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A Hierarchical Spatiotemporal Statistical Model Motivated by Glaciology

  • Giri GopalanEmail author
  • Birgir Hrafnkelsson
  • Christopher K. Wikle
  • Håvard Rue
  • Guðfinna Aðalgeirsdóttir
  • Alexander H. Jarosch
  • Finnur Pálsson
Article
  • 64 Downloads

Abstract

In this paper, we extend and analyze a Bayesian hierarchical spatiotemporal model for physical systems. A novelty is to model the discrepancy between the output of a computer simulator for a physical process and the actual process values with a multivariate random walk. For computational efficiency, linear algebra for bandwidth limited matrices is utilized, and first-order emulator inference allows for the fast emulation of a numerical partial differential equation (PDE) solver. A test scenario from a physical system motivated by glaciology is used to examine the speed and accuracy of the computational methods used, in addition to the viability of modeling assumptions. We conclude by discussing how the model and associated methodology can be applied in other physical contexts besides glaciology.

Keywords

Model discrepancy Uncertainty quantification Emulation 

Notes

References

  1. Baum, L. E. and Petrie, T. (1966), “Statistical Inference for Probabilistic Functions of Finite State Markov Chains,” Annals of Mathematical Statistics, 37, 1554–1563,  https://doi.org/10.1214/aoms/1177699147.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Berliner, L. M. (1996), “Hierarchical Bayesian Time Series Models,” in Hanson, K. M. and Silver, R. N. (editors), Maximum Entropy and Bayesian Methods, Dordrecht: Springer Netherlands.Google Scholar
  3. — (2003), “Physical-statistical modeling in geophysics,” Journal of Geophysical Research: Atmospheres, 108, n/a–n/a,  https://doi.org/10.1029/2002JD002865. 8776.
  4. Berrocal, V., Gelfand, A., and Holland, D. (2014), “Assessing exceedance of ozone standards: a space-time downscaler for fourth highest ozone concentrations,” Environmetrics, 25, 279–291.MathSciNetCrossRefGoogle Scholar
  5. Björnsson, H. and Pálsson, F. (2008), “Icelandic glaciers,” Jökull, 58, 365–386.Google Scholar
  6. Breiman, L. (2001), “Random Forests,” Machine Learning, 45, 5–32,  https://doi.org/10.1023/A:1010933404324.CrossRefzbMATHGoogle Scholar
  7. Brinkerhoff, D. J., Aschwanden, A., and Truffer, M. (2016), “Bayesian Inference of Subglacial Topography Using Mass Conservation,” Frontiers in Earth Science, 4, 8, http://journal.frontiersin.org/article/10.3389/feart.2016.00008.
  8. Brynjarsdóttir, J. and O’Hagan, A. (2014), “Learning about physical parameters: the importance of model discrepancy,” Inverse Problems, 30, 114007, http://stacks.iop.org/0266-5611/30/i=11/a=114007.
  9. Bueler, E., Lingle, C. S., Kallen-Brown, J. A., Covey, D. N., and Bowman, L. N. (2005), “Exact solutions and verification of numerical models for isothermal ice sheets,” Journal of Glaciology, 51, 291–306.CrossRefGoogle Scholar
  10. Calderhead, B., Girolami, M., and Lawrence, N. D. (2008), “Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes,” in Proceedings of the 21st International Conference on Neural Information Processing Systems, NIPS’08, USA: Curran Associates Inc., http://dl.acm.org/citation.cfm?id=2981780.2981808.
  11. Chkrebtii, O. A., Campbell, D. A., Calderhead, B., Girolami, M. A., et al. (2016), “Bayesian Solution Uncertainty Quantification for Differential Equations,” Bayesian Analysis, 11, 1239–1267.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Conrad, P. R., Girolami, M., Särkkä, S., Stuart, A., and Zygalakis, K. (2017), “Statistical analysis of differential equations: introducing probability measures on numerical solutions,” Statistics and Computing, 27, 1065–1082.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Cressie, N. and Wikle, C. K. (2011), Statistics for Spatio-Temporal Data, John Wiley & Sons.Google Scholar
