= 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.<Geometric Data Analysis by Brigitte Le Roux and Henry Rouanet>[Chapters 1–4, overview, correspondence analysis, Euclidean clouds and principal component analysis]{ref}
For a column-centered data <matrix> with <singular value decomposition> $X=U\Sigma V^T$, the first $k$ <principal components> give coordinates $XV_k=U_k\Sigma_k$. 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.
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