- 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:

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

- 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 centre 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 centre fJ between observations i and i 0is written as follows in two different notations:

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

- 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 centred 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(1)
- 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.