  14. Cuffey, K. M. and Paterson, W. (2010), The Physics of Glaciers, Academic Press, 4 edition.Google Scholar
  15. Flowers, G. E., Marshall, S. J., Björnsson, H., and Clarke, G. K. (2005), “Sensitivity of Vatnajökull ice cap hydrology and dynamics to climate warming over the next 2 centuries,” Journal of Geophysical Research: Earth Surface, 110.Google Scholar
  16. Fowler, A. C. and Larson, D. A. (1978), “On the Flow of Polythermal Glaciers. I. Model and Preliminary Analysis,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 363, 217–242, http://www.jstor.org/stable/79748.
  17. Friedman, J., Hastie, T., and Tibshirani, R. (2001), The Elements of Statistical Learning, volume 1, Springer series in statistics. New York, NY, NY, USA.Google Scholar
  18. Geirsson, Ó. P., Hrafnkelsson, B., and Simpson, D. (2015), “Computationally efficient spatial modeling of annual maximum 24-h precipitation on a fine grid,” Environmetrics, 26, 339–353, https://onlinelibrary.wiley.com/doi/abs/10.1002/env.2343.
  19. Golub, G. H. and Van Loan, C. F. (2012), Matrix Computations, volume 3, Johns Hopkins University Press.Google Scholar
  20. Gopalan, G., Hrafnkelsson, B., Adalgeirsdóttir, G., Jarosch, A. H., and Pálsson, F. (2018), “A Bayesian hierarchical model for glacial dynamics based on the shallow ice approximation and its evaluation using analytical solutions,” The Cryosphere, 12, 2229–2248.CrossRefGoogle Scholar
  21. Guan, Y., Haran, M., and Pollard, D. (2016), “Inferring Ice Thickness from a Glacier Dynamics Model and Multiple Surface Datasets,” ArXiv e-prints.Google Scholar
  22. Gupta, A. and Kumar, V. (1994), “A scalable parallel algorithm for sparse Cholesky factorization,” in Proceedings of the 1994 ACM/IEEE Conference on Supercomputing, Supercomputing ’94, Los Alamitos, CA, USA: IEEE Computer Society Press, http://dl.acm.org/citation.cfm?id=602770.602898.
  23. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Hooten, M. B., Leeds, W. B., Fiechter, J., and Wikle, C. K. (2011), “Assessing First-Order Emulator Inference for Physical Parameters in Nonlinear Mechanistic Models,” Journal of Agricultural, Biological, and Environmental Statistics, 16, 475–494,  https://doi.org/10.1007/s13253-011-0073-7.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Hutter, K. (1982), “A mathematical model of polythermal glaciers and ice sheets,” Geophysical & Astrophysical Fluid Dynamics, 21, 201–224,  https://doi.org/10.1080/03091928208209013.CrossRefzbMATHGoogle Scholar
  27. Kennedy, M. C. and O’Hagan, A. (2001), “Bayesian calibration of computer models,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63, 425–464.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kusnierczyk, W. (2012), rbenchmark: Benchmarking routine for R, https://CRAN.R-project.org/package=rbenchmark. R package version 1.0.0.
  29. Lehmann, E. and Casella, G. (2003), Theory of Point Estimation, Springer Texts in Statistics, Springer New York, https://books.google.com/books?id=0q-Bt0Ar-sgC.
  30. Liaw, A. and Wiener, M. (2002), “Classification and Regression by randomForest,” R News, 2, 18–22, https://CRAN.R-project.org/doc/Rnews/.
  31. Lindgren, F., Rue, H., and Lindström, J. (2011), “An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 423–498.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Liu, F. and West, M. (2009), “A dynamic modelling strategy for Bayesian computer model emulation,” Bayesian Analysis, 4, 393–411,  https://doi.org/10.1214/09-BA415.MathSciNetCrossRefzbMATHGoogle Scholar
  33. Madsen, H. (2007), Time Series Analysis, Chapman and Hall/CRC.Google Scholar
  34. Murray, I., Adams, R. P., and MacKay, D. J. (2010), “Elliptical slice sampling”, Journal of Machine Learning Research W&CP, 9, 541–548.Google Scholar
  35. Owhadi, H. and Scovel, C. (2017), “Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis”, ArXiv e-prints.Google Scholar
  36. Pagendam, D., Kuhnert, P., Leeds, W., Wikle, C., Bartley, R., and Peterson, E. (2014), “Assimilating catchment processes with monitoring data to estimate sediment loads to the Great Barrier Reef,” Environmetrics, 25, 214–229.MathSciNetCrossRefGoogle Scholar
