Ensemble Kalman methods

Data Assimilation; Data fusion; Sloppy filters for over-ambitious models



\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\cov}{\operatorname{Cov}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} \renewcommand{\one}{\unicode{x1D7D9}}\]

A random-sampling variant/generalisation of the Kalman-Bucy filter. That also describes particle filters, but the randomisation is different than those. We can do both types of randomisation. This sent has a few tweaks that make it more tenable in tricky situations with high dimensional state spaces or nonlinearities in inconvenient places. A popular data assimilation method for spatiotemporal models.

Ensemble Kalman filters make it somewhat easier to wring estimates out of data.

Tutorial introductions

Katzfuss, Stroud, and Wikle (2016); Roth et al. (2017); Fearnhead and KΓΌnsch (2018), are all pretty good. Schillings and Stuart (2017) has been recommended by Haber, Lucka, and Ruthotto (2018) as the canonical modern version. Wikle and Berliner (2007) presents it in a broader data assimilation context, although it is too curt to be helpful for me. Mandel (2009) is slightly longer. The inventor of the method explains it in Geir Evensen (2003), but I could make neither head nor tail of that, since it uses too much oceanography terminology. Roth et al. (2017) is probably the best for my background. Let us copy their notation.

We start from the discrete-time state-space models; the basic one is the linear system \[ \begin{aligned} x_{k+1} &=F x_{k}+G v_{k}, \\ y_{k} &=H x_{k}+e_{k}, \end{aligned} \] with state \(x\in\mathbb{R}^n\) and the measurement \(y\in\mathbb{R}^m\). The initial state \(x_{0}\), the process noise \(v_{k}\), and the measurement noise \(e_{k}\) are mutually independent such that \[\begin{aligned} \Ex x_{0}&=\hat{x}_{0}\\ \Ex v_{k}&=0\\ \Ex e_{k}&=0\\ \cov x_{0} &=P_{0}\\ \cov v_{k} & =Q\\ \cov e_{k}&=R \end{aligned}\] and all are Gaussian.

The Kalman filter propagates state estimates \(\hat{x}_{k \mid k}\) and covariance matrices \(P_{k \mid k}\) for this model. The KF update or prediction or forecast is given by the step \[ \begin{aligned} &\hat{x}_{k+1 \mid k}=F \hat{x}_{k \mid k} \\ &P_{k+1 \mid k}=F P_{k \mid k} F^{\top}+G Q G^{\top} \end{aligned} \] We predict the observations forward using these state estimates via \[ \begin{aligned} \hat{y}_{k \mid k-1} &=H \hat{x}_{k \mid k-1}, \\ S_{k} &=H P_{k \mid k-1} H^{\top}+R . \end{aligned} \] Given these and an actual observation, we update the state estimates using a gain matrix, \(K_{k}\) \[ \begin{aligned} \hat{x}_{k \mid k} &=\hat{x}_{k \mid k-1}+K_{k}\left(y_{k}-\hat{y}_{k \mid k-1}\right) \\ &=\left(I-K_{k} H\right) \hat{x}_{k \mid k-1}+K_{k} y_{k}, \\ P_{k \mid k} &=\left(I-K_{k} H\right) P_{k \mid k-1}\left(I-K_{k} H\right)^{\top}+K_{k} R K_{k}^{\top}. \end{aligned} \] in what geoscience types refer to as an analysis update. The variance-minimising gain is given \[ K_{k}=P_{k \mid k-1} H^{\top} S_{k}^{-1}=M_{k} S_{k}^{-1}, \] where \(M_{k}\) is the cross-covariance between the state and output predictions.

In the Ensemble Kalman filter, we approximate some of these quantities of interest using samples; this allows us to relax the assumption of Gaussianity and gets us computational savings in certain problems of interest. That does sound very similar to particle filters, and indeed there is a relation.

Various extensions of Kalman filters as per Katzfuss, Stroud, and Wikle (2016).

