Editing Morphometrics (section)

====Procrustes-based geometric morphometrics====
[[File:Procrustes superimposition.png|thumb|Procrustes superimposition]]
Shape analysis begins by removing the information that is not about shape. By definition, shape is not altered by translation, scaling or rotation.<ref>{{cite journal|last=Kendall|first=D.G.|title=The diffusion of shape|journal=Advances in Applied Probability|year=1977|volume=9|pages=428–430|doi=10.2307/1426091|issue=3|jstor=1426091|s2cid=197438611 }}</ref>  Thus, to compare shapes, the non-shape information is removed from the coordinates of landmarks.  There is more than one way to do these three operations. One method is to fix the coordinates of two points to (0,0) and (0,1), which are the two ends of a baseline. In one step, the shapes are translated to the same position (the same two coordinates are fixed to those values), the shapes are scaled (to unit baseline length) and the shapes are rotated.<ref name=Bookstein1991 />  An alternative, and preferred method, is [[Procrustes superimposition]].  This method translates the centroid of the shapes to (0,0); the ''x'' coordinate of the centroid is the average of the ''x'' coordinates of the landmarks, and the ''y'' coordinate of the centroid is the average of the ''y''-coordinates. Shapes are scaled to unit centroid size, which is the square root of the summed squared distances of each landmark to the centroid.  The configuration is rotated to minimize the deviation between it and a reference, typically the mean shape. In the case of semi-landmarks, variation in position along the curve is also removed. Because shape space is curved, analyses are done by projecting shapes onto a space tangent to shape space.  Within the tangent space, conventional multivariate statistical methods such as multivariate analysis of variance and multivariate regression, can be used to test statistical hypotheses about shape.

Procrustes-based analyses have some limitations.  One is that the Procrustes superimposition uses a least-squares criterion to find the optimal rotation; consequently, variation that is localized to a single landmark will be smeared out across many.  This is called the 'Pinocchio effect'. Another is that the superimposition may itself impose a pattern of covariation on the landmarks.<ref>{{cite journal|last=Rohlf|first=F. J.|author2=Slice, D.|title=Extensions of the Procrustes method for the optimal superimposition of landmarks|journal=Systematic Zoology|year=1990|volume=39|pages=40–59|doi=10.2307/2992207|issue=1|jstor=2992207|citeseerx=10.1.1.547.626}}</ref><ref>{{cite journal|last=Walker|first=J.|title=The ability of geometric morphometric methods to estimate a known covariance matrix|journal=Systematic Biology|year=2000|volume=49|pages=686–696|doi=10.1080/106351500750049770|pmid=12116434|issue=4|doi-access=free}}</ref> Additionally, any information that cannot be captured by landmarks and semilandmarks cannot be analyzed, including classical measurements like "greatest skull breadth".  Moreover, there are criticisms of Procrustes-based methods that motivate an alternative approach to analyzing landmark data.