
## Input: Cloud Of Points Endowed With The Chi Squared Metric

The cloud of points consists of the couples: (multidimensional) profile coordinate and (scalar) mass. We have NJ (I) = {(f i J
, fi);i ∈ I} ⊂ RJ , and again similarly for NI (J). Included in this expression is the fact that the cloud of observations, NJ (I), is a subset of the real space of dimensionality |J| where |.| denotes cardinality of the attribute set, J.

The overall inertia is as follows:

$$M^{2}(N_{J}(I))=M^{2}(N_{I}(J))=\|f_{I J}-f_{I}f_{J}\|_{f_{I}f_{J}}^{2}$$
$\sum_{i\in I,j\in J}(f_{ij}-f_{i}f_{j})^{2}/f_{i}f_{j}$  $\sum_{i\in I,j\in J}(f_{ij}-f_{i}f_{j})^{2}/f_{i}f_{j}$ 
$$\left(1\right)$$

The term kfIJ −fI fJ k 2 fI fJ
is the χ 2 metric between the probability distribution fIJ and the product of marginal distributions fI fJ , with as center of the metric the product fI fJ . Decomposing the moment of inertia of the cloud NJ (I) - or of NI (J) since both analyses are inherently related - furnishes the principal axes of inertia, defined from a singular value decomposition.

## 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:

$$d(i,i^{\prime})=\|f_{J}^{i}-f_{J}^{i^{\prime}}\|_{f_{J}}^{2}=\sum_{j}\frac{1}{f_{j}}\left(\frac{f_{i j}}{f_{i}}-\frac{f_{i^{\prime}j}}{f_{i^{\prime}}}\right)^{2}\tag{2}$$
$$\left({\boldsymbol{3}}\right)$$

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.