Instead of maintaining the \(n\)-dimensional estimate \(\hat{x}_{k \mid k}\) and the \(n \times n\) covariance \(P_{k \mid k}\) as such, we maintain an ensemble of \(N<n\) sampled state realizations \[X_{k}:=\left[x_{k}^{(i)}\right]_{i=1}^{N}.\] This notation is intended to imply that we are treating these realisations as an \(n \times N\) matrix \(X_{k \mid k}\) with columns \(x_{k}^{(i)}\). We introduce the following notation for ensemble moments: \[ \begin{aligned} &\bar{x}_{k \mid k}=\frac{1}{N} X_{k \mid k} \one \\ &\bar{P}_{k \mid k}=\frac{1}{N-1} \widetilde{X}_{k \mid k} \widetilde{X}_{k \mid k}^{\top}, \end{aligned} \] where \(\one=[1, \ldots, 1]^{\top}\) is an \(N\)-dimensional vector and \[ \widetilde{X}_{k \mid k}=X_{k \mid k}-\bar{x}_{k \mid k} \one^{\top}=X_{k \mid k}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right) \] is an ensemble of anomalies/deviations from \(\bar{x}_{k \mid k}\), which I would call it the centred version. We attempt to match the moments of the ensemble with those realised by a true Kalman filter, in the sense that \[ \begin{aligned} &\bar{x}_{k \mid k}:=\frac{1}{N} \sum_{i=1}^{N} x_{k}^{(i)} \approx \hat{x}_{k \mid k}, \\ &\bar{P}_{k \mid k}:=\frac{1}{N-1} \sum_{i=1}^{N}\left(x_{k}^{(i)}-\bar{x}_{k \mid k}\right)\left(x_{k}^{(i)}-\bar{x}_{k \mid k}\right)^{\top} \approx P_{k \mid k} . \end{aligned} \] The forecast step computes \(X_{k+1 \mid k}\) such that its moments are close to \(\hat{x}_{k+1 \mid k}\) and \(P_{k+1 \mid k}\). An ensemble of \(N\) independent process noise realizations \(V_{k}:=\left[v_{k}^{(i)}\right]_{i=1}^{N}\) with zero mean and covariance \(Q\), is used in \[ X_{k+1 \mid k}=F X_{k \mid k}+G V_{k}. \]

Next the \(X_{k \mid k-1}\) is adjusted to obtain the filtering ensemble \(X_{k \mid k}\) by applying an update to each ensemble member: With some gain matrix \(\bar{K}_{k}\) the KF update is applied to the ensemble by the update \[ X_{k \mid k}=\left(I-\bar{K}_{k} H\right) X_{k \mid k-1}+\bar{K}_{k} y_{k} \one^{\top} . \] This does not yet approximate the update of the full Kalman observation β€” there is no term \(\bar{K}_{k} R \bar{K}_{k}^{\top}\); We have a choice how to implement that.

Stochastic EnKF update

In the stochastic method, we use artificial zero-mean measurement noise realizations \(E_{k}:=\left[e_{k}^{(i)}\right]_{i=1}^{N}\) with covariance \(R\). \[ X_{k \mid k}=\left(I-\bar{K}_{k} H\right) X_{k \mid k-1}+\bar{K}_{k} y_{k} \one^{\top}-\bar{K}_{k} E_{k} . \] The resulting \(X_{k \mid k}\) has the correct ensemble mean and covariance, \(\hat{x}_{k \mid k}\) and \(P_{k \mid k}\).

If we define a predicted output ensemble \[ Y_{k \mid k-1}=H X_{k \mid k-1}+E_{k} \] that evokes the classic Kalman update (and encapsulates information about) \(\hat{y}_{k \mid k-1}\) and \(S_{k}\), we can rewrite this update into one that resembles the Kalman update: \[ X_{k \mid k}=X_{k \mid k-1}+\bar{K}_{k}\left(y_{k} \one^{\top}-Y_{k \mid k-1}\right) . \]

Now, the gain matrix \(\bar{K}_{k}\) in the classic KF is computed from the covariance matrices of the predicted state and output. In the EnKF, the required \(M_{k}\) and \(S_{k}\) must be estimated from the prediction ensembles. The obvious way of doing that is to once again centre the ensemble, \[ \begin{aligned} &\widetilde{X}_{k \mid k-1}=X_{k \mid k-1}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right) \\ &\widetilde{Y}_{k \mid k-1}=Y_{k \mid k-1}\left(I_{N}-\frac{1}{N} \one \one^{\top}\right) \end{aligned} \] and use the sample covariances \[ \begin{aligned} \bar{M}_{k} &=\frac{1}{N-1} \widetilde{X}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top}, \\ \bar{S}_{k} &=\frac{1}{N-1} \widetilde{Y}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top} . \end{aligned} \] The gain \(\bar{K}_{k}\) is then the solution to the system of linear equations, \[ \bar{K}_{k} \widetilde{Y}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top}=\widetilde{X}_{k \mid k-1} \widetilde{Y}_{k \mid k-1}^{\top} \]

Deterministic update

Resemblance to unscented/sigma-point filtering also apparent. TBD.

