每年出版 6 期
ISSN 打印: 2152-5080
ISSN 在线: 2152-5099
Indexed in
REPLICATION-BASED EMULATION OF THE RESPONSE DISTRIBUTION OF STOCHASTIC SIMULATORS USING GENERALIZED LAMBDA DISTRIBUTIONS
摘要
Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family−the generalized lambda distribution−to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.
-
Rasmussen, C.E. and Williams, C.K.I., Gaussian Processes for Machine Learning, Adaptive Computation and Machine Learning Series, Cambridge, MA: The MIT Press, 2006.
-
Xiu, D. and Karniadakis, G.E., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput, 24(2):619-644,2002.
-
Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, 2nd ed., Mineola, NY: Courier Dover Publications, 2003.
-
McCullagh, P. and Nelder, J., Generalized Linear Models, Monographs on Statistics and Applied Probability, Boca Raton, FL: Chapman and Hall/CRC, 1989.
-
Iooss, B. and Ribatet, M., Global Sensitivity Analysis of Computer Models with Functional Inputs, Reliab. Eng. Sys. Saf., 94:1194-1204, 2009.
-
Ankenman, B., Nelson, B., and Staum, J., Stochastic Kriging for Simulation Metamodeling, Oper. Res, 58:371-382, 2009.
-
Dacidian, M. and Carroll, R.J., Variance Function Estimation, J. Am. Stat. Assoc., 400:1079-1091,1987.
-
Fan, J. and Yao, Q.W., Efficient Estimation of Conditional Variance Functions in Stochastic Regression, Biometrika, 85:645-660, 1998.
-
Marrel, A., Iooss, B., Da Veiga, S., and Ribatet, M., Global Sensitivity Analysis of Stochastic Computer Models with Joint Metamodels, Stat. Comput, 22:833-847, 2012.
-
Bhattacharya, P.K. and Gangopadhyay, A.K., Kernel and Nearest-Neighbor Estimation of a Conditional Quantile, Ann. Stat., 18(3):1400-1415, 1990.
-
Plumlee, M. and Tuo, R., Building Accurate Emulators for Stochastic Simulations via Quantile Kriging, Technometrics, 56:466-473,2014.
-
Koenker, R., Quantile Regression: 40 Years on, Annu. Rev. Econ., 9:155-176,2017.
-
Hastie, T. and Tibshirani, R., Generalized Additive Models, Monographs on Statistics and Applied Probability, Vol. 43, Boca Raton, FL: Chapman and Hall/CRC, 1990.
-
Fan, J. and Gijbels, I., Local Polynomial Modelling and Its Applications, Monographs on Statistics and Applied Probability, Vol. 66, Boca Raton, FL: Chapman and Hall/CRC, 1996.
-
Hall, P., Racine, J., and Li, Q., Cross-Validation and the Estimation of Conditional Probability Densities, J. Am. Stat. Assoc., 99:1015-1026, 2004.
-
Efromovich, S., Dimension Reduction and Adaptation in Conditional Density Estimation, J. Am. Stat. Assoc., 105:761-774, 2010.
-
Tsybakov, A.B., Introduction to Nonparametric Estimation, Springer Series in Statistics, Cambridge, MA: Springer, 2009.
-
Moutoussamy, V., Nanty, S., and Pauwels, B., Emulators for Stochastic Simulation Codes, ESAIM: Math. Model. Numer. Anal, 48:116-155,2015.
-
Browne, T., Iooss, B., Le Gratiet, L., Lonchampt, J., and Remy, E., Stochastic Simulators based Optimization by Gaussian Process Metamodels-Application to Maintenance Investments Planning Issues, Quality Reliab. Eng. Int., 32(6):2067-2080, 2016.
-
Karian, Z.A. and Dudewicz, E.J., Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Boca Raton, FL: Chapman and Hall/CRC, 2000.
-
Xiu, D., Numerical Methods for Stochastic Computations-A Spectral Method Approach, Princeton, NJ: Princeton University Press, 2010.
