Geometric data analysis

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.

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.