scholarly journals Parametric covariance dynamics for the nonlinear diffusive Burgers' equation

2018 ◽  
Author(s):  
Olivier Pannekoucke ◽  
Marc Bocquet ◽  
Richard Ménard
Keyword(s):  
Kalman Filter ◽  
Burgers Equation ◽  
Diffusion Tensor ◽  
Error Variance ◽  
Covariance Model ◽  
Computationally Efficient ◽  
Closure Model ◽  
Error Covariance ◽  
Efficient Alternative ◽  
Linear Covariance

Abstract. The parametric Kalman filter (PKF) is a computationally efficient alternative method to the ensemble Kalman filter (EnKF). The PKF relies on an approximation of the error covariance matrix by a covariance model with space-time evolving set of parameters. This study extends the PKF to nonlinear dynamics using the diffusive Burgers' equation as an application, focusing on the forecast step of the assimilation cycle. The covariance model considered is based on the diffusion equation, with the diffusion tensor and the error variance as evolving parameter. An analytical derivation of the parameter dynamics highlights a closure issue. Therefore, a closure model is proposed based on the so-called kurtosis of the local correlation functions. Numerical experiments compare the PKF forecast with the statistics obtained from an large ensemble of nonlinear forecasts. These experiments strengthen the closure model and demonstrate the ability of the PKF to reproduce the tangent-linear covariance dynamics, at a low numerical cost.

scholarly journals Parametric covariance dynamics for the nonlinear diffusive Burgers equation

2018 ◽  
Vol 25 (3) ◽  
pp. 481-495 ◽  
Author(s):  
Olivier Pannekoucke ◽  
Marc Bocquet ◽  
Richard Ménard
Keyword(s):  
Kalman Filter ◽  
Burgers Equation ◽  
Diffusion Tensor ◽  
Error Variance ◽  
Covariance Model ◽  
Computationally Efficient ◽  
Closure Model ◽  
Error Covariance ◽  
Efficient Alternative ◽  
Linear Covariance

Abstract. The parametric Kalman filter (PKF) is a computationally efficient alternative method to the ensemble Kalman filter. The PKF relies on an approximation of the error covariance matrix by a covariance model with a space–time evolving set of parameters. This study extends the PKF to nonlinear dynamics using the diffusive Burgers equation as an application, focusing on the forecast step of the assimilation cycle. The covariance model considered is based on the diffusion equation, with the diffusion tensor and the error variance as evolving parameters. An analytical derivation of the parameter dynamics highlights a closure issue. Therefore, a closure model is proposed based on the kurtosis of the local correlation functions. Numerical experiments compare the PKF forecast with the statistics obtained from a large ensemble of nonlinear forecasts. These experiments strengthen the closure model and demonstrate the ability of the PKF to reproduce the tangent linear covariance dynamics, at a low numerical cost.

scholarly journals Microstructure Diffusion Scalar Measures from Reduced MRI Acquisitions

2019 ◽  
Author(s):  
Santiago Aja-Fernández ◽  
Rodrigo de Luis-García ◽  
Maryam Afzali ◽  
Malwina Molendowska ◽  
Tomasz Pieciak ◽  
...  
Keyword(s):  
Diffusion Tensor ◽  
State Of The Art ◽  
B Value ◽  
Computationally Efficient ◽  
Huge Amount ◽  
Average Diffusion ◽  
Efficient Alternative ◽  
Alternative Techniques ◽  
Dense Sampling ◽  
Diffusion Properties

AbstractIn diffusion MRI, the Ensemble Average diffusion Propagator (EAP) provides relevant microstructural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like the Diffusion Tensor. The direct estimation of the EAP, however, requires a dense sampling of the Cartesian q-space. Due to the huge amount of samples needed for an accurate reconstruction, more efficient alternative techniques have been proposed in the last decade. Even so, all of them imply acquiring a large number of diffusion gradients with different b-values. In order to use the EAP in practical studies, scalar measures must be directly derived, being the most common the return-to-origin probability (RTOP) and the return-to-plane and return-to-axis probabilities (RTPP, RTAP).In this work, we propose the so-called “Apparent Measures Using Reduced Acquisitions” (AMURA) to drastically reduce the number of samples needed for the estimation of diffusion properties. AMURA avoids the calculation of the whole EAP by assuming the diffusion anisotropy is roughly independent from the radial direction. With such an assumption, and as opposed to common multi-shell procedures based on iterative optimization, we achieve closed-form expressions for the measures using information from one single shell. This way, the new methodology remains compatible with standard acquisition protocols commonly used for HARDI (based on just one b-value). We report extensive results showing the potential of AMURA to reveal microstructural properties of the tissues compared to state of the art EAP estimators, and is well above that of Diffusion Tensor techniques. At the same time, the closed forms provided for RTOP, RTPP, and RTAP-like magnitudes make AMURA both computationally efficient and robust.

scholarly journals SymPKF (v1.0): a symbolic and computational toolbox for the design of parametric Kalman filter dynamics

