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H.j. Woltring, Fax/tel +31.40.413 744
07-14-1992, 12:45 AM
Dear Biomch-L readers,

With Richard Reyment's kind consent, the following book review is being
cross-posted from MorphMet@cunyvm.bitnet.

Regards -- hjw.

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Date: Sat, 2 May 1992 15:19:00 CET
From: PALRR@SEUDAC21.BITNET "Richard Reyment"
Sender: Biological Morphometrics Mailing List
Subject: Review of Bookstein

Hoping that Fred Bookstein does not object, I distribute herewith
my review of the "Orange Book", requested by the editors of BIOMETRICS.

The concept of shape is a difficult one to define verbally in
satisfactory terms, even more so mathematically. The word has a
vernacular significance charged with overtones - shape, shapeless,
shapely, shape up, etc. The part synonym "form" may be more
precise, but not always. The translation of "shape" into other
languages also poses a problem. In the following, a novel and
biologically relevant approach to the definition of shape and the
analysis of ontogenetic and phylogenetic changes in shape is
reviewed; the newness of the concepts has necessitated a longer
presentation than is usual.

In 1917, Sir D'Arcy Wentworth Thompson published his celebrated
book on growth and form. Among the many splendid ideas he
summarized in this text, one, that of depicting changes in shape
by reference to deformations of an organism superimposed on a grid,
has long resisted attempts at satisfactory quantification. Thompson
did not provide an explicit solution, nor did he say how he made
his diagrams. We now know that they were the combined product of
inspired imagination and freehand drawing. Over the years since the
idea first appeared in print, many people have tried to produce a
solution. None of these attempts, which include a trend-surface
representation, can be claimed to be successful.

It was not until 1978 that the signs of a general approach to the
problem began to appear when F. L. Bookstein published his doctoral
thesis. He has continued his research into the quantification of
shape over the last 13 years, the results of which now appear in
the volume being reviewed. First a little background information.

Many readers will know about the principal component factor
analytical method of analyzing size and shape that was originally
proposed by Teissier in 1938, and its subsequent development in the
hands of numerous workers, to wit, the application of the method of
principal component analysis to a covariance matrix computed from
the logarithms of measurements made on an organism - length,
height, breadth - distances between hopefully diagnostic points on
its surface. This procedure has been used now for 20 years; it has
been accepted by serious statisticians and it is to be found in
most texts on applied multivariate analysis as an established fact
- but, for the actual purposes it is invoked, it is usually not
very informative. This statement should not be understood to imply
that standard multivariate procedures applied to all morphometrical
problems are ineffectual, only that they are unsuitable for a
reasonable quantification of change in shape.

In order to explain why the principal component method is
inadequate, it is necessary to understand what size and shape are.
In any of the length measures (palaeo)biologists take, size and
shape are inextricably confounded. There is no way in which a
global transformation of the principal component variety is going
to be able to separate one from the other, despite computational
artifacts that sometimes appear to support the idea. What is
required is a method that will successfully delineate shape as such
and size as such, be able to put the one back into the other at
will, and which will permit different analyses of the two. Several
people have thought about this over the years, and one good way of
being able to gain control of what you are doing is to provide
interesting points with locations specified in an x-y-coordinate
system. These interesting points are called, landmarks, using a
term borrowed most nearly, and not unreasonably, from craniometry
(the illogicality of the original borrowing need not concern us
here). Landmarks link the geometry of the organism, the mathematics
of deformation and biological inference.

The pairs of coordinates, localizing each landmark, can be
studied by Bookstein's method of shape-coordinates, which considers
landmarks three at a time, two of which are constrained to form a
baseline of unit length. These have now been officially baptized
Bookstein shape-variables by other workers in the field. A
consideration of all possible combinations of landmarks leads to a
detailed cartography of the shape variability of a sample of some
species. Size can be easily introduced as centroid size, if
required. Shape coordinates are invaluable for studying shape-
change in, for example, shell-bearing protozoans.

