Structures of the covariance matrix: An overview

Anderson T. W. (1973): Asymptotically efficient estimation of covariance matrices with linear structure. Ann. Stat. 1(1): 135–141.10.1214/aos/1193342389Search in Google Scholar

Andrews D. W. K., Ploberger, W. (1996): Testing for Serial Correlation Against an ARMA(1,1) Process. J. Am. Stat. Assoc, 91(435): 1331–1342.10.1080/01621459.1996.10477002Search in Google Scholar

Chakraborty M. (1998): An efficient algorithm for solving general periodic Toeplitz systems, in IEEE Transactions on Signal Processing, vol. 46, no. 3: 784-787.10.1109/78.661347Search in Google Scholar

Cui X., Lin C., Zhao J., Zeng L., Zhang D., Pan J. (2016): Covariance structure regularization via Frobenius norm discrepancy. Linear Algebra Appl. 510: 124–145.10.1016/j.laa.2016.08.013Search in Google Scholar

Devijver E., Gallopin M. (2018): Block-Diagonal Covariance Selection for High-Dimensional Gaussian Graphical Models. J. Am. Stat. Assoc. 113(4): 306-314.10.1080/01621459.2016.1247002Search in Google Scholar

Filipiak K., John M., Markiewicz A. (2020): Comments on Maximum Likelihood Estimation and Projections Under Multivariate Statistical Models. In: Holgersson, T., Singull, M. (eds.) Recent Developments in Multivariate and Random Matrix Analysis, Springer, 51-66.10.1007/978-3-030-56773-6_4Search in Google Scholar

Filipiak K., Klein D. (2018): Approximation with a Kronecker product structure with one component as compound symmetry or autoregression. Electronic J. Linear Algebra. 559: 11-33.10.1016/j.laa.2018.08.031Search in Google Scholar

Filipiak K., Klein D. (2021): Estimation and testing the covariance structure of doubly multivariate data. In: Filipiak, K., Markiewicz, A., von Rosen, D. (Eds.) Multivariate, Multilinear and Mixed Linear Models, Springer, 131-156.10.1007/978-3-030-75494-5_6Search in Google Scholar

Filipiak K., Klein D., Markiewicz A., Mokrzycka M. (2020): Approximation with a Kronecker product structure with one component as compound symmetry or autoregression via entropy loss function. Linear Algebra Appl. 610: 625-646.10.1016/j.laa.2020.10.013Search in Google Scholar

Filipiak K., Klein D., Mokrzycka M. (2018a): Estimators comparison of separable covariance structure with one component as compound symmetry matrix. Electronic J. Linear Algebra. 33: 83-98.10.13001/1081-3810.3740Search in Google Scholar

Filipiak K., Klein D., Mokrzycka M. (2021): Separable covariance structure identification for doubly multivariate data. In: Filipiak, K., Markiewicz, A., von Rosen, D. (Eds.) Multivariate, Multilinear and Mixed Linear Models, Springer, 113-130.10.1007/978-3-030-75494-5_5Search in Google Scholar

Filipiak K., Markiewicz A., Mieldzioc A., Sawikowska A. (2018b): On projection of a positive definite matrix on a cone of nonnegative definite Toeplitz matrices. Electron. J. Linear Algebra. 33: 74-82.10.13001/1081-3810.3750Search in Google Scholar

Fuglede B, Jensen S. T. (2013): Positive projections of symmetric matrices and Jordan algebras. Expo. Math. 31: 295-303.10.1016/j.exmath.2013.01.005Search in Google Scholar

Gilson M., Dahmen D., Moreno-Bote R., Insabato A., Helias A. (2019): The covariance perceptron: A new framework for classification and processing of time series in recurrent neural networks. bioRxiv. https://doi.org/10.1101/562546.10.1101/562546Search in Google Scholar

Hannart A, Naveau P. (2014): Estimating high dimensional covariance matrices: A new look at the Gaussian conjugate framework. J Multivar Anal. 131: 149-162.10.1016/j.jmva.2014.06.001Search in Google Scholar

Ikeda Y, Kubokawa T, Srivastava MS.(2016): Comparison of linear shrinkage estimators of a large covariance matrix in normal and non-normal distributions. Comput Stat Data Anal. 95: 95-108.10.1016/j.csda.2015.09.011Search in Google Scholar

Janiszewska M., Markiewicz A. (2022): Estimation of block covariance matrix with linearly structured blocks. Submitted.Search in Google Scholar

Janiszewska M, Markiewicz A., Mokrzycka M. (2020): Block matrix approximation via entropy loss function. Appl. Math. 65 (6): 829-844.10.21136/AM.2020.0023-20Search in Google Scholar

Janiszewska M, Markiewicz A., Mokrzycka M. (2022): Quasi shrinkage estimation of block structured covariance matrix. J. Stat. Comput. Simul. Submitted.Search in Google Scholar

