Good questions. The topic can be confusing and difficult to visualize - especially for 3D landmark data.
The 1999 paper in the Journal of Classification that Adams mentions was my attempt to describe its practical relevance in morphometrics. My 1999 paper in Hystrix may also be of interest to you. You may wish to play with my tpsTri software also as it was used for some of the figures in those papers. Easiest to at first just think of variation in shapes of triangles in 2 dimensions. As you mention, Kendall's shape space for triangles can be visualized as the surface of a sphere of radius 1/2. What convinced me of the importance of Kendall's shape space was that Kendall showed that the distribution of all possible triangles was a uniform distribution in Kendall's shape space. After a GPA the distribution of shapes is on a hemisphere of radius 1 (corresponding to centroid size of 1). I am not sure if anyone has given a good name for this hemisphere corresponding to all possible triangles Procrustes aligned to any single shape. In 1999 I called it a "preshape space of triangles aligned to a reference triangle". Not very snappy! Perhaps I should have called it something like the "Slice hemisphere" as Dennis Slice first showed it to me and was puzzled why it was not a surface of a sphere. The distribution of triangles is not, however, uniform on the surface of the hemisphere. As you mention, conventional multivariate methods assume linear spaces and use linear matrix algebra. The tangent space is as you mention the projection of points from the surface of the hemisphere onto a plane that passes just through the point on the hemisphere that corresponds to the reference triangle (not really a "reference" just the mean shape in practical applications). I have called it Kendall's tangent space but I probably should have named it after John Kent as he influenced my understanding. Something I found fascinating was that the distribution of all possible triangles was again uniform in the projection within the circular distribution (Kendall showed that also). The importance, to me, of being uniform was that if I see some pattern in the distribution (clusters, covaniance, etc.) then it implies something about the distribution of shapes not just an artifact of the mathematical operations used to create the projection (as in the case of EDMA and some other earlier statistical approaches suggested for analyzing shape variation). Of course, for very small variation in shape the points will be close to their mean so that distances on the surface of the hemisphere (thus close to their projections) and distances in the tangent space will be very similar (though distance in the tangent space will be a little larger due to the projection). Might be good enough for some studies but if one does not do the projection then you will find that a PCA of your GPA aligned data (assuming large n > 2p-4) will not yield 4 zero eigenvalues as it should with centering, rotation, and size removed. Only 3 will be zero (i.e., computationally numbers like 10^-15 or so). The 4th smallest might be "only" 10^-8 or so. That is a result of the curved shape of the hemisphere. If you do the projection then the last 4 eigenvalues will be essentially zero as the curvature is now gone. An alternative is to perform the multivariate analysis directly in Kendall's shape space. Kent (I cannot locate the references right now but it was in early 2000s) showed one could, for example, perform a generalization of a PCA directly in the curved surface. Some odd properties as eigenvectors were great circles on the curved surfaces as I remember). One can generalize, of course from triangles to shapes with more landmarks and to landmarks in 3 dimensions. The 3-dimensional case is more complicated to try to visualize because the simplest case requires 5 dimensions to represent not just 3 so one cannot just look at the space. It also has some more complicated properties. You could look at the book "Shape & shape theory" by Kendall, Barden, and Le (1999). It presents a way to visualize variation in the 5-dimensional shape space. Was not an easy read for me but perhaps I should try again! This distinction between shape space and tangent space is not of much importance in practical applications where biological variation tends to be small compared to all possible variation among p landmarks and because one usually only looks at the distribution along the first few eigenvectors with the largest eigenvalues but I prefer to have computations match what one expects theoretically rather than just being good approximations. When programming being just "close enough" could hide subtle bugs. Getting rid of that known artifact also allows one to try to possibly interpret those eigenvectors with the smallest eigenvalues as they correspond to the most stable aspects of possible shape variation (i.e., least varying due to development, environment, etc.). Does this help or confuse more? F. James Rohlf Distinguished Professor, Emeritus and Research Professor Depts: Anthropology and Ecology & Evolution Stony Brook University On 9/8/2021 2:26:38 PM, Adams, Dean [EEOB] <dcad...@iastate.edu> wrote: Karolin, A reading of Rohlf 1999 may help. Dean Rohlf, F.J. 1999. Shape statistics: Procrustes superimpositions and tangent spaces. Journal of Classification. 16:197-223. Dr. Dean C. Adams Distinguished Professor of Evolutionary Biology Director of Graduate Education, EEB Program Department of Ecology, Evolution, and Organismal Biology Iowa State University https://faculty.sites.iastate.edu/dcadams/ [https://faculty.sites.iastate.edu/dcadams/] phone: 515-294-3834 From: morphmet2@googlegroups.com <morphmet2@googlegroups.com> On Behalf Of karolin....@gmail.com Sent: Tuesday, September 7, 2021 7:04 AM To: Morphmet <morphmet2@googlegroups.com> Subject: [MORPHMET2] Questions about Kendall’s shape space and tangent space projection 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 [mailto: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 [https://groups.google.com/d/msgid/morphmet2/ac177d3d-b28a-41b2-bfb6-a7a9f8cd73d0n%40googlegroups.com?utm_medium=email&utm_source=footer]. -- 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. 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