More recently, broader developments of theory and practice have
been brought into play. In effect, the algebra of latent roots and
vectors has been applied to landmark variables by Bookstein by
means of a theoretical development that hails from the French
mathematicians Duchon and Meinguet and their theory of thin-plate
spline interpolation, which can be used as a tool for modelling
shape-change as the deformation of a thin metal plate. The flexing
of the plate is going to take energy to bring about the flexing,
and a quantity can be calculated which is a figurative
representation of this, just as in the case of a real metal plate.
There will be a uniform, or affine, component to the deformation
for which parallel lines remain parallel (think of a square which
you deform into a parallelogram ). There is also an irregular non-
uniform or non-affine part to such a transformation. The non-
affine part can, by the decomposition afforded by latent roots and
vectors pertaining to the bending energy of the plate, be broken
down into successively more local regions of the organism. Thus
growth can be interpreted on a global scale right down to small
differentials occurring between and around closely spaced landmarks
(with the greatest bending energies). There is, of course, no
bending energy attaching to the affine part of the deformation.

Bookstein's thin-plate model gives us then a conceptually
attractive way of analyzing, both in numbers and pictures, the most
intricate shifts and relationships in shape in an organism, both
geographically and temporally. No doubt as the outcome of
Bookstein's energetic engagement with his subject, several
mathematical statisticians have taken up the study of shape as a
statistical problem. As mentioned in the first paragraph, shape
seems to mean different things to different people, not always with
biometrical relevance, and notwithstanding that the mathematical
results accruing from this work are rich in interest, their
applicability to the biology of shape seems to me to be restricted.
The most recent development of geometric morphometrics has been to
the construction of an atlas for human brains. This replaces the
single idealized illustration of a brain by an averaged specimen in
which information on mean and variability is contained. This marks
a giant step forward in our ability to extract quantitative
observations from images of organisms. The application of this
advance to the quantitative appraisal of the evolution of life in
terms of computer-based visualization techniques is immediate and
obvious and the step to a generalized Linnaean taxonomy is short.
We can now develop our evolutionary analysis in terms of landmarks
and pixels, condensed into averaged pictures that can be subjected
to any kind of rigorous analysis we want. It is hoped that grant-
processing bodies in Europe realize this, the sooner the better.
However, my personal experiences make me pessimistic.

The first two chapters deal with principles and definitions and
should be studied carefully if later sections are to make sense,
for it is here that the line is drawn between "traditional"
multivariate morphometrics (as introduced by Robert Blackith and
myself 20 years ago) and the geometric treatment required by
landmark data. This is unfamiliar ground to even the most
professional of biometricians. The detailed treatment of the topics
begins in Chapter 3, which examines the subject of landmarks and
the usual distances of "traditional" morphometrics. Chapter 4
compares and contrasts standard methods of multivariate analysis,
centroid size and the concept of multivariate allometry, and
demonstrates the logical brittleness of the "standard explanation"
of shape-change. The concept of Shape Coordinates, the building
blocks of geometric morphometry, forms the main material of Chapter
5. A shape variable is any measurement of the configuration of
landmarks that does not change when your ruler stretches or
shrinks, a definition that has its roots in Mosimann's fundamental
theorem of 1970. This chapter finishes with an account of
"Kendall's shape-space", in which any shape of a set of landmarks
is a single point in shape-space of relevant structure. These
results by a group comprising D. G. Kendall, Mardia, Goodall,
Dryden and Kent are of much mathematical interest but I cannot see
their being of direct biological applicability, at least in their
present state of development.

In Chapter 6, the principal axes of shape are taken up. A
biological shape-change can be represented as a symmetric tensor.
Why is this a useful rendition and not just "for show"? Recall that
a tensor is a mathematical operator upon one or several vectors
that supplies the same answer regardless of the coordinate system,
a requisite for a general descriptor of shape. Some use has been
made by evolutionists of descriptive finite elements for the
"tensor analysis" of variation in shape. It is demonstrated that
the method is flawed with respect to the uses to which it is often
put, not least because of there being a deficiency in the number of
descriptors required for statistical purposes: 2k - 3 for k
landmarks in two dimensions and 3k - 6 in three dimensions.

