References
- Anandkumar, A., V. Y. Tan, F. Huang, and A. S. Willsky. 2012. High-dimensional Gaussian graphical model selection: Walk summability and local separation criterion. Journal of Machine Learning Research 13:2293–337.
- Antti, K., and S. Puntanen. 1983. A connection between the partial correlation coefficient and the correlation coefficient of certain residuals. Communications in Statistics - Simulation and Computation 12 (5):639–41. doi:https://doi.org/10.1080/03610918308812348.
- Baba, K., R. Shibata, and M. Sibuya. 2004. Partial correlation and conditional correlation as measures of conditional independence. Australian & New Zealand Journal of Statistics 46 (4):657–64. doi:https://doi.org/10.1111/j.1467-842X.2004.00360.x.
- Böhm, W., and K. Hornik. 2014. Generating random correlation matrices by the simple rejection method: Why it does not work. Statistics & Probability Letters 87:27–30. doi:https://doi.org/10.1016/j.spl.2013.12.012.
- Cramer, H. 1946. Mathematical methods of statistics, 500. Princeton, NJ: Princeton University Press.
- Epskamp, S., L. J. Waldorp, R. Mõttus, and D. Borsboom. 2018. The Gaussian graphical model in cross-sectional and time-series data. Multivariate Behavioral Research 53 (4):453–80. doi:https://doi.org/10.1080/00273171.2018.1454823.
- Erdős, P., and A. Rényi. 1960. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 5 (1):17–60.
- Gershgorin, S. A. 1931. Über die Abgrenzung der Eigenwerte einer Matrix. Bulletin de L’Académie Des Sciences de l’URSS. Classe Des Sciences Mathématiques et na 6:749–54.
- Ha, M. J., and W. Sun. 2014. Partial correlation matrix estimation using ridge penalty followed by thresholding and re-estimation. Biometrics 70 (3):762–70. doi:https://doi.org/10.1111/biom.12186.
- Harville, D. A. 1998. Matrix algebra from a statistician’s perspective. New York City, NY: Springer.
- Joe, H. 2006. Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis 97 (10):2177–89. doi:https://doi.org/10.1016/j.jmva.2005.05.010.
- Knol, D. L., and J. M. ten Berge. 1989. Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika 54 (1):53–61. doi:https://doi.org/10.1007/BF02294448.
- Lafit, G., F. Tuerlinckx, I. Myin-Germeys, and E. Ceulemans. 2019. A partial correlation screening approach for controlling the false positive rate in sparse gaussian graphical models. Scientific Reports 9 (1):1–24. doi:https://doi.org/10.1038/s41598-019-53795-x.
- Lauritzen, S. L. 1996. Graphical models. Oxford, England: Clarendon Press.
- Lewandowski, D., D. Kurowicka, and H. Joe. 2009. Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100 (9):1989–2001. doi:https://doi.org/10.1016/j.jmva.2009.04.008.
- Lewis, M. C., and G. P. Styan. 1981. Equalities and inequalities for conditional and partial correlation coefficients. In Statistics and related topics: Proceedings of the International Symposium on Statistics and Related Topics: Ottawa, May 1980, ed. M. Csorgo, D. A. Dawson, J. N. K. Rao, and A. K. Saleh, 57–65. Amsterdam: North–Holland.
- Marrelec, G., A. Krainik, H. Duffau, M. Pélégrini-Issac, S. Lehéricy, J. Doyon, and H. Benali. 2006. Partial correlation for functional brain interactivity investigation in functional MRI. Neuroimage 32 (1):228–37. doi:https://doi.org/10.1016/j.neuroimage.2005.12.057.
- Noschese, S., L. Pasquini, and L. Reichel. 2013. Tridiagonal toeplitz matrices: Properties and novel applications. Numerical Linear Algebra with Applications 20 (2):302–26. doi:https://doi.org/10.1002/nla.1811.
- Peña, D., and J. Rodrı́guez. 2003. Descriptive measures of multivariate scatter and linear dependence. Journal of Multivariate Analysis 85 (2):361–74. doi:https://doi.org/10.1016/S0047-259X(02)00061-1.
- Pourahmadi, M., and X. Wang. 2015. Distribution of random correlation matrices: Hyperspherical parameterization of the cholesky factor. Statistics & Probability Letters 106:5–12. doi:https://doi.org/10.1016/j.spl.2015.06.015.
- R Core Team. 2018. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
- Rousseeuw, P. J., and G. Molenberghs. 1994. The shape of correlation matrices. The American Statistician 48 (4):276–79. doi:https://doi.org/10.2307/2684832.
- Schäfer, J., and K. Strimmer. 2005. An empirical Bayes approach to inferring large-scale gene association networks. Bioinformatics (Oxford, England) 21 (6):754–64. doi:https://doi.org/10.1093/bioinformatics/bti062.
- Seber, G. A. 2008. A matrix handbook for statisticians. Hoboken, NJ: John Wiley & Sons.
- Stifanelli, P. F., T. M. Creanza, R. Anglani, V. C. Liuzzi, S. Mukherjee, F. P. Schena, and N. Ancona. 2013. A comparative study of covariance selection models for the inference of gene regulatory networks. Journal of Biomedical Informatics 46 (5):894–904. doi:https://doi.org/10.1016/j.jbi.2013.07.002.
- Strang, G. 2016. Introduction to linear algebra. Wellesley, MA: Wellesley-Cambridge Press.
- Sylvester, J. J. 1852. XIX. A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4 (23):138–42. doi:https://doi.org/10.1080/14786445208647087.
- Tenenhaus, A., V. Guillemot, X. Gidrol, and V. Frouin. 2010. Gene association networks from microarray data using a regularized estimation of partial correlation based on PLS regression. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 7 (2):251–62. doi:https://doi.org/10.1109/TCBB.2008.87.
- Wong, F., C. K. Carter, and R. Kohn. 2003. Efficient estimation of covariance selection models. Biometrika 90 (4):809–30. doi:https://doi.org/10.1093/biomet/90.4.809.