12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Simulation of Strongly Non-Gaussian Non-stationary StochasticProcesses utilizing Karhunen-Loeve ExpansionHwanpyo KimGraduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USAMichael D. ShieldsAssistant Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore,USAABSTRACT: The simulation of non-stationary and non-Gaussian stochastic processes is a challengingproblem of considerable practical interest. Recently, Shields et al. have developed a class of concep-tually simple and efficient methods for simulation of non-Gaussian processes using translation processtheory (collectively referred to as the Iterative Translation Approximation Method - ITAM) that itera-tively upgrades the underlying Gaussian power spectral density function for simulation using the spectralrepresentation method. However, the currently existing ITAM method for generation of non-stationaryand non-Gaussian processes requires additional approximations in the estimation of the evolutionaryspectrum. An extension of the ITAM is proposed that utilizes the K-L expansion. The developed methoditeratively upgrades the covariance function directly and, in so doing, avoids the complex and non-uniqueinverse problem of estimating the evolutionary spectrum from the non-stationary autocorrelation. The ap-plication of the method for a strongly non-Gaussian and non-stationary process with a prescribed targetnon-Gaussian correlation function is demonstrated.1. INTRODUCTIONProbabilistic analysis of complex problems involv-ing strong non-linearities and uncertainties repre-sented by non-Gaussian stochastic processes/fieldstypically requires Monte Carlo simulation. MonteCarlo simulation, which is the most robust methodavailable for these problems, requires accurate sim-ulation of stochastic processes. Well establishedmethods are readily available for the simulation ofstationary Gaussian random processes. Unfortu-nately, these stationary and Gaussian processes areoften not representative of reality. Instead, realiza-tions of non-Gaussian and non-stationary stochasticprocess are necessary for accurate analysis. How-ever, simulation of these processes present numer-ous challenges and consequently, the developmentof methods for their simulation has lagged consid-erably behind the stationary and Gaussian models.The Karhunen-Loeve (K-L) expansion and theSpectral Representation Method (SRM) are the twomost commonly employed methods for generationof random processes in mechanics. The K-L ex-pansion (Ghanem and Spanos (1991), Huang et al.(2001)) affords an optimally succinct representa-tion of the stationary and non-stationary processesfrom a specified covariance function and its im-plementation is straightforward for Gaussian pro-cesses. Phoon et al. (2002) and Phoon et al.(2005) developed K-L-based methods for the sim-ulation of non-Gaussian and non-stationary pro-cesses that iteratively update the probability densityfunction’s of the K-L random variables. Further-more, Sakamoto and Ghanem (2002a,b) proposeda technique for simulating non-stationary and non-Gaussian stochastic processes that utilizes a poly-nomial chaos expansion (PCE) of the K-L randomvariables. Another class of algorithms for generat-ing non-Gaussian processes couples the SRM, (Shi-nozuka and Jan (1972); Shinozuka and Deodatis(1991)) with translation process theory, (Grigoriu(1995)). Several developments along these lineshave been proposed over the past 25 years includ-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ing works by Yamazaki and Shinozuka (1988), De-odatis and Micaletti (2001), and Bocchini and De-odatis (2008) among others. Most recently, Shieldset al. (2011) and Shields and Deodatis (2013a)proposed the Iterative Translation ApproximationMethod (ITAM) for simulation of stationary non-Gaussian processes that upgrades the underlyingGaussian power spectral density function in a sim-ple and efficient manner. This method has beenextended to the simulation of non-stationary andnon-Gaussian processes by Shields and Deodatis(2013b).For strongly non-Gaussian and non-stationaryprocesses, each of the aforementioned