A large returns panel can contain strongly correlated assets. Risk models and research pipelines compress the panel by retaining dominant directions. Small singular directions may contain noise, useful hedges or economically important risks; small variance alone does not identify noise. This note combines the singular value decomposition (SVD), principal component analysis (PCA), and the Eckart–Young theorem at statement level — with one argument why the truncated SVD is optimal — and reads the result as compressing a covariance or a returns panel into a few factors.
Learning objectives. By the end of this note you should be able to:
- write a thin / full SVD and the rank- truncation ;
- state Eckart–Young (Frobenius and spectral) and compute the truncation error from discarded singular values;
- relate PCA on a centred panel to the eigendecomposition of the sample covariance;
- choose for a target explained variance or approximation error in a finance setting.
In this lesson
- SVD and rank- truncation
- PCA and the covariance eigenbasis
Before you start
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