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Geometric morphometrics, a new analytical approach to comparison of
digitized images
I.Ya. Pavlinov. In: Information technologies in biodiversity research.
St.Petersburg, 2001. P. 41-64.

Introduction

The demarcation between shape and size of morphological structures is
fundamental for many of biological investigations. Till recent times, two
approaches have been existing and developing in parallel in such
investigations, quantitative analytical and qualitative geometrical.
In the first approach, the exploratory protocol was deduced to
«extract» a size component from the overall form diversity using one or
another technique and to obtain, as a «residual», a shape component
(Bookstein, 1989a). Among these techniques were calculation of indices,
logarithm recoding, principal component analysis, regression analysis, etc.
Such methods evidently provide an indirect estimates of diversity of shapes
which are biologically sound inasmuch as the respective underlying
presumptions are correct. However, the shapes proper, mutual
transformations thereof remain aside. Besides, application of many
statistical tests is correct in case of rather large datasets with normal
(or multinormal) distributions of variable which is hold true not
frequently.
Somewhat apart stands a quantitative approach which is based on
description of the shape diversity by a set of discrete morphotypes.
Calculating their frequencies and comparison of samples by these
frequencies provide some numerical estimate of similarities among the
samples. However, two problems there exist. First, variation of shapes is,
a as rule, continuous, so recognition of discrete morphotypes is more or
less arbitrary. Besides, particular shapes are not being compared directly,
only some average estimates of among-samples differences result from the
analyses.
Unlike the analytical approaches, the geometric one is aimed at
comparison of the shapes themselves. In the modern science it can be traced
back to pioneered investigations of d'Arcy Thompson (1917; the last
edition is Thompson, 1992) who was the first to apply transformation grid
to illustrate interrelations of various shapes (Fig. 1). Although his
results were reproduces not a once in different manuals on biometrics, it
did not won wide recognition. The cause was simple enough: it didn't
contain any numerical tool, it was pure «figurative» approach allowing only
pairwise visual comparisons. Earlier attempts to unite the method of
transformation grid with standard statistical tools (see review in
Bookstein, 1991) had no success: the general cause was that before
searching for an appropriate numerical approach to geometrical analyses of
shapes an appropriate approach to description of the shape was needed. That
is, it was first requested to define what are the data and the variables
for such an analyses (Bookstein, 1993).
The break-through in this field became an idea of using, as initial
variables, not standard linear measurements but Cartesian coordinates of
the landmarks placed on the morphological objects being compared.
Respectively, the differences among the latters by their shapes was
determined as the differences among landmark configurations. This
descriptive approach added by a special, so called Procrustes distance gave
birth to a new branch of biometrics which became known as «geometric
morphometrics» (Bookstein, 1991).
In short, the latter may be defined as a set of methods of
multivariate eigenanalysis of the landmark coordinates that describe
configuration of the morphological objects in the shape space (see below).
It is elaborated as an analytical toolkit allowing to extract size effect
from results of investigation of the form of morphological structures which
is achieved by a set of specific algebraic techniques (Rohlf, Marcus, 1993;
Pavlinov, 1995). By these, geometric morphometrics differs significantly
from other quantitative approaches to analysis of morphological shapes such
as Fourier analysis or fractal analysis (e.g. Ferson et al., 1985; Hartwig,
1991; Slice, 1993a; Costa, Cesar, 2000).
Geometric morphometrics was burn just several decades ago when first
fundamental ideas was formulated (Kendall, 1984; Bookstein, 1986). At
present, its mathematical apparatus is thought to be developed quite enough
in order to allow resolving many applied problems (Bookstein, 1991, 1996).
