I would like to clarify a few points. There is no 'shape space projection'. Shapes are points in Kendall's shape space. As Thomas points out it is a curved space. Standard multivariate analyses assume linear spaces so one usually tries to approximate shape space by mapping from shape space to a linear space tangent to Kendall shape space. It still has the same number of dimensions as the Kendall shape space (for 3D data it is 3*p - 7, for 2D data 2*p - 4, not just 3*p or 2*p because variation in location, orientation, and scale have been removed). The approximation is such that shapes further from the reference (point of tangency that defines the tangent space) will not be represented correctly. For that reason one uses a mean shape as the reference so that the shapes in sample are relatively similar to the reference shape. The distances among shapes similar to the reference are represented quite accurately.
The help files of most of my tps programs have a diagram showing the relationships between Kendall shape space, the space of GPA aligned shapes, and the tangent space. It shows a cross-section of the space for p=3 landmarks in a 2D physical space. Difficult to describe in a text-only email. One may sometimes use Mahalanobis D within the tangent space when one wishes to measure dissimilarity between shapes in terms of how much samples from different populations overlap statistically rather than in terms of how much they differ in shape per se. It represents a very different concept of distance. Its use implies a linear space such as the tangent space. Note that the number of dimensions stays the same as noted above. I hope these comments help. ------------------------ F. James Rohlf, Distinguished Professor Ecology & Evolution, Stony Brook University www: http://life.bio.sunysb.edu/ee/rohlf > -----Original Message----- > From: morphmet [mailto:[EMAIL PROTECTED] > Sent: Saturday, December 08, 2007 6:44 AM > To: morphmet > Subject: Tangent Space vs. Shape Space > > Hello all, > > While introducing the analysis of shape to a colleague, I came up > against a bit of difficulty explaining the differences, and relative > merits, of a tangent space vs shape space analysis. I think I found > that > my attempts to explain this only highlighted gaps in my own > understanding. So, I provide here my explanation of the differences. I > invite comments and criticisms to help us or, or at least me, better > understand these differences. > > > > Tangent space is a shorthand term for the projection and simple > analysis > of a shape within multivariate space. In geometric morphometrics each > Cartesian coordinate value is treated as a position on a unique axis. > For a 3D tetrahedron shape there are four points and each is described > by three coordinate values, hence the shape is treated as a single > point > in a twelve dimensional space. Tetrahedrons that are similarly shaped > will be located close to each other in this multidimensional space. > Shapes within a natural group will form a cluster of point locations. > Within group variance is an expression of the distance between > multidimensional points within a cluster. Statistical analysis compares > within group variance to the variance between groups, and thereby > determines if one group is statistically, or probably, different from > another. In the simplest analysis the straight-line, Euclidean, > distance > between points can be used. This is the analytical perspective of > tangent space. However, the multivariate points are actually being > plotted on the surface of a manifold. A manifold is a multivariate > object where distances in the local environment can be reasonably > evaluated as straight line Euclidean distance - but this distance > measure actually lays tangent to the manifold. When distances become > great enough, the multidimensional curvature of the manifold reduces > the > utility of tangent space projection. A distance measure must be used > that is measured along the surface of the manifold. This is the > concept > of the "shape space" projection. The data transformation into shape > space reduces the dimensionality of the manifold and allows for more > accurate distance measures between points, which becomes an expression > of the D2 Mahalanobis distances that are commonly provided in > multivariate statistical analyses. The question remains, when is shape > space projection required or when is the tangent space projection > sufficient? The answer really depends upon the research question and > the > judgement of the researcher. The tangent space projection is more > conservative, distances among points are always shorter. This means > that > tangent space projections are more likely to overemphasize the > similarity of shapes - a type II statistical error. When an analysis of > a tangent space projection finds a significant difference between > groups, it is unlikely that the analysis of the shape space projection > will produce a different result. > > > > > > > */Thomas M. Greiner, Ph.D./* > Anatomist and Physical Anthropologist > Dept. of Health Professions > University of Wisconsin - La Crosse > 1725 State Street > La Crosse, WI 54601 USA > > Phone: (608) 785-8476 > Fax: (608) 785-8460 > > > > -- > Replies will be sent to the list. > For more information visit http://www.morphometrics.org -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