The additive measurement noise model we have used the \(e_{k}\) for should not affect the cross covariance \(M_k\). Thus it is reasonable to make the substitution \[ \widetilde{Y}_{k \mid k-1}\longrightarrow \widetilde{Z}_{k \mid k-1}=H \widetilde{X}_{k \mid k-1} \] to get a less noisy update \[ \begin{aligned} \bar{M}_{k} &=\frac{1}{N-1} \widetilde{X}_{k \mid k-1} \widetilde{Z}_{k \mid k-1}^{\top} \\ \bar{S}_{k} &=\frac{1}{N-1} \widetilde{Z}_{k \mid k-1} \widetilde{Z}_{k \mid k-1}^{\top}+R \end{aligned} \] The Kalman gain \(\bar{K}_{k}\) is then computed as in the KF. Or we can take it to be a matrix square-root \(R^{\frac{1}{2}}\) with \(R^{\frac{1}{2}} R^{\frac{\top}{2}}=R\) and then factorize \[ \bar{S}_{k}=\left[\begin{array}{cc}\frac{1}{\sqrt{N-1}} \widetilde{Z}_{k \mid k-1}\quad R^{\frac{1}{2}}\end{array}\right] \left[\begin{array}{c}\frac{1}{\sqrt{N-1}} \widetilde{Z}^{\top}_{k \mid k-1} \\ R^{\frac{\top}{2}}\end{array}\right]. \]

TBD: EAKF and ETKF (Tippett et al. 2003) which deterministically propagate an estimate \[ P_{k \mid k}^{\frac{1}{2}} P_{k \mid k}^{\frac{\top}{2}}=P_{k \mid k} \] which introduces less sampling noise. Roth et al. (2017) explain it as rewriting the measurement update to use a square root \(P_{k \mid k-1}^{\frac{1}{2}}\) and in particular the ensemble approximation \(\frac{1}{N-1} \widetilde{X}_{k \mid k-1}\) : \[ \begin{aligned} P_{k \mid k} &=\left(I-K_{k} H\right) P_{k \mid k-1} \\ &=P_{k \mid k-1}^{\frac{1}{2}}\left(I-P_{k \mid k-1}^{\frac{\top}{2}} H^{\top} S_{k}^{-1} H P_{k \mid k-1}^{\frac{1}{2}}\right) P_{k \mid k-1}^{\frac{\top}{2}} \\ & \approx \frac{1}{N-1} \widetilde{X}_{k \mid k-1}\left(I-\frac{1}{N-1} \widetilde{Z}_{k \mid k-1}^{\top} \bar{S}_{k}^{-1} \widetilde{Z}_{k \mid k-1}\right) \widetilde{X}_{k \mid k-1}^{\top}. \end{aligned} \] Factorising, \[ \left(I-\frac{1}{N-1} \widetilde{Z}_{k \mid k-1}^{\top} \bar{S}_{k}^{-1} \widetilde{Z}_{k \mid k-1}\right)=\Pi_{k}^{\frac{1}{2}} \Pi_{k}^{\frac{\top}{2}}, \] The \(\Pi_{k}^{\frac{1}{2}}\in\mathbb{R}^{N\times N}\) can be used to create a deviation ensemble \[ \tilde{X}_{k \mid k}=\tilde{X}_{k \mid k-1} \Pi_{k}^{\frac{1}{2}} \] that correctly encodes \(P_{k \mid k}\) without using random perturbations. The actual filtering is achieved by updating each sample according to \[ \bar{x}_{k \mid k}=\left(I-\bar{K}_{k} H\right) F_{x_{k-1 \mid k-1}}+\bar{K}_{k} y_{k}, \] where \(\bar{K}_{k}\) is computed from the deviation ensembles.

