Score matching filters

2021 ◽  
Author(s):  
Marie Turčičová ◽  
Jan Mandel ◽  
Kryštof Eben
Keyword(s):  
Kalman Filter ◽  
Covariance Matrix ◽  
Markov Random Fields ◽  
Computational Cost ◽  
Time Step ◽  
Sample Covariance ◽  
Monte Carlo Approximation ◽  
Gauss Markov Random Fields ◽  
Ensemble Filtering ◽  
Non Gaussian

<p>A widely popular group of data assimilation methods in meteorological and geophysical sciences is formed by filters based on Monte-Carlo approximation of the traditional Kalman filter, e.g. <span>E</span><span>nsemble Kalman filter </span><span>(EnKF)</span><span>, </span><span>E</span><span>nsemble </span><span>s</span><span>quare-root filter and others. Due to the computational cost, ensemble </span><span>size </span><span>is </span><span>usually </span><span>small </span><span>compar</span><span>ed</span><span> to the dimension of the </span><span>s</span><span>tate </span><span>vector. </span><span>Traditional </span> <span>EnKF implicitly uses the sample covariance which is</span><span> a poor estimate of the </span><span>background covariance matrix - singular and </span><span>contaminated by </span><span>spurious correlations. </span></p><p><span>W</span><span>e focus on modelling the </span><span>background </span><span>covariance matrix by means of </span><span>a linear model for its inverse. This is </span><span>particularly </span><span>useful</span> <span>in</span><span> Gauss-Markov random fields (GMRF), </span><span>where</span> <span>the inverse covariance matrix has </span><span>a banded </span><span>structure</span><span>. </span><span>The parameters of the model are estimated by the</span><span> score matching </span><span>method which </span><span>provides</span><span> estimators in a closed form</span><span>, cheap to compute</span><span>. The resulting estimate</span><span> is a key component of the </span><span>proposed </span><span>ensemble filtering algorithms. </span><span>Under the assumption that the state vector is a GMRF in every time-step, t</span><span>he Score matching filter with Gaussian resamplin</span><span>g (SMF-GR) </span><span>gives</span><span> in every time-step a consistent (in the large ensemble limit) estimator of mean and covariance matrix </span><span>of the forecast and analysis distribution</span><span>. Further, we propose a filtering method called Score matching ensemble filter (SMEF), based on regularization of the EnK</span><span>F</span><span>. Th</span><span>is</span><span> filter performs well even for non-Gaussian systems with non-linear dynamic</span><span>s</span><span>. </span><span>The performance of both filters is illustrated on a simple linear convection model and Lorenz-96.</span></p>

scholarly journals High-Dimensional Mahalanobis Distances of Complex Random Vectors

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1877
Author(s):  
Deliang Dai ◽  
Yuli Liang
Keyword(s):  
Covariance Matrix ◽  
Fixed Ratio ◽  
Asymptotic Distributions ◽  
Random Vectors ◽  
Mahalanobis Distances ◽  
Random Samples ◽  
Sample Covariance ◽  
Independent Variable ◽  
Non Gaussian ◽  
Leave One Out

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c=p/n→∞. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F=S2−1S1—the product of a sample covariance matrix S1 (from the independent variable array (be(Zi)1×n) with the inverse of another covariance matrix S2 (from the independent variable array (Zj≠i)p×n)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S1 and S2 of the F-matrix is not required.