2021 ◽  
Author(s):  
Olivier Pannekoucke ◽  
Philippe Arbogast
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Burgers Equation ◽  
Covariance Matrices ◽  
Covariance Model ◽  
Local Anisotropy ◽  
Code Generator ◽  
Univariate Statistics ◽  
Automatic Code ◽  
Python Package

Abstract. Recent researches in data assimilation lead to the introduction of the parametric Kalman filter (PKF): an implementation of the Kalman filter, where the covariance matrices are approximated by a parameterized covariance model. In the PKF, the dynamics of the covariance during the forecast step relies on the prediction of the covariance parameters. Hence, the design of the parameter dynamics is crucial while it can be tedious to do this by hand. This contribution introduces a python package, SymPKF, able to compute PKF dynamics for univariate statistics and when the covariance model is parameterized from the variance and the local anisotropy of the correlations. The ability of SymPKF to produce the PKF dynamics is shown on a non-linear diffusive advection (Burgers equation) over a 1D domain and the linear advection over a 2D domain. The computation of the PKF dynamics is performed at a symbolic level, but an automatic code generator is also introduced to perform numerical simulations. A final multivariate example illustrates the potential of SymPKF to go beyond the univariate case.

scholarly journals SymPKF (v1.0): a symbolic and computational toolbox for the design of parametric Kalman filter dynamics

2021 ◽  
Vol 14 (10) ◽  
pp. 5957-5976
Author(s):  
Olivier Pannekoucke ◽  
Philippe Arbogast
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Burgers Equation ◽  
Covariance Matrices ◽  
Covariance Model ◽  
Local Anisotropy ◽  
Code Generator ◽  
Univariate Statistics ◽  
Automatic Code ◽  
Python Package

Abstract. Recent research in data assimilation has led to the introduction of the parametric Kalman filter (PKF): an implementation of the Kalman filter, whereby the covariance matrices are approximated by a parameterized covariance model. In the PKF, the dynamics of the covariance during the forecast step rely on the prediction of the covariance parameters. Hence, the design of the parameter dynamics is crucial, while it can be tedious to do this by hand. This contribution introduces a Python package, SymPKF, able to compute PKF dynamics for univariate statistics and when the covariance model is parameterized from the variance and the local anisotropy of the correlations. The ability of SymPKF to produce the PKF dynamics is shown on a nonlinear diffusive advection (the Burgers equation) over a 1D domain and the linear advection over a 2D domain. The computation of the PKF dynamics is performed at a symbolic level, but an automatic code generator is also introduced to perform numerical simulations. A final multivariate example illustrates the potential of SymPKF to go beyond the univariate case.

Empirical determination of the covariance of forecast errors: an empirical justification and reformulation of Hybrid covariance models

2021 ◽  
Author(s):  
Diego Saul Carrio Carrio ◽  
Craig Bishop ◽  
Shunji Kotsuki
Keyword(s):  
Covariance Matrix ◽  
Error Variance ◽  
Separation Distance ◽  
Covariance Model ◽  
Ensemble Size ◽  
True Error ◽  
Error Covariance ◽  
Covariance Models ◽  
Small Ensemble ◽  
Increasing Function

<p>The replacement of climatological background error covariance models with Hybrid error covariance models that linearly combine a localized ensemble covariance matrix and a climatological error covariance matrix has led to significant forecast improvements at several forecasting centres. To deepen understanding of why the Hybrid’s superficially ad-hoc mix of ensemble based covariances and climatological covariances yielded such significant improvements, we derive the linear state estimation equations that minimize analysis error variance given an imperfect ensemble covariance. For high dimensional models, the computational cost of the very large sample sizes required to empirically estimate the terms in these equations is prohibitive. However, a reasonable and computationally feasible approximation to these equations can be obtained from empirical estimates of the true error covariance between two model variables given an imperfect ensemble covariance between the same two variables.   Here, using a Data Assimilation (DA) system featuring a simplified Global Circulation Model (SPEEDY), pseudo-observations of known error variance and an ensemble data assimilation scheme (LETKF),  we quantitatively demonstrate that the traditional Hybrid used by many operational centres is a much better approximation to the true covariance given the ensemble covariance than either the static climatological covariance or the localized ensemble covariance. These quantitative findings help explain why operational centres have found such large forecast improvements when switching from a static error covariance model to a Hybrid forecast error covariance model. Another fascinating finding of our empirical study is that the form of current Hybrid error covariance models is fundamentally incorrect in that the weight given to the static covariance matrix is independent of the separation distance of model variables. Our results show that this weight should be an increasing function of variable separation distance.  It is found that for ensemble covariances significantly different to zero, the true error covariance of spatially separated variables is an approximately linear function of the corresponding ensemble covariance, However, for small ensemble sizes and ensemble covariances near zero, the true covariance is an increasing function of the magnitude of the ensemble covariance and reaches a local minimum at the precise point where the ensemble covariance is equal to zero. It is hypothesized that this behaviour is a consequence of small ensemble size and, specifically, associated spurious fluctuations of the ensemble covariances and variances. Consistent with this hypothesis, this local minimum is almost eliminated by quadrupling the ensemble size.</p>

scholarly journals A Nonvariational Consistent Hybrid Ensemble Filter

2015 ◽  
Vol 143 (12) ◽  
pp. 5073-5090 ◽  
Author(s):  
Craig H. Bishop ◽  
Bo Huang ◽  
Xuguang Wang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Forecast Error ◽  
Model Space ◽  
High Rank ◽  
Covariance Model ◽  
Ensemble Data Assimilation ◽  
Error Covariance ◽  
Ensemble Data ◽  
Ensemble Transform Kalman Filter