As least-squares

TBD. Permits calculating the operations without forming covariance matrices.

Going nonlinear

TBD

Monte Carlo moves in the ensemble

The ensemble is rank deficient. Question: When can we sample other states from the ensemble to improve the rank by stationary posterior moves?

Managing overconfidence

TBD

Ensemble methods in smoothing

Katzfuss, Stroud, and Wikle (2016) claims there are two major approaches to smoothing: Stroud et al. (2010) -type reverse methods, and the EnKS (Geir Evensen and van Leeuwen 2000) which augments the states with lagged copies rather than doing a reverse pass.

Here are some other papers I saw N. K. Chada, Chen, and Sanz-Alonso (2021); Luo et al. (2015); White (2018); Zhang et al. (2018).

System identification in

Can we use ensemble methods for online parameter estimation? Apparently. G. Evensen (2009); Malartic, Farchi, and Bocquet (2021); Moradkhani et al. (2005); Fearnhead and KΓΌnsch (2018).

Theoretical basis for probabilists

Bishop and Del Moral (2020); Del Moral, Kurtzmann, and Tugaut (2017); Garbuno-Inigo et al. (2020); Kelly, Law, and Stuart (2014); Le Gland, Monbet, and Tran (2009); Taghvaei and Mehta (2019).

Lanczos trick in precision estimates

Pleiss et al. (2018),Ubaru, Chen, and Saad (2017).

Relation to particle filters

Intimate. See particle filters.

Schilling’s filter

Claudia Schilling’s filter (Schillings and Stuart 2017) is an elegant version which looks somehow more general than the original but also simpler. Haber, Lucka, and Ruthotto (2018) use it to train neural nets (!) and show a rather beautiful connection to stochastic gradient descent in section 3.2.

See Neural nets by ensemble Kalman filtering.

