Principal Component Analysis
Principal component analysis, PCA, is the first dimension-reduction method most researchers reach for, and interviewers ask about it in two ways. The mechanical version is "how does PCA work, step by step?". The research version is "you ran PCA on stock returns and the top component explains 30%. What is it, and what would you do with it?". Both follow from the eigen-decomposition in the previous lesson.
The method
PCA replaces a set of correlated variables with uncorrelated components, ordered from the most variance to the least.
- Centre each variable by subtracting its mean. Standardise it as well if the variables are in different units.
- Compute the covariance matrix of the prepared data, or the correlation matrix if you standardised.
- Find its eigenvectors and eigenvalues, and sort them from the largest eigenvalue down.
- Project the data onto the leading eigenvectors. The -th component is , with variance .
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