FCA-$2$modes

Correspondence analysis of a two-way contingency table with cells $n_{ij}, \quad i=1\cdots I, \quad j=1\cdots J$ can be described as follows. The usual notations are:

$$n_{i.}=\sum_j n_{ij}, \quad n_{.j}=\sum_i n_{ij}, \quad n_{..}=N=\sum_{ij} n_{ij}$$

and then the observed proportions are defined as $p_{ij}=n_{ij}/N$. Diagonal metrics containing vector margins $P_{I.}={}^t(\cdots p_i.\cdots)$ and $P_{.J}$ used thereafter are noted $D_I$ and $D_J$. Correspondence analysis provides a decomposition of the measure of lack of independence between the two categorical variables indexed respectively by $i$ and $j$ in performing the PCA (or generalised PCA) of the following triple ([5]):

$$(D_I^{-1}PD_J^{-1}\quad -{\it 1}\!\!I_{IJ}, \quad D_I, \quad D_J)$$ (21)

where the triple is defined as ( data, metric on $I\!\!R^I$, metric on $I\!\!R^J$). The measure of lack of independence can be written :

$$\frac{\chi^2}{N}=\sum_{ij}\frac{(p_{ij}-p_{i.}p_{.j})^2}{... ....}p_{.j}(\frac{p_{ij}}{ p_{i.}p_{.j}} -1)^2=\sum_s \sigma_s^2$$ (22)

where the $\sigma_s$ are the singular values of the PCA of the triple given above. From the data reconstruction formula, one can write for $r\leqslant min(I-1,J-1)$:

$$\frac{\widehat{p}_{ij}}{ p_{i.}p_{.j}}=1+\sum_{s=1}^r\sigma_s\psi_{is}\varphi_{sj}$$ (23)

or equivalently in a tensor form:

$$D_I^{-1}\widehat{P}D_J^{-1}\quad ={\it 1}\!\!I_{IJ}+ \sum_{s... ...otimes\varphi_s =\sum_{s=0}^r\sigma_s\psi_s\otimes\varphi_s$$ (24)

where $\sigma_0=1$, $\psi_0={\it 1}\!\!I_I$, and $\varphi_0={\it 1}\!\!I_J$. If $r= min(I-1,J-1)$ the approximation is exact i.e. $\widehat{P}$ is $P$. From equation (24) and $\sum_{ij} p_{ij}=1$ (which implies the solution $s=0$) it is possible to perform the PCA of the triple:

$$(D_I^{-1}PD_J^{-1}, \quad D_I, \quad D_J) \mbox{ or in tensor form } ((D_I^{-1}\otimes D_J^{-1})P, \quad D_I, \quad D_J).$$ (25)

This last equation generalised for $k > 2$ enables to look at lack of marginal independence through associated solutions of the first Principal Tensor ([10]).