Dear Morphometricians, I am currently trying to understand the mathematical backgrounds of landmark-based geometric morphometrics. Some questions arose that we could not answer during discussions in our lab which is why I hope you can help - many thanks in advance!
The first question is: What exactly is “Kendall’s shape space”? If I understand Kendall’s (1984) statement in Eq. 4 correctly, the shape space is a quotient space; the elements are equivalence classes of pre-shapes (a fiber on the pre-shape sphere becomes one element in shape space). The elements of the equivalence classes have less “coordinates” (vector elements) than the original landmark configuration and lie on a hyperdimensional sphere with a radius of 1. In Theorem 2 Kendall (1984) states that the shape space for triangles is isometric to a three-dimensional sphere with a radius of 0.5. The triangles on this sphere with a radius of 0.5 are represented by three Cartesian coordinates that are calculated from the original landmark configuration (Kendall 1984, section 5), whereas the triangles are represented by equivalence classes in shape space. In several publications I now find illustrations of a hemisphere of radius 1 and a sphere of radius 0.5 (both share one point at the pole); those publications usually use the full landmark set. The sphere of radius 0.5 is often termed “Kendall’s shape space” (sometimes with a reference to triangles, sometimes not). So, how does this fit with the definitions and statements in Kendall (1984)? Is there a publication that extends Kendall (1984) to the use of full landmark configurations and explains how they are (mathematically) related to the sphere with radius 0.5 (for all numbers and dimensions of landmarks)? Related to this question: what do the points on the sphere of radius 0.5 in those publications look like? Are they equivalence classes, full landmark configurations, or 3 cartesian coordinates representing triangles? Are they really scaled to unit centroid size as the shapes on the pre-shape sphere [= elements of equivalence classes in shape space]? The second question is: Why do we need a tangent space projection? I understand that the superimposed landmark configurations lie on a hyper-hemisphere and I know the argument that standard statistical procedures need a linear space. Yet, the superimposed landmark configurations are matrices or vectors, depending on how they are formatted, for which we can compute Euclidean distances. Where exactly do the statistical tests go wrong if we use the superimposed landmark configurations without tangent space projection and calculate Euclidean distances? If I, for example, think about MANOVAs as suggested by Anderson (2001, Austral Ecology), I guess that the mean shapes of the groups need to be calculated to be able to calculate the different sums of squares. If the mean “shape” is calculated by group-wisely simply calculating the mean of each of the coordinates, the resulting mean “shape” of each group lies within the hyper-hemisphere of radius 1. So the mean “shape” is not a shape because the centroid size is not standardized. Yet, if I got all distance calculations correctly (see attached R-script “Compare_distance_measures_in_original_and_tangent_space.R”), I find that the Euclidean distances between the mean “shapes” inside the hyper-hemisphere are slightly closer to the corresponding Procrustes distances than the Euclidean distances in tangent space; the Procrustes distances have been calculated by rescaling the mean “shapes” to unit centroid size followed by determining the arc length between them. If the mean “shapes” inside the hyper-hemisphere are rescaled to unit centroid size, then the Euclidean distances between them are even closer to the Procrustes distances. In addition, if I simulate groups of landmark configurations, superimpose them without and with tangent space projection, and test for significant differences between the groups, I feel that the decision on the significance of the group differences is correct slightly more often if it is based on the superimposed landmark sets without tangent space projection (not exhaustively or formally tested; see R-script “compare_ProcrustesMANOVA_in_original_and_tangent_space.R”). And one last, more general question: If all landmark configurations are superimposed onto a common mean shape, does this also minimize the Procrustes distances (measured as arc length) between all pairs of landmark configurations and between the mean shapes of sub-groups of landmark configurations? Thanks a lot for your insights! Kind regards Karo -- Dr. Karolin Engelkes Institute of Evolutionary Biology and Animal Ecology University of Bonn Phone: +49 (0) 228 73 5481 An der Immenburg 1 53121 Bonn Germany -- You received this message because you are subscribed to the Google Groups "Morphmet" group. To unsubscribe from this group and stop receiving emails from it, send an email to morphmet2+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/morphmet2/ac177d3d-b28a-41b2-bfb6-a7a9f8cd73d0n%40googlegroups.com.
compare_ProcrustesMANOVA_in_original_and_tangent_space.R
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Compare_distance_measures_in_original_and_tangent_space.R
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