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
