Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix. (English) Zbl 1478.62282
Summary: We present an ensemble filtering method based on a linear model for the precision matrix (the inverse of the covariance) with the parameters determined by Score Matching Estimation. The method provides a rigorous covariance regularization when the underlying random field is Gaussian Markov. The parameters are found by solving a system of linear equations. The analysis step uses the inverse formulation of the Kalman update. Several filter versions, differing in the construction of the analysis ensemble, are proposed, as well as a Score matching version of the Extended Kalman Filter.
MSC:
| 62M40 | Random fields; image analysis |
| 62F12 | Asymptotic properties of parametric estimators |
| 62H12 | Estimation in multivariate analysis |
| 62P12 | Applications of statistics to environmental and related topics |
Software:
GMRFLibReferences:
| [1] | T. W. Anderson, Asymptotically efficient estimation of covariance matrices with linear structure, Ann. Statist., 1, 135-141 (1973) · Zbl 0296.62022 · doi:10.1214/aos/1193342389 |
| [2] | O. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1978. · Zbl 0387.62011 |
| [3] | M. Bukal, I. Marković and I. Petrović, Score matching based assumed density filtering with the von Mises-Fisher distribution, 20th International Conference on Information Fusion (Fusion), Xi’an, China, 2017. |
| [4] | G. Burgers; P. J. van Leeuwen; G. Evensen, Analysis scheme in the ensemble Kalman filter, Monthly Weather Review, 126, 1719-1724 (1998) · doi:10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2 |
| [5] | A. P. Dempster, Covariance selection, Biometrics, 28, 157-175 (1972) · doi:10.2307/2528966 |
| [6] | P. G. M. Forbes; S. Lauritzen, Linear estimating equations for exponential families with application to Gaussian linear concentration models, Linear Algebra Appl., 473, 261-283 (2015) · Zbl 1312.62068 · doi:10.1016/j.laa.2014.08.015 |
| [7] | R. Furrer; T. Bengtsson, Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants, J. Multivariate Anal., 98, 227-255 (2007) · Zbl 1105.62091 · doi:10.1016/j.jmva.2006.08.003 |
| [8] | J. E. Gentle, Matrix Algebra. Theory, Computations and Applications in Statistics, \(2^nd\) edition, Springer Texts in Statistics, Springer, Cham, 2017. · Zbl 1386.15002 |
| [9] | A. Hyvärinen, Estimation of non-normalized statistical models by score matching, J. Mach. Learn. Res., 6, 695-709 (2005) · Zbl 1222.62051 |
| [10] | A. Hyvärinen, Some extensions of score matching, Comput. Statist. Data Anal., 51, 2499-2512 (2007) · Zbl 1161.62326 · doi:10.1016/j.csda.2006.09.003 |
| [11] | I. Kasanický; J. Mandel; M. Vejmelka, Spectral diagonal ensemble Kalman filters, Nonlin. Processes Geophys., 22, 485-497 (2015) · doi:10.5194/npg-22-485-2015 |
| [12] | M. Katzfuss; J. R. Stroud; C. K. Wikle, Understanding the ensemble Kalman filter, Amer. Statist., 70, 350-357 (2016) · Zbl 07665895 · doi:10.1080/00031305.2016.1141709 |
| [13] | K. Law, A. Stuart and K. Zygalakis, Data assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015. · Zbl 1353.60002 |
| [14] | K. Law, H. Tembine and R. Tempone, Deterministic mean-field ensemble Kalman filtering, SIAM J. Sci. Comput., 38 (2016), A1251-A1279. · Zbl 1351.60047 |
| [15] | F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 598-631. · Zbl 1225.93108 |
| [16] | E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, \(3^{rd}\) edition, Springer Texts in Statistics, Springer, New York, 2005. · Zbl 1076.62018 |
| [17] | F. Lindgren; H. Rue; J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 73, 423-498 (2011) · Zbl 1274.62360 · doi:10.1111/j.1467-9868.2011.00777.x |
| [18] | D. M. Livings; S. L. Dance; N. K. Nichols, Unbiased ensemble square root filters, Phys. D, 237, 1021-1028 (2008) · Zbl 1147.93413 · doi:10.1016/j.physd.2008.01.005 |
| [19] | E. N. Lorenz, Predictability - A problem partly solved, in Predictability of Weather and Climate, Cambridge University Press, 2006, 40-58. |
| [20] | J. Mandel; L. Cobb; J. D. Beezley, On the convergence of the ensemble Kalman filter, Appl. Math., 56, 533-541 (2011) · Zbl 1248.62164 · doi:10.1007/s10492-011-0031-2 |
| [21] | E. D. Nino-Ruiz; A. Sandu, Ensemble Kalman filter implementations based on shrinkage covariance matrix estimation, Ocean Dynamics, 65, 1423-1439 (2015) · doi:10.1007/s10236-015-0888-9 |
| [22] | O. Pannekoucke; L. Berre; G. Desroziers, Filtering properties of wavelets for local background-error correlations, Quart. J. Roy. Meterol. Soc., 133, 363-379 (2007) · doi:10.1002/qj.33 |
| [23] | J. A. Rozanov, Markov random fields, and stochastic partial differential equations, Mat. Sb. (N.S.), 103, 590-613 (1977) · Zbl 0364.60100 |
| [24] | H. Rue and L. Held, Gaussian Markov Random Fields. Theory and Applications, Monographs on Statistics and Applied Probability, 104, Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1093.60003 |
| [25] | P. Sakov; P. Oke, A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters, Tellus A, 60, 361-371 (2008) · doi:10.1111/j.1600-0870.2007.00299.x |
| [26] | D. Simpson; F. Lindgren; H. Rue, Think continuous: Markovian Gaussian models in spatial statistics, Spatial Statistics, 1, 16-29 (2012) · doi:10.1016/j.spasta.2012.02.003 |
| [27] | A. Spantini, R. Baptista and Y. Marzouk, Coupling techniques for nonlinear ensemble filtering, preprint, arXiv: 1907.00389. |
| [28] | M. K. Tippett; J. L. Anderson; C. H. Bishop; T. M. Hamill; J. S. Whitaker, Ensemble square root filters, Monthly Weather Review, 131, 1485-1490 (2003) · doi:10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2 |
| [29] | F. Tronarp, R. Hostettler and S. Särkkä, Continuous-discrete von Mises-Fisher filtering on \(S^2\) for reference vector tracking, 21st International Conference on Information Fusion (FUSION), Cambridge, UK, 2018. |
| [30] | M. Turčičová, Covariance Estimation for Filtering in High Dimension, Ph.D thesis, Charles University, 2021. Available from: http://hdl.handle.net/20.500.11956/136400. |
| [31] | G. Ueno; T. Tsuchiya, Covariance regularization in inverse space, Quart. J. Roy. Meterol. Soc., 135, 1133-1156 (2009) · doi:10.1002/qj.445 |
| [32] | X. Wang; C. H. Bishop; S. J. Julier, Which is better, an ensemble of positive-negative pairs or a centered spherical simplex ensemble?, Monthly Weather Review, 132, 1590-1605 (2004) · doi:10.1175/1520-0493(2004)132<1590:WIBAEO>2.0.CO;2 |
| [33] | A. Weaver; P. Courtier, Correlation modelling on the sphere using a generalized diffusion equation, Quart. J. Roy. Meterol. Soc., 127, 1815-1846 (2001) · doi:10.1002/qj.49712757518 |
| [34] | A. T. Weaver; I. Mirouze, On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation, Quart. J. Roy. Meterol. Soc., 139, 242-260 (2013) · doi:10.1002/qj.1955 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
