Principal components

Objective. Reduce dimension through spectral decomposition.

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Learn: the key idea

Principal components analysis reduces the dimension of correlated data by finding new orthogonal axes that capture the most variance. Each principal component is an eigenvector of the covariance matrix Σ, and its associated eigenvalue gives the variance explained along that direction. Components are ordered by decreasing eigenvalue, so the first few often summarise most of the data's structure, allowing lower-dimensional approximation with minimal information loss. For example, if Σ has eigenvalues 8 and 2, the first principal component explains 8/(8+2) = 80% of the total variance, making it the dominant direction of spread.

Worked example

A covariance matrix for principal components analysis has eigenvalues 12 and 3. What proportion of total variance is explained by the first component?

1 $Total variance = sum of eigenvalues = 12 + 3 = 15$
2 $Proportion explained by first = λ₁/total$
3 $Compute: 12/15$
4 $Simplify: 4/5 (80%)$

Answer: 4/5 (80%)

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Common mistake. Students perform principal components analysis on raw variables with very different scales without standardising; large-scale variables then dominate the eigenvectors artificially. Always standardise variables first unless scale differences are genuinely meaningful.

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