September 4, 2021
Dear Morphmetters,
I'm writing to draw your attention to a paper of mine
published last week in Benedikt Hallgrimsson's journal
Evolutionary Biology. You can get a copy by clicking
"Explore online first articles" on the journal home page
https://www.springer.com/journal/11692.
The paper has the rather recondite title
"A new method for landmark-based studies of the
dynamic stability of growth, with
implications for evolutionary analyses," and
it indulges my customary prolixity, running 30 double-column
pages including 18 figures, most of them multipanel.
The paper was stimulated by a paper
Philipp Mitteroecker and Ekaterina Stansfield published
earlier this year ("A model of developmental
canalization, applied to human cranial form,"
PLoS Computational Biology, 17(2), e1008381) suggesting
that longitudinal landmark data deserve a better protocol
for analysis than our usual approaches, which are based
on the covariance matrix of transformed landmark coordinates
pooled over ages. Both papers argue
that the matrix being analyzed should not
be that pooled version, but instead the series of
covariance matrices for each time-specific set of
those coordinates against their changes to the next age
of observation. M. and S. suggested a Partial Least
Squares analysis of the Procrustes shape coordinates of
such growth data sets. My paper instead recommends and demonstrates
a version relying on what I've been calling Boas coordinates
(Procrustes without the size-standardization
step) and, instead of PLS, a true eigenanalysis, meaning,
one using the same R call "eigen" that we
use for PCA. The difference is that instead of
eigen(t(X)%*%X) we ask for, diagram, scatter, and
interpret eigen(t(X)%*%Y) where X is the form data matrix
at time i and Y is the changes of form from there to time i+1
after both have been mean-centered by columns.
What makes this small code change unexpectedly challenging is
that the eigenanalysis of those growth-by-form covariance matrices
can result in complex eigenvalues and eigenvectors (quantities
that involve terms in the square root of minus 1).
The paper takes a lot of space to explain an intuitive
version of that complex arithmetic that, in effect, replaces
a complex-valued thin plate spline by a pair of real-valued
ones. The engineers have already named these _canonical
vectors,_ a label I'm suggesting here as well, even though
these vectors are different from the vectors you get from
canonical correlations, from multivariate discriminant analysis,
and so on. (The other multivariate statisticians
have no copyright on the word "canonical.")
The reviewers thought my argument was logical but the
requirement of complex numbers would situate the technique
out of reach for most organismal biologists. I argued back
basically that (i) many biologists already do indeed master
complex arithmetic, especially in bioengineering applications,
and (ii) the resulting findings are worth the cognitive
stretch. Also, I've included a sort of tutorial, not on
the square root of minus 1, exactly, but on the meaning of
those complex eigenvalues and their eigenvectors as expressed
in our usual scatterplots of scores and deformation grids.
This, in turn, suggests an experiment of the sort most of you
already had to navigate back when you were taught that the struggle
to learn enough about matrix algebra, matrix inverses, and
principal component analysis was worth the effort. I've argued
for years that the benefits of matching more advanced mathematics
than that to appropriate biological questions
are worth the cost of mastering that
advanced math. So I'm hoping (as are the reviewers of the
second version, and also, I presume, the journal editor)
that some of you out there, particularly the more
mathematically adventuresome, might suspend your disbelief
in the square root of minus 1 long enough to enjoy dipping into
the arguments of my paper and its extended 8-landmark, 8-age,
18-animal example. The paper is free at
https://doi.org/10.1007/s11692-021-09548-8
or, as I mentioned already, you can get to it from
the journal home page.
Thanks for considering this challenge, Fred Bookstein
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