Appraisals on Resolving Singularity Problem of Covariance matrix in High Dimension
This is an expository essay that reviews the recent developments on resolving the singularity problem in the variance/covariance matrix in high dimension. Furthermore, the interrelated and multidimensional linear relationships between columns or rows in the covariance matrix structure result into zero determinants in its computations and are singular. This also means that the inverse of the underlying matrix becomes inflexible, that leads to a computational problem in the likelihood. Therefore, this paper intent to review some of the literatures in resolving the singularity problems and also to evaluate some of the methods used. This study is very vital since it will resolve major problems in many areas such as functional magnetic resonance imaging analysis of gene expression arrays, risk management and portfolio allocation.
Anderberg, M. R. (2014). Cluster analysis for applications: probability and mathematical statistics: a series of monographs and textbooks (Vol. 19). Academic press.
Bickel, P. J., & Levina, E. (2008). Covariance regularization by thresholding. The Annals of Statistics, 2577-2604.
Dai, C., Lu, K., & Xiu, D. (2017). Knowing Factors or Factor Loadings, or Neither? Evaluating Estimators of Large Covariance Matrices with Noisy and Asynchronous Data.
Dai, W., Wang, S., Xiong, H., & Jiang, X. (2018). Privacy Preserving Federated Big Data Analysis. In Guide to Big Data Applications (pp. 49-82). Springer International Publishing.
Davoudi, A., Ghidary, S. S., & Sadatnejad, K. (2017). Dimensionality reduction based on distance preservation to local mean for symmetric positive definite matrices and its application in brain–computer interfaces. Journal of Neural Engineering, 14(3), 036019.
Engel, J., Buydens, L., & Blanchet, L. (2017). An overview of large‐dimensional covariance and precision matrix estimators with applications in chemometrics. Journal of Chemometrics, 31(4).
Fan, J., Han, F., & Liu, H. (2014). Challenges of big data analysis. National science review, 1(2), 293-314.
Fan, J., Liao, Y., & Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 603-680.
Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Annals of statistics, 295-327.
Jolliffe, I. T., & Cadima, J. (2016). Principal component analysis: a review and recent developments. Phil. Trans. R. Soc. A, 374(2065), 20150202.
Kazem, S., & Hatam, A. (2017). A modification on strictly positive definite RBF-DQ method based on matrix decomposition. Engineering Analysis with Boundary Elements, 76, 90-98.
Kuismin, M. O., Kemppainen, J. T., & Sillanpää, M. J. (2017). Precision Matrix Estimation with ROPE. Journal of Computational and Graphical Statistics, (just-accepted).
Lan, L., Zhang, K., Ge, H., Cheng, W., Liu, J., Rauber, A., ...& Zha, H. (2017). Low-rank decomposition meets kernel learning: A generalized Nyström method. Artificial Intelligence.
Lee, J. O., & Schnelli, K. (2016). Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population. The Annals of Applied Probability, 26(6), 3786-3839.
Li, Z., Wang, Q., & Yao, J. (2017). Identifying the number of factors from singular values of a large sample auto-covariance matrix. The Annals of Statistics, 45(1), 257-288.
Liu, S., Maljovec, D., Wang, B., Bremer, P. T., & Pascucci, V. (2017). Visualizing high- dimensional data: Advances in the past decade. IEEE transactions on visualization and computer graphics, 23(3), 1249-1268.
Paul, D., & Wang, L. (2016). Discussion of “Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation”. Electronic Journal of Statistics, 10(1), 74-80.
Puccio, S., Grillo, G., Licciulli, F., Severgnini, M., Liuni, S., Bicciato, S., ...& Peano, C. (2017). WoPPER: Web server for Position Related data analysis of gene Expression in Prokaryotes. Nucleic Acids Research.
Stewart, S., Ivy, M. A., & Anslyn, E. V. (2014). The use of principal component analysis and discriminant analysis in differential sensing routines. Chemical Society Reviews, 43(1), 70-84.
Sun, J., Frees, E. W., & Rosenberg, M. A. (2008). Heavy-tailed longitudinal data modeling using copulas. Insurance: Mathematics and Economics, 42(2), 817-830.
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