Fundamental limitations of ad hoc linear and quadratic multi-level regression models for physical systems

Discrete and Continuous Dynamical Systems - Series B

Volumn 17, Issue 4, 2012, Pages 1333-1363

  Majda, Andrew J ^a   Yuan, Yuan ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Instability; Model error; Nonlinear regression models; Unphysical blow up

Indexed keywords

EID: 84861676871     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2012.17.1333     Document Type: Article

References (26)

1. A. J. Majda, I. Timofeyev and E. Vanden Eijnden, Models for stochastic climate prediction, PNAS, 96 (1999), 14687-14691. (Pubitemid 30019700)
2. A. J. Majda, I. Timofeyev and E. Vanden Eijnden, A mathematical framework for stochastic climate models, Commun. Pure Appl. Math., 54 (2001), 891-974.
3. A. J. Majda, C. Franzke and B. Khouider, An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2429-2455.
4. A. J. Majda, R. Abramov and M. Grote, "Information Theory and Stochastics for Multiscale Nonlinear Systems," CRM Monograph Series, 25, American Mathematical Society, Providence, RI, 2005.
5. I. Horenko, Finite element approach to clustering of multidimensional time series, SIAM J. Sci. Comput., 32 (2010), 62-83.
6. I. Horenko, On the identification of nonstationary factor models and their application to atmospheric data analysis, J. Atmos. Sci., 67 (2010), 1559-1574.
7. D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains, J. Atmos. Sci., 65 (2008), 2661-2675.
8. A. J. Majda, C. Franzke and D. Crommelin, Normal forms for reduced stochastic climate models, PNAS, 106 (2009), 3649-3653.
9. A. J. Majda, C. Franzke, A. Fischer and D. Crommelin, Distinct metastable atmospheric regimes despite nearly Gaussian statistics: A paradigm model, PNAS, 103 (2006), 8309-8314. (Pubitemid 43838451)
10. C. Franzke, A. J. Majda and E. Vanden-Eijnden, Low-order stochastic mode reduction for a realistic barotropic model climate, J. Atmos. Sci., 62 (2005), 1722-1745. (Pubitemid 41135963)
11. T. Delsole, Stochastic models of quasigeostrophic turbulence, Surv. Geophys., 25 (2004), 107-149. (Pubitemid 38783891)
12. A. J. Majda, J. Harlim and B. Gershgorin, Mathematical strategies for filtering turbulent dynamical systems, Discrete and Continuous Dynamical Systems, 27 (2010), 441-486.
13. A. J. Majda, B. Gershgorin and Y. Yuan, Low frequency response and fluctuation-dissipation Theorems: Theory and practice, J. Atmos. Sci., 67 (2010), 1186-1201.
14. A. J. Majda, R. Abramov and B. Gershgorin, High skill in low-frequency climate response through fluctuation dissipation theorems despite structural instability, PNAS, 107 (2010), 581-586.
15. S. Kravtsov, D. Kondrashov and M. Ghil, Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability, J. Climate, 18 (2005), 4404-4424. (Pubitemid 41805480)
16. D. Kondrashov, S. Kravtsov, A. W. Robertson and M. Ghil, A hierarchy of data-based ENSO models, J. Climate, 18 (2005), 4425-4444. (Pubitemid 41805481)
17. D. Kondrashov, S. Kravtsov and M. Ghil, Empirical mode reduction in a model of extratropical low-frequency variability, J. Atmos. Sci, 63 (2006), 1859-1877. (Pubitemid 44184030)
18. G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions, J. Stat. Phys., 127 (2007), 741-781.
19. R. Azencott, A. Beri and I. Timofeyev, Adaptive sub-sampling for parametric estimation of Gaussian diffusions, J. Stat. Phys., 139 (2010), 1066-1089.
20. R. Azencott, A. Beri and I. Timofeyev, Sub-sampling and parametric estimation for multiscale dynamics, submitted, 2010.
21. Y. Yuan and A. J. Majda, Invariant measures and asymptotic gaussian bounds for normal forms of stochastic climate model, Chinese Annals of Mathematics, Series B, 32 (2011), 343-368.
22. A. J. Majda and B. Gershgorin, Quantifying uncertainty in climate change science through empirical information theory, PNAS, 107 (2010), 14958-14963.
23. C. Franzke, I. Horenko, A. J. Majda and R. Klein, Systematic metastable atmospheric regime identification in a AGCM, J. Atmos. Sci., 66 (2009), 1997-2012.
24. J. Berner and G. Branstator, Linear and nonlinear signatures in planetary wave dynamics of an AGCM: Probability density functions, J. Atmos. Sci., 64 (2007), 117-136. (Pubitemid 46152679)
25. R. Z. Has'minskiǐ, A limit theorem for solutions of Differential equations with a random right-hand part, Teor. Verojatnost. i Primenen, 11 (1966), 444-462.
26. T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis, 12 (1973), 55-67.