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  • MorphoMetrics Book Review (XPost)

    Dear Biomch-L readers,

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

    Regards -- hjw.

    - = - = - = - = - = - = - = - = - = - = -

    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
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