7.1 Normalized Principal Component Analysis

Active variables: these are continuous (or quantitative) variables.

• Distance between variables: based on correlation

\[ d^2(j,l) = 2(1 - cor(j,l)) \]

The more associated two variables are, the smaller the angle between them.

• Distance between individuals: classic (squared) euclidean distance (sum of the squares of the differences of standardized data).

Interpretation

Factorial Planes for Variables: This has to do with the representation of the linear association between variables. We can interpret the directions in the plane in terms of the correlations between the variables and the axes.

Factorial Planes for Individuals: This has to do with the representation of the similarities between individuals. We interpret the positions of the individuals on the factorial plane in terms of the meaning assigned to the axes.

Supplementary Variables

• Continuous: correlations with the axes; they are positioned in the factorial plane of the variables.

• Categorical: representation of each category as the center of gravity of the individuals having such category. We use the v-test to assess their “characterization power.”