methodspossesses notable drawbacks. K-L based simula-tion methods are subject to the Central Limit The-orem. Therefore, in the limit as the number of K-L random variables grows large the process tendstoward Gaussian regardless of the distribution ofthe K-L variables themselves. Moreover, even fora small number of K-L variables, the process of-ten cannot maintain the strongly non-Gaussian fea-tures that are desired. SRM-based methods, on theother hand, require significant approximations inorder to produce non-stationary and non-Gaussianprocesses (Shields and Deodatis (2013b)). Mostnotably, no unique inverse exists to compute theevolutionary spectrum (ES) as defined by Priest-ley (1965) from the non-stationary autocorrelationfunction. Thus, approximations must be introducedthat can cause significant errors for strongly non-Gaussian and non-stationary processes.In this work, an extension of the ITAM that uti-lizes the K-L expansion for simulation is proposed.The method developed in this study has the fol-lowing advantages for simulation of non-stationaryand non-Gaussian processes. First, estimation ofthe evolutionary spectrum is avoided; thus allevi-ating the associated approximations and allowingdirect simulation from the covariance matrix. Sec-ond, generated realizations are produced directlyfrom Gaussian K-L random variables with the re-sulting Gaussian process translated to match thetarget marginal non-Gaussian probability functionexactly. Lastly, It is conceptually simple, straight-forward to implement, and converges rapidly (usu-ally in 10 iterations or less). The applicationof the method for a strongly non-Gaussian andnon-stationary process with prescribed target non-Gaussian covariance is demonstrated.2. KARHUNEN-LOEVE EXPANSIONConsider a random process A(t,θ) defined onthe probability space (Ω,σ ,P) over the domains, t ∈ [a,b] with zero mean and covariance functionC(s, t) possessing finite variance. The process canbe expressed according to the K-L expansion asA(t,θ) =∞∑i=1√λiζi(θ) fi(t) (1)where λi and fi(x) are the eigenvalues and eigen-functions of the covariance C(s, t) determined bysolving the Fredholm integral equation of the sec-ond kind given by∫ baC(s, t) fi(s)ds = λi fi(t) (2)The parameters ζi are a set of uncorrelated ran-dom variables with zero mean E[ζi] = 0 and cor-relation E[ζiζ j] = δi j. The K-L expansion can beconveniently implemented for simulation by trun-cating a countable number, M, of its terms asA˜(t,θ) =M∑i=1√λiζi(θ) fi(x) (3)For Gaussian processes, the random variables ζifollow the standard Gaussian distribution. How-ever, the distribution of ζi for non-Gaussian pro-cesses can be difficult to determine and may requirecomplex dependence structure in general (Grigoriu(2010)).3. TRANSLATION PROCESS THEORYTranslation process theory (Grigoriu (1995)) repre-sents a non-Gaussian process Y (t) through the non-linear transformation of a Gaussian process, X(t),as follows:Y (t) = g(X(t)) (4)where g(·) is a general nonlinear transformation.The so-called standard translation is defined suchthat g(·) = F−1 · Φ(·) where F(·) and Φ(·) are212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the arbitrary non-Gaussian and standard Gaussianmarginal cumulative distribution functions, respec-tively.The extension of translation process theory fornon-stationary processes was proposed by Ferranteet al. (2005) where the mapping in Eq. (4) is gener-alized to incorporate a time-dependent transforma-tion asY (t) = g(X(t), t) = F−1 · {Φ(X(t)), t} (5)where X(t) is a non-stationary Gaussian processand Y (t) is a non-stationary and non-Gaussian pro-cess with time-varying marginal cumulative distri-bution function F(·, t).Utilizing non-stationary translation process the-ory, the correlation R(s, t) of the non-stationary andnon-Gaussian process can be computed from thestandard Gaussian correlation asR(s, t) = µ(s)µ(t)+√C(s,s)C(t, t)ξ (s, t)=∞∫−∞∞∫−∞g(u,s)g(v, t)φ{u,v;ρ(s, t)}dudv(6)where µ(t), C(t, t), and ξ (s, t) are the mean,variance, and correlation coefficient of the non-Gaussian process Y (t) and φ(s, t) denotes the jointGaussian PDF with correlation coefficient, ρ(s, t).A