Function returns the positive semidefinite projection of a symmetric matrix using the eigenvalue method.
makePsd(S, method = "covariance")A matrix containing the positive semi definite matrix.
a non-PSD matrix.
character, indicating whether the negative eigenvalues of the correlation or covariance should be replaced by zero. Possible values are "covariance" and "correlation".
Jonathan Cornelissen, Kris Boudt, and Emil Sjoerup.
We use the eigenvalue method to transform \(S\) into a positive semidefinite covariance matrix (see, e.g., Barndorff-Nielsen and Shephard, 2004, and Rousseeuw and Molenberghs, 1993). Let \(\Gamma\) be the orthogonal matrix consisting of the \(p\) eigenvectors of \(S\). Denote \(\lambda_1^+,\ldots,\lambda_p^+\) its \(p\) eigenvalues, whereby the negative eigenvalues have been replaced by zeroes. Under this approach, the positive semi-definite projection of \(S\) is \( S^+ = \Gamma' \mbox{diag}(\lambda_1^+,\ldots,\lambda_p^+) \Gamma\).
If method = "correlation", the eigenvalues of the correlation matrix corresponding to the matrix \(S\) are transformed, see Fan et al. (2010).
Barndorff-Nielsen, O. E. and Shephard, N. (2004). Measuring the impact of jumps in multivariate price processes using bipower covariation. Discussion paper, Nuffield College, Oxford University.
Fan, J., Li, Y., and Yu, K. (2012). Vast volatility matrix estimation using high frequency data for portfolio selection. Journal of the American Statistical Association, 107, 412-428
Rousseeuw, P. and Molenberghs, G. (1993). Transformation of non positive semidefinite correlation matrices. Communications in Statistics - Theory and Methods, 22, 965-984.