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 
    1. is at right angles to every preceeding principal component. Usually wesay "are orthogonal to" rather than "at right angles", but they mean thesame thing.
    2. fits the variables as well as possible in a least squares sense, giventhe orthogonality constraint. 
    3. The equation becomes:

    4. Y1, Y2,... Yn = a + b1X1+b2X2  +... + R
      where R are the residual difference between themodel and the data:

    Residuals and Additional Components

    1. The residuals are represented by R in the equation.
    2. Each additional component can be thought of as the first principal componentof the residuals that remain after the previous components have been computed. 
    3. Each additional principal component line identifies the "central tendency"of the set of the residuals.

    The Model

    1. The several  principal components provide a simplified description--- a model --- of the set of  variables.
    2. There are a maximum of N principal components when there are N variables.They will provide a perfect, but imparsimonious model of the data.

    3. Usually we decide on a "small" number of components which fit"enough" variance, what ever that means.