It is being developed pretty actively: general essays, collective books and
review papers issued that expose adequately its theory, methods and
applications in biological investigations (Rohlf, Bookstein, 1990a;
Bookstein, 1991, 1996; Marcus et al., 1993, 1996; Rohlf, 1996, 1999;
Dryden, Mardia, 1998; Monteiro, Reis, 1999; Costa, Cesar, 2000). The
software for geometric morphometrics is also quite developed to allow
analyses of a lot of biological tasks. There is an Internet network uniting
more than four hundred «geometricians» and accessible by the address
MORPHMET@LISTSERV.CUNY.EDU (supported by L. Marcus), and a site where
computer programs are available (see chapter «Software» below) along with
nearly exhaustive bibliography on geometric morphometrics including more
than 500 references (the web-address is http://www.public.asu.edu/~jmlynch-
geomorph-index.html, supported by J. Lynch).
Below a short review of basic ideas and methods of geometric
morphometrics is provided, along with its computer programs, a brief
glossary, and some results of their use. Taking into consideration that the
issue is addressed to biologists, as well as specialty of the author, the
mathematical backgrounds of the approach in question is omitted.

Basic formalisms

From a formal standpoint, the object of geometric morphometrics is a
set of x-,y- or x-,y-,z- Cartesian coordinates, for 2- dimensional or 3-
dimensional bodies, respectively. The object is described by a p?k matrix
where p is number of landmarks and k is its dimensionality as a physical
body.
Being digitized, the object can undergone the following
transformations reflected in landmark coordinates: isometric
transformations which are translation, rotation, reflection (only position
changes), isometric scaling (only the size changes), and non-isometric
stretching/shearing (only the shape changes). It is evident that landmark
coordinates contain initially redundant information about the object
(position, size and shape are all incorporated), so the first task is to
elaborate some secondary shape variables which would contain only
information about shape proper. The latter, by definition, is determined by
configuration (net location) of the landmarks.
It is to be stressed that these shape variables are initially tensors.
This means, first, that each of them is defined by a pair (or three) of
coordinate axes that has no separate sense. Second, these axes are
equivalent: replacing of x by y and vice versa makes no changes in the
properties of the objects being compared. This puts certain limitation on
application of geometric morphometrics: if the variables defining axes of a
«morphospace» in which these objects are distributed are not tensors (and
thus, in particular, are not equivalent in the above sense), the objects
cannot not seemingly be studied by the approach in question.
Shape transformations studied by geometric morphometrics are
decomposed into two components, uniform (affine, linear) and non-uniform
(non-affine, non-linear) (Bookstein, 1991). The first one corresponds to
nonlocalizable transformations: these include total stretching and shearing
that are the same at all landmarks. It is «modeled» by a rectangle turning
into a rhomb when parallel lines remain parallel (Fig. 2a). The second
component includes local stretching, shearing, twisting, bending etc that
are different at different landmarks and thus localizable. This component
can be represented by non-linear bendings of lines of a rectangle (Fig.
2b).
In order to eliminate the «size factor» from an object description, it
is required to express it using landmark coordinates only. For this, a
centroid size is calculated as a sum of squared distances between all
landmarks; or, alternatively, as a square root of sum of squared distances
between all landmarks and the object centroid (Bookstein, 1991). This
variable is not correlated with any one of shape variables and is used for
specimens alignment (see below).
All procedures of geometric morphometrics implies using a reference
object: its landmark configuration defines geometry of the shape space (see
below) and all specimens in the sample are compared relative to it. In some
instances it may be a real or a hypothetical object that depicts, for
instance, beginning of a morphological transformation series. In others,
the reference is calculated (by least square analysis or its analogies) as
a consensus (average, mean) configuration so that its overall differences
from all specimens in a sample by all (or by some, see below) landmarks
would be minimized.
The total of morphological objects described by landmark coordinates
constitutes the figure space of pk dimensionality (Goodall, 1991; Rohlf,
1996). It is to be stressed that this space has nothing in commong with a
physical space «filed in» with morphological objects. As a matter of fact,
it is a special case of phenetic hyperspace, so it is a mathematical
artifact being construed for a given set of digitized objects by
application of some mathematical operations to a set of landmark
coordinates.
The procedure of geometric morphometrics starts with the objects being
centered on the origin (their centroids are superimposed by translations).
