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

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