Abstract
We present an approach for estimating physical parameters in nonlinear models that relies on an approximation to the mechanistic model itself for computational efficiency. The proposed methodology is validated and applied in two different modeling scenarios: (a) Simulation and (b) lower trophic level ocean ecosystem model. The approach we develop relies on the ability to predict right singular vectors (resulting from a decomposition of computer model experimental output) based on the computer model input and an experimental set of parameters. Critically, we model the right singular vectors in terms of the model parameters via a nonlinear statistical model. Specifically, we focus our attention on first-order models of these right singular vectors rather than the second-order (covariance) structure.
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Bayarri, M. J., Berger, J. O., Paulo, R., Sacks, J., Cafeo, J. A., Cavendish, J., Lin, C. H., and Tu, J. (2007a), “A Framework for Validation of Computer Models,” Technometrics, 49, 138–154.
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.
Bliznyuk, N., Ruppert, D., Shoemaker, C. A., Regis, R., Wild, S., and Mugunthan, P. (2008), “Bayesian Calibration of Computationally Expensive Models Using Optimization and Radial Basis Function Approximation,” Journal of Computational and Graphical Statistics, 17, 270–294.
Breiman, L. (2001), “Random Forests,” Machine Learning, 45, 5–32.
Cangelosi, A. R., and Hooten, M. B. (2009), “Models for Bounded Systems with Continuous Dynamics,” Biometrics, 65, 850–856.
Conti, S., Gosling, J. P., Oakley, J. E., and O’Hagan, A. (2009), “Gaussian Process Emulation and Dynamic Computer Codes,” Biometrika, 96, 663–676.
Craig, P. S., Goldstein, M., Rougier, J. C., and Seheult, A. H. (2001), “Bayesian Forecasting for Complex Systems using Computer Simulators,” Journal of the American Statistical Association, 96, 717–729.
Crawford, W. R., Brickley, P. J., and Thomas, A. C. (2007), “Mesoscale Eddies Determine Phytoplankton Distribution in Northern Gulf of Alaska,” Progress in Oceanography, 75, 287–303.
Cutler, D. R., Edwards, T. C., Beard, K. H., Cutler, A., Hess, K. T., Gibson, J., and Lawler, J. J. (2007), “Random Forests for Classification in Ecology,” Ecology, 88, 2783–2792.
Drignei, D. (2008), “Fast Statistical Surrogates for Dynamical 3D Computer Models of Brain Tumors,” Journal of Computational and Graphical Statistics, 17, 859–884.
Fiechter, J., and Moore, A. M. (2009), “Interannual Spring Bloom Variability and Ekman Pumping in the Coastal Gulf of Alaska,” Journal Geophysical Research, 114, C06004.
Fiechter, J., Moore, A. M., Edwards, C. A., Bruland, K. W., Di Lorenzo, E., Lewis, C. V. W., Powell, T. M., Curchitser, E. N., and Hedstrom, K. (2009), “Modeling Iron Limitation of Primary Production in the Coastal Gulf of Alaska,” Deep Sea Research II, 56, 2503–2519.
Frolov, S., Baptista, A. M., Leen, T. K., Lu, Z., and van der Merwe, R. (2009), “Fast Data Assimilation Using a Nonlinear Kalman Filter and a Model Surrogate: An Application to the Columbia River Estuary,” Dynamics of Atmospheres and Oceans, 48, 16–45.
Gentle, J. E. (2007), Matrix Algebra: Theory, Computations, and Applications in Statistics, New York: Springer.
Haidvogel, D. B., Arango, H., Budgell, W. P., Cornuelle, B. D., Curchitser, E. N., Di Lorenzo, E., Fennel, K., Geyer, W. R., Hermann, A. J., Lanerolle, L., Levin, J., McWilliams, J. C., Miller, A. J., Moore, A. M., Powell, T. M., Shchepetkin, A. F., Sherwood, C. R., Signell, R. P., Warner, J. C., and Wilkin, J. (2008), “Ocean Forecasting in Terrain-Following Coordinates: Formulation and Skill Assessment of the Regional Ocean Modeling System,” Journal of Computational Physics, 227, 3595–3624.
Hastie, T., Tibshirani, R., and Friedman, J. (2009), Elements of Statistical Learning (2nd ed.), New York: Springer.
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 on Scientific Computing, 26, 448–466.
Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008), “Computer Model Calibration Using High-Dimensional Output,” Journal of the American Statistical Association, 103, 570–583.
Kennedy, M. C., and O’Hagan, A. (2001), “Bayesian Calibration of Computer Models,” Journal of the Royal Statistical Society: Series B, 63, 425–464.
Liu, F., and West, M. (2009), “A Dynamic Modelling Strategy for Bayesian Computer Model Emulation,” Bayesian Analysis, 4, 393–412.
O’Hagan, A. (2006), “Bayesian Analysis of Computer Code Outputs: A Tutorial,” Reliability Engineering and System Safety, 91, 1290–1300.
Powell, T. M., Lewis, C. V. W., Curchitser, E. N., Haidvogel, D. B., Hermann, A. J., and Dobbins, E. L. (2006), “Results from a Three-Dimensional, Nested, Biological-Physical Model of the California Current System and Comparisons with Statistics from Satellite Imagery,” Journal of Geophysical Research, 111 (C0), 7018.
Roberts, G. O., and Rosenthal, J. S. (2009), “Examples of Adaptive MCMC,” Journal of Computational and Graphical Statistics, 18, 349–367.
Rougier, J. C. (2008), “Efficient Emulators for Multivariate Deterministic Functions,” Journal of Computational and Graphical Statistics, 17, 827–843.
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989), “Design and Analysis of Computer Experiments,” Statistical Science, 4, 409–423.
Stabeno, P. J., Bond, N. A., Hermann, A. J., Kachel, N. B., Mordy, C. W., and Overland, J. E. (2004), “Meteorology and Oceanography of the Northern Gulf of Alaska,” Continental Shelf Research, 24, 859–897.
van der Merwe, R., Leen, T. K., Lu, Z., Frolov, S., and Baptista, A. M. (2007), “Fast Neural Network Surrogates for Very High Dimensional Physics-Based Models in Computational Oceanography,” Neural Networks, 20, 462–478.
Ward, B. A., Friedrichs, M. A. M., Anderson, T. R., and Oschlies, A. (2010), “Parameter Optimisation Techniques and the Problem of Underdetermination in Marine Biogeochemical Models,” Journal of Marine Systems, 81, 34–43.
Xiao, X., White, E. P., Hooten, M. B., and Durham, S. L. (2011), “On the Use of Log-Transformation Versus Nonlinear Regression for Analyzing Biological Power-Laws,” Ecology, 92, 1887–1894.
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Hooten, M.B., Leeds, W.B., Fiechter, J. et al. Assessing First-Order Emulator Inference for Physical Parameters in Nonlinear Mechanistic Models. JABES 16, 475–494 (2011). https://doi.org/10.1007/s13253-011-0073-7
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DOI: https://doi.org/10.1007/s13253-011-0073-7