general challenge with translation process the-ory is that, given a prescribed covariance functionand associated non-Gaussian marginal probabilitydensity function, it is not always possible to iden-tify a corresponding underlying Gaussian correla-tion function. In other words, the inverse of Eq. (6)does not always exist. In such cases, the prescribedcovariance function and marginal probability den-sity function are said to be incompatible with thetranslation process model. However, given the con-venience of the translation model and the lack ofcomparable methods for modeling non-Gaussianprocesses, it is often desirable to identify an un-derlying Gaussian process that, when mapped tothe non-Gaussian distribution using Eq. (6), pro-duces a non-Gaussian correlation function that isas close as possible to the prescribed non-Gaussiancorrelation function. This is the motivation for thepreviously developed Iterative Translation Approx-imation Method (ITAM) described in the followingsection and its improvement in this work.4. ITERATIVE TRANSLATION APPROXI-MATION METHODThe ITAM, originally developed by Shields et al.(2011) and extended for non-Gaussian and non-stationary processes by Shields and Deodatis(2013b), utilizes a simple and efficient iterativemethod to upgrade the underlying Gaussian powerspectral density (stationary) or evolutionary spec-trum (non-stationary). In the non-stationary casethe ITAM upgrades the underlying Gaussian evolu-tionary spectrum asS(i+1)G (ω, t) =[STN(ω, t)S(i)N (ω, t)]βS(i)G (ω, t) (7)where S(i)G (ω, t) and S(i)N (ω, t) are the underlyingGaussian and the computed non-Gaussian ES at it-eration i, respectively and β is parameter utilized tooptimize the convergence.The challenge associated with this method isthat it requires estimation of the evolutionary spec-trum from the non-stationary correlation function.However, in general the evolutionary spectrum isnot uniquely defined for a given non-stationaryautocorrelation (Priestley (1965); Benowitz et al.(2014)). Thus, the ITAM proposed by Shields andDeodatis (2013b) requires an estimation that maycause significant errors in the case of strongly non-Gaussian and non-stationary processes. Recently,Benowitz et al. (2014) has investigated the unique-ness of this inversion and the results indicate thatit may be possible to obtain a unique evolutionaryspectrum under specific conditions, but the com-putational cost required to estimate this evolution-ary spectrum with reasonable accuracy is extremelyhigh.5. PROPOSED ITAM WITH K-L EXPAN-SIONA new ITAM utilizing the K-L expansion for sim-ulation is proposed in this work. This new ITAMiteratively upgrades the underlying non-stationaryautocorrelation function (ACF) directly (removing312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015approximations associated with evolutionary spec-trum estimation). The following procedure de-scribes the proposed methodology for finding thecompatible underlying Gaussian ACF.1. Initialize underlying Gaussian ACF, R(0)G (s, t).2. Upgrade the underlying Gaussian ACF.3. Find the nearest positive semi-definite ACF tothe upgraded ACF.4. Compute the non-Gaussian ACF at iteration i,R(i)N (s, t), by non-stationary translation processtheory.5. Check the difference between the computedand target non-Gaussian ACF. If convergence,proceed to the next step. Otherwise, iterateback to step 2.6. Simulate the processes using the K-L expan-sion and translate process theory.When an incompatible pair of marginal non-Gaussian non-stationary PDF and ACF is pre-scribed, the initial underlying Gaussian ACF canbe defined arbitrarily - as long as it is a properlydefined autocorrelation function (i.e. is symmetricand positive semi-definite). In practice, it is use-ful to reduce calculation time by setting the initialunderlying Gaussian ACF equal to the target non-Gaussian ACF.The proposed upgrading method in step 2 is simi-lar to that of the SRM-based ITAM given in Eq. (7).Specifically, the underlying Gaussian normalizedACF, ρ(s, t) is upgraded asρ(i+1)(s, t) =[ξ T (s, t)ξ (i)(s, t)]ρ(i)(s, t) (8)where ξ (i)(s, t) is the computed non-Gaussian ACFat iteration