Improving the condition number of estimated covariance matrices

Jemima M. Tabeart; Sarah L. Dance; Amos S. Lawless; Nancy K. Nichols; Joanne A. Waller

doi:10.1080/16000870.2019.1696646

Improving the condition number of estimated covariance matrices

Tellus

Volume 72 (2020): Issue 1

By: Jemima M. Tabeart, Sarah L. Dance, Amos S. Lawless, Nancy K. Nichols and Joanne A. Waller

Open Access

|Jan 2020

Figures & Tables

Table 1

Change in standard deviation for the SOAR covariance matrix for both methods of reconditioning.

κ_max

σ

σ_{R R}

α corr. RR

σ_{M E}

α corr. ME10002.236062.264711.0132.254391.0085002.236062.293401.0262.275991.0181002.236062.513061.1242.457371.099

[i] Columns 4 and 6 show α, the multiplicative inflation factor corresponding to the values for $σ_{R R}$ and $σ_{M E},$ respectively.

Changes to correlations between the original SOAR matrix and the reconditioned matrices for $κ_{\max} = 100 .$ (a) shows $C (100, :) = C_{MVI} (100, :)$ (black solid), $C_{R R} (100, :)$ (red dashed), $C_{M E} (100, :)$ (blue dot-dashed) (b) shows $100 \times \frac{C (100, :) - C_{R R} (100, :)}{C (100, :)}$ (red dashed) and $100 \times \frac{C (100, :) - C_{M E} (100, :)}{C (100, :)}$ (blue dot-dashed). As the SOAR matrix is symmetric, we only plot the first 100 entries for (b).

Standard deviations for the IASI covariance matrix $Σ$ (black solid), $Σ_{R R}$ (red dashed), $Σ_{M E}$ (blue dot-dashed) for $κ_{\max} = 100 .$

Difference in correlations for IASI (a) $(C - C_{R R}) ° sign (C),$ (b) $(C - C_{M E}) ° sign (C),$ and (c) $(C_{M E} - C_{R R}) ° sign (C),$ where $°$ denotes the Hadamard product. Red indicates that the absolute correlation is decreased by reconditioning and blue indicates the absolute correlation is increased. The colourscale is the same for (a) and (b) but different for (c). Condition numbers of the corresponding covariance matrices are given by $κ (R) = 2005.98, κ (R_{R R}) = 100$ and $κ (R_{M E}) = 100 .$

Change in pointwise difference of discrete Fourier transform (DFT) from $x_{est}$ to $x_{\mod}$ where *a_est* denotes the vector of coefficients of the imaginary part of $DFT (x_{est}) .$ A positive (negative) value indicates that $x_{\mod}$ is closer to (further from) $x_{true}$ than $x_{est}$ and the amplitude shows how large this change is. Vertical dashed lines show the locations of non-zero values for the true signal.

Difference between *a_true* and *a_RR* for different choices of *κ_max*. Vertical dashed lines show the locations of non-zero values for the true signal.

Table 2

Changes to convergence of RR, MI and MVI for different values of κ_max.

κ_max10, 0001, 0001005010RR24524417014173ME24023919314576Infl RR244244238233199

[i] For all choices of κ_max, convergence for $R_{true}$ occurs in 17 iterations and $R_{est}$ occurs in 244 iterations.

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DOI: https://doi.org/10.1080/16000870.2019.1696646 | Journal eISSN: 3035-9554

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Language: English

Page range: 1696646 - 1696646

Published on: Jan 1, 2020

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

condition number,

covariance approximation,

observation error covariance matrix,

data assimilation,

reconditioning

© 2020 Jemima M. Tabeart, Sarah L. Dance, Amos S. Lawless, Nancy K. Nichols, Joanne A. Waller, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 72 (2020): Issue 1

Improving the condition number of estimated covariance matrices

Figures & Tables

Table 1

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Table 2

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