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PCA and Dimensionality Reduction

"You run PCA on two years of daily returns for 100 stocks. What is the first component, and how many of the components are real?" A complete answer turns the components into portfolios, explains why the later ones are unstable, and compares each eigenvalue with the noise level that is left once the market is removed.

What the basics lesson covers

The principal component analysis lesson covers the method step by step, the choice between covariance and correlation, the market component of an equity panel and the Marchenko-Pastur noise range. In short, PCA is the eigen-decomposition Σ=VΛV⊤\Sigma = V \Lambda V^\top of the covariance or correlation matrix. For NN independent assets over TT periods, with q=N/Tq = N / T, the sample correlation eigenvalues fill the range from (1−q)2(1 - \sqrt{q})^2 to (1+q)2(1 + \sqrt{q})^2.

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