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## Principal Components

A quick summary, all is detailed below:
• centered ,
• SVD best approximation to ,
• Cols of new variables,
• Principal Components ,
• Principal axes ,
• Supplementary points ,
• Distance between two points,

• Transition Formulae ,
• Scale problem solution: changes the metric,
• Generalized PCA 2nd generalized svd,
• , , ,
• Special case: correspondence analysis,
• , , , ,
• ,
• Number of Components. PRESS(M) = .

Maximization of with

• for eigenvector of ,
• .

1) centered, all points (observations) same weight.

variables can be replaced by the columns of .

is the best (optimal) approximation of rank for .

The columns of are the directions along which the variances are maximal.

Definition: Principal components are the coordinates of the observations on the basis of the new variables (namely the columns of ) and they are the rows of . The components are orthogonal and their lengths are the singular values .

In the same way the principal axes are defined as the rows of the matrix . Coordinates of the variables on the basis of columns of .

However, this decomposition will be highly dependent upon the unity of measurement (scale) on which the variables are measured. It is only used, in fact, when the are all of the same order''. Usually what is done is that a weight is assigned to each variable that takes into account its overall scale. This weight is very often inversely proportional to the variable's variance. So we define a different metric between observations instead of

The same can be said of the observations; some may be more important'' than others (resulting from a mean of a group which is larger).

2) General PCA (X,Q,D)

centered with respect to :.

Generalized singular value decomposition.

Practical Computation:

Principal Components are the columns of: .

eigenvectors of .

eigenvectors of correlation matrix.

Next: Discriminant Analysis Up: Principal Components Previous: Example of the Power   Index
Susan Holmes 2002-01-12