PCA Explained Variance Ratio¶

Formula¶

\[ \mathrm{EVR}_k = \frac{\lambda_k}{\sum_{j=1}^{p}\lambda_j} \]

Plot¶

type: bars
xs: 1 | 2 | 3 | 4 | 5
ys: 0.55 | 0.22 | 0.11 | 0.07 | 0.05
xmin: 0.5
xmax: 5.5
ymin: 0
ymax: 0.6
height: 280
title: Example explained variance ratios

Parameters¶

\(\lambda_k\): \(k\)-th eigenvalue of covariance matrix
\(p\): number of features/components

What it means¶

Explained variance ratio tells how much total variance is captured by each principal component.

What it's used for¶

Choosing the number of components in PCA.
Communicating compression tradeoffs.

Key properties¶

Ratios sum to 1 across all components.
Cumulative explained variance is often more useful than per-component values.

Common gotchas¶

Large variance is not always equal to task-relevant signal.
Scaling features changes the covariance spectrum.

Example¶

If the first two components explain 85% variance, a 2D projection may preserve much of the spread.

How to Compute (Pseudocode)¶

Input: PCA eigenvalues lambda[1..p] (usually sorted descending)
Output: explained variance ratios EVR[1..p]

total <- sum_{j=1..p} lambda[j]
for k from 1 to p:
  EVR[k] <- lambda[k] / total

return EVR

Complexity¶

Time: \(O(p)\) once PCA eigenvalues are available
Space: \(O(p)\) for the ratio vector
Assumptions: \(p\) is the number of components; the cost of fitting PCA/eigendecomposition is excluded