References

Anderson, Jeffrey L. 2009. β€œEnsemble Kalman Filters for Large Geophysical Applications.” IEEE Control Systems Magazine 29 (3): 66–82.
Bishop, Adrian N., and Pierre Del Moral. 2020. β€œOn the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering.” arXiv:2006.08843 [Math, Stat], June.
Chada, Neil K., Yuming Chen, and Daniel Sanz-Alonso. 2021. β€œIterative Ensemble Kalman Methods: A Unified Perspective with Some New Variants.” Foundations of Data Science 3 (3): 331.
Chada, Neil, and Xin Tong. 2022. β€œConvergence Acceleration of Ensemble Kalman Inversion in Nonlinear Settings.” Mathematics of Computation 91 (335): 1247–80.
Chen, Chong, Yixuan Dou, Jie Chen, and Yaru Xue. 2022. β€œA Novel Neural Network Training Framework with Data Assimilation.” The Journal of Supercomputing, June.
Chen, Yuming, Daniel Sanz-Alonso, and Rebecca Willett. 2021. β€œAuto-Differentiable Ensemble Kalman Filters.” arXiv:2107.07687 [Cs, Stat], July.
Del Moral, P., A. Kurtzmann, and J. Tugaut. 2017. β€œOn the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman–Bucy Filters.” SIAM Journal on Control and Optimization 55 (1): 119–55.
Dubrule, Olivier. 2018. β€œKriging, Splines, Conditional Simulation, Bayesian Inversion and Ensemble Kalman Filtering.” In Handbook of Mathematical Geosciences: Fifty Years of IAMG, edited by B.S. Daya Sagar, Qiuming Cheng, and Frits Agterberg, 3–24. Cham: Springer International Publishing.
Evensen, G. 2009. β€œThe Ensemble Kalman Filter for Combined State and Parameter Estimation.” IEEE Control Systems 29 (3): 83–104.
Evensen, Geir. 2003. β€œThe Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation.” Ocean Dynamics 53 (4): 343–67.
β€”β€”β€”. 2004. β€œSampling Strategies and Square Root Analysis Schemes for the EnKF.” Ocean Dynamics 54 (6): 539–60.
β€”β€”β€”. 2009. Data Assimilation - The Ensemble Kalman Filter. Berlin; Heidelberg: Springer.
Evensen, Geir, and Peter Jan van Leeuwen. 2000. β€œAn Ensemble Kalman Smoother for Nonlinear Dynamics.” Monthly Weather Review 128 (6): 1852–67.
Fearnhead, Paul, and Hans R. KΓΌnsch. 2018. β€œParticle Filters and Data Assimilation.” Annual Review of Statistics and Its Application 5 (1): 421–49.
Finn, Tobias Sebastian, Gernot Geppert, and Felix Ament. 2021. β€œEnsemble-Based Data Assimilation of Atmospheric Boundary Layerobservations Improves the Soil Moisture Analysis.” Preprint. Catchment hydrology/Modelling approaches.
Garbuno-Inigo, Alfredo, Franca Hoffmann, Wuchen Li, and Andrew M. Stuart. 2020. β€œInteracting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler.” SIAM Journal on Applied Dynamical Systems 19 (1): 412–41.
Grooms, Ian, and Gregor Robinson. 2021. β€œA Hybrid Particle-Ensemble Kalman Filter for Problems with Medium Nonlinearity.” PLOS ONE 16 (3): e0248266.
Guth, Philipp A., Claudia Schillings, and Simon Weissmann. 2020. β€œEnsemble Kalman Filter for Neural Network Based One-Shot Inversion.” arXiv.
Haber, Eldad, Felix Lucka, and Lars Ruthotto. 2018. β€œNever Look Back - A Modified EnKF Method and Its Application to the Training of Neural Networks Without Back Propagation.” arXiv:1805.08034 [Cs, Math], May.
Hou, Elizabeth, Earl Lawrence, and Alfred O. Hero. 2016. β€œPenalized Ensemble Kalman Filters for High Dimensional Non-Linear Systems.” arXiv:1610.00195 [Physics, Stat], October.
Kantas, Nikolas, Arnaud Doucet, Sumeetpal S. Singh, Jan Maciejowski, and Nicolas Chopin. 2015. β€œOn Particle Methods for Parameter Estimation in State-Space Models.” Statistical Science 30 (3): 328–51.
Katzfuss, Matthias, Jonathan R. Stroud, and Christopher K. Wikle. 2016. β€œUnderstanding the Ensemble Kalman Filter.” The American Statistician 70 (4): 350–57.
Kelly, D. T. B., K. J. H. Law, and A. M. Stuart. 2014. β€œWell-Posedness and Accuracy of the Ensemble Kalman Filter in Discrete and Continuous Time.” Nonlinearity 27 (10): 2579.
Kovachki, Nikola B., and Andrew M. Stuart. 2019. β€œEnsemble Kalman Inversion: A Derivative-Free Technique for Machine Learning Tasks.” Inverse Problems 35 (9): 095005.
Kuzin, Danil, Le Yang, Olga Isupova, and Lyudmila Mihaylova. 2018. β€œEnsemble Kalman Filtering for Online Gaussian Process Regression and Learning.” 2018 21st International Conference on Information Fusion (FUSION), July, 39–46.
Lakshmivarahan, S., and David J. Stensrud. 2009. β€œEnsemble Kalman Filter.” IEEE Control Systems Magazine 29 (3): 34–46.
