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Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

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  • Furrer, Reinhard
  • Bengtsson, Thomas

Abstract

This work studies the effects of sampling variability in Monte Carlo-based methods to estimate very high-dimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample-based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induce considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article, we quantify this variability with mean squared error measures for two Monte Carlo-based Kalman filter variants: the ensemble Kalman filter and the ensemble square-root Kalman filter. Expressions of the error measures are derived under weak assumptions and show that sample sizes need to grow proportionally to the square of the system dimension for bounded error growth. To reduce necessary ensemble size requirements and to address rank-deficient sample covariances, covariance-shrinking (tapering) based on the Schur product of the prior sample covariance and a positive definite function is demonstrated to be a simple, computationally feasible, and very effective technique. Rules for obtaining optimal taper functions for both stationary as well as non-stationary covariances are given, and optimal taper lengths are given in terms of the ensemble size and practical range of the forecast covariance. Results are also presented for optimal covariance inflation. The theory is verified and illustrated with extensive simulations.

Suggested Citation

  • Furrer, Reinhard & Bengtsson, Thomas, 2007. "Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 227-255, February.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:2:p:227-255
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    References listed on IDEAS

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    1. Gneiting, Tilmann, 2002. "Compactly Supported Correlation Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 493-508, November.
    2. Furrer, Reinhard, 2005. "Covariance estimation under spatial dependence," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 366-381, June.
    3. Gneiting, Tilmann, 1999. "Radial Positive Definite Functions Generated by Euclid's Hat," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 88-119, April.
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    Citations

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    Cited by:

    1. Frei, Marco & Künsch, Hans R., 2013. "Mixture ensemble Kalman filters," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 127-138.
    2. Lam, Clifford, 2008. "Estimation of large precision matrices through block penalization," LSE Research Online Documents on Economics 31543, London School of Economics and Political Science, LSE Library.
    3. Yi, Feng & Zou, Hui, 2013. "SURE-tuned tapering estimation of large covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 339-351.
    4. Zhao, Junguang & Xu, Xingzhong, 2016. "A generalized likelihood ratio test for normal mean when p is greater than n," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 91-104.
    5. Chen, Bei & Gel, Yulia R., 2010. "Autoregressive frequency detection using Regularized Least Squares," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1712-1727, August.
    6. repec:eee:csdana:v:114:y:2017:i:c:p:12-25 is not listed on IDEAS
    7. Jon Sætrom & Henning Omre, 2013. "Uncertainty Quantification in the Ensemble Kalman Filter," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 868-885, December.
    8. Xue, Lingzhou & Zou, Hui, 2013. "Minimax optimal estimation of general bandable covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 45-51.
    9. Zvi Bodie & Jérôme Detemple & Marcel Rindisbacher, 2009. "Life-Cycle Finance and the Design of Pension Plans," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 249-286, November.

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