I don't think the issue is really the choice of Procrustes vs other metrics. Unless one is comfortable just looking at numbers, one needs to make plots to see what patterns, groupings, etc. there may be in ones data. I agree that the points I raised apply to most any multivariate analysis for whatever metric one chooses. The problem of seeing what is going on in high-dimensional spaces is quire general. Once one gets past the various tests of significance in a multivariate analysis one usually wants to try to _see_ what is going on in ones data and that usually requires looking at some sort of low-dimensional plots of the data. This does not imply that one wishes to find clusters as in some of the early numerical taxonomy applications. It is just a question of trying to see what is going on in high-dimensional data from looking at a low-dimensional summary.
It is routine now to perform an analysis of relative warps (perhaps as a PCA of partial warp scores + uniform) and retain just the first two or three dimensions because that is all one can plot at one time. These plots are examined in order to look for patterns or trends of variation in shape. In these ordination plots distances between points no longer match the Procrustes distances between the corresponding shapes since the space is just a projection from a higher-dimensional space. In a PCA, shapes that are very different will be shown as far apart in the ordination but some points shown as close together may actually be rather different in shape. For that reason, statistical tests etc. should be performed in the original high-dimensional space and not in the low-dimensional ordination (as one sometimes sees in Applications). The question is what information would you prefer to lose when you prepare a low-dimensional plot for visual examination. PCA has one choice built into it of what information should be preserved and what should be discarded. Non-metric MDS is based on another choice. A good-fitting non-metric MDS may preserve more useful information about the original Procrustes distances than does PCA. Often it is pretty obvious which shapes are most different so it seems to me, to be more useful to have ordinations that preserve more subtle differences. It is often instructive to plot distances in the 2 or 3-dimensional ordination against the distances in the full data. One can then see at a glance the distances that are faithfully reproduced vs. those that are not. Sometimes the differences are very interesting and help one decide for a particular dataset whether one would lose too much by going from a metric method such as PCA to a non-metric MDS. Jim ----------------------- F. James Rohlf, Distinguished Professor & Graduate Program Director State University of New York, Stony Brook, NY 11794-5245 www: http://life.bio.sunysb.edu/ee/rohlf > -----Original Message----- > From: morphmet [mailto:[EMAIL PROTECTED] > Sent: Saturday, October 01, 2005 12:49 PM > To: morphmet > Subject: RE: PWS and MDS? > > October 1, 2005 > > I thank Jim Rohlf for his prompt and very interesting comment. > He is correct, of course, that any MDS technique that > relaxes the requirement of matching original distances must > at the same time relax the identity of the resulting > ordinations with PCOORD plots of the variables contributing > to the original distances. It is a tribute to his longevity > in our field that the primary paper he cites is dated 1972. > At that time I was still a graduate student of social > theory. It was years before I became a statistician, and in > going back to pick up the literature I'd overlooked in a > misspent youth I never encountered Rohlf's indeed quite > pertinent paper before yesterday's post. > But that dating of 1972 is also a useful cue for what > is (I hope) a useful continuing comment. 1972 was long > before the onset of useful shape coordinates -- indeed > before the original Kendall publication of 1977 (that none > of us saw for years) that set the Procrustes metric on the > road to its multivariate implementation. Is it appropriate > to apply a technique from 1972 to shape variables invented > in the 1990's? What assumptions does the 1972 method make > about the origins of the data to which it is being applied? > Jim's comment would apply to _any_ multivariate sum > of squares, arising from any vector-valued data matrix > -- shape coordinates, measured distances, measured angles, > log-distances. It does not even assume that the metric _is_ > a sum of squares. Like the rest of the MDS family of > ordinations, it can work (meaning: it can produce useful > ordination plots) when applied to Manhattan metrics, string > matching measures, perceived similarity from subjects in > psychology experiments, whatever. > But shape coordinates are not like that. They arise from > an optimizing construction in the original Cartesian > geometry of the image data, and relaxing the distance alters > the conceptual context according to which that choice of > distance itself came to be justified. > Remember that the Procrustes family of techniques has > two exactly equipotent purposes: sorting the organisms (the > task we call ordination), but also ordinating the space of > possible measurements (the space of "shape variables" dual > to the coordinate space). The underlying beauty of the > Procrustes toolkit is its reinterpretation of the original > Kendall geometry in a multivariate context that makes this > duality possible. Remember, too, that MDS itself arose > originally in the social sciences, where the data to which > it applied do not have any geometry of their own. > There is no natural match between the two approaches > to the meaning of "distances," which correspond to two > completely different styles of statistical science. > The relaxation of the distance metric that Jim published in > 1972, as applied to Procrustes shape coordinates (or, for > that matter, to the new size-shape coordinates), breaks the > connection between those two purposes. While the result of > a metric MDS on shape coordinates are interpretable as > rotations of the Procrustes shape coordinates, the axes of a > nonmetric MDS are not interpretable as shape variables at > all. One arrives at an ordination, yes, wherein nearby > specimens are more similar, are perhaps usefully clustered -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
