Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems
Abstract
:1 Introduction | 3 |
2 Information Theory and Information Barriers with Model Error and Some Instructive Stochastic Models | 5 |
2.1 An Information-Theoretic Framework of Quantifying Model Error and Model Sensitivity | 5 |
2.2 Information Barriers in Capturing Model Fidelity . . . . . . . . . . . . . . . . . . . . . . | 7 |
2.2.1 First Information Barrier: Using Gaussian Approximation in Non-Gaussian Models | 8 |
2.2.2 Second Information Barrier: Using Single Point Correlation to Approximate Full CorrelationMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 13 |
2.3 Intrinsic Information Barrier in Predicting Mean Response to the Change of Forcing . . | 15 |
2.4 Slow-Fast System and Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 17 |
2.5 Fitting Autocorrelation Function of Time Series by a Spectral Information Criteria . . . . | 19 |
3 Quantifying Model Error with Information Theory in State Estimation and Prediction | 23 |
3.1 Kalman Filter, State Estimation and Linear Stochastic Model Prediction . . . . . . . . . . | 23 |
3.2 Asymptotic Behavior of Prediction and Filtering in One-Dimensional Linear Stochastic | |
Models withModel Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 26 |
3.2.1 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 26 |
3.2.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 27 |
3.2.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 27 |
3.3 An Information Theoretical Framework for State Estimation and Prediction . . . . . . . | 28 |
3.3.1 Motivation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 28 |
3.3.2 Assessing the Skill of Estimation and Prediction Using Information Theory . . . | 30 |
3.4 State Estimation and Prediction for Complex Scalar Forced Ornstein–Uhlenbeck (OU) | |
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 31 |
3.5 State Estimation and Prediction for Multiscale Slow-Fast Systems . . . . . . . . . . . . . | 36 |
3.5.1 A 3 × 3 Linear Coupled Multiscale Slow-Fast System . . . . . . . . . . . . . . . . | 37 |
3.5.2 ShallowWater Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 43 |
4 Information, Sensitivity and Linear Statistical Response—Fluctuation–Dissipation Theorem (FDT) | 47 |
4.1 Fluctuation–Dissipation Theorem (FDT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 48 |
4.1.1 The General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 48 |
4.1.2 Approximate FDT Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 49 |
4.2 Information Barrier for Linear Reduced Models in Capturing the Response in the Second Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 50 |
4.3 Information Theory for Finding the Most Sensitive Change Directions . . . . . . . . . . | 54 |
5 Given Time Series, Using Information Theory for Physics-Constrained Nonlinear Stochastic Model for Prediction | 59 |
5.1 A General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 59 |
5.2 Model Calibration via Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . | 60 |
5.3 Applications: Assessing the Predictability Limits of Time Series Associated with Tropical Intraseasonal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 61 |
6 Reduced-Order Models (ROMs) for Complex Turbulent Dynamical Systems | 63 |
6.1 Strategies for Reduced-Order Models for Predicting the Statistical Responses and UQ . | 63 |
6.1.1 Turbulent Dynamical System with Energy-Conserving Quadratic Nonlinearity . | 63 |
6.1.2 Modeling the Effect of Nonlinear Fluxes . . . . . . . . . . . . . . . . . . . . . . . . | 65 |
6.1.3 A Reduced-Order Statistical Energy Model with Optimal Consistency and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 66 |
6.1.4 Calibration Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 67 |
6.2 Physics-Tuned Linear Regression Models for Hidden (Latent) Variables . . . . . . . . . . | 68 |
6.3 Predicting Passive Tracer Extreme Events . . . . . . . . . . . . . . . . . . . . . . . . . . . | 71 |
6.3.1 Approximating Nonlinear Advection Flow Using Physics-Tuned Linear Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 73 |
6.3.2 Predicting Passive Tracer Extreme Events with Low-Order Stochastic Models . . | 76 |
7 Conclusions | 82 |
A Derivations of Fisher Information from Relative Entropy | 83 |
B Details of the Canonical Model for Low Frequency Atmospheric Variability | 84 |
C Augmented System for Prediction and Filtering Distributions | 85 |
C.1 Augmented System for Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 85 |
C.2 Augmented System for Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 86 |
D Possible Non-Gaussian PDFs of a Linear Model with Time-Periodic Forcing Based on the Sample Points in a Single Trajectory | 87 |
References | 91 |
1. Introduction
- How to measure the skill (i.e., the statistical accuracy) of a given imperfect model in reproducing the present states and predicting the future states in an unbiased fashion?
