Principal Components
Forrest Young's Notes
Copyright © 1999 by Forrest W. Young.
Overview of Principal Components

Principal Components

Principal Components is a statistical technique that is used to summarize
the variance in a group of variables by linear combinations of the variables.
The linear combinations are computed to maximize the variance accounted
for.


Purpose

To summarize the variation in several numeric variables by a smaller number
of variables called components.


Linear Combinations

The components are the linear combinations of the original variables that
explain the maximum variance.

The First Principal Component
The first principal component summarizes as much variance in the variables
as can be summarized by any single linear combination.
Additional Principal Components
Additional principal compoents are at right angles to all preceeding
ones (are "orthogonal" to them) and account for as much variance
as is possible.
The first several principal components

explain as much variation in the raw data as can be explained by that many
orthogonal linear combinations.

represent orthogonal directions in the raw data that are the longest directions
that are mutually at right angles to each other.

form a rigid, orthogonal rotation of the original raw data into an orientation
where the new dimensions have maximum variance.