-
Freimer, M., Kollia, G., Mudholkar, G.S., and Lin, C.T., A Study of the Generalized Tukey Lambda Family, Commun. Stat. Theory Methods, 17:3547-3567, 1988.
-
Chalabi, Y., Scott, D.J., and Wiirtz, D., The Generalized Lambda Distribution as an Alternative to Model Financial Returns, Eidgenossische Technische Hochschule, Zurich, and University of Auckland, New Zealand, No. 2009-01, 2011.
-
Karian, Z.A. andDudewicz, E.J., Handbook of Fitting Statistical Distributions withR, New York: Taylor & Francis, 2010.
-
Corlu, C.G. and Meterelliyoz, M., Estimating the Parameters of the Generalized Lambda Distribution: Which Method Performs Best?, Commun. Stat. Simul. Comput., 45(7):2276-2296,2016.
-
Lakhany, A. and Massuer, H., Estimating the Parameters of the Generalised Lambda Distribution, Algo Res. Q., 3(3):47-58, 2000.
-
Su, S., Numerical Maximum Log Likelihood Estimation for Generalized Lambda Distributions, Comput. Stat. Data Anal., 51(8):3983-3998, 2007.
-
Burden, R.L., Faires, J.D., and Burden, A.M., Numerical Analysis, Boston: Cengage Learning, 2015.
-
Ernst, O.G., Mugler, A., Starkloff, H.J., and Ullmann, E., On the Convergence of Generalized Polynomial Chaos Expansions, ESAIM: Math. Modell. Numer. Anal., 46:317-339,2012.
-
Sudret, B., Polynomial Chaos Expansions and Stochastic Finite Element Methods, in Risk and Reliability in Geotechnical Engineering, Chap. 6, New York: Taylor and Francis, pp. 265-300, 2015.
-
Berveiller, M., Sudret, B., and Lemaire, M., Stochastic Finite Elements: A Non-Intrusive Approach by Regression, Eur. J. Comput. Mech., 15(1-3):81-92, 2006.
-
Blatman, G. and Sudret, B., Adaptive Sparse Polynomial Chaos Expansion based on Least Angle Regression, J. Comput. Phys, 230:2345-2367, 2011.
-
Blatman, G. and Sudret, B., An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis, Probab. Eng. Mech, 25:183-197, 2010.
-
Marelli, S. and Sudret, B., UQLab User Manual-Polynomial Chaos Expansions, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, Tech. Rep. UQLab-V1.3-104, 2019.
-
Marelli, S. and Sudret, B., UQLAB: A Framework for Uncertainty Quantification in Matlab, Vulnerability, Uncertainty, and Risk, Proc. of 2nd Int. Conf. on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, pp. 2554-2563,2014.
-
Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R., Least Angle Regression, Ann. Stat., 32:407-499, 2004.
-
Tropp, J.A. and Gilbert, A.C., Signal Recovery from Random Measurements via Orthogonal Matching Pursuit, IEEE Trans. Inf. Theory, 53(12):4655-4666, 2007.
-
Torre, E., Marelli, S., Embrechts, P., and Sudret, B., Data-Driven Polynomial Chaos Expansion for Machine Learning Regression, J. Comput. Phys, 388:601-623, 2019.
-
Steihaug, T., The Conjugate Gradient Method and Trust Regions in Large Scale Optimization, SIAMJ. Num. Anal, 20(3):626-637, 1983.
-
Arnold, D.V. and Hansen, N., A (1+1)-CMA-ES for Constrained Optimisation, Proc. of the Genetic and Evolutionary Computation Conference 2012 (GECCO2012), T. Soule and J.H. Moore, Eds., pp. 297-304, 2012.
-
Moustapha, M., Lataniotis, C., Wiederkehr, P., Wagner, P.-R., Wicaksono, D., Marelli, S., and Sudret, B., UQLab User Manual, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, Tech. Rep. UQLab-V1.3-201, 2019.
