Seattle, March 25, 2023
Dear MorphMetters,
I wrote you back in January about a downloadable posting of mine
over at arxiv.org recommending two major changes to GMM that make it much
easier to describe how grid features lie upon the organism you're studying.
Now I'm writing to alert you to another item of mine, posted yesterday as
DOI:10.1101/2023.03.23.533997
at biorxiv.org, that extends the simplest version of that exploratory method,
the analysis of quadratic trends (including one version of growth-
gradients), to an entire new GMM toolkit. The article shows how
the new tools directly respond to D'Arcy Thompson's and then
Peter Sneath's long-frustrated hope (1917 and 1967, respectively)
for a way of analyzing transformation grids that could sometimes result in
explicit characters of a new type for phylogenetic or evo-devo analyses.
This new preprint, "Quadratic trends: a morphometric tool both old and new"
(58 pages, 18000 words, 32 figures, plus Supplement),
resurrects Sneath's idea of polynomial deformation models but, digging
deeply into the algebra of the quadratic case, finds an important new
application of the most familiar shape in our linear multivariate
statistics, the ellipse, that greatly enhances the import for organismal
biology of the NONlinear aspects of our familiar deformation models.
There's a little 19th-century calculus, some unfamiliar formulas,
and a spate of brand-new data graphics.
As the arxiv.org posting did, the central argument
of the new piece evinces my usual late-life iconoclasm: to persist
as a source of useful insights for evolutionary or developmental
biology, GMM needs to supersede the Procrustes toolkit in toto and
should complement the interpolating thin-plate spline by a regression-
based modeling tool not pinned to exact landmark locations.
Also, we should all change immediately from Cartesian coordinates to
polar coordinates for all our organismal plots.
After introducing two formalisms (second derivatives as regression
coefficients, cardinal diameters of the resulting ellipses) using
the familiar textbook resource of the Vilmann neurocranial octagons,
the paper's main example is a reanalysis of Les Marcus's
classic data set (Marcus et al., Hystrix, 2000) of mammal
skulls. This choice added a full 30 centimeters of books to my
library -- all five volumes of Grzimek's Encyclopedia of Mammals, 1987.
Whether you agree, disagree, claim priority for yourself,
Peter Sneath, or somebody else, or have modifications or extensions
to suggest, I'd be pleased to receive any comments on the argument here,
which I think will be an important contribution once it is disseminated.
And if I use your comments in any revision I will acknowledge you by name
if you wish me to do so. You can find this biorxiv preprint at
https://biorxiv.org/cgi/content/short/2023.03.23.533997v1
It hasn't been submitted to a journal yet because I think comments
from this community will greatly improve it prior to that formal step.
If you would like to cite it before it is (I hope) ultimately
accepted at some journal, you can use that DOI.
And if somebody would program this up in R for general use, that
would make it a lot easier for any of you who are interested to
give it a try. Also, what journal(s) might tolerate a manuscript
this long and eccentric? Or should I try dividing it into two parts,
breaking between Section III and Section IV, but say that they need
to be reviewed jointly and, when accepted, published consecutively?
Many thanks to Joe Felsenstein, Norm Macleod, and Jim Rohlf, who
took the trouble to comment on several earlier drafts of this material,
and to Erika Hingst-Zaher, who sent me the mammal data ten years ago.
Fred Bookstein
[email protected],[email protected]
========
ABSTRACT: The original exposition of the method
of ``Cartesian transformations'' in
D'Arcy Thompson's great essay
On Growth and Form of 1917 is still its most cited.
But generations of theoretical biologists
have struggled ever since to invent a biometric method
aligning that approach with the comparative anatomist's ultimate goal of
inferring biologically meaningful hypotheses from empirical
geometric patterns. Thirty years ago our community
converged on a common data resource, samples of landmark configurations,
and a currently popular biometric toolkit for this purpose,
the ``morphometric synthesis,'' that combines
Procrustes shape coordinates with thin-plate spline renderings of
their various multivariate statistical comparisons.
But because both tools algebraically disarticulate
the landmarks in the course of a linear
multivariate analysis, they have no
access to the actual anatomical information conveyed by the
arrangements and adjacencies of these locations as they combine
in pairs or higher numbers into substructures.
This paper explores a geometric approach circumventing
these fundamental difficulties: an explicit
statistical methodology for the simplest nonlinear patterning
of these comparisons at their largest scale,
their fits by what Sneath (1967) called
quadratic trend surfaces. After an initial quadratic regression
of target configurations on a template,
the proposed method ignores individual shape
coordinates completely, replacing them by a close reading of
the regression coefficients accompanied by several new
diagrams, notably the exhaustive summary of each regression
by an unfamiliar biometric ellipse, its circuit of second-order
directional derivatives.
These novel trend coordinates, directly visualizable
in their own coordinate plane, do not reduce
to any of the usual Procrustes or thin-plate summaries.
The geometry and algebra of these second-derivative ellipses seem a
serviceable first approximation for applications in evo-devo
studies and elsewhere. Two examples are offered, one the
classic growth data set of Vilmann neurocranial octagons and the
other the Marcus group's data set of midsagittal cranial landmarks over
most of the orders of the mammals. Each analysis
yields startling new findings inaccessible to the
current GMM toolkit. A closing discussion
suggests a variety of ways by which innovations in this spirit
might burst the current straitjacket of Procrustes coordinates
and thin-plate splines that together so severely constrain the conversion
of landmark locations into understanding across our science.
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