×

Uncertainty analysis for computationally expensive models with multiple outputs. (English) Zbl 1302.62279

Summary: Bayesian MCMC calibration and uncertainty analysis for computationally expensive models is implemented using the SOARS (Statistical and Optimization Analysis using Response Surfaces) methodology. SOARS uses a radial basis function interpolator as a surrogate, also known as an emulator or meta-model, for the logarithm of the posterior density. To prevent wasteful evaluations of the expensive model, the emulator is built only on a high posterior density region (HPDR), which is located by a global optimization algorithm. The set of points in the HPDR where the expensive model is evaluated is determined sequentially by the GRIMA algorithm described in detail in another paper but outlined here. Enhancements of the GRIMA algorithm were introduced to improve efficiency. A case study uses an eight-parameter SWAT2005 (Soil and Water Assessment Tool) model where daily stream flows and phosphorus concentrations are modeled for the Town Brook watershed which is part of the New York City water supply. A Supplemental Material file available online contains additional technical details and additional analysis of the Town Brook application.

MSC:

62P12 Applications of statistics to environmental and related topics

Software:

CONDOR; SemiPar; ORBIT
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arnold, J. G., Srinivasan, R., Muttiah, R. R., and Williams, J. R. (1998), ”Large Area Hydrologic Modeling and Assessment. Part I: Model Development,” Journal of the American Water Resources Association, 34, 73–89.
[2] Bates, D. M., and Watts, D. G. (1988), Nonlinear Regression Analysis and its Applications, New York: Wiley. · Zbl 0728.62062
[3] Bayarri, M. J., Berger, J. O., Paulo, R., Sacks, J., Cafeo, J. A., Cavendish, C.-H., and Tu, J. (2007a), ”A Framework for Validation of Computer Models,” Technometrics, 49, 138–154.
[4] Bayarri, M. J., Berger, J. O., Cafeo, J., Garcia-Donato, G., Liu, F., Palomo, J., Parthasarathy, R. J., Paulo, R., Sacks, J., and Walsh, D. (2007b), ”Computer Model Validation with Functional Output,” The Annals of Statistics, 35, 1874–1906. · Zbl 1144.62368
[5] Bliznyuk, N., Ruppert, D., and Shoemaker, C. A. (2011), ”Bayesian Inference Using Efficient Interpolation of Computationally Expensive Densities With Variable Parameter Costs,” Journal of Computational and Graphical Statistics, 20, 636–655.
[6] – (2012), ”Local Derivative-Free Approximation of Computationally Expensive Posterior Densities,” Journal of Computational and Graphical Statistics. doi: 10.1080/10618600.2012.681255 .
[7] Bliznyuk, N., Ruppert, D., Shoemaker, C. A., Regis, R., Wild, S., and Mugunthan, P. (2008), ”Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation,” Journal of Computational and Graphical Statistics, 17, 270–294.
[8] Box, G. E. P., and Cox, D. R. (1964), ”An Analysis of Transformations,” Journal of the Royal Statistical Society, Series B, 26, 211–246. · Zbl 0156.40104
[9] Buhmann, M. D. (2003), Radial Basis Functions, New York: Cambridge University Press. · Zbl 1038.41001
[10] Carroll, R. J., and Ruppert, D. (1984), ”Power Transformation When Fitting Theoretical Models to Data,” Journal of the American Statistical Association, 79, 321–328.
[11] – (1988), Transformation and Weighting in Regression, New York: Chapman & Hall. · Zbl 0666.62062
[12] Cumming, J. A., and Goldstein, M. (2009), ”Small Sample Bayesian Designs for Complex High-Dimensional Models Based on Information Gained Using Fast Approximations,” Journal of the American Statistical Association, 51, 377–388.
[13] Eckhardt, K., Haverkamp, S., Fohrer, N., and Frede, H. G. (2002), ”SWAT-G, A Version of SWAT99.2 Modified for Application to Low Mountain Range Catchments,” Physics and Chemistry of the Earth, 27, 641–644.
[14] Grizzetti, B., Bouraoui, F., Granlund, K., Rekolainen, S., and Bidoglio, G. (2003), ”Modelling Diffuse Emission and Retention of Nutrients in the Vantaanjoki Watershed (Finland) Using the SWAT Model,” Ecological Modelling, 169, 25–38.
[15] Hamilton, J. D. (1994), Time Series Analysis, Princeton: Princeton University Press. · Zbl 0831.62061
