## 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:
- 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.
Y 1, Y2,... Yn = a + b1X1+b2X2 +... + Rwhere R are the residual difference between themodel and the data:
Usually we decide on a " small" number of components which fit"enough" variance, what ever that means. |