The Ensemble Kalman Filter for Continuous Updating of Reservoir Simulation Models

2005 ◽  
Vol 128 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Yaqing Gu ◽  
Dean S. Oliver
Keyword(s):  
Kalman Filter ◽  
Covariance Matrix ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Water Saturation ◽  
State Variables ◽  
Time Step ◽  
Two Phase ◽  
One Dimensional ◽  
Reservoir State

This paper reports the use of ensemble Kalman filter (EnKF) for automatic history matching. EnKF is a Monte Carlo method, in which an ensemble of reservoir state variables are generated and kept up-to-date as data are assimilated sequentially. The uncertainty of reservoir state variables is estimated from the ensemble at any time step. Two synthetic problems are selected to investigate two primary concerns with the application of the EnKF. The first concern is whether it is possible to use a Kalman filter to make corrections to state variables in a problem for which the covariance matrix almost certainly provides a poor representation of the distribution of variables. It is tested with a one-dimensional, two-phase waterflood problem. The water saturation takes large values behind the flood front, and small values ahead of the front. The saturation distribution is bimodal and is not well modeled by the mean and variance. The second concern is the representation of the covariance via a relatively small ensemble of state vectors may be inadequate. It is tested by a two-dimensional, two-phase problem. The number of ensemble members is kept the same as for the one-dimensional problem. Hence the number of ensemble members used to create the covariance matrix is far less than the number of state variables. We conclude that EnKF can provide satisfactory history matching results while requiring less computation work than traditional history matching methods.

scholarly journals The Diffusion Kernel Filter Applied to Lagrangian Data Assimilation

2009 ◽  
Vol 137 (12) ◽  
pp. 4386-4400 ◽  
Author(s):  
Paul Krause ◽  
Juan M. Restrepo
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Time Series Data ◽  
Computational Cost ◽  
Series Data ◽  
Diffusion Kernel ◽  
Test Bed ◽  
Error Statistics ◽  
Lagrangian Data ◽  
Non Gaussian

Abstract The diffusion kernel filter is a sequential particle-method approach to data assimilation of time series data and evolutionary models. The method is applicable to nonlinear/non-Gaussian problems. Within branches of prediction it parameterizes small fluctuations of Brownian-driven paths about deterministic paths. Its implementation is relatively straightforward, provided a tangent linear model is available. A by-product of the parameterization is a bound on the infinity norm of the covariance matrix of such fluctuations (divided by the grid model dimension). As such it can be used to define a notion of “prediction” itself. It can also be used to assess the short time sensitivity of the deterministic history to Brownian noise or Gaussian initial perturbations. In pure oceanic Lagrangian data assimilation, the dynamics and the statistics are nonlinear and non-Gaussian, respectively. Both of these characteristics challenge conventional methods, such as the extended Kalman filter and the popular ensemble Kalman filter. The diffusion kernel filter is proposed as an alternative and is evaluated here on a problem that is often used as a test bed for Lagrangian data assimilation: it consists of tracking point vortices and passive drifters, using a dynamical model and data, both of which have known error statistics. It is found that the diffusion kernel filter captures the first few moments of the random dynamics, with a computational cost that is competitive with a particle filter estimation strategy. The authors also introduce a clustered version of the diffusion kernel filter (cDKF), which is shown to be significantly more efficient with regard to computational cost, at the expense of a slight degradation in the description of the statistics of the dynamical history. Upon parallelizing branches of prediction, cDKF can be computationally competitive with EKF.

Filtering Turbulent Sparsely Observed Geophysical Flows

2010 ◽  
Vol 138 (4) ◽  
pp. 1050-1083 ◽  
Author(s):  
John Harlim ◽  
Andrew J. Majda
Keyword(s):  
Kalman Filter ◽  
Stochastic Models ◽  
Large Scale ◽  
Baroclinic Instability ◽  
Computational Cost ◽  
Model Errors ◽  
Test Bed ◽  
Stochastic Filtering ◽  
Zonal Jets ◽  
Non Gaussian