- How to make the best possible estimate of model sensitivity to changes in external or internal parameters by utilizing the imperfect knowledge available of the present state? What are the most sensitive parameters for the change of the model status given uncertain knowledge of the present state?
- How to design cheap and practical reduced models that are nevertheless able to capture both the main statistical features of nature and the correct response to external/internal perturbations?
- How to develop a systematic data-driven nonlinear modeling and prediction framework that provides skillful forecasts and allows accurate quantifications of the forecast uncertainty?
- How to build effective models, efficient algorithms and unbiased quantification criteria for online data assimilation (state estimation or filtering) and prediction especially in the presence of model error?
2. Information Theory and Information Barriers with Model Error and Some Instructive Stochastic Models
2.1. An Information-Theoretic Framework of Quantifying Model Error and Model Sensitivity
2.2. Information Barriers in Capturing Model Fidelity
2.2.1. First Information Barrier: Using Gaussian Approximation in Non-Gaussian Models
2.2.2. Second Information Barrier: Using Single Point Correlation to Approximate Full Correlation Matrix
2.3. Intrinsic Information Barrier in Predicting Mean Response to the Change of Forcing
2.4. Slow-Fast System and Reduced Model
2.5. Fitting Autocorrelation Function of Time Series by a Spectral Information Criteria
3. Quantifying Model Error with Information Theory in State Estimation and Prediction
3.1. Kalman Filter, State Estimation and Linear Stochastic Model Prediction
3.2. Asymptotic Behavior of Prediction and Filtering in One-Dimensional Linear Stochastic Models with Model Error
3.2.1. Prediction
3.2.2. Filtering
3.2.3. Comparison
3.3. An Information Theoretical Framework for State Estimation and Prediction
3.3.1. Motivation Examples
- The root-mean-square error (RMSE):
- The pattern correlation (PC):
3.3.2. Assessing the Skill of Estimation and Prediction Using Information Theory
- The Shannon entropy residual,
- The mutual information,
- The relative entropy,
- The Shannon entropy residual (Gaussian framework),
- The mutual information (Gaussian framework),
- The relative entropy (Gaussian framework),
3.4. State Estimation and Prediction for Complex Scalar Forced Ornstein–Uhlenbeck (OU) Processes
3.5. State Estimation and Prediction for Multiscale Slow-Fast Systems
3.5.1. A Linear Coupled Multiscale Slow-Fast System
- Full observations, full forecast model (F/F). The observational operator g is an identity such thatThe forecast model is the same as in (105). Although this straightforward setup may not be practical (see below) and can be expensive when a much larger dimension of the system is considered (see next subsection), the results from such a setup can be used as a baseline for testing various modifications and reduced models as will be presented below.
- Partial observations, full forecast model (P/F). The real observations typically involve the superposition of different wave components. It is usually impossible to artificially separate these components from the noisy observations. Therefore, here we let the observational operator be , namely the observation is the combination of the three variables,The forecast model remains the same as that in (105).
- Partial observations, reduced forecast model (P/R). In practice, only part of the state variables are of particular interest in filtering and prediction. These state variables usually lie in large or resolved scales, such as the GB flow. Therefore, simple reduced forecast models are typically designed to reduce the computational cost and retain the key features in filtering and predicting these variables. To this end, the following reduced forecast model is used
- Partial observations, reduced forecast model and tuned observational noise level with inflation (P/R tuned). It is easy to notice that in the previous setup (P/R), the signals of and actually become part of the observational noise in filtering and predicting . This is known as the representation error [53,100,150,151,152,153,154]. However, if the original observational noise level is still used in updating the Kalman gain, then the filtering and prediction skill may be affected by the representation error. To resolve this issue, we utilize an inflated in the analysis step to compute the Kalman gain while the other setups remain the same as in the P/R case. Here, the inflated is given by
3.5.2. Shallow Water Flows
- One geostrophically balanced (GB) mode with eigenvalueThe GB mode is incompressible.
