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 -- but
the mathematical basis for the selection of the distance
measure being _sent_ for MDS has been removed, and with
it the possibility that the resulting coordinates have further
formal properties independent of the name and the author of the
computer program that generated them.
It is not solely a matter of style or taste, then,
but also a matter of fundamental scientific methodology, to
ask: what is the impact upon ultimate scientific interpretation
of this abandonment of the duality between specimens and variables
that is woven into the foundation of the Procrustes techniques?
The original question to which Jim and I both responded asked
(among other things) about the correspondence of shape
distance with geographical distance. The Procrustes
method allows, among other things, the coordinates of a PCOORD plot to be
correlated with latitude and longitude, for description
of actual gradients or clines (the approach we call
"singular warps"). But this separate
optimization, the PLS analysis of map coordinates
against shape coordinates, no longer is interpretable when
the input shape representation is a nonmetric MDS output instead of
an exact re-projection of the initial Procrustes metric geometry.
It is no longer talking about explaining shape covariance;
it is, in fact, no longer talking about "explaining" anything at all
in the sense of "explanation" that we all borrow from the world
of linear models.
We have prior experience in
a similar topic, the relation of the Procrustes coordinates
to the resistant-fit version of superposition. Resistant fit produces
occasionally
interpretable diagrams (yes, Virginia, there really is a
Pinocchio), but at considerable cost in the
blocking of any subsequent multivariate analysis. It didn't
matter in the original (1982) publications of the resistant-fit
method, but it came to matter shortly afterwards, as the
synthesis emerged of which I am speaking, in which shape
coordinates serve two roles, not only one. By now, in 2005,
I am not aware of any scientific uses of the resistant-fit
methods -- the price is simply too high. In my view
the road through nonmetric MDS is likewise a biometrical
dead-end. If there is any further interpretation of the
resulting diagrams, it is not via statistical properties
conveyed by the actual plotted point locations. Those
properties have been dissolved by the software. And it is in
fact the same price: the breaking of the tie between the
coordinates of the final diagrams and their interpretation
as shape variables.
Yes, the technique of nonmetric MDS, which does indeed
downplay the weight of the most different shapes vis-a-vis those
closer together, furthers one of the core purposes
of numerical taxonomy, namely, cluster-based classification.
This is at the cost of several other purposes for which no
techniques existed in 1972 but that are now available
_simultaneously_ in the Procrustes toolkit. Jim appropriately
mentions ordination as one purpose that can be served even when the
formal symmetries of the Procrustes approach are relaxed
in this particular way. But he did not go on to indicate
as well the costs of the practice he recommended, which is
to say, the breaking of the formal geometric tie between
object coordinates and measureable variables that gives the toolkit as a whole
its power. Once this tie is broken, it can't be restored
by any subsequent multivariate maneuver. In 1972, there
was no such tie to worry about. The intervening third
of a century has given us additional multivariate power
that should not be set aside lightly.
But then (my final comment) this is a strange place
to start relinquishing the Procrustes approach. If you are
willing to relax Procrustes distance against a general
monotone function, why on earth are you using Procrustes
distance in the first place? Why relax _that_ assumption,
instead of the symmetry over landmarks and directions
(the assumption of the offset isotropic Gaussian model)
that is so more more obviously violated in realistic
data sets? The Procrustes metric is "what we mean
by shape distance" only if you accept the stringent
symmetries that made Kendall's original insights
feasible. If you are going to start altering assumptions,
monotone transformations of a still totally symmetric
formalism are a strange, I would say non-biological,
place to begin. Better to consider the original landmark
scheme itself (something that went unmentioned and of
course unfigured in the original post to which we are
both responding), and ask instead the kinds of questions
for which we _do_ have answers within the perseverating
Procrustes framework: questions about anisotropy of landmark
variability, about landmarks vs. semilandmarks, and the
other tools that are compatible with the original
mathematics. The nonmetric MDS, which predates the whole
Procrustes approach, has no handles by which to pick up
these additional tools.
I look forward to additional comments on this
theme. Last year my Vienna group and I published a comment on the vicissitudes
of morphometrics, arising in numerical taxonomy but now
centered (at least in Europe)
in physical anthropology, evolutionary biology and evo-devo.
The innovation of which I'm speaking here might be
intrinsic to this translation: the emergence of a duality between
descriptors of specimens and descriptors of quantitative
measures per se, a duality for which classic numerical
taxonomy never seemed to have much use. This is
not a criticism of the taxonomy itself, of course, only
a comment on the corresponding limitations of subsequent
quantitative scientific context.
Fred Bookstein
[EMAIL PROTECTED]
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