- The eigenvalues associated with the factors, identically in the space of observations indexed by set I, and in the space of attributes indexed by set J, are given by the eigenvalues associated with the decomposition of the inertia.
- The decomposition of the inertia is a principal axis decomposition, which is arrived at through a singular value decomposition.

## Supplementary Elements: Information Space Fusion Dual Spaces And Transition Formulas

•
$F_{\alpha}(i)=\lambda_{\alpha}^{-\frac{1}{2}}\sum_{j\in J}f_{j}^{i}G_{\alpha}(j)$ for $\alpha=1,2,\ldots N;i\in I$  $G_{\alpha}(j)=\lambda_{\alpha}^{-\frac{1}{2}}\sum_{i\in I}f_{i}^{j}F_{\alpha}(i)$ for $\alpha=1,2,\ldots N;j\in J$
- *Transition formulas*: The coordinate of element i ∈ I is the barycentre of the coordinates of the elements j ∈ J, with associated masses of value given by the coordinates of f i j of the profile f i J
. This is all to within the λ
− 12 α constant.
- In the output display, the barycentric principle comes into play: this allows us to simultaneously view and interpret observations and attributes.

## Supplementary Elements

Overly-preponderant elements (i.e. row or column profiles), or exceptional elements (e.g. a sex attribute, given other performance or behavioural attributes)
may be placed as supplementary elements. This means that they are given zero mass in the analysis, and their projections are determined using the transition formulas. This amounts to carrying out a Correspondence Analysis first, without these elements, and then projecting them into the factor space following the determination of all properties of this space.

Here too we have a new approach to fusion of information spaces focusing the projection.

## Hierarchical Clustering

Background on hierarchical clustering in general, and the particular algorithm used here, can be found in [5].

Consider the projection of observation i onto the set of all factors indexed by α, {Fα(i)} for all α, which defines the observation i in the new coordinate frame.

This new factor space is endowed with the (unweighted) Euclidean distance, d. We seek a hierarchical clustering that takes into account the observation