Dear Prof. Greiner, I leave answers to your interesting questions to statisticians.
I have a certainly simplistic view of the tangent space projection. The way I see it is just as a way to have data in an Euclidean space where standard statistics can be performed. Most of the time this is perfectly appropriate as biological variation tends to be small and the approximation is excellent. If one either has a bad tangent space approximation (this may happens for instance in Procrustes based locomotion analysis) or wants to work in the shape space for specific reasons, then I guess one needs a different statistics developed for curved spaces. Geographers have sometimes similar issues (much simpler however as they mirror the case of triangles in landmark based geometric morphometrics) and I suspect they have some ad hoc methods for doing the stats without need to project localities on a flat map. In geometric morphometrics, the only examples I can recall of statistical tests which did not need a tangent space projection are permutation tests (or other resampling stats) using Procrustes distances. About the tangent space 'underestimating' distances compared to Procrustes distances in the full shape space, this is true only if one is using an orthogonal projection. If one uses a stereographic projection, distances would be larger in the tangent space. For the kind of morphometric analyses most of us do, this generally does not make any practical difference as the approximation of the tangent space is so good that either projection (orthogonal or stereographic) produces exactely the same outcome. I look forward to reading what the statisticians have to say about this. Cheers Andrea At 12:43 08/12/2007 +0100, you wrote: >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
