Ensemble Kalman Filtering without a Model

Hamilton, Franz; Berry, Tyrus; Sauer, Tim

doi:10.1103/physrevx.6.011021

Cited by 71 publications

(103 citation statements)

References 47 publications

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“…To form a stable estimate of Q , the noisy estimate

{boldQ}_{t - 1}^{e}

is combined using an exponentially weighted moving average,

Q_{t} = Q_{t - 1} + false({boldQ}_{t - 1}^{e} - Q_{t - 1} false) false/ τ,

where τ is the window of the moving average. Berry and Sauer () & Hamilton et al () provide additional details on the estimation of noise covariance.…”

Section: Methodsmentioning

confidence: 99%

“…The known

{boldz}_{t + 1}^{1}, {boldz}_{t + 1}^{2}, \dots, {boldz}_{t + 1}^{N}

(based on

{x^{o}}_{t + 1}^{1}, {x^{o}}_{t + 1}^{2}, \dots, {x^{o}}_{t + 1}^{N}

), are used in a local model to predict z t + 1 . The local model

\overset{f}{true}

, which can be generated using a weighted average of the nearest neighbors (Hamilton et al, ; Lagergren et al, ) can be written as,

z_{t + 1} = ω_{1} {boldz}_{t + 1}^{1} + ω_{2} {boldz}_{t + 1}^{2} + \dots + ω_{N} {boldz}_{t + 1}^{N},

ω_{i} = \frac{e^{- {false(d_{i} false/ σ false)}^{2}}}{\sum_{j = 1}^{N} e^{- {false(d_{j} false/ σ false)}^{2}}},

where d i is the distance of the j th neighbor to z t and σ is a bandwidth parameter, which controls the contribution of each neighbor in the local model (here σ = 2). The above prediction is applied to estimate the delay coordinate vector at t + 1.…”

Section: Methodsmentioning

confidence: 99%

“…A number of studies employ data‐driven (nonparametric) approaches to produce accurate statistical simulations (e.g., Dreano et al, ; Hamilton et al, ; Lguensat et al, ; Sauer, ; Tandeo et al, ). Hamilton et al () and Berry and Harlim () considered the case when models are partially known.…”

Section: Introductionmentioning

confidence: 99%

“…The main idea of Takens' theorem is that the model equations can be replaced by a data‐driven nonparametric reconstruction of the system's dynamics. The filter implements Takens' method for attractor reconstruction within the Kalman filtering framework, allowing for a model‐free approach to filter noisy data (Hamilton et al, ). Takens method has been used in various studies for nonparametric time series predictions (see, e.g., Packard et al, ; Sauer et al, , ; Takens, ).…”

Section: Introductionmentioning

confidence: 99%

“…Takens method has been used in various studies for nonparametric time series predictions (see, e.g., Packard et al, ; Sauer et al, , ; Takens, ). This technique replaced the model with a delay coordinate embedding scheme and has been shown by Hamilton et al () to not only obtain comparable results to a standard Kalman filter‐based framework but also may perform better when model errors are significant. A similar idea has been used by Tandeo et al () and Lguensat et al () to simulate the dynamics of complex systems using a nonparametric sampler.…”

Section: Introductionmentioning

confidence: 99%

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Nonparametric Data Assimilation Scheme for Land Hydrological Applications

Khaki

Hamilton

Forootan

et al. 2018

Water Resources Research

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Data assimilation, which relies on explicit knowledge of dynamical models, is a well-known approach that addresses models' limitations due to various reasons, such as errors in input and forcing data sets. This approach, however, requires intensive computational efforts, especially for high-dimensional systems such as distributed hydrological models. Alternatively, data-driven methods offer comparable solutions when the physics underlying the models are unknown. For the first time in a hydrological context, a nonparametric framework is implemented here to improve model estimates using available observations. This method uses Takens delay coordinate method to reconstruct the dynamics of the system within a Kalman filtering framework, called the Kalman-Takens filter. A synthetic experiment is undertaken to fully investigate the capability of the proposed method by comparing its performance with that of a standard assimilation framework based on an adaptive unscented Kalman filter (AUKF). Furthermore, using terrestrial water storage (TWS) estimates obtained from the Gravity Recovery And Climate Experiment mission, both filters are applied to a real case scenario to update different water storages over Australia. In situ groundwater and soil moisture measurements within Australia are used to further evaluate the results. The Kalman-Takens filter successfully improves the estimated water storages at levels comparable to the AUKF results, with an average root-mean-square error reduction of 37.30% for groundwater and 12.11% for soil moisture estimates. Additionally, the Kalman-Takens filter, while reducing estimation complexities, requires a fraction of the computational time, that is, ∼8 times faster compared to the AUKF approach.