-
Sobol', I.M., Distribution of Points in a Cube and Approximate Evaluation of Integrals, U.S.S.R Comput. Math. Math. Phys, 7:86-112, 1967.
-
McNeil, A.J., Frey, R., and Embrechts, P., Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton, NJ: Princeton University Press, 2005.
-
Gray, A., Greenhalgh, D., Hu, L., Mao, X., and Pan, J., A Stochastic Differential Equation SIS Epidemic Model, SIAMJ. Appl. Math, 71(3):876-902,2011.
-
Iversen, E.B., Morales, J.M., Moller, J.K., Mao, X., and Madsen, H., Short-Term Probabilistic Forecasting of Wind Speed Using Stochastic Differential Equations, Int. J. Forecast., 32:981-990,2015.
-
Jimenez, M.N., Le Maitre, O.P., and Knio, O.M., Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations, SIAMJ. Uncertainty Quantif, 5:212-239, 2017.
-
Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Berlin: Springer, 1992.
-
Wand, M. and Jones, M.C., Kernel Smoothing, Boca Raton, FL: Chapman and Hall, 1995.
-
Jonkman, B.J., TurbSim User's Guide: Version 1.50, National Renewable Energy Laboratory, Tech. Rep. NREL/TP-500-46198,2009.
-
Abdallah, I., Lataniotis, C., and Sudret, B., Parametric Hierarchical Kriging for Multi-Fidelity Aero-Servo-Elastic Simulators-Application to Extreme Loads on Wind Turbines, Probab. Eng. Mech., 55:67-77, 2019.
-
Jonkman, J., Butterfield, S., Musial, W., and Scott, G., Definition of a 5-MW Reference Wind Turbine for Offshore System Development, National Renewable Energy Laboratory, Tech. Rep., 2009.
-
McKay, M.D., Beckman, R.J., and Conover, W. J., A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 21(2):239-245, 1979.
-
Slot, R., Sorensen, J.D., Sudret, B., Svenningsen, L., and Thogersen, M., Surrogate Model Uncertainty in Wind Turbine Reliability Assessment, Renew. Energy, vol. 151, pp. 1150-1162,2020.
-
Rosenblatt, M., Remarks on a Multivariate Transformation, Ann. Math. Stat., 23:470-472,1952.
-
Zhu, X. and Sudret, B., Emulation of Stochastic Simulators Using Generalized Lambda Models, SIAMJ. Uncertainty Quantif., 2020, (submitted).
-
Tsokanas Nikolaos, Zhu Xujia, Abbiati Giuseppe, Marelli Stefano, Sudret Bruno, Stojadinović Božidar, A Global Sensitivity Analysis Framework for Hybrid Simulation with Stochastic Substructures, Frontiers in Built Environment, 7, 2021. Crossref
-
Zhu Xujia, Sudret Bruno, Emulation of Stochastic Simulators Using Generalized Lambda Models, SIAM/ASA Journal on Uncertainty Quantification, 9, 4, 2021. Crossref
-
Singh D, Dwight R P, Laugesen K, Beaudet L, Viré A, Probabilistic surrogate modeling of offshore wind-turbine loads with chained Gaussian processes, Journal of Physics: Conference Series, 2265, 3, 2022. Crossref
-
Abbiati Giuseppe, Marelli Stefano, Ligeikis Connor, Christenson Richard, Stojadinović Božidar, Training of a Classifier for Structural Component Failure Based on Hybrid Simulation and Kriging, Journal of Engineering Mechanics, 148, 1, 2022. Crossref
-
Zhu Xujia, Sudret Bruno, Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models, Reliability Engineering & System Safety, 214, 2021. Crossref
-
Lüthen Nora, Marelli Stefano, Sudret Bruno, Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark, SIAM/ASA Journal on Uncertainty Quantification, 9, 2, 2021. Crossref