Standard Formula for the Uniform Shape Component in Landmark Data
In describing how the shapes of landmark configurations vary, an important role is played by the uniform component of shape variation. Geometrically, the uniform component parameterizes shape changes that leave parallel lines parallel throughout the form and so show neither a spatial gradient nor any spatial localization. Algebraically, the shapes produced from a mean form in this way make up a linearized subspace, U, of our multivariate shape space in the vicinity of that mean, so that the particular component that corresponds to a particular specimen or group difference can be computed as a pair of scores (for two-dimensional data) or a quintet of scores (for three-dimensional data) by a Procrustes projection onto U. The versions of this crucial projection that are offered by currently available program packages are inconsistent in several important aspects. This paper suggests a resolution of the contradictions among these approaches. I introduce a nearly exact Procrustes version of this component and show that it is identical to two separate versions previously suggested in the literature and that it represents a more precise estimate for yet another. A somewhat different version of this component is useful for multivariate statistical testing. The computations are demonstrated using the growing rat skull data set familiar from several other canonical demonstrations of the new geometric methods.
KeywordsReference Configuration Shape Space Reference Form Centroid Size Procrustes Distance
Unable to display preview. Download preview PDF.
- Bookstein, F. L. 1991. Morphometric tools for landmark data: Geometry and Biology. Cambridge University Press: New York. ( The Orange Book. )Google Scholar
- Bookstein, F. L. 1993. A brief history of the morphometric synthesis. In: L. F. Marcus, E. Bello, and A. Garcia-Valdecasas, (eds.), Contributions to morphometrics. Monografias del Museo Nacional de Ciencias Naturales 8, Madrid. Pages 15–40.Google Scholar
- Bookstein, F. L. 1996. Biometrics, biomathematics, and the morphometric synthesis. Bulletin of Mathematical Biology 58.Google Scholar
- Goodall, C. R. 1991. Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society, Series B, 53: 285–339.Google Scholar
- Mardia, K. V. 1995. Shape advances and future perspectives. In: Current Issues in Statistical Shape Analysis. Leeds University Press: Leeds, U.K. Pages 57–75.Google Scholar
- Rohlf, F. J. 1993. Relative warp analysis and an example of its application to mosquito wings. In: L. F. Marcus, E. Bello, and A. García-Valdecasas, (eds.), Contributions to morphometrics. Monografias del Museo Nacional de Ciencias Naturales 8, Madrid. Pages 131–159.Google Scholar
- Rohlf, F. J., and F. L. Bookstein, (eds.), 1990. Proceedings of the Michigan morphometrics workshop. University of Michigan Museum of Zoology Special Publican 2.Google Scholar
- Thompson, D’A.W. 1961. On growth and form (1917 edition abridged by John Tyler Bonner.) Cambridge University Press: London.Google Scholar