Chapter 7 presents the main intellectual achievements of
geometric morphometrics. It begins with Procrustean superposition
(known to multivariate workers from the results of P. Sch¦nemann
and J. Gower) - the computation of best-fitting overlays by various
criteria. There is, I think, a role for Procrustes in morphometry,
namely, for ad hoc confirmation of phylogenetic comparisons where,
perhaps, a hundred pores, hursts and hollows are to be scanned
for evidence of evolutionary shifts. This was, after all, the
reason for R. G. Benson's introduction of the idea into
phylogenetic analysis.
The thin-plate spline interpolation and its ramifications
constitute the remainder of this part of the book. The thin
plate spline of the bending energy matrix has turned out to
have very useful properties which, incidentally, permit its
elucidation in the usual terminology of multivariate statistics.
It is zero for a large class of shape-changes, to wit, the uniform
ones that can be measured by anisotropy and it weights movements of
landmarks differently. Students of structural geology will
recognize this as the criteria sought by the geologist Ernst Cloos
some 60 years ago (he only really succeeded in treating the uniform
class).
There is an essential dichotomy in the philosophy of geometric
morphometrics here. You can examine the deformation resulting from
transforming from one set of average landmarks to the second set of
interest, and you can look at what happens in a sample of forms on
k landmarks, referred to an average configuration of landmarks,
just as in the fixed case of principal component (factor) analysis.
In the first situation, one obtains "principal warps" (translation
of the original "flexion" of Duchon), which are the latent vectors
of the bending energy matrix and which denote the displacement of
points irrespective of global affine transformations. These warps
can be back-transformed via an appropriate vector multiple into the
original Cartesian plane of the data to yield what Bookstein calls
"partial warps", the picture of the grid (sensu D'Arcy Thompson)
deformed by the vector multiple. This is a valuable technique for
expressing complex evolutionary changes in features that otherwise
might not even be suspected and which from my experience do not
even manifest themselves under the stereoscan microscope.
In the second situation, one obtains Bookstein's "relative
warps" for the sample of forms, the coordinate pairs of which are
constrained to a selected baseline (the shape coordinates) and
which are obtained from the latent roots and vectors of the bending
energy matrix via a series of steps. The method of relative warps
pictures the vectors of displacement of each of the landmarks and
is, therefore, helpful for probing subtle morphological
polymorphisms, ontogenetic patterns, and ecophenotypic
differentiation in a sample of some taxon.
The final Chapter 8 bears the title Retrospect and Prospect. It
summarizes the main ideology of Geometric Morphometrics and points
the reader in new and exciting directions of research in which the
techniques of computer visualization and powerful Work Stations
will play a vital part. This, I believe, is where the most
spectacular advances can be awaited.
Apart from the main theme of Bookstein's development of the
analysis of shape, there are ample sections dealing with the path-
analysis of Sewall Wright, for a time relegated to the biometrical
curiosity closet, but now undergoing a renaissance in the life and
earth sciences, including the construction of informative
geochemical models. The treatment of a proper model for anagenesis
and stasis is included in an appendix; unless you have been
following the right literature, it may come as something of a jolt
to your well-being to learn that the punctuation-gradualism
confrontation of evolutionary biology depends on ideas that are not
unchallengeable in that morphological distances in time series that
might seem to be trending significantly could really be doing no
such thing. The very comprehensive Appendix also contains
mathematical details and instructions for making some of the
diagrams. There are lists of data used to exemplify the methods.
Morphometric Tools for Landmark Data cannot be claimed to be an
easy book to read, but that is hardly the fault of the author, who
profiles himself as an ambitious stylist, perhaps at times
mesmerized by mellifluence. I commend the book to anybody who is
concerned with evolutionary biology, particularly the study of
evolutionary series of morphologies of fossils, the evolutionary
interrelationships of the often bizarre shapes of Precambrian-
Cambrian life, and also structural geologists should be able to
find much of value and interest for their work. The ground covered
is unfamiliar to biometricians and statisticians and requires
preparatory reading in the geometry of figures under transformation
(as was developed by Felix Klein) for anybody wanting to become
seriously involved with the theory. It is to be hoped most
sincerely that the book will be made compulsory reading for
University biometrical courses.
The new Geometric Morphometry, as epitomized in Bookstein's text,
is a fully rounded scientific achievement. This degree of
completeness, encompassing both philosophy and method, is seldom
found in the quantification of the Earth and Life Sciences, where
results are often isolated accomplishments, incomplete and
impossible to pursue to a logical end, more often than not arising
by accident rather than design.

Richard Reyment