Shape variables are usually constructed by a Procrustes superimposition. The raw coordinates are superimposed by translating the configurations to a common centroid, scaling to unit centroid size, and rotating until the sum of the squared distances between corresponding landmarks is minimized. The remaining Procrustes coordinates describe shape per se.
A particular advantage of geometric morphometrics is that
multivariate statistical results like PCA can be visualized as shapes
or shape changes. Similarly, multivariate regressions of Procrustes
shape coordinates on some external variable (such as centroid size,
behavioural or ecological variables) - which is called shape regression
- can be visualized as shape deformation. Another very useful tool is
Partial Least Squares (PLS) analysis (called singular warp
analysis in geometric morphometrics) allowing one to assess the pattern
of covariation between two or more blocks of variables.
A recent addition to the morphometric synthesis is PCA of size-shape space. The augmentation of the shape variables by the logarithm of centroid size allows one to study patterns of size and shape (i.e. form) together. This has turned out to be of particular importance for developmental studies or discrimination studies.