An Information-Theoretic Framework for Improving Imperfect Dynamical Predictions Via Multi-Model Ensemble Forecasts

Branicki, Michał; Majda, Andrew J.

doi:10.1007/s00332-015-9233-1

Cited by 20 publications

(5 citation statements)

References 83 publications

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“…In (Branicki and Majda 2014), a systematic information-theoretic approach was developed to quantify the statistical accuracy of Kalman filters with model error and the optimality of the im-perfect Kalman filters in terms of three information measures was presented. Another application of information theory is illustrated in (Branicki and Majda 2015) for improving imperfect predictions via multi-model ensemble forecasts.…”

Section: Conditional Gaussian Nonlinear Systemsmentioning

confidence: 99%

Filtering Nonlinear Turbulent Dynamical Systems through Conditional Gaussian Statistics

Chen

Majda

2016

Monthly Weather Review

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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.

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Section: Conditional Gaussian Nonlinear Systemsmentioning

confidence: 99%

Filtering Nonlinear Turbulent Dynamical Systems through Conditional Gaussian Statistics

Chen

Majda

2016

Monthly Weather Review

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“…Therefore significant model errors always occur due to the high wavenumber truncation in the imperfect model approximations. A systematic information-theoretic framework has been shown useful to improve model fidelity and sensitivity [58,59,10] for complex systems including perturbation formulas and multimodel ensembles that can be utilized to improve model error. In many applications to complex systems with model error such as the climate change science [22,74], it is crucially important to provide guidelines to improve the predictive skill of imperfect models for their responses to changes in various external forcing perturbations.…”

Section: Reduced-order Model Predictions For Responses In Various Dynmentioning

confidence: 99%

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Majda¹,

Qi²

2018

SIAM Rev.

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Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering including climate, material, and neural science. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial values, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. In the development of a proper UQ scheme for systems of high or infinite dimensionality with instabilities, significant model errors compared with the true natural signal are always unavoidable due to both the imperfect understanding of the underlying physical processes and the limited computational resources available through direct Monte-Carlo integration. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity).A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. There are generally three stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical response for th...

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“…For example, the multiphysics (or multimodel) ensemble approach was proposed for simulating surrogate statistics for the model error such as the forecast bias (see the review article [64]). Recent mathematical justification of such approaches was given through an information theoretic framework [65]. Another method, proposed by [66], simultaneously estimates a certain parametric form of mean model error estimator and applies an empirical choice of additive covariance inflation.…”

Section: Simultaneous Estimations Of Bias and Model Error Covariancesmentioning

confidence: 99%

Model Error in Data Assimilation

Harlim¹

2016

Nonlinear and Stochastic Climate Dynamics

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Data assimilation (or Bayesian filtering) is a statistical method to find the conditional distribution of the hidden variables of interest given noisy observations from nature. In application, the hidden variables of interest can be the state variables that are directly or indirectly observed or can even be some unobserved parameters in the models. In practice, data assimilation is typically realized by numerical schemes that produce conditional statistics of the state variables of interests, accounting for the information from the observations, rather than the corresponding conditional distribution; this gives a reasonable justification why we called it a "statistical method". When observations are available at discrete times, Bayesian filtering is an iterative predictor-corrector scheme that adjusts the prior forecast (background) statistical estimates from a predictor (or dynamical model) to be more consistent with the current observations. This correction step is referred to as analysis in the atmospheric and ocean science (AOS) community. Subsequently, the posterior (corrected or analysis) statistical estimates are fed into the model as initial conditions for future time prior statistical estimates.While the typical problems of interest are nonlinear, non-Gaussian, and high-dimensional, many practical data assimilation schemes that are currently used rely on Gaussian and/or linear assumptions. In particular, most practical data assimilation schemes are some type of approximation of the celebrated Kalman filter [1], which is the optimal solution (in the least squares sense) of the Bayesian filtering problem under linear and Gaussian assumptions. Essentially, all of these approximations were introduced to reduce the computational cost and to improve the statistical predictions. For example, in the AOS data assimilation community, two important schemes are: (i) the ensemble Kalman filtering methods [2,3,4,5,6,7,8,9, 10] which rely on empirical statistical estimates from ensemble forecasts; (ii) variational-based methods [11,12,13,14] that rely on linear tangent and adjoint models. Operationally, most of the weather prediction centers, including the European Center for Medium-range Weather Forecasts (ECMWF), the UK Met Office, and the National Centers for Environmental Prediction (NCEP), are adopting hybrid approaches, taking advantages from both the ensemble and variational based methods [15,16,17,18,19]. While these approximations were introduced for practical consideration, theoretical understanding of the convergence of these methods in idealistic settings were established [20,21,22] for ensemble Kalman filter and for variational based methods [23,24,25]. We should also mention that while these approximate methods can provide reasonable estimates of the first-order statistics, recent study [26] suggested that one should be cautious in interpreting their second order statistical estimates.In the numerical weather forecasting applications, these approximate filters are routinely used to assimilate observation...

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An Information-Theoretic Framework for Improving Imperfect Dynamical Predictions Via Multi-Model Ensemble Forecasts

Cited by 20 publications

References 83 publications

Filtering Nonlinear Turbulent Dynamical Systems through Conditional Gaussian Statistics

Filtering Nonlinear Turbulent Dynamical Systems through Conditional Gaussian Statistics

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Model Error in Data Assimilation

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