I. (1988). "Principles of Statistical Radiophysics, vol. 2, Correlation Theory of Random Processes”. New York: Springer-Verlag. Shields, M.D., Deodatis, G. and Bocchini, P. (2011). "A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process”, Probabilistic Engineering Mechanics, 110 (4), 511–519. Shinozuka, M. and Deodatis, G. (1991). "Simulation of stochastic processes by spectral representation”, Applied Mechanics Reviews, 44 (4), 191–204. Shinozuka, M. and Jan, C.M. (1972). "Digital simulations of random processes and its applications”, Journal of Sound and Vibration, 25 (1), 111–128. Yamazaki, F. and Shinozuka, M. (1988). "Digital generation of non-Gaussian stochastic fields”, Journal of Engineering Mechanics, 114 (7), 1183–1197. Gardiner, C. (2009). "Stochastic Methods: A Hand-book for the Natural and Social Sciences 4th ed”, Springer. 

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