Assessing the Quality of a Positive Semidefinite Covariance Correction

Summary

The document raises a practical problem in covariance estimation: missing observations can produce a sample covariance matrix that is not positive semidefinite, which prevents some downstream methods from using it directly. It asks how to judge a correction that makes the matrix positive semidefinite while preserving as much of the original covariance information as possible.

No correction method, quality metric, dataset, or empirical comparison is supplied, so the document does not establish which repair is preferable. It frames the tradeoff that a useful assessment would need to capture: mathematical feasibility for methods such as PCA or Markowitz optimization, alongside distortion from the original estimates. Any assessment also depends on the missing-data pattern and on what properties matter for the intended analysis.

Key ideas

• Missing observations can lead to a sample covariance matrix that is not positive semidefinite.
• A corrected matrix may be needed for PCA or Markowitz portfolio optimization.
• Assessing a correction involves balancing positive semidefiniteness against preservation of the original covariances.
• The document poses the metric question but gives no proposed metric or validation evidence.

Tags

# Covariance Matrix: Calculating Error


# Covariance Matrix: Calculating Error












I have a sample covariance matrix that is non positive-semi definite (due to missing data points). I am looking at a number of techniques to 'fix' my covariance matrix and make it positive semi-definite so that I can use PCA, Markowitz portfolio optimisation, etc.

I was wondering, is there a way to check the quality of my correction to the covariance matrix? You'd want to keep as much of the original covariance as possible, but at the same time ensure it was positive-semi definite. Is there some potential metric based on that?

Any ideas are well appreciated.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.