# Filtering smallest eigenvalues

In Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors , which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on pages 9-10, Section 6, Case study: security-based investment.

[...] We estimate every month the covariance matrix of the of the [sic] S&P stock returns $\Sigma_{F}$ using a one-year rolling window of daily observations, and filtering the smallest eigenvalues to ensure positive definiteness.

What is meant by "filtering the smallest eigenvalues"? What methods are there to transform the covariance matrix in such a way that the smallest eigenvalues (and corresponding eigenvectors) get removed, but the rest stay intact, and which one is used here in particular?

Edit: This SE question  is highly relevant to this one.

• I assume Truncated Singular Value Decomposition  was meant, but I am happy to give the answer to someone with a more informed input.  en.wikipedia.org/wiki/… – hps Jul 2 '18 at 15:10
• Meucci typically provides Matlab code for all his papers, so one can go to the code and investigate. With respect to hps's point, sometimes people also set the eigenvalues to the average of the smallest eigenvalues. This works in some cases, but if your covariance matrix contains variables with significantly different scales, then you may have issues. Random matrix theory can also motivate the choice of selecting the eigenvalues that are noise. – John Jul 2 '18 at 15:17
• By "filtering" he just means removing the corresponding eigenvectors/eigenvalues. A common approach for fixing non-pos def matrices. By smalles he means the eigenvalues whose absolute value is smaller than some threshold. – noob2 Jul 2 '18 at 16:17
• @John I have previously downloaded and looked at Meucci's Matlab code, but that part of the example is not part of the code. – hps Jul 2 '18 at 18:53
• @noob2 You mean removing the eigenvalues and corresponding from the Eigendecomposition and using the resulting matrix? – hps Jul 2 '18 at 21:28