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“…To form a stable estimate of Q , the noisy estimate

{boldQ}_{t - 1}^{e}

is combined using an exponentially weighted moving average,

Q_{t} = Q_{t - 1} + false({boldQ}_{t - 1}^{e} - Q_{t - 1} false) false/ τ,

where τ is the window of the moving average. Berry and Sauer () & Hamilton et al () provide additional details on the estimation of noise covariance.…”

Section: Methodsmentioning

confidence: 99%

“…The known

{boldz}_{t + 1}^{1}, {boldz}_{t + 1}^{2}, \dots, {boldz}_{t + 1}^{N}

(based on

{x^{o}}_{t + 1}^{1}, {x^{o}}_{t + 1}^{2}, \dots, {x^{o}}_{t + 1}^{N}

), are used in a local model to predict z t + 1 . The local model

\overset{f}{true}

, which can be generated using a weighted average of the nearest neighbors (Hamilton et al, ; Lagergren et al, ) can be written as,

z_{t + 1} = ω_{1} {boldz}_{t + 1}^{1} + ω_{2} {boldz}_{t + 1}^{2} + \dots + ω_{N} {boldz}_{t + 1}^{N},

ω_{i} = \frac{e^{- {false(d_{i} false/ σ false)}^{2}}}{\sum_{j = 1}^{N} e^{- {false(d_{j} false/ σ false)}^{2}}},

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonparametric Data Assimilation Scheme for Land Hydrological Applications

Khaki

Hamilton

Forootan

et al. 2018

Water Resources Research

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Q‐learning based adaptive Kalman filtering for partial model‐free dynamic systems

Tang,

Luan,

Ding

et al. 2024

Adaptive Control & Signal

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SummaryIn this article, we propose an adaptive Kalman filtering based on Q‐learning for partial model‐free dynamic systems. First, a cost function is defined to iteratively update the prior state value when the model parameters are unknown. Then, the observations in a period of time are utilized to improve the accuracy and updating speed of the prior state estimation by means of the multi‐innovation least squares. Next, considering that the weight matrix in the cost function will change due to external noise noise and model mismatch, the innovation‐based adaptive estimation algorithm is presented to adjust the weight matrix by using the covariance of the information sequence. Finally, the proposed algorithms are applied to estimate the water level of a quadruple water tank system.

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Machine learning modeling and predictive control of nonlinear processes using noisy data

Rincón

Liu

et al. 2021

AIChE Journal

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This work focuses on machine learning modeling and predictive control of nonlinear processes using noisy data. We use long short‐term memory (LSTM) networks with training data from sensor measurements corrupted by two types of noise: Gaussian and non‐Gaussian noise, to train the process model that will be used in a model predictive controller (MPC). We first discuss the LSTM training with noisy data following a Gaussian distribution, and demonstrate that the standard LSTM network is capable of capturing the underlying process dynamic behavior by reducing the impact of noise. Subsequently, given that the standard LSTM performs poorly on a noisy data set from industrial operation (i.e., non‐Gaussian noisy data), we propose an LSTM network using Monte Carlo dropout method to reduce the overfitting to noisy data. Furthermore, an LSTM network using co‐teaching training method is proposed to further improve its approximation performance when noise‐free data from a process model capturing the nominal process state evolution is available. A chemical process example is used throughout the manuscript to illustrate the application of the proposed modeling approaches and demonstrate their open‐ and closed‐loop performance under a Lyapunov‐based model predictive controller with state measurements corrupted by industrial noise.

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Ensemble Kalman Filtering without a Model

Cited by 71 publications

References 47 publications

Nonparametric Data Assimilation Scheme for Land Hydrological Applications

Nonparametric Data Assimilation Scheme for Land Hydrological Applications

Q‐learning based adaptive Kalman filtering for partial model‐free dynamic systems

Machine learning modeling and predictive control of nonlinear processes using noisy data

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