A modified Procrustes analysis for bilaterally symmetrical outlines, with an application to microevolution in Baculites

In quantitative paleobiology, one common format for image-derived information is the closed two-dimensional outline curve. Before the end of the last century a great variety of morphometric tools were on offer to deal with this data type, including diverse analyses of distance from a central point, “eigenshape analysis” of a carefully normalized tangent angle function, and Procrustes analysis of configurations of sliding semilandmarks. For the special case of outline forms that are nominally bilaterally symmetric, this paper offers a hybrid approach that fuses the Procrustes toolkit with a variant of a much older method, analysis of radial distance according to angles out of a center. The synthesis represents each outline as a regularly resampled data set of radially aligned shape coordinates analogous to bilateral Procrustes shape coordinates. When the center is iteratively located in one particular way, this representation has the same Procrustes distance statistics as a conventional radial representation, while the formalism of shape coordinates permits symmetrization, localizability, and thin-plate spline visualizations. We explain the method and sketch some of its advantages and limitations in the course of an example involving 99 Baculites inornatus outlines spanning 90 m of deposits from a high-precision measured section in the Campanian-age Rosario Formation cropping out at Punta San Jose, Baja California. This sequence shows substantial changes of pole curvature over time, changes that may involve a punctuation event regarding hydrostatics, hydrodynamics, or defense against predators. A concluding comment deals with implications of the developments here for Baculites microevolution and for morphometrics.

Fred L. Bookstein and Peter D. Ward