Filtering nonlinear turbulent dynamical systems through conditional Gaussian statistics

Nan Chen, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

In this paper, a general conditional Gaussian framework for filtering complex turbulent systems is introduced. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the filter allows closed analytical formulas for updating the posterior states and is thus computationally efficient. An information-theoretic framework is developed to assess the model error in the filter estimates. Three types of applications in filtering conditional Gaussian turbulent systems with model error are illustrated. First, dyad models are utilized to illustrate that ignoring the energy-conserving nonlinear interactions in designing filters leads to significant model errors in filtering turbulent signals from nature. Then a triad (noisy Lorenz 63) model is adopted to understand the model error due to noise inflation and underdispersion. It is also utilized as a test model to demonstrate the efficiency of a novel algorithm, which exploits the conditional Gaussian structure, to recover the time-dependent probability density functions associated with the unobserved variables. Furthermore, regarding model parameters as augmented state variables, the filtering framework is applied to the study of parameter estimation with detailed mathematical analysis. A new approach with judicious model error in the equations associated with the augmented state variables is proposed, which greatly enhances the efficiency in estimating model parameters. Other examples of this framework include recovering random compressible flows from noisy Lagrangian tracers, filtering the stochastic skeleton model of the Madden-Julian oscillation (MJO), and initialization of the unobserved variables in predicting the MJO/monsoon indices.

Original languageEnglish (US)
Pages (from-to)4885-4917
Number of pages33
JournalMonthly Weather Review
Volume144
Issue number12
DOIs
StatePublished - Jan 1 2016

Fingerprint

Madden-Julian oscillation
filter
statistics
compressible flow
mathematical analysis
probability density function
inflation
model test
skeleton
monsoon
tracer
energy
parameter
parameter estimation
index

Keywords

  • Bayesian methods
  • Error analysis
  • Filtering techniques
  • Statistics
  • Stochastic models
  • Time series

ASJC Scopus subject areas

  • Atmospheric Science

Cite this

Filtering nonlinear turbulent dynamical systems through conditional Gaussian statistics. / Chen, Nan; Majda, Andrew J.

In: Monthly Weather Review, Vol. 144, No. 12, 01.01.2016, p. 4885-4917.

Research output: Contribution to journalArticle

@article{76518831771544f1b2f8173ff73ab8c7,
title = "Filtering nonlinear turbulent dynamical systems through conditional Gaussian statistics",
abstract = "In this paper, a general conditional Gaussian framework for filtering complex turbulent systems is introduced. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the filter allows closed analytical formulas for updating the posterior states and is thus computationally efficient. An information-theoretic framework is developed to assess the model error in the filter estimates. Three types of applications in filtering conditional Gaussian turbulent systems with model error are illustrated. First, dyad models are utilized to illustrate that ignoring the energy-conserving nonlinear interactions in designing filters leads to significant model errors in filtering turbulent signals from nature. Then a triad (noisy Lorenz 63) model is adopted to understand the model error due to noise inflation and underdispersion. It is also utilized as a test model to demonstrate the efficiency of a novel algorithm, which exploits the conditional Gaussian structure, to recover the time-dependent probability density functions associated with the unobserved variables. Furthermore, regarding model parameters as augmented state variables, the filtering framework is applied to the study of parameter estimation with detailed mathematical analysis. A new approach with judicious model error in the equations associated with the augmented state variables is proposed, which greatly enhances the efficiency in estimating model parameters. Other examples of this framework include recovering random compressible flows from noisy Lagrangian tracers, filtering the stochastic skeleton model of the Madden-Julian oscillation (MJO), and initialization of the unobserved variables in predicting the MJO/monsoon indices.",
keywords = "Bayesian methods, Error analysis, Filtering techniques, Statistics, Stochastic models, Time series",
author = "Nan Chen and Majda, {Andrew J.}",
year = "2016",
month = "1",
day = "1",
doi = "10.1175/mwr-d-15-0437.1",
language = "English (US)",
volume = "144",
pages = "4885--4917",
journal = "Monthly Weather Review",
issn = "0027-0644",
publisher = "American Meteorological Society",
number = "12",

}

TY - JOUR

T1 - Filtering nonlinear turbulent dynamical systems through conditional Gaussian statistics

AU - Chen, Nan

AU - Majda, Andrew J.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - In this paper, a general conditional Gaussian framework for filtering complex turbulent systems is introduced. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the filter allows closed analytical formulas for updating the posterior states and is thus computationally efficient. An information-theoretic framework is developed to assess the model error in the filter estimates. Three types of applications in filtering conditional Gaussian turbulent systems with model error are illustrated. First, dyad models are utilized to illustrate that ignoring the energy-conserving nonlinear interactions in designing filters leads to significant model errors in filtering turbulent signals from nature. Then a triad (noisy Lorenz 63) model is adopted to understand the model error due to noise inflation and underdispersion. It is also utilized as a test model to demonstrate the efficiency of a novel algorithm, which exploits the conditional Gaussian structure, to recover the time-dependent probability density functions associated with the unobserved variables. Furthermore, regarding model parameters as augmented state variables, the filtering framework is applied to the study of parameter estimation with detailed mathematical analysis. A new approach with judicious model error in the equations associated with the augmented state variables is proposed, which greatly enhances the efficiency in estimating model parameters. Other examples of this framework include recovering random compressible flows from noisy Lagrangian tracers, filtering the stochastic skeleton model of the Madden-Julian oscillation (MJO), and initialization of the unobserved variables in predicting the MJO/monsoon indices.

AB - In this paper, a general conditional Gaussian framework for filtering complex turbulent systems is introduced. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the filter allows closed analytical formulas for updating the posterior states and is thus computationally efficient. An information-theoretic framework is developed to assess the model error in the filter estimates. Three types of applications in filtering conditional Gaussian turbulent systems with model error are illustrated. First, dyad models are utilized to illustrate that ignoring the energy-conserving nonlinear interactions in designing filters leads to significant model errors in filtering turbulent signals from nature. Then a triad (noisy Lorenz 63) model is adopted to understand the model error due to noise inflation and underdispersion. It is also utilized as a test model to demonstrate the efficiency of a novel algorithm, which exploits the conditional Gaussian structure, to recover the time-dependent probability density functions associated with the unobserved variables. Furthermore, regarding model parameters as augmented state variables, the filtering framework is applied to the study of parameter estimation with detailed mathematical analysis. A new approach with judicious model error in the equations associated with the augmented state variables is proposed, which greatly enhances the efficiency in estimating model parameters. Other examples of this framework include recovering random compressible flows from noisy Lagrangian tracers, filtering the stochastic skeleton model of the Madden-Julian oscillation (MJO), and initialization of the unobserved variables in predicting the MJO/monsoon indices.

KW - Bayesian methods

KW - Error analysis

KW - Filtering techniques

KW - Statistics

KW - Stochastic models

KW - Time series

UR - http://www.scopus.com/inward/record.url?scp=84995502691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995502691&partnerID=8YFLogxK

U2 - 10.1175/mwr-d-15-0437.1

DO - 10.1175/mwr-d-15-0437.1

M3 - Article

VL - 144

SP - 4885

EP - 4917

JO - Monthly Weather Review

JF - Monthly Weather Review

SN - 0027-0644

IS - 12

ER -