Additional Principal Components
Forrest Young's Notes
Copyright © 1999 by Forrest W. Young.
- Each Additional Principal Component
- Each additional principal component is the linearfunction of the set of variables which
- is at right angles to every preceeding principal component. Usually wesay "are orthogonal to" rather than "at right angles", but they mean thesame thing.
- fits the variables as well as possible in a least squares sense, giventhe orthogonality constraint.
- The equation becomes:
Y1, Y2,... Yn = a + b1X1+b2X2 +... + R
where R are the residual difference between themodel and the data:
Residuals and Additional Components
- The residuals are represented by R in the equation.
- Each additional component can be thought of as the first principal componentof the residuals that remain after the previous components have been computed.
- Each additional principal component line identifies the "central tendency"of the set of the residuals.
- The several principal components provide a simplified description--- a model --- of the set of variables.
- There are a maximum of N principal components when there are N variables.They will provide a perfect, but imparsimonious model of the data.
Usually we decide on a "small" number of components which fit"enough" variance, what ever that means.