---
tag: ABXD13M340
kind: concept
name: Principal component analysis
msc: [62H25, 15A18]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_component_analysis
same_as:
  wikidata: Q2873
  wikipedia: Principal component analysis
---

## Summary

Principal component analysis replaces correlated variables by new uncorrelated variables, the principal components, ordered by decreasing variance.
The components are the coordinates along the [[eigenvectors|0N6960W0BV]] of the [[covariance matrix|0V7G6WMFK0]].
Keeping only the first few components gives the best low-dimensional linear summary of the data in the least squares sense.
It is the most common method of dimension reduction.

## Explanations

### L1 human

Picture a cloud of data points in the plane shaped like a tilted cigar.
The first principal component is the direction along the cigar, where the points spread the most.
The second is the direction across it, at a right angle to the first.
If two test scores have variances $2$ and $2$ and covariance $1$, the first direction is $(1, 1)/\sqrt{2}$, along which the variance is $3$.
Describing each point by its position along the cigar alone loses little information.