Then their centroid sizes are scaled to unit due to which the objects
become aligned and the «size factor» is eliminated from all subsequent
comparisons. The alignment is fulfiled by minimizing differences among
respective centroid sized calculated using all landmarks (procrustes
method) or a pair of priory defined landmarks (baseline method).
As a result of these manipulations, the above figure space turns into
the shape space, or the Kendall's shape space, a keystone of the entire
geometric morphometrics. Its metrics is called Procrustes metrics as it is
formed basically by the Procrustes distance which is a specific analogy of
Euclidian distance (see below). Coordinate axes of a Kedall's space could
be assumed tensros. Its dimensionality is pk-k-k(k-1): for 2-dimensional
objects its has 2p-4 dimensions, while for 3-dimensional objects its has 3p-
7 dimensions. This space is fundamentally non-Euclidean: its geometry, for
a simplest case of a plane objects described by three landmarks, can be
visualized by a hypersphere surface (Fig. 3). The morphological objects
(shapes) are points on this surface. For more complex morphological
objects, the geometry of the shape space appears to be substantially more
complicated and cannot be so easily visualized (Goodall, 1991).
Metric properties of this space allow to assess differences among
shapes as a distances between them. The basic dissimilarity measure is the
above Procrustes distance. It was first suggested as early as in 60-ies
(e.g. Sneath, 1967), but it was geometric morphometrics that estimated its
true worth. There are several versions of this distance with pretty simple
interrelations (Dryden, Mardia, 1998; Rohlf, 1999a). The angle Procrustes
distance r is defined as an angle (in radians) between radii connecting two
points on the hyperspace surface to its center (see Fig. 3), its numerical
expression is arccosine of the square root of sum of squared distances of
landmark coordinates of aligned objects (Bookstein, 1993). It can be
defined also as the geodesic distance as it provides estimation of distance
by the hyperspace surface. The chord Procrustes distance dp is defined as
length of the chord connecting the same points and is calculated as the
square root of sum of squared distances of landmark coordinates. These two
distances are related by the formula r=2sin-1(dp /2). As the Euclidean
distance, the Procrustes distance is a metrics and thus can be used in
cluster and ordination standard routines (see «Application of standard
routines» below).
Non-Euclidean geometry of the shape space makes it difficult to use
multivariate methods based on assumption of orthonormality of covariance
matrix and on linear combinations of variables. It is evident that the more
dissimilar are particular shapes, the more distantly are situated the
respective points on the hyperspace, and hence the more prominent are non-
linear effects. Therefore the statistical methods adequately describe
shapes differences (distribution of points on the hyperspace) if the
latters are «small» (Bookstein, 1991).
To avoid non-linearity effects, the Kendall's space is approximated by
a tangent space that has a Euclidian geometry. It is visualized by a
hyperplane tangent to the Kendall's hypervolume (see Fig. 3). Respective
tangent point to sphere point defines position of an consensus (for the
given dataset) landmark configuration. The shapes can be defined now as
projections of the initial points onto this hyperplane, so all shape
variations are imbedded without significant loss of information. Their
dissimilarities are now evaluated as distances in the tangent space, which
are Euclidean distances de. It is evident that de and dp are related by
monotonous linear function, so that the scale of dissimilarities is not
much disturbed, especially if they are not especially large.
Math properties of the shape space allow to postulate that the
differences among morphological objects actually studied by biologists are
generally «small»: respective points are distributed in the environments of
the tangent point (Rohlf, 1996, 1999). Empirically, the above disturbance
is estimated by comparison of two distances calculated for the given
dataset, Procrustes (in form of geodesic on the hypervolume) and Euclidean
(on the hyperplane) using programm TPSmall. The linearity assumption is met
in nearly all the cases studied, except comparison of an object with its
reflection (Rohlf, 1999).
It is to be stress that geometry of both Kendall's and tangent spaces
is defined not by interrelation among the specimens studied (in particular,
by their landmark covariation) but by the references configuration only.