- Two gravity modes with eigenvaluesThe gravity modes are compressible.
4. Information, Sensitivity and Linear Statistical Response—Fluctuation–Dissipation Theorem (FDT)
4.1. Fluctuation–Dissipation Theorem (FDT)
4.1.1. The General Framework
4.1.2. Approximate FDT Methods
4.2. Information Barrier for Linear Reduced Models in Capturing the Response in the Second Order Statistics
4.3. Information Theory for Finding the Most Sensitive Change Directions
5. Given Time Series, Using Information Theory for Physics-Constrained Nonlinear Stochastic Model for Prediction
5.1. A General Framework
5.2. Model Calibration via Information Theory
5.3. Applications: Assessing the Predictability Limits of Time Series Associated with Tropical Intraseasonal Variability
6. Reduced-Order Models (ROMs) for Complex Turbulent Dynamical Systems
6.1. Strategies for Reduced-Order Models for Predicting the Statistical Responses and UQ
6.1.1. Turbulent Dynamical System with Energy-Conserving Quadratic Nonlinearity
- is a linear operator representing dissipation and dispersion. Here, L is skew symmetric representing dispersion and D is a negative definite symmetric operator representing dissipative process such as surface drag, radiative damping, viscosity, etc.
- is a bilinear term and it satisfies energy conserving property with .
- The linear dynamical operator expressing energy transfers between the mean field and the stochastic modes (effect due to B), as well as energy dissipation (effect due to D) and non-normal dynamics (effect due to L)
- The positive definite operator expressing energy transfer due to the external stochastic forcing
- The energy flux between different modes due to non-Gaussian statistics (or nonlinear terms) modeled through third-order moments
6.1.2. Modeling the Effect of Nonlinear Fluxes
- Quasilinear Gaussian closure model: The simplest approximation for the closure methods at the first stage should be simply neglecting the nonlinear part entirely [182,183,184]. That is, setThus, the nonlinear energy transfer mechanism will be entirely neglected in this Gaussian closure model. This is the similar idea in the eddy-damped Markovian model where the moment hierarchy is closed at the level of second moments with Gaussian assumption and a much larger eddy-damped parameter is introduced to replace the molecular viscosity [121,185]. Obviously, this crude Gaussian approximation will not work well in general due to the cutoff of the energy flow when strong nonlinear interactions between modes occur. Actually, the deficiency of this crude approximation has been shown under the Lorenz 96 framework, and in a final equilibrium state, there exists only one active mode with a critical wavenumber [11,50]. Such closures are only useful in the weakly nonlinear case where the quasi-linear effects are dominant.
- Models with consistent equilibrium statistics: The next strategy is to construct the simplest closure model with consistent equilibrium statistics. Thus, the direct way is to choose constant damping and noise term scaled with the total variance. We propose two possible choices as in [50] for the damping and noise in (185) below.Gaussian closure 1 (GC1): letGaussian closure 2 (GC2): letAbove, only two scalar model parameters are introduced, and represents the identity matrix. GC1 is the familiar strategy of adding constant damping and white noise forcing to represent nonlinear interaction; GC2 scales with the total variance (or total statistical energy) so that the model sensitivity can be further improved as the system is perturbed. From both GC1 and GC2, we introduce uniform additional damping rate for each spectral mode controlled by a single scalar parameter , while the additional noise with variance is added to make sure climate fidelity in equilibrium.The statistical model closure is used to approximate the third-order moments in the true dynamics, thus the exponents of the total energy in GC2 should be consistent in scaling dimension. In the positive-definite part , it calibrates the rate of energy injected into the spectral mode due to nonlinear effect in the order . The factor scales with the total energy with exponent so that the corrections keep consistent with the third-order moment approximations; In the negative damping rate , the scaling function is used to characterize the amount of energy that flows out the spectral mode due to nonlinear interactions. Scaling factor with a square-root of the total energy with exponent is applied for this damping rate multiplying the variance in order to make it consistent in scaling dimension with third moments.However, the damping and noise are chosen empirically without consideration about the true dynamical features in each mode. A more sophisticated strategy with slightly more complexity in computation is to introduce the damping and noise judiciously according to the linearized dynamics. Then, climate consistency for each mode can be satisfied automatically.