  37. Payne, A. J., Huybrechts, P., Abe-Ouchi, A., Calov, R., Fastook, J. L., Greve, R., Marshall, S. J., Marsiat, I., Ritz, C., Tarasov, L., and Thomassen, M. P. A. (2000), “Results from the EISMINT model intercomparison: the effects of thermomechanical coupling,” Journal of Glaciology, 46, 227–238.CrossRefGoogle Scholar
  38. Robert, C. (2007), The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, Springer Texts in Statistics, Springer New York, https://books.google.com/books?id=NQ5KAAAAQBAJ.
  39. Rue, H. (2001), “Fast sampling of Gaussian Markov random fields,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63, 325–338.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Rue, H. and Held, L. (2005), Gaussian Markov Random Fields: Theory and Applications, CRC press.Google Scholar
  41. Salter, J. M., Williamson, D. B., Scinocca, J., and Kharin, V. (2019), “Uncertainty Quantification for Computer Models With Spatial Output Using Calibration-Optimal Bases,” Journal of the American Statistical Association, 0, 1–24,  https://doi.org/10.1080/01621459.2018.1514306.
  42. Shen, X. and Wasserman, L. (2001), “Rates of convergence of posterior distributions,” Annals of Statistics, 29, 687–714,  https://doi.org/10.1214/aos/1009210686.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Sigurdarson, A. N. and Hrafnkelsson, B. (2016), “Bayesian prediction of monthly precipitation on a fine grid using covariates based on a regional meteorological model,” Environmetrics, 27, 27–41, https://ideas.repec.org/a/wly/envmet/v27y2016i1p27-41.html.
  44. Solin, A. and Särkkä, S. (2014), “Explicit Link Between Periodic Covariance Functions and State Space Models,” in Kaski, S. and Corander, J. (editors), Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, volume 33 of Proceedings of Machine Learning Research, Reykjavik, Iceland: PMLR, http://proceedings.mlr.press/v33/solin14.html.
  45. van der Vaart, A. (2000), Asymptotic Statistics, Asymptotic Statistics, Cambridge University Press, https://books.google.com/books?id=UEuQEM5RjWgC.
  46. van der Veen, C. (2013), Fundamentals of Glacier Dynamics, CRC Press, 2 edition.Google Scholar
  47. Whittle, P. (1954), “ON STATIONARY PROCESSES IN THE PLANE,” Biometrika, 434–449.Google Scholar
  48. — (1963), “Stochastic processes in several dimensions,” Bulletin of the International Statistical Institute, 40, 974–994.MathSciNetzbMATHGoogle Scholar
  49. Wikle, C. K. (2016), Hierarchical Models for Uncertainty Quantification: An Overview, Springer International Publishing, 1–26.Google Scholar
  50. Wikle, C. K., Berliner, L. M., and Cressie, N. (1998), “Hierarchical Bayesian space-time models,” Environmental and Ecological Statistics, 5, 117–154,  https://doi.org/10.1023/A:1009662704779.CrossRefGoogle Scholar
  51. Wikle, C. K., Milliff, R. F., Nychka, D., and Berliner, L. M. (2001), “Spatiotemporal Hierarchical Bayesian Modeling Tropical Ocean Surface Winds,” Journal of the American Statistical Association, 96, 382–397.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Zammit-Mangion, A., Rougier, J., Bamber, J., and Schön, N. (2014), “Resolving the Antarctic contribution to sea-level rise: a hierarchical modelling framework,” Environmetrics, 25, 245–264.MathSciNetCrossRefGoogle Scholar

Copyright information

© International Biometric Society 2019

Authors and Affiliations

  • Giri Gopalan
    • 1
    Email author
  • Birgir Hrafnkelsson
    • 1
  • Christopher K. Wikle
    • 2
  • Håvard Rue
    • 3
  • Guðfinna Aðalgeirsdóttir
    • 1
  • Alexander H. Jarosch
    • 4
  • Finnur Pálsson
    • 1
  1. 1.University of IcelandReykjavíkIceland
  2. 2.University of MissouriColumbiaUSA
  3. 3.King Abdullah University of Science and TechnologyThuwalSaudi Arabia
  4. 4.University of InnsbruckInnsbruckAustria

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