# Combining the Tools of Geometric Morphometrics

• Fred L. Bookstein
Chapter
Part of the NATO ASI Series book series (NSSA, volume 284)

## Abstract

The greatest strength of the new geometric morphometrics is the system of interrelated multivariate and graphical procedures it offers for a variety of analytic questions involving landmark data. A typical analysis will begin with the conversion of landmark data into a multivariate statistical representation of shape, will continue with a series of broadly familiar multivariate matrix manipulations, and will conclude by inspection of a considerable variety of diagrams that represent the findings in both the space of shape coordinates per se and the space of the two-or three-dimensional image of the organism. The choices under the first heading, the passage to a multivariate representation of shape, include two-point shape coordinates, partial warp scores, and Procrustes residuals. Each of these except the partial warp scores is unsuitable for some subset of the reasonable matrix manipulations; for instance, shape coordinates do not supply sensible principal components analyses, and Procrustes residuals cannot lead to sound canonical variate analyses without modification. The modes of diagramming data include thin-plate splines, partial warp splines and scatters, Procrustes residual scatters, and resistant-fit scatters, among others. Most analyses benefit greatly from exploiting more than one of these.

Not every combination of shape coordinates, multivariate maneuvers, and diagram styles makes sense. Underlying the multivariate geometry of any sample is an a-priori Procrustes geometry of shape per se, and the three parts of any analysis must be mutually consistent in the use they make of this geometry as well as the usual linear geometry of multivariate analysis. Some formal structures serve two roles in this synthesis; for instance, the affine term of a generalized affine least-squares fit is equivalent to the uniform component of the relation between a specimen shape and the sample mean. Within the realm of multivariate computations per se, some analyses are automatically equivalent: for instance, relative warps analysis with α = −1 is a principal coordinates analysis of bending energy. Familiar statistical tests (e.g. two-sample comparisons by Hotelling’s T 2 ) come in several approximate versions for sample sizes smaller than those for which the standard procedure is appropriate. Some combinations are usually inappropriate; for instance, resistant-fit residuals are not shape coordinates and should not be used as input for any linear multivariate computation. And interpretation of certain computations requires the prior inspection of others; for instance, any relative warps analysis of the nonaffine space should be preceded by analysis of the full shape space. There are modifications of all the basic tools for applications to symmetric forms, and all come in one version for two-dimensional data and a different version for three-dimensional data.

This chapter explains the entire toolkit in terms of the logic by which diverse choices are combined into complete analyses that circumvent a variety of popular pitfalls.

## Keywords

Shape Space Centroid Size Canonical Variate Analysis Relative Warp Procrustes Distance
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.