John M., Mieldzioc A. (2020): The comparison of the estimators of banded Toeplitz covariance structure under the highdimensional multivariate model. Comm. Statist. Simulation Comput. 734–752.10.1080/03610918.2019.1614622Search in Google Scholar

Judge, G.G., Hill, R.C., Griffiths, W.E., Lutkepohl, H. and Lee, T.-C. (1982): Introduction to the theory and practice of econometrics, John Wiley & Sons: New York.Search in Google Scholar

Krishnaiah P. R. and Lee Jack C. (1974): On Covariance Structures. Sankhya. 38 (4): 357-371.Search in Google Scholar

Lee K, Baek C, Daniels MJ. 2017: ARMA Cholesky Factor Models for the Covariance Matrix of Linear Models. Comput Stat Data Anal. 115: 267-280.10.1016/j.csda.2017.05.001566906029109594Search in Google Scholar

Liang Y., Rosen von D., Rosen von T. (2020): On Properties of Toeplitz-type Covariance Matrices in Models with Nested Random Effects. Statistical Papers. 62 (6): 2509-2528.10.1007/s00362-020-01202-3Search in Google Scholar

Lin L., Higham N. J., Pan J. (2014): Covariance structure regularization via entropy loss function. Comput. Statist. Data Anal. 72: 315-327.10.1016/j.csda.2013.10.004Search in Google Scholar

Lu N., Zimmerman D. L. (2005): The likelihood ratio test for a separable covariance matrix. Stat. Probab. Lett. 73: 449-457.10.1016/j.spl.2005.04.020Search in Google Scholar

Markiewicz A, Mieldzioc A. (2022): Improved estimators of linearly structured covariance matrices. Submitted.10.2478/bile-2022-0011Search in Google Scholar

Mieldzioc A., Mokrzycka M., Sawikowska A. (2021): Identification of Block-Structured Covariance Matrix on an Example of Metabolomic Data. Separations. 8(11):205: 734-752.10.3390/separations8110205Search in Google Scholar

Mokrzycka M. (2021): Aproksymacja macierzy kowariancji wybranymi strukturami w modelach podwójnie wielowymiarowych. [Doctoral dissertation thesis, Adam Mickiewicz University in Poznań, Poland].Search in Google Scholar

Ning L., Jiang X., Gourgiou T. (2006): Geometric methods for estimation of structured covariances. Biometrica, 93(4): 927-941.Search in Google Scholar

Pan J., Fang K. (2002): Growth Curve Models and Statistical Diagnostics. Springer-Verlag, New York.10.1007/978-0-387-21812-0Search in Google Scholar

Rao, C. R. (2005): Score test: historical review and recent developments. In Advances in Ranking and Selection, Multiple Comparisons, and Reliability, (Eds., N. Balakrishnan, N. Kannan, H.N. Nagaraja), Chapter 1, Birkhauser, Boston, 3-20.10.1007/0-8176-4422-9_1Search in Google Scholar

Roy A., Zmyślony R., Fonseca M., Leiva R. (2016): Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure. J. Multivar. Anal. 144: 81–90.10.1016/j.jmva.2015.11.001Search in Google Scholar

Seely J. (1971): Quadratic Subspaces and Completeness. Ann. Statist. 42(2): 710-721.10.1214/aoms/1177693420Search in Google Scholar

Shi M., Xu Li., Sol'e P. (2021): On isodual double Toeplitz codes. Computer Science. ArXiv.Search in Google Scholar

Srivastava M., Rosen von T, Rosen von D. (2008): Models with a Kronecker product covariance structure: Estimation and testing. Math. Methods Statist. 17: 357-370.10.3103/S1066530708040066Search in Google Scholar

Szatrowski T. H. (1976): Estimation and testing for block compound symmetry and other patterned covariance matrices with linear and non-linear structure. Technical report No. 107, Dept. of Statistics, Stanford University.Search in Google Scholar

Szatrowski T. H. (1982): Testing and estimation in the block compound symmetry problem. J. Educ. Stat. 7(1): 3-18.10.3102/10769986007001003Search in Google Scholar

Timm, N.H. (2002): Applied Multivariate Analysis. Springer-Verlag Inc., New York.Search in Google Scholar

Tsay, S. R. (2002): Analysis of Financial Time Series. Canada: John Wiley & Sons Inc.10.1002/0471264105Search in Google Scholar

Woods J, (1972): Two-dimensional discrete Markovian fields IEEE Transactions on Information Theory. 18 (2): 232-240.10.1109/TIT.1972.1054786Search in Google Scholar

Wilks, S. S. (1946): Sample Criteria for Testing Equality of Means, Equality of Variances, and Equality of Covariances in a Normal Multivariate Distribution. Ann. Math. Stat. 17, no. 3: 257–81.10.1214/aoms/1177730940Search in Google Scholar