Figuratively speaking, the geometry of the shape space is first established
using this configuration and then the objects are projected on its
hypersurface. This means, among others, that its geometry (in particular,
orthonormality) is pre-established by Procrustes metrics prior to any
analyses of the data and does not depend on interrelations among objects
themselves (unlike what is presumed by factor analysis) (Bookstein, 1996).
It follows from the immediate above that the reference configuration
plays very important role in explorations using methods of geometric
morphometrics. Defining reference one or another way and/or changing its
landmark coordinates means changing geometry of the shape space. Therefore
it seems reasonable to use the consensus configuration as a reference: it
corresponds to the tangent point and is situated in an average position
relative to distribution of the objects studied, so alignment of the
latters relative to the reference thus defined causes less disturbance of
initial (on the Kendall's space surface) similarity relations among the
shapes as compared to any «peripheral» configuration (Rohlf, 1996).
Besides, the properties of the shape space become dependent, at least in
part, on geometry of the objects for which consensus reference is
calculated.
Such a way of construing the shape spaces implies an important
property thereof: they turn to be, that is to say, «local». This means that
each shape space based on a particular reference configuration appeares to
be «isolated» from all other shape spaces based on any other references.
This is a pure mathematical property followed from the non-monotonicity
theorem (N. MacLeod, in litt.), and it makes such spaces a kind of «close
axiomatic systems», so some «translators» are requested for their mutual
interpretation. This problem only begins to be recognized, but hardly yet
completely. As a matter of fact, incommensurability of results obtained for
different shape spaces (operationally, for different datasets) contradicts
to the vary nature of the comparative method without which not any
biological research is even possible. Thus, the problem in question
requests some general resolution without which possibilities of geometric
morphometrics appeares to be quite «local».
It follows from fundamental properties of the shape space that methods
of geometric morphometrics do not generally bounded by distribution laws.
Therefore most of limitations related to those laws are not in effect for
them. As a consequence, this makes unnecessary calculating confidence
intervals for the shape variables and, subsequently, estimation of
statistical significance of the results (but see chapter «Quantitative
shape comparisons» below).

The Data

The research field of geometric morphometrics is a diversity of
morphological structures, that is of physical bodies on which landmarks can
be reasonably placed and Cartesian coordinates thereof can be taken. This
diversity may be uncertain individual variation; differences among any
discrete groups - taxa, sexes, ages, insect casts, biomorphs, ecotones and
others; series of postures or locomotory phases (MacLeod, Rose, 1993);
fluctuating asymmetry (Klingenber, McIntyre, 1998); etc.
The morphological structure is actually 3-dimensional. However, some
technical limitations, precision of digitizing devices in particular, make
acquiring of 3D data for small (up to several centimeters) objects at
present rather problematic (Dean, 1996). Therefore the more common practice
now is to work with 2D objects: these may be either 2D projections of
original 3D objects or such plane structures for which third dimension has
no special biological meaning. The latters are exemplified by insect wing,
plant leaf, chewing surface of vole tooth, etc.
Such a reduction may lead to loss of some relevant information. As a
half-measure, a method of «pseudo-3D» presentation of an object is
developed which is based on analysis of a set of 2D projections placed at
strictly fixed angles (Fadda et al., 1997).
At present, the question of «nature» of the objects to which geometric
morphometric methods could be applied is nearly uninvestigated. It is
accepted by a default that they are to be «classic» morphological entities
such as a skull. However, as far as the analysis is applied to not an
object itself but to it digitized screen image, the above question is not
as simple as it may look. Its general decision being unavailable, two
particular but significant restrictions are to be indicated. First: the
variables defining a «morphospace» are to be tensors in order to the latter
could be treated as a shape space (see above chapter «Basic formalisms»).
Consequently, if this condition is not met then an object could not
probably be studied by means of geometric morphometrics. Second, followed
from the first: as far as alignment of the objects involves their rotation,
then if such an operation changes position of these objects relative to the
non-equivalent axes of a respective «morphospace», the vary alignment
becomes nonsense, so the entