Seattle, May 13, 2021
Dear MorphMetters,
I have been following with interest the current morphmet thread,
which began with a posting by Andrea and continued with comments from
Philipp, Chris, and David (apologies if I've missed
any other contributors). But
it may be time to change the level of the conversation.
In the following I'm not saying anything I haven't already said in
my textbook and several of my recent articles, so I won't take up
space here to particularize the citations -- anyway, the articles
are mostly available open-access from Evolutionary Biology and
Biological Theory, and that 2018 textbook is in print from Cambridge.
Let's focus, like David said right at the end of his post,
on the inconvenience that the papers
that get these distinctions right are those that tend to be
"more esoteric" in their methods. The following is not intended
to be critical of David's point in any way, but to elaborate
on it. First, go to the dictionary:
the relevant definition of "esoteric" says it means
"designed for or understood by the especially
initiated alone." But isn't that what the biometrics component
of our graduate curricula is _for_? Can you imagine
a structural engineer, for instance, being criticized
for calculating the strength of a column that holds up a bridge by
a formula that combines length, radius, the strength of the
concrete, and the density of the incorporated rebar,
on the grounds that it is "esoteric"? Or telling the people
who use Reynolds numbers for everything from bird and fish
biometrics to stream flow to naval architecture that exploiting
this famous parameter is "esoteric"? Of course not --
those "more esoteric" papers are in fact more _precise_, in a way
understood by those who have invested the training time to
master exactly that "esoteric" jargon that frightens the
outsiders. In other words, the esoteric language is there for
a good reason: for speeding scientific progress.
The situation with the word "relative," on which our thread here has
converged, is something like that. We have, in general, two
choices: either to accept the ("esoteric") discipline that comes
with the purposes that the word "relative" is trying (but
failing) to fence around -- usually, equivalence classes with
respect to some recognized transformation group --
or to go on using ordinary language but prefacing
every empirical claim with the word "relative" and hoping somehow
that our audience understands the precise import of that prefix.
I think this second option, though it is the one embraced in many
of our applied papers, is the one that is failing
these days: I mean, the fond hope (taking the word "fond" in its
root sense, meaning, "foolish") that the typical users of GMM,
and beyond them the typical readers of those papers, will somehow
align their intuitions about the meaning of words like "local"
or "relative" with the actual arithmetic (and its
underlying math and graphics) producing the coordinates they are
looking at. (It is the careful alignment of the arithmetic with
those words, originally for shape and lately for form, that concerns my
recent papers. Procrustes shape coordinates don't do what they must do to
be interpreted the way we usually interpret them.
This point is worth bruiting about more widely.)
So, given that mismatch in the hands of all but a minority
fraction of our community, isn't it time to consider the first
fork above, the recourse to a better mathematics that, although
requiring some discipline and training (i.e., "esoteric"),
makes the word "relative" unnecessary?
The mathematics per se is no longer new --
back in 1989 I offered a method for this purpose,
the method of partial warps (PW's), as a substitute for _both_ the language
of localization and the language of relativization. The partial warps
are relative (in the appropriate sense, which is the mathematical one)
by explicit construction -- you don't have to repeat the word
"relative" over and over -- and you can see in my last few papers
on integration how they handle the rhetoric of gradients, localization,
and modules as well. PW's are automatically "relative to"
the uniform term (which itself is a relative descriptor, i.e. relative
to the similarity group). Also, the PW's are automatically geometrically
orthogonal (e.g., "relative") to one another, and when taken
all together they let the investigator understand the balance between
integration and localization in a very useful way that has
interesting implications for allometry in particular.
In other words, if there is a general suspicion out there that
the current language of landmark-specific shape coordinates is
unsuitable for many (perhaps most) of
our explanatory purposes, why not consider this
long-available alternative that circumvents that specific problem
entirely? Instead of precising more theorems, let me just refer you
to my book and my last few nethodological papers.
And if this solution doesn't suit your findings,
feel free to experiment with other solutions, other bases for the
space of shape coordinates or Boas coordinates, that might work.
In other words, don't complain that the exoteric world doesn't use its
words correctly -- ("exoteric" is a real word, the opposite
of "esoteric") -- instead, put in the time to master the appropriate
biomathematics (it is not that deep, and it is now nearly a third of a
century old) that lets you get on with your work in an admittedly
esoteric language that fits both the arithmetic and the understanding.
By the way, these solutions probably aren't going to rely on
principal components much. There's another topic that our
students are going to have to master much more "esoterically." The
fallacies of PCA deserve many threads of their own, that would
be launched the same way this one was, by a comment noting how
badly their actual arithmetic, here in our context of
GMM, has drifted from our verbal reports of that arithmetic.
The psychologists beat us to this realization nearly a century ago,
when they switched from principal component analysis to factor
analysis, and Sewall Wright talked us through the transition way
back in 1954. We would do well to inject a week or two of discussion
of factor analysis into our graduate multivariate core. Again, there
are Bookstein papers you could start with, also papers by
Mitteroecker as well. And, in general, our
curricula should be spending much less time teaching students how to
compute things by prescripted button presses,
and much more time teaching them how to be constructively
skeptical of what they and their colleagues are computing. Right?
Yours with best regards, from temporarily sunny Seattle,
Fred Bookstein
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