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Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix

  • * Corresponding author: Marie Turčičová

    * Corresponding author: Marie Turčičová 

This research was partially supported by TA CR grant TL01000238, NSF grant ICER-1664175 and by the project TO01000219 funded by Norway Grants and TA CR

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  • 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.

    Mathematics Subject Classification: 62F12, 62H12, 62P12.

    Citation:

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  • Figure 1.  Block band-diagonal structure of inverse covariance matrix for two-dimensional Markov fields of type $ 10\times10 $ points. There are assumed 4, 8, 12 neighbours of any gridpoint

    Figure 2.  The sparsity structure of the precision matrix (a) and of the matrix $ \tilde{D} $ of the linear system for the estimation of $ \boldsymbol{\beta} $ (b)

    Figure 3.  Performance illustration of EnKF, ETKF, SMEF (Algorithm 2), SMETKF (Section 4.3) and SMEF-GR (Algorithm 1) for the Lorenz 96 model, for selected combinations of ensemble size and observation error. $ N = 10, \rho = 3 $ (a), $ N = 20, \rho = 2 $ (b), $ N = 30, \rho = 1 $ (c). Observation step 2

    Figure 4.  RMSE of non-ensemble filters, averaged from 50 replications. $ \rho = 2 $ (a), $ \rho = 3 $ (b). Observation step 2

    Figure 5.  Trajectories of RMSE of all 50 replications, $ \rho = 2 $. ExKF(a), SMExKF (Algorithm 3)(b)

    Figure 6.  RMSE, averaged over 1000 replications, $ N = 50 $, observation step 4, $ \rho = 3 $ (a), $ \rho = 4 $ (b)

    Table 1.  Lorenz 96: RMSE of different filtering algorithms averaged from 1000 time steps and 50 replications of the run, initialized from different randomly selected states. The "div" columns show the number of divergent replications

    EnKF ETKF SMEF SMETKF SMEF-GR
    N $ \rho $ RMSE div RMSE div RMSE div RMSE div RMSE div
    10 1 4.31 50 3.54 50 1.00 1 0.55 0 0.80 0
    10 2 4.30 50 3.63 50 0.89 0 0.84 0 1.18 13
    10 3 4.27 50 3.85 50 1.52 2 1.42 0 1.59 35
    20 1 2.79 48 0.45 0 0.43 0 0.40 0 0.61 0
    20 2 3.55 48 0.74 0 0.62 1 0.55 0 0.87 0
    20 3 3.37 48 0.97 0 1.15 8 0.82 2 1.16 3
    30 1 0.44 0 0.45 0 0.41 0 0.40 0 0.56 0
    30 2 0.69 0 0.68 0 0.55 0 0.55 0 0.75 0
    30 3 0.95 0 0.88 0 0.74 0 0.72 2 0.94 0
     | Show Table
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    Table 2.  Lorenz 96: RMSE of the two non-ensemble filtering algorithms averaged from 1000 time steps and 50 replications of the run, initialized from different randomly selected states. The "div" columns show the number of divergent replications

    ExKF SMExKF
    Obs. step $ \rho $ RMSE div RMSE div
    2 1 0.34 0 0.54 0
    2 2 0.53 1 0.71 0
    2 3 1.71 16 0.86 0
     | Show Table
    DownLoad: CSV

    Table 3.  Lorenz 96: RMSE of different filtering algorithms averaged from 1000 time steps and 50 replications of the run, initialized from different randomly selected states. The "div" columns show the number of divergent replications. Observation step 4

    EnKF ETKF SMEF SMETKF SMEF-GR
    N r RMSE div RMSE div RMSE div RMSE div RMSE div
    30 3.0 3.46 46 3.35 40 2.49 30 3.51 34 2.53 1
    30 4.0 3.56 40 3.50 41 2.63 45 3.66 43 2.67 0
    40 3.0 3.55 36 3.32 24 2.05 23 3.35 39 2.34 0
    40 4.0 3.75 30 3.58 23 2.42 47 3.56 42 2.49 0
    50 3.0 3.85 33 3.53 15 2.02 19 3.25 35 2.18 0
    50 4.0 3.98 39 3.75 28 2.22 41 3.39 44 2.36 1
     | Show Table
    DownLoad: CSV
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