advanced practice · coding problem
PCA variance explained
2D data with cov matrix [[4,3],[3,9]]. Eigenvalues are 10 and 3. What fraction of variance do the top 1 and top 2 components explain? Why center before PCA?
The mechanism
PCA finds an orthogonal basis ordered by variance. Top-k projection is the best k-dimensional summary in squared-error sense. Center first.
Expected evidence
PC1: 10/13≈77%; both: 100%. Centering removes the mean offset so variance is spread, not dominated by the origin.
Try this drill in the practice workspace →
Hints when you need them
- Variance explained = λ_i / Σλ.
- Uncentered PCA chases the mean.
Watch for these mistakes
- PCA on uncentered data
- Components without domain meaning
- PCA for prediction without care