
## Output: Cloud Of Points Endowed With The Euclidean Metric In Factor Space

The χ 2 distance with center fJ between observations i and i 0is written as follows in two different notations:

1 fj fij fi − fi 0j fi 0 2(2) d(i, i0) = kf i J − f i 0 J k 2 fJ = X j
$$\left({\boldsymbol{3}}\right)$$
In the factor space this pairwise distance is identical. The coordinate system and the metric change. For factors indexed by α and for total dimensionality N (N = min {|I| − 1, |J| − 1}; the subtraction of 1 is since the χ 2 distance is centered and hence there is a linear dependency which reduces the inherent dimensionality by 1) we have the projection of observation i on the αth factor, Fα, given by Fα(i):

$$d(i,i^{\prime})=\sum_{\alpha=1..N}\left(F_{\alpha}(i)-F_{\alpha}(i^{\prime})\right)^{2}$$
0))2(3)
In Correspondence Analysis the factors are ordered by decreasing moments of inertia. The factors are closely related, mathematically, in the decomposition of the overall cloud, NJ (I) and NI (J), inertias. 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.

## Dual Spaces And Transition Formulas

The projection of observation i on the αth factor is Fα(i). Likewise, the projection of attribute j on the αth factor in its associated space is Gα(j). The following relationship holds.

$$F_{\alpha}(i)=\lambda_{\alpha}^{-\frac{1}{2}}\sum_{j\in J}f_{j}^{j}G_{\alpha}(j)\mbox{for}\alpha=1,2,\ldots N;i\in I$$  $$G_{\alpha}(j)=\lambda_{\alpha}^{-\frac{1}{2}}\sum_{i\in I}f_{i}^{j}F_{\alpha}(i)\mbox{for}\alpha=1,2,\ldots N;j\in J\tag{4}$$

Relation (4) gives us the *transition formulas*: The coordinate of element i ∈ I is the barycenter 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.