Estimating model‐error covariances in nonlinear state‐space models using Kalman smoothing and the expectation–maximization algorithm

Dreano, Denis; Tandeo, Pierre; Pulido, Manuel; Ait‐El‐Fquih, Boujemaa; Chonavel, Thierry; Hoteit, Ibrahim

doi:10.1002/qj.3048

Cited by 63 publications

(84 citation statements)

References 29 publications

(54 reference statements)

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“…The EM algorithm was able to estimate subgrid-scale parameters with good accuracy while standard ensemble Kalman filter techniques failed. It has also been applied to the Lorenz-63 system to estimate model error covariance (Dreano et al, 2017).…”

mentioning

confidence: 99%

Stochastic parameterization identification using ensemble Kalman filtering combined with maximum likelihood methods

Pulido¹,

Bocquet²,

Carrassi³

et al. 2018

Tellus A: Dynamic Meteorology and Oceanography

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Manuscript received xx xxxx xx; in final form xx xxxx xx) A B S T R A C TFor modelling geophysical systems, large-scale processes are described through a set of coarse-grained dynamical equations while small-scale processes are represented via parameterizations. This work proposes a method for identifying the best possible stochastic parameterization from noisy data. State-the-art sequential estimation methods such as Kalman and particle filters do not achieve this goal succesfully because both suffer from the collapse of the parameter posterior distribution. To overcome this intrinsic limitation, we propose two statistical learning methods. They are based on the combination of two methodologies: the maximization of the likelihood via Expectation-Maximization (EM) and Newton-Raphson (NR) algorithms which are mainly applied in the statistic and machine learning communities, and the ensemble Kalman filter (EnKF). The methods are derived using a Bayesian approach for a hidden Markov model. They are applied to infer deterministic and stochastic physical parameters from noisy observations in coarse-grained dynamical models. Numerical experiments are conducted using the Lorenz-96 dynamical system with one and two scales as a proof-of-concept. The imperfect coarse-grained model is modelled through a one-scale Lorenz-96 system in which a stochastic parameterization is incorpored to represent the small-scale dynamics. The algorithms are able to identify an optimal stochastic parameterization with a good accuracy under moderate observational noise. The proposed EnKF-EM and EnKF-NR are promising statistical learning methods for developing stochastic parameterizations in high-dimensional geophysical models.

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confidence: 99%

Stochastic parameterization identification using ensemble Kalman filtering combined with maximum likelihood methods

Pulido¹,

Bocquet²,

Carrassi³

et al. 2018

Tellus A: Dynamic Meteorology and Oceanography

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“…Q ; Nakabayashi and Ueno, ; Pulido et al ) and/or diagonal (Ueno and Nakamura, ; Dreano et al ) covariance parametrizations, some success has been noted. However, the scope of this paper is restricted to the estimation of a single multiplicative inflation factor, β , for

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.…”

Section: Overview: Adaptive Inflationmentioning

confidence: 99%

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

Raanes¹,

Bocquet

Carrassi³

2019

Quart J Royal Meteoro Soc

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This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the “finite‐size” refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF‐N to make a hybrid that is suitable for contexts where model error is present and imperfectly parametrized. Benchmarks are obtained from experiments with the two‐scale Lorenz model and its slow‐scale truncation. The proposed hybrid EnKF‐N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree.

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“…A Gaussian additive noise might not even be the best statistical model for such error. The optimal value of q could be determined using an empirical Bayes approach based on, for instance, the expectation-maximisation technique in order to determine the maximum a posteriori of the conditional density of 5 q (see e.g., Dreano et al, 2017;Pulido et al, 2018). However, this would make us deviate too much from the objective of this work.…”

Section: Inferring the Dynamics From Partial And Noisy Observationsmentioning

confidence: 99%

Data assimilation as a deep learning tool to infer ODE representations of dynamical models

Bocquet

Brajard

Carrassi

et al. 2019

Preprint

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Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a 5 Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to cope with high-dimensional models.It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considera-10 tions on stability shed light on the assets and limitations of the method.The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal to identify or improve the model dynamics, build a surrogate or reduced model, or produce forecasts from mere observations of the physical model.

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Estimating model‐error covariances in nonlinear state‐space models using Kalman smoothing and the expectation–maximization algorithm

Cited by 63 publications

References 29 publications

Stochastic parameterization identification using ensemble Kalman filtering combined with maximum likelihood methods

Stochastic parameterization identification using ensemble Kalman filtering combined with maximum likelihood methods

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

Data assimilation as a deep learning tool to infer ODE representations of dynamical models

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