Shape Dimensionality Metrics for Landmark Data
AbstractThe study of modularity in geometric morphometric landmark data has focused attention on an underlying question, that of whole-shape modularity, or the pattern and strength of covariation among all landmarks. Measuring whole-shape modularity allows measurement of the dimensionality of the shape, but current methods used to measure this dimensionality are limited in application. This paper proposes a metric for measuring the “effective dimensionality”, De, of geometric morphometric landmark data based on the Shannon entropy of the eigenvalue vector of the covariance matrix of GPA landmark data. A permutation test to establish null rank deficiency is developed to allow standardization for comparing dimensionality metrics between data sets, and a bootstrap test is employed for measures of dispersion. These novel methods are applied to a data set of 14 landmarks taken from 119 dire wolf jaws from Rancho La Brea. Comparison with the current test based on eigenvalue dispersion demonstrates that the new metric is more sensitive to detecting population differences in whole-shape modularity. The effective dimensionality metric is extended, in the dense semilandmark case, to a measure of “latent dimensionality”, Dl. Latent dimensionality should be comparable among landmark spaces, whether they are homologous or not.