Also, if you have a mathematics background, I would recommend Goodall (1991) (https://doi.org/10.1111/j.2517-6161.1991.tb01825.x), besides the papers from Kendall in the 1980s. I have colleagues that are mathematicians that were able to understand what it was all about in a couple of days after reading this material. Leandro
Em quinta-feira, 9 de setembro de 2021 às 10:36:19 UTC-3, f.jame...@stonybrook.edu escreveu: > I was a little vague there. Unfortunately there are several relevant > distances. Usually best to think of Procrustes distances in terms of angles > or great circle distances. Max between two shapes is pi/2. Distances in > the tangent space are linear but with a maximum of 2 for shapes with > projections maximally far apart - but misleading because they are actually > the same shape (Procrustes distance of almost zero). Points around the > outer limits of the hemisphere have a Euclidean distance of 1 to the > reference triangle but a Procrustes distance of pi/2 = 1.57. Problem of > trying to project surface of a sphere onto a flat map. I shouldn't have > tried to simplify so much. Looking at pictures may be safer than words. > > It is less confusing for the range found in most biological data where > shape variation is fortunately usually "small". > > > *F. James Rohlf * > Distinguished Professor, Emeritus and Research Professor > Depts: Anthropology and Ecology & Evolution > Stony Brook University > > On 9/9/2021 5:36:16 AM, Karolin Engelkes <karolin....@gmail.com> wrote: > Thank you all so much for the literature recommendations and, Prof. Rohlf, > for the explanations! Especially the part on the PCA and the patterns of > distribution in the different spaces is very helpful for me, but also gives > me a lot to think about. So please give me some time to go through all of > it and through the literature and digest it. > > @ Prof. Rohlf: One question immediately comes to my mind that is related > to your statement that the “distance in the tangent space will be a little > larger [than the distances on the hemisphere] due to the projection“. Does > this refer to an orthogonal or a stereographic projection? I would expect > the opposite for an orthogonal projection (at least from taking a ruler and > measuring the linear distances between the “reference” [intersection of the > hemisphere with the y-axis] and, respectively, the points B, C, and D in > your Fig. 4 in the paper from 1999). > > Thanks again and best wishes, > Karo > > Am Do., 9. Sept. 2021 um 09:54 Uhr schrieb mahendiran mylswamy < > mahe...@gmail.com>: > >> Another nice piece of writing 'on Shape' Theory by Prof. Mac leod, pdf >> attached, may help to understand and appreciate the concepts well. >> >> On Thu, Sep 9, 2021 at 7:35 AM f.jame...@stonybrook.edu < >> f.jame...@stonybrook.edu> wrote: >> >>> 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] <dca...@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/ >>> >>> phone: 515-294-3834 <(515)%20294-3834> >>> >>> >>> >>> *From:* morp...@googlegroups.com <morp...@googlegroups.com> *On Behalf >>> Of *karolin....@gmail.com >>> *Sent:* Tuesday, September 7, 2021 7:04 AM >>> *To:* Morphmet <morp...@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 <+49%20228%20735481> >>> >>> 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+...@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+...@googlegroups.com. >>> To view this discussion on the web visit >>> https://groups.google.com/d/msgid/morphmet2/Mailbird-a92a6485-db7d-4c94-82a2-397208a82e0f%40stonybrook.edu >>> >>> <https://groups.google.com/d/msgid/morphmet2/Mailbird-a92a6485-db7d-4c94-82a2-397208a82e0f%40stonybrook.edu?utm_medium=email&utm_source=footer> >>> . >>> >> >> >> -- >> *************************************** >> M Mahendiran, Ph D >> Sr. Scientist - Division of Wetland Ecology >> Salim Ali Centre for Ornithology and Natural History (SACON) >> Anaikatti (PO), Coimbatore - 641108, TamilNadu, India >> Tel: 0422-2203100 (Ext. 122), 2203122 (Direct), Mob: 09787320901 >> Fax: 0422-2657088 >> http://www.sacon.in/staff/dr-m-mahendiran/ >> >> P Please consider the environment before printing this email >> > -- You received this message because you are subscribed to the Google Groups "Morphmet" group. 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