- Modified quasi-Gaussian (MQG) closure with equilibrium statistics: In this modified quasi-Gaussian closure model originally proposed in [11,45], we exploit more about the true nonlinear energy transfer mechanism from the equilibrium statistical information. Thus, the additional damping and noise proposed as before are calibrated through the equilibrium nonlinear flux by lettingis the effective damping from equilibrium, and is the effective noise from the positive-definite component. Unperturbed equilibrium statistics in the nonlinear flux are used to calibrate the higher-order moments as additional energy sink and source. The true equilibrium higher-order flux can be calculated without error from first and second order moments in from the unperturbed true dynamics (180) in steady state following the steady state statistical solution relation:
6.1.3. A Reduced-Order Statistical Energy Model with Optimal Consistency and Sensitivity
- (i).
- Higher-order corrections from equilibrium statistics: In the first part of the correction using the damping and noise operator as , unperturbed equilibrium statistics in the nonlinear flux are used to calibrate the higher-order moments as additional energy sink and source following the procedure in (189). Therefore, the equilibrium statistics can be guaranteed to be consistent with the truth, and the true energy mechanism can be restored.
- (ii).
- Additional damping and noise to model changes in nonlinear flux: The above corrections in step (i) by using equilibrium information for nonlinear flux is found to be insufficient for accurate prediction in the reduced-order methods since the scheme is only marginally stable and the energy transferring mechanism may change with large deviation from the equilibrium case when external perturbations are applied. Thus, we also introduce the additional damping and noise as from (187). is just a constant scalar parameter to add uniform dissipation on each mode, and is the further correction as an additional energy source to maintain climate fidelity.
- (iii).
- Statistical energy scaling to improve model sensitivity: Still note that these additional parameters are added regardless of the true nonlinear perturbed energy mechanism where only unperturbed equilibrium statistics are used. To capture the responses to a specific perturbation forcing, it is better to make the imperfect model parameters change adaptively according to the total energy structure. Considering this, the additional damping and noise corrections are scaled with factors related with the total statistical variance as>
6.1.4. Calibration Strategy
6.2. Physics-Tuned Linear Regression Models for Hidden (Latent) Variables
6.3. Predicting Passive Tracer Extreme Events
- Whether a linear Gaussian dynamics in approximating the advection flow is able to capture tracer non-Gaussian statistical structures?
- How to design an unambiguous reduced-order stochastic modeling strategy with high prediction skill of the tracer field?
6.3.1. Approximating Nonlinear Advection Flow Using Physics-Tuned Linear Regression Model
6.3.2. Predicting Passive Tracer Extreme Events with Low-Order Stochastic Models
- Properly reflecting the nonlinear energy mechanism from the true system.
- Imperfect stochastic model consistency in equilibrium statistics and autocorrelation functions.
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Derivations of Fisher Information from Relative Entropy
Appendix B. Details of the Canonical Model for Low Frequency Atmospheric Variability
Appendix C. Augmented System for Prediction and Filtering Distributions
Appendix C.1. Augmented System for Prediction
Appendix C.2. Augmented System for Filtering
Appendix D. Possible Non-Gaussian PDFs of a Linear Model with Time-Periodic Forcing Based on the Sample Points in a Single Trajectory
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---|---|---|---|---|
Filter | √ | √ | √ | √ |
Pred. | √ | √ | √ | √ |
Filter | small and moderate | small and moderate | N/A | N/A |
Pred. | small and moderate | small | N/A | N/A |
Filter | √ | small to moderate | moderate | small to moderate |
Pred. | √ | small to moderate | moderate | small to moderate |
Filter | small to moderate | small for | N/A | N/A |
Pred. | small to moderate for | small for | N/A | N/A |
and small for |
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Majda, A.J.; Chen, N. Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems. Entropy 2018, 20, 644. https://doi.org/10.3390/e20090644
Majda AJ, Chen N. Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems. Entropy. 2018; 20(9):644. https://doi.org/10.3390/e20090644
Chicago/Turabian StyleMajda, Andrew J., and Nan Chen. 2018. "Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems" Entropy 20, no. 9: 644. https://doi.org/10.3390/e20090644