| Title:
|
Block matrix approximation via entropy loss function (English) |
| Author:
|
Janiszewska, Malwina |
| Author:
|
Markiewicz, Augustyn |
| Author:
|
Mokrzycka, Monika |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
65 |
| Issue:
|
6 |
| Year:
|
2020 |
| Pages:
|
829-844 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of the paper is to present a procedure for the approximation of a symmetric positive definite matrix by symmetric block partitioned matrices with structured off-diagonal blocks. The entropy loss function is chosen as approximation criterion. This procedure is applied in a simulation study of the statistical problem of covariance structure identification. (English) |
| Keyword:
|
approximation |
| Keyword:
|
block covariance structure |
| Keyword:
|
entropy loss function |
| MSC:
|
15A30 |
| MSC:
|
15B99 |
| MSC:
|
62H20 |
| MSC:
|
65F99 |
| idZBL:
|
Zbl 07285959 |
| idMR:
|
MR4191371 |
| DOI:
|
10.21136/AM.2020.0023-20 |
| . |
| Date available:
|
2020-11-18T09:40:17Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148399 |
| . |
| Reference:
|
[1] Cui, X., Li, C., Zhao, J., Zeng, L., Zhang, D., Pan, J.: Covariance structure regularization via Frobenius-norm discrepancy.Linear Algebra Appl. 510 (2016), 124-145. Zbl 1352.62090, MR 3551623, 10.1016/j.laa.2016.08.013 |
| Reference:
|
[2] Dey, D. K., Srinivasan, C.: Estimation of a covariance matrix under Stein's loss.Ann. Stat. 13 (1985), 1581-1591. Zbl 0582.62042, MR 0811511, 10.1214/aos/1176349756 |
| Reference:
|
[3] Fackler, P. L.: Notes on matrix calculus.Available at https://studyres.com/doc/15927579/notes-on-matrix-calculus (2005), 14 pages. |
| Reference:
|
[4] Filipiak, K., Klein, D.: Approximation with a Kronecker product structure with one component as compound symmetry or autoregression.Linear Algebra Appl. 559 (2018), 11-33. Zbl 1402.62117, MR 3857535, 10.1016/j.laa.2018.08.031 |
| Reference:
|
[5] Filipiak, K., Klein, D., Markiewicz, A., Mokrzycka, M.: Approximation with a Kronecker product structure with one component as compound symmetry or autoregression via entropy loss function.(to appear) in Linear Algebra Appl. (2020). MR 3857535, 10.1016/j.laa.2020.10.013 |
| Reference:
|
[6] Filipiak, K., Klein, D., Mokrzycka, M.: Estimators comparison of separable covariance structure with one component as compound symmetry matrix.Electron. J. Linear Algebra 33 (2018), 83-98. Zbl 1406.15020, MR 3962256, 10.13001/1081-3810.3740 |
| Reference:
|
[7] Filipiak, K., Markiewicz, A., Mieldzioc, A., Sawikowska, A.: On projection of a positive definite matrix on a cone of nonnegative definite Toeplitz matrices.Electron. J. Linear Algebra 33 (2018), 74-82. Zbl 1406.15028, MR 3962255, 10.13001/1081-3810.3750 |
| Reference:
|
[8] Gilson, M., Dahmen, D., Moreno-Bote, R., Insabato, A., Helias, M.: The covariance perceptron: A new paradigm for classication and processing of time series in recurrent neuronal networks.Available at https://www.biorxiv.org/content/10.1101/562546v4 (2020), 39 pages. 10.1101/562546 |
| Reference:
|
[9] James, W., Stein, C.: Estimation with quadratic loss.Proc. 4th Berkeley Symp. Math. Stat. Probab. Vol. 1 University of California Press, Berkeley (1961), 361-379. Zbl 1281.62026, MR 0133191 |
| Reference:
|
[10] John, M., Mieldzioc, A.: The comparison of the estimators of banded Toeplitz covariance structure under the high-dimensional multivariate model.Commun. Stat., Simulation Comput. 49 (2020), 734-752. MR 4068485, 10.1080/03610918.2019.1614622 |
| Reference:
|
[11] Kollo, T., Rosen, D. von: Advanced Multivariate Statistics with Matrices.Mathematics and Its Applications 579. Springer, Dordrecht (2005). Zbl 1079.62059, MR 2162145, 10.1007/1-4020-3419-9 |
| Reference:
|
[12] Lin, L., Higham, N. J., Pan, J.: Covariance structure regularization via entropy loss function.Comput. Stat. Data Anal. 72 (2014), 315-327. Zbl 06983908, MR 3139365, 10.1016/j.csda.2013.10.004 |
| Reference:
|
[13] Lu, N., Zimmerman, D. L.: The likelihood ratio test for a separable covariance matrix.Stat. Probab. Lett. 73 (2005), 449-457. Zbl 1071.62052, MR 2187860, 10.1016/j.spl.2005.04.020 |
| Reference:
|
[14] Magnus, J. R., Neudecker, H.: Symmetry, 0-1 matrices and Jacobians: A review.Econom. Theory 2 (1986), 157-190. 10.1017/S0266466600011476 |
| Reference:
|
[15] Magnus, J. R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics.Wiley Series in Probability and Statistics. Wiley, Chichester (1999). Zbl 0912.15003, MR 1698873, 10.1002/9781119541219 |
| Reference:
|
[16] Mieldzioc, A., Mokrzycka, M., Sawikowska, A.: Covariance regularization for metabolomic data on the drought resistance of barley.Biometrical Lett. 56 (2019), 165-181. 10.2478/bile-2019-0010 |
| Reference:
|
[17] Pan, J.-X., Fang, K.-T.: Growth Curve Models and Statistical Diagnostics.Springer Series in Statistics. Springer, New York (2002). Zbl 1024.62025, MR 1937691, 10.1007/978-0-387-21812-0 |
| Reference:
|
[18] Srivastava, M. S., Rosen, T. von, Rosen, D. von: Models with a Kronecker product covariance structure: Estimation and testing.Math. Methods Stat. 17 (2008), 357-370. Zbl 1231.62101, MR 2483463, 10.3103/S1066530708040066 |
| Reference:
|
[19] Szatrowski, T. H.: Estimation and testing for block compound symmetry and other patterned covariance matrices with linear and non-linear structure.Technical Report OLK NSF 107, Stanford University, Stanford (1976), 187 pages. MR 2626007 |
| Reference:
|
[20] Szatrowski, T. H.: Testing and estimation in the block compound symmetry problem.J. Educ. Behavioral Stat. 7 (1982), 3-18. 10.3102/10769986007001003 |
| Reference:
|
[21] Szczepańska-Álvarez, A., Hao, C., Liang, Y., Rosen, D. von: Estimation equations for multivariate linear models with Kronecker structured covariance matrices.Commun. Stat., Theory Methods 46 (2017), 7902-7915. Zbl 1373.62262, MR 3660028, 10.1080/03610926.2016.1165852 |
| . |
