6.4.1 Supplementary elements

The most distinguishing feature of descriptive principal components analysis is the possibility of introducing supplementary elements (variables or individuals) into factor graphics.

If p_i columns are added to matrix R, we get matrix R⁺ with (p+p₁) columns. If n₁ rows are added to n rows, we get the matrix R₊with n+n₁ rows and p columns.

The matrices R⁺ and R₊ are transformed respectively into matrices X⁺ and X₊ in order to make the new rows and columns comparable to those of X.

In  ⁿ, we have p₁ supplementary variable points. For consistency in the interpretation of inter-variable distances in terms of correlations, the following transformation must be performed (normalized principal axes analysis).

(18)

We compute new means and new standard deviations that incorporate the supplementary variables in order to place these supplementary variables on the sphere of the unit radius.

The projection operator on the axis 3 ⁿ is the unit vector

(19)

The abscissae of the p₁ supplementary variables on these axes are therefore the p₁ components of the vector X⁺Va

In  ^p, the introduction of the supplementary points consists of positioning them relative to the centroid of the points (which has already been computed) and then dividing the coordinates by the standard deviations of the variables (which have already been calculated for the n individuals).

Therefore the following transformation is performed

(20)

The projection operator on axis a of 3 ^p is the unit vector Ua. The abscissae of the n₁ supplementary individuals are therefore the n₁ components of the vector X₊Ua.

Let X_s be the matrix

The product X₊Ua simultaneously yields the n+n₁ coordinates of the original individuals plus the supplementary individuals.

The variables or individuals involved directly in fitting the data are also called active elements. The supplementary elements are also called passive or illustrative elements.