i and ξ T (s, t) is the prescribed targetstandardized non-Gaussian ACF. However, thereare a few noteworthy differences with the SRM-based ITAM. Most notably, the evolutionary spec-trum is a strictly positive quantity while the ACF isnot. This has two significant consequences. First,the exponent β is removed in the proposed methodbecause iterations using a non-integer exponent willproduce imaginary numbers for negative ratios inthe parentheses. Second, the resulting ACF fromthe iterations in Eq. (8) is not necessary positivesemi-definite and is therefore not a valid autocor-relation function. To correct this, it is necessaryto identify the nearest positive semi-definite ACF(step 3). This is achieved using the method pro-posed by Qi and Sun (2006) who use a quadrati-cally convergent newton method whose quadraticconvergence has been proven.Finally, in step 5 the relative difference be-tween the computed and target non-Gaussian non-stationary ACF asε(i) = 100√√√√∑N−1n=0 ∑N−1m=0[ξ (i)(sn, tm)−ξ T (sn, tm)]2∑N−1n=0 ∑N−1m=0[ξ T (sn, tm)]2(9)When the value of the relative difference stabi-lizes to a constant value, the iterations are stoppedand the K-L expansion is used to generate theGaussian process from the converged underlyingGaussian ACF using Eq. (3). Then, the generatedGaussian sample function is translated to the non-Gaussian distribution using Eq. (5).6. NUMERICAL EXAMPLETo demonstrate the capabilities of the improvedITAM, consider the simulation of the strongly non-Gaussian and non-stationary process with targetnon-stationary covariance in Figure 1 is given byC(s, t) = min(s, t) · cos[4pi(s− t)] (10)Figure 1: Target non-stationary covariance.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The process possesses a non-stationary "U-shaped" beta marginal distribution with cumulativedistribution function given byF(y; p,q) =Γ(p+q)Γ(p)Γ(q)∫ u0zp−1(1− z)q−1dz (11)where u = y−yminymax−ymin with upper and lower boundsymin = µb(t)+σb(t)√p(p+q+1)q and ymax = µb(t)−σb(t)√q(p+q+1)p , and Γ(·) is the gamma functionwith parameters p = 0.342 and q = 0.528. Themean µb(t) = 0 and the variance σ2b (t) =C(t, t) = t.The shape of this “U-shaped” beta PDF is shown inFigure 2 along with the standard normal pdf.x-6 -4 -2 0 2 4 6PDF00.10.20.30.40.5U Beta at t=0.25U Beta at t=0.5U Beta at t=0.75U Beta at t=1NormalFigure 2: Strongly non-Gaussian and non-stationary"U-shaped" beta marginal PDF shown with the stan-dard normal distributionThe converged non-Gaussian covariance at sev-eral time instants are shown in Figure 3. Despite thestrongly non-Gaussian and non-stationary nature ofthe target process, the computed non-Gaussian andnon-stationary covariance converges to the targetwith high accuracy for all values of t. The con-verged results are calculated in only 5 iterations.Across the entire time domain, the maximum rela-tive difference is 5.58% with the converged resultsmaintaining the shape and magnitude of the targetcovariance very well.s0 0.2 0.4 0.6 0.8 1Covariance-1-0.500.51 TargetComputed ϵ=2.866%(a) t = 1s0 0.2 0.4 0.6 0.8 1Covariance-1-0.500.51 TargetComputed ϵ=2.675%(b) t = 0.75s0 0.2 0.4 0.6 0.8 1Covariance-1-0.500.51 TargetComputed ϵ=3.309%(c) t = 0.5s0 0.2 0.4 0.6 0.8 1Covariance-1-0.500.51 TargetComputed ϵ=5.581%(d) t = 0.25Figure 3: Comparison of the computed non-Gaussianand non-stationary covariance with the target at differ-ent times7. COMPARISON WITH THE ORIGINALSRM-ITAMThe performance of the SRM-based ITAM and theproposed ITAM is demonstrated using the strongly512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015non-stationary target ES given asS(t,ω) = 14√piexp[−(ω−ω0(t)2)2](12)where ω0(t) = 10+40t. Plots of the target ES andthe corresponding covariance are presented in Fig-ure 4. The prescribed PDF is the "U-shaped" betaPDF in Eq. (11) with constant mean µb(t) = 0 andvariance σ2b (t) = 1.