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Brigitte Le Roux studies geometric data analysis, including correspondence analysis and principal component analysis. Her research account describes investigations of how principal axes change when observations, weights or the underlying Euclidean geometry are modified. This stability question asks which apparent structures survive a change in the analysis, rather than treating every fitted axis as intrinsic to the data.^[ref]

Her joint book with Henry Rouanet, Geometric Data Analysis by Brigitte Le Roux and Henry Rouanet, develops the weighted-cloud approach through correspondence analysis, Euclidean clouds and principal component analysis. It connects the linear algebra of projections with explicit choices about representation of multivariate observations.^[ref]

Correspondence analysis represents a nonnegative contingency table geometrically, displaying how row and column profiles depart from independence. Let $P$ be the table divided by its grand total, and let $r=P1$, $c=P^{T}1$ be positive marginal proportions. Its standardized residual matrix isA singular value decomposition of $S$ provides principal axes; the sum of squared singular values equals the Pearson Pearson chi-squared statistic for contingency tables divided by the grand total. Thus weighted departures from independence become geometric variation.^[ref]

For counts $\left(\begin{array}{cc}30& 10\\ 10& 30\end{array}\right)$, the total is 80 and both margins are $(1/2,1/2)$. Then $S=\left(\begin{array}{cc}1/4& -1/4\\ -1/4& 1/4\end{array}\right)$ has one nonzero singular value $1/2$. The total inertia is therefore $1/4$ and the Pearson chi-squared statistic for contingency tables is 20. The association has one geometric dimension. Zero margins require removal or separate treatment before using the inverse square roots.

Geometric data analysis represents multivariate observations as weighted clouds of points, then studies their Euclidean geometry through distances, orthogonal projections and principal components. Principal component analysis applies this approach to numerical variables; correspondence analysis adapts it to a table of counts with its own weights and metric. Choosing the weights and metric is part of the analysis, since these choices decide which differences between observations count as large.^[ref]

For a column-centered data matrix with singular value decomposition $X=U\Sigma V^T$, the first $k$ principal components give coordinates $X{V}_k=U_k{\Sigma }_k$. They are the orthogonal projection onto the chosen principal directions. Rescaling one variable changes the distances and can change those directions: measurements in metres and millimetres should not receive different influence merely because the units were changed. The geometry must therefore be interpreted together with the scaling convention, rather than treating the resulting plot as an intrinsic picture of the observations.

Henry Rouanet worked on statistics for researchers in psychology and the social sciences, emphasizing interpretable methods for analyzing actual data. His own account describes that aim and organizes his work by statistical problem rather than by an abstract sequence of prerequisites.^[ref]

With Brigitte Le Roux, he wrote Geometric Data Analysis by Brigitte Le Roux and Henry Rouanet. Its treatment of correspondence analysis and principal component analysis expresses multivariate data as weighted clouds in Euclidean geometry. The choice of weights and metric specifies which departures count as large; the resulting picture is part of the statistical model, not simply an illustration added afterward.^[ref]