Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency

Michal Branicki, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation-dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable perfect system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.

Original languageEnglish (US)
Pages (from-to)2543-2578
Number of pages36
JournalNonlinearity
Volume25
Issue number9
DOIs
StatePublished - Sep 2012

Fingerprint

Model Error
Intermittency
intermittency
Climate
Imperfect
climate
Uncertainty
Prediction
predictions
Fluctuation-dissipation Theorem
sensitivity
dynamical systems
Moment Closure
Dynamical systems
Complex Dynamical Systems
dissipation
theorems
Coarse-graining
Materials Science
Model

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency. / Branicki, Michal; Majda, Andrew J.

In: Nonlinearity, Vol. 25, No. 9, 09.2012, p. 2543-2578.

Research output: Contribution to journalArticle

@article{3c19fa3d49884b13a755bd5ee7b1426f,
title = "Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency",
abstract = "This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation-dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable perfect system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.",
author = "Michal Branicki and Majda, {Andrew J.}",
year = "2012",
month = "9",
doi = "10.1088/0951-7715/25/9/2543",
language = "English (US)",
volume = "25",
pages = "2543--2578",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing Ltd.",
number = "9",

}

TY - JOUR

T1 - Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency

AU - Branicki, Michal

AU - Majda, Andrew J.

PY - 2012/9

Y1 - 2012/9

N2 - This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation-dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable perfect system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.

AB - This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation-dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable perfect system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.

UR - http://www.scopus.com/inward/record.url?scp=84865527649&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865527649&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/25/9/2543

DO - 10.1088/0951-7715/25/9/2543

M3 - Article

AN - SCOPUS:84865527649

VL - 25

SP - 2543

EP - 2578

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 9

ER -