[16] Higdon, D., Kennedy, M., Cavendish, J. C., Cafeo, J. A., and Ryne, R. D. (2004), ”Combining Field Data and Computer Simulations for Calibration and Prediction,” SIAM Journal of Scientific Computation, 26, 448–466. · Zbl 1072.62018
[17] Kennedy, M. C., and O’Hagan, A. (2001), ”Bayesian Calibration of Computer Models,” Journal of the Royal Statistical Society, Series B, 63, 425–464. · Zbl 1007.62021
[18] Levy, S., and Steinberg, D. M. (2010), ”Computer Experiments: A Review,” AStA Advances in Statistical Analysis, 94, 311–324.
[19] Qian, P. Z. G. (2009), ”Nested Latin Hypercube Design,” Biometrika, 96, 957–970. · Zbl 1179.62103
[20] Qian, P. Z. G., and Wu, C. F. J. (2008), ”Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments,” Technometrics, 50, 192–204.
[21] Rasmussen, C. E. (2003), ““Gaussian Processes to Speed up Hybrid Monte Carlo for Expensive Bayesian Integrals,” in Bayesian Statistics 7, eds. J. M. Bernardo, J. O. Berger, A. P. Berger, and A. F. M. Smith, pp. 651–659.
[22] Regis, R. G., and Shoemaker, C. A. (2009), ”Parallel Stochastic Global Optimization Using Radial Basis Functions,” INFORMS Journal on Computing, 21, 411–426. · Zbl 1243.90160
[23] Ruppert, D., Wand, M. P., and Carroll, R. J. (2003), Semiparametric Regression, Cambridge: Cambridge University Press. · Zbl 1038.62042
[24] Santner, T. J., Williams, B. J., and Notz, W. I. (2010), The Design and Analysis of Computer Experiments, New York: Springer. · Zbl 1041.62068
[25] Shoemaker, C. A., Regis, R., and Fleming, R. (2007), ”Watershed Calibration Using Multistart Local Optimization and Evolutionary Optimization With Radial Basis Function Approximation,” Journal of Hydrologic Science, 52, 450–465.
[26] Singh, A. (2011), ”Global Optimization of Computationally Expensive Hydrologic Simulation Models,” Ph.D. Thesis in Civil and Environmental Engineering, Cornell University.
[27] Tierney, L. (1994), ”Markov Chains for Exploring Posterior Distributions,” The Annals of Statistics, 22, 1701–1786. · Zbl 0829.62080
[28] Tierney, L., and Kadane, J. (1986), ”Accurate Approximations for Posterior Moments and Marginal Densities,” Journal of the American Statistical Association, 81, 82–86. · Zbl 0587.62067
[29] Tjelmeland, H., and Hegstad, B. K. (2001), ”Model Jumping Proposals in MCMC,” Scandinavian Journal of Statistics, 28, 205–223. · Zbl 0972.65005
[30] Tolson, B. (2005), ”Automatic Calibration, Management and Uncertainty Analysis: Phosphorus Transport in the Cannonsville Watershed, ” Ph.D. dissertation, School of Civil and Environmental Engineering, Cornell University.
[31] Tolson, B., and Shoemaker, C. A. (2004), ”Watershed Modeling of the Cannonsville Basin Using SWAT2000: Model Development, Calibration and Validation for the Prediction of Flow, Sediment and Phosphorus Transport to the Cannonsville Reservoir, Version 1,” Technical Report, School of Civil and Environmental Engineering, Cornell University. Available at http://ecommons.library.cornell.edu/handle/1813/2710 .
[32] – (2007a), ”The Dynamically Dimensioned Search Algorithm for Computationally Efficient Automatic Calibration of Environmental Simulation Models,” Water Resources Research, 43, W01413. doi: 10.1029/2005WR004723 .
[33] – (2007b), ”Cannonsville Reservoir Watershed SWAT2000 Model Development, Calibration and Validation,” Journal of Hydrology, 337, 68–89 doi: 10.1016/j.jhydrol.2007.01.017 .
[34] Vanden Berghen, F., and Bersini, H. (2005), ”CONDOR, a New Parallel, Constrained Extension of Powell’s UOBYQA Algorithm: Experimental Results and Comparison With the DFO Algorithm,” Journal of Computational and Applied Mathematics, 181, 157–175. · Zbl 1072.65088
[35] Wild, S. M., and Shoemaker, C. A. (2011), ”Global Convergence of Radial Basis Functions Trust Region Derivative-Free Algorithms,” SIAM Journal on Optimization, 20, 387–415. · Zbl 1397.65024
[36] Wild, S. M., Regis, R. G., and Shoemaker, C. A. (2007), ”ORBIT: Optimization by Radial Basis Function Interpolation in Trust-Regions,” SIAM Journal on Scientific Computing, 30, 3197–3219. · Zbl 1178.65065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.