makePsd: Returns the positive semidinite projection of a symmetric matrix using the eigenvalue method
Description
Function returns the positive semidinite projection of a symmetric matrix using the eigenvalue method.
Usage
makePsd(S,method="covariance")
Arguments
S
matrix.
method
character, indicating whether the negative eigenvalues of the correlation or covariance should be replaced by zero. Possible values are "covariance" and "correlation".
Value
An xts object containing the aggregated trade data.
Details
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\(\lambda_1^+,\ldots,\lambda_p^+\) its \(p\) eigenvalues, whereby the negative eigenvalues have been replaced by zeroes.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).
References
Barndorff-Nielsen, O. and N. Shephard (2004). Measuring the impact of
jumps in multivariate price processes using bipower covariation. Discussion
paper, Nuffield College, Oxford University. Fan, J., Y. Li, and K. Yu (2010). Vast volatility matrix estimation using high frequency data for portfolio selection. Working paper. Rousseeuw, P. and G. Molenberghs (1993). Transformation of non positive semidefinite correlation matrices. Communications in Statistics - Theory and Methods 22, 965-984.