Law, Kody J. H., Hamidou Tembine, and Raul Tempone. 2016. β€œDeterministic Mean-Field Ensemble Kalman Filtering.” SIAM Journal on Scientific Computing 38 (3).
Le Gland, FranΓ§ois, Valerie Monbet, and Vu-Duc Tran. 2009. β€œLarge Sample Asymptotics for the Ensemble Kalman Filter,” 25.
Lei, Jing, Peter Bickel, and Chris Snyder. 2009. β€œComparison of Ensemble Kalman Filters Under Non-Gaussianity.” Monthly Weather Review 138 (4): 1293–1306.
Luo, Xiaodong, Andreas S. Stordal, Rolf J. Lorentzen, and Geir NΓ¦vdal. 2015. β€œIterative Ensemble Smoother as an Approximate Solution to a Regularized Minimum-Average-Cost Problem: Theory and Applications.” SPE Journal 20 (05): 962–82.
Malartic, Quentin, Alban Farchi, and Marc Bocquet. 2021. β€œState, Global and Local Parameter Estimation Using Local Ensemble Kalman Filters: Applications to Online Machine Learning of Chaotic Dynamics.” arXiv:2107.11253 [Nlin, Physics:physics, Stat], July.
Mandel, Jan. 2009. β€œA Brief Tutorial on the Ensemble Kalman Filter.” arXiv:0901.3725 [Physics], January.
Moradkhani, Hamid, Soroosh Sorooshian, Hoshin V. Gupta, and Paul R. Houser. 2005. β€œDual State–Parameter Estimation of Hydrological Models Using Ensemble Kalman Filter.” Advances in Water Resources 28 (2): 135–47.
Pleiss, Geoff, Jacob R. Gardner, Kilian Q. Weinberger, and Andrew Gordon Wilson. 2018. β€œConstant-Time Predictive Distributions for Gaussian Processes.” In. arXiv.
Popov, Andrey Anatoliyevich. 2022. β€œCombining Data-Driven and Theory-Guided Models in Ensemble Data Assimilation.” ETD. Virginia Tech.
Reich, Sebastian, and Simon Weissmann. 2019. β€œFokker-Planck Particle Systems for Bayesian Inference: Computational Approaches,” November.
Roth, Michael, Gustaf Hendeby, Carsten Fritsche, and Fredrik Gustafsson. 2017. β€œThe Ensemble Kalman Filter: A Signal Processing Perspective.” EURASIP Journal on Advances in Signal Processing 2017 (1): 56.
Schillings, Claudia, and Andrew M. Stuart. 2017. β€œAnalysis of the Ensemble Kalman Filter for Inverse Problems.” SIAM Journal on Numerical Analysis 55 (3): 1264–90.
Stroud, Jonathan R., Michael L. Stein, Barry M. Lesht, David J. Schwab, and Dmitry Beletsky. 2010. β€œAn Ensemble Kalman Filter and Smoother for Satellite Data Assimilation.” Journal of the American Statistical Association 105 (491): 978–90.
Taghvaei, Amirhossein, and Prashant G. Mehta. 2019. β€œAn Optimal Transport Formulation of the Ensemble Kalman Filter,” October.
β€”β€”β€”. 2021. β€œAn Optimal Transport Formulation of the Ensemble Kalman Filter.” IEEE Transactions on Automatic Control 66 (7): 3052–67.
Tippett, Michael K., Jeffrey L. Anderson, Craig H. Bishop, Thomas M. Hamill, and Jeffrey S. Whitaker. 2003. β€œEnsemble Square Root Filters.” Monthly Weather Review 131 (7): 1485–90.
Ubaru, Shashanka, Jie Chen, and Yousef Saad. 2017. β€œFast Estimation of \(tr(f(A))\) via Stochastic Lanczos Quadrature.” SIAM Journal on Matrix Analysis and Applications 38 (4): 1075–99.
White, Jeremy T. 2018. β€œA Model-Independent Iterative Ensemble Smoother for Efficient History-Matching and Uncertainty Quantification in Very High Dimensions.” Environmental Modelling & Software 109 (November): 191–201.
Wikle, Christopher K., and L. Mark Berliner. 2007. β€œA Bayesian Tutorial for Data Assimilation.” Physica D: Nonlinear Phenomena, Data Assimilation, 230 (1): 1–16.
Yang, Biao, Jonathan R. Stroud, and Gabriel Huerta. 2018. β€œSequential Monte Carlo Smoothing with Parameter Estimation.” Bayesian Analysis 13 (4): 1137–61.
Yegenoglu, Alper, Kai Krajsek, Sandra Diaz Pier, and Michael Herty. 2020. β€œEnsemble Kalman Filter Optimizing Deep Neural Networks: An Alternative Approach to Non-Performing Gradient Descent.” In Machine Learning, Optimization, and Data Science, edited by Giuseppe Nicosia, Varun Ojha, Emanuele La Malfa, Giorgio Jansen, Vincenzo Sciacca, Panos Pardalos, Giovanni Giuffrida, and Renato Umeton, 12566:78–92. Cham: Springer International Publishing.
Zhang, Jiangjiang, Guang Lin, Weixuan Li, Laosheng Wu, and Lingzao Zeng. 2018. β€œAn Iterative Local Updating Ensemble Smoother for Estimation and Uncertainty Assessment of Hydrologic Model Parameters With Multimodal Distributions.” Water Resources Research 54 (3): 1716–33.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.