(a) Target non-stationary evolutionary spectrum(b) Target non-stationary covarianceFigure 4: Target non-stationary evolutionary spectrumand covarianceNumerical results based on the two differentITAMs are presented on Table 1. As the originalTable 1: Relative difference of the computed non-Gaussian non-stationary covariance between the SRM-ITAM and KL-ITAMRelative difference (%)Time SRM-ITAM KL-TAMs = 0 17.81 1.086s = 1 12.11 5.317ITAM with SRM upgrades the underlying Gaus-sian ES, which is difficult to estimate for stronglynon-stationary cases and requires computation of apseudo-ES (Shields and Deodatis (2013a)), it haslarger relative difference throughout time. How-ever, the presented KL-ITAM, which upgrades theunderlying correlation function directly, approxi-mates the target covariance better than the previousmethod. The relative difference of the KL-basedITAM does not exceed 6%, the other difference isalmost 15% over the time. These differences willbe more significant for stronger non-Gaussian andnon-stationary targets.8. CONCLUSIONSAn improvement to the Iterative Translation Ap-proximation Method for generation of stronglynon-Gaussian and non-stationary process is pro-posed which utilizes direct upgrading on the non-stationary autocorrelation function and simulationwith the K-L expansion. The method improvesupon existing K-L based simulation methods bypairing them with translation process theory in or-der to match marginal non-Gaussian probabilitydensity functions exactly. It also improves uponthe existing SRM-based ITAM simulation methodsby removing the need to estimate the evolutionaryspectrum which can produce significant errors. Themethod is shown to be highly accurate, straightfor-ward to implement, and converge rapidly - even forprocesses that are very difficult to simulate.9. REFERENCESBenowitz, B., Shields, M., and Deodatis, G. (2014).“Determining evolutionary spectra from non-stationary autocorrelation functions.” ProbabilisticEngineering Mechanics.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015t0 0.5 1 1.5 2Covariance-1-0.500.51TargetSRM-ITAMKL-ITAM(a) s = 0t0 0.5 1 1.5 2Covariance-1-0.500.51 TargetSRM-ITAMKL-ITAM(b) s = 1Figure 5: Comparison of the computed non-Gaussianand non-stationary covariance with the target betweenSRM-ITAM and KL-ITAMBocchini, P. and Deodatis, G. (2008). “Critical re-view and latest developments of a class of simu-lation algorithms for strongly non-gaussian randomfields.” Probabilistic Engineering Mechanics, 23(4),393–407.Deodatis, G. and Micaletti, R. C. (2001). “Simulationof highly skewed non-gaussian stochastic processes.”Journal of engineering mechanics, 127(12), 1284–1295.Ferrante, F., Arwade, S., and Graham-Brady, L. (2005).“A translation model for non-stationary, non-gaussianrandom processes.” Probabilistic engineering me-chanics, 20(3), 215–228.Ghanem, R. G. and Spanos, P. D. (1991). Stochasticfinite elements: a spectral approach, Vol. 387974563.Springer.Grigoriu, M. (1995). Applied non-Gaussian processes:Examples, theory, simulation, linear random vibra-tion, and MATLAB solutions. PTR Prentice Hall Up-per Saddle River, NJ.Grigoriu, M. (2010). “Probabilistic models for stochas-tic elliptic partial differential equations.” Journal ofComputational Physics, 229(22), 8406–8429.Huang, S., Quek, S., and Phoon, K. (2001). “Conver-gence study of the truncated karhunen–loeve expan-sion for simulation of stochastic processes.” Interna-tional journal for numerical methods in engineering,52(9), 1029–1043.Phoon, K., Huang, H., and Quek, S. (2005). “Simulationof strongly non-gaussian processes using karhunen–loève expansion.” Probabilistic engineering mechan-ics, 20(2), 188–198.Phoon, K., Huang, S., and Quek, S. (2002). “Simula-tion of second-order processes using karhunen–loeveexpansion.” Computers & structures, 80(12), 1049–1060.Priestley, M. B. (1965). “Evolutionary spectra and non-stationary processes.” Journal of the Royal StatisticalSociety. Series B (Methodological), 204–237.Qi, H. and Sun, D. (2006). “A quadratically convergentnewton method for computing the nearest correlationmatrix.” SIAM journal on matrix analysis and appli-cations, 28(2), 360–385.Sakamoto, S. and Ghanem, R. (2002a). “Polyno-mial chaos decomposition for the simulation of non-gaussian nonstationary stochastic processes.” Journalof engineering mechanics, 128(2), 190–201.Sakamoto, S. and Ghanem, R. (2002b). “Simulation ofmulti-dimensional