Abstract Filtering sparsely turbulent signals from nature is a central problem of contemporary data assimilation. Here, sparsely observed turbulent signals from nature are generated by solutions of two-layer quasigeostrophic models with turbulent cascades from baroclinic instability in two separate regimes with varying Rossby radius mimicking the atmosphere and the ocean. In the “atmospheric” case, large-scale turbulent fluctuations are dominated by barotropic zonal jets with non-Gaussian statistics while the “oceanic” case has large-scale blocking regime transitions with barotropic zonal jets and large-scale Rossby waves. Recently introduced, cheap radical linear stochastic filtering algorithms utilizing mean stochastic models (MSM1, MSM2) that have judicious model errors are developed here as well as a very recent cheap stochastic parameterization extended Kalman filter (SPEKF), which includes stochastic parameterization of additive and multiplicative bias corrections “on the fly.” These cheap stochastic reduced filters as well as a local least squares ensemble adjustment Kalman filter (LLS-EAKF) are compared on the test bed with 36 sparse regularly spaced observations for their skill in recovering turbulent spectra, spatial pattern correlations, and RMS errors. Of these four algorithms, the cheap SPEKF algorithm has the superior overall skill on the stringent test bed, comparable to LLS-EAKF in the atmospheric regime with and without model error and far superior to LLS-EAKF in the ocean regime. LLS-EAKF has special difficulty and high computational cost in the ocean regime with small Rossby radius, which creates stiffness in the perfect dynamics. The even cheaper mean stochastic model, MSM1, has high skill, which is comparable to SPEKF for the oceanic case while MSM2 has significantly worse filtering performance than MSM1 with the same inexpensive computational cost. This is interesting because MSM1 is based on a simple new regression strategy while MSM2 relies on the conventional regression strategy used in stochastic models for shear turbulence.

Centered error entropy Kalman filter with application to satellite attitude determination

2021 ◽  
pp. 014233122110198
Author(s):  
Baojian Yang ◽  
Lu Cao ◽  
Dechao Ran ◽  
Bing Xiao
Keyword(s):  
Kalman Filter ◽  
Gaussian Noise ◽  
Attitude Determination ◽  
Complexity Analysis ◽  
Iteration Algorithm ◽  
Robust Algorithms ◽  
Convergence Conditions ◽  
Satellite Attitude ◽  
Heavy Tailed ◽  
Non Gaussian

Due to unavoidable factors, heavy-tailed noise appears in satellite attitude estimation. Traditional Kalman filter is prone to performance degradation and even filtering divergence when facing non-Gaussian noise. The existing robust algorithms have limited accuracy. To improve the attitude determination accuracy under non-Gaussian noise, we use the centered error entropy (CEE) criterion to derive a new filter named centered error entropy Kalman filter (CEEKF). CEEKF is formed by maximizing the CEE cost function. In the CEEKF algorithm, the prior state values are transmitted the same as the classical Kalman filter, and the posterior states are calculated by the fixed-point iteration method. The CEE EKF (CEE-EKF) algorithm is also derived to improve filtering accuracy in the case of the nonlinear system. We also give the convergence conditions of the iteration algorithm and the computational complexity analysis of CEEKF. The results of the two simulation examples validate the robustness of the algorithm we presented.

Analysis of grid operation state based on improved maximum eigenvalue sample covariance matrix algorithm

2021 ◽  
pp. 095965182110012
Author(s):  
Dinghui Wu ◽  
Juan Zhang ◽  
Bo Wang ◽  
Tinglong Pan
Keyword(s):  
Covariance Matrix ◽  
Signal To Noise Ratio ◽  
Poor Performance ◽  
Sample Covariance Matrix ◽  
Maximum Eigenvalue ◽  
Signal To Noise ◽  
Matrix Algorithm ◽  
Sample Covariance ◽  
Noise Ratio ◽  
Application Fields

Traditional static threshold–based state analysis methods can be applied to specific signal-to-noise ratio situations but may present poor performance in the presence of large sizes and complexity of power system. In this article, an improved maximum eigenvalue sample covariance matrix algorithm is proposed, where a Marchenko–Pastur law–based dynamic threshold is introduced by taking all the eigenvalues exceeding the supremum into account for different signal-to-noise ratio situations, to improve the calculation efficiency and widen the application fields of existing methods. The comparison analysis based on IEEE 39-Bus system shows that the proposed algorithm outperforms the existing solutions in terms of calculation speed, anti-interference ability, and universality to different signal-to-noise ratio situations.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

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