Abstract A consistent hybrid ensemble filter (CHEF) for using hybrid forecast error covariance matrices that linearly combine aspects of both climatological and flow-dependent matrices within a nonvariational ensemble data assimilation scheme is described. The CHEF accommodates the ensemble data assimilation enhancements of (i) model space ensemble covariance localization for satellite data assimilation and (ii) Hodyss’s method for improving accuracy using ensemble skewness. Like the local ensemble transform Kalman filter (LETKF), the CHEF is computationally scalable because it updates local patches of the atmosphere independently of others. Like the sequential ensemble Kalman filter (EnKF), it serially assimilates batches of observations and uses perturbed observations to create ensembles of analyses. It differs from the deterministic (no perturbed observations) ensemble square root filter (ESRF) and the EnKF in that (i) its analysis correction is unaffected by the order in which observations are assimilated even when localization is required, (ii) it uses accurate high-rank solutions for the posterior error covariance matrix to serially assimilate observations, and (iii) it accommodates high-rank hybrid error covariance models. Experiments were performed to assess the effect on CHEF and ESRF analysis accuracy of these differences. In the case where both the CHEF and the ESRF used tuned localized ensemble covariances for the forecast error covariance model, the CHEF’s advantage over the ESRF increased with observational density. In the case where the CHEF used a hybrid error covariance model but the ESRF did not, the CHEF had a substantial advantage for all observational densities.

Effects of Accounting-Method Choices on Subjective Performance-Measure Weighting Decisions: Experimental Evidence on Precision and Error Covariance

2005 ◽  
Vol 80 (4) ◽  
pp. 1163-1192 ◽  
Author(s):  
Ranjani Krishnan ◽  
Joan L. Luft ◽  
Michael D. Shields
Keyword(s):  
Performance Measures ◽  
Error Variance ◽  
Spillover Effects ◽  
Performance Measure ◽  
Incentive System ◽  
Experimental Setting ◽  
Error Covariance ◽  
Psychology Theory ◽  
Optimal Weighting ◽  
Similarity And Difference

Performance-measure weights for incentive compensation are often determined subjectively. Determining these weights is a cognitively difficult task, and archival research shows that observed performance-measure weights are only partially consistent with the predictions of agency theory. Ittner et al. (2003) have concluded that psychology theory can help to explain such inconsistencies. In an experimental setting based on Feltham and Xie (1994), we use psychology theories of reasoning to predict distinctive patterns of similarity and difference between optimal and actual subjective performance-measure weights. The following predictions are supported. First, in contrast to a number of prior studies, most individuals' decisions are significantly influenced by the performance measures' error variance (precision) and error covariance. Second, directional errors in the use of these measurement attributes are relatively frequent, resulting in a mean underreaction to an accounting change that alters performance measurement error. Third, individuals seem insufficiently aware that a change in the accounting for one measure has spillover effects on the optimal weighting of the other measure in a two-measure incentive system. In consequence, they make performance-measure weighting decisions that are likely to result in misallocations of agent effort.

Maximum Correntropy Criterion Constrained Kalman Filter

2017 ◽  
Author(s):  
Seyed Fakoorian ◽  
Mahmoud Moosavi ◽  
Reza Izanloo ◽  
Vahid Azimi ◽  
Dan Simon
Keyword(s):  
Kalman Filter ◽  
Gaussian Noise ◽  
State Constraints ◽  
Statistical Information ◽  
Second Order ◽  
System Model ◽  
Error Covariance ◽  
Inequality State ◽  
Maximum Correntropy Criterion ◽  
Non Gaussian

Non-Gaussian noise may degrade the performance of the Kalman filter because the Kalman filter uses only second-order statistical information, so it is not optimal in non-Gaussian noise environments. Also, many systems include equality or inequality state constraints that are not directly included in the system model, and thus are not incorporated in the Kalman filter. To address these combined issues, we propose a robust Kalman-type filter in the presence of non-Gaussian noise that uses information from state constraints. The proposed filter, called the maximum correntropy criterion constrained Kalman filter (MCC-CKF), uses a correntropy metric to quantify not only second-order information but also higher-order moments of the non-Gaussian process and measurement noise, and also enforces constraints on the state estimates. We analytically prove that our newly derived MCC-CKF is an unbiased estimator and has a smaller error covariance than the standard Kalman filter under certain conditions. Simulation results show the superiority of the MCC-CKF compared with other estimators when the system measurement is disturbed by non-Gaussian noise and when the states are constrained.

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