non-gaussian non-stationary ran-dom fields.” Probabilistic Engineering Mechanics,17(2), 167–176.Shields, M. and Deodatis, G. (2013a). “Estimation ofevolutionary spectra for simulation of non-stationaryand non-gaussian stochastic processes.” Computers &Structures, 126, 149–163.Shields, M. and Deodatis, G. (2013b). “A simple andefficient methodology to approximate a general non-gaussian stationary stochastic vector process by atranslation process with applications in wind veloc-ity simulation.” Probabilistic Engineering Mechan-ics, 31, 19–29.Shields, M., Deodatis, G., and Bocchini, P. (2011). “Asimple and efficient methodology to approximate a712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015general non-gaussian stationary stochastic process bya translation process.” Probabilistic Engineering Me-chanics, 26(4), 511–519.Shinozuka, M. and Deodatis, G. (1991). “Simulation ofstochastic processes by spectral representation.” Ap-plied Mechanics Reviews, 44(4), 191–204.Shinozuka, M. and Jan, C.-M. (1972). “Digital simula-tion of random processes and its applications.” Jour-nal of sound and vibration, 25(1), 111–128.Yamazaki, F. and Shinozuka, M. (1988). “Digital gen-eration of non-gaussian stochastic fields.” Journal ofEngineering Mechanics, 114(7), 1183–1197.8
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International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)
Simulation of strongly non-Gaussian non-stationary stochastic processes utilizing Karhunen-Loeve expansion Kim, Hwanpyo; Shields, Michael D. Jul 31, 2015
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Title | Simulation of strongly non-Gaussian non-stationary stochastic processes utilizing Karhunen-Loeve expansion |
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Kim, Hwanpyo Shields, Michael D. |
Contributor | International Conference on Applications of Statistics and Probability (12th : 2015 : Vancouver, B.C.) |
Date Issued | 2015-07 |
Description | The simulation of non-stationary and non-Gaussian stochastic processes is a challenging problem of considerable practical interest. Recently, Shields et al. have developed a class of conceptually simple and efficient methods for simulation of non-Gaussian processes using translation process theory (collectively referred to as the Iterative Translation Approximation Method - ITAM) that iteratively upgrades the underlying Gaussian power spectral density function for simulation using the spectral representation method. However, the currently existing ITAM method for generation of non-stationary and non-Gaussian processes requires additional approximations in the estimation of the evolutionary spectrum. An extension of the ITAM is proposed that utilizes the K-L expansion. The developed method iteratively upgrades the covariance function directly and, in so doing, avoids the complex and non-unique inverse problem of estimating the evolutionary spectrum from the non-stationary autocorrelation. The application of the method for a strongly non-Gaussian and non-stationary process with a prescribed target non-Gaussian correlation function is demonstrated. |
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Conference Paper |
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Text |
Language | eng |
Notes | This collection contains the proceedings of ICASP12, the 12th International Conference on Applications of Statistics and Probability in Civil Engineering held in Vancouver, Canada on July 12-15, 2015. Abstracts were peer-reviewed and authors of accepted abstracts were invited to submit full papers. Also full papers were peer reviewed. The editor for this collection is Professor Terje Haukaas, Department of Civil Engineering, UBC Vancouver. |
Date Available | 2015-05-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0076040 |
URI | http://hdl.handle.net/2429/53180 |
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Non UBC |
Citation | Haukaas, T. (Ed.) (2015). Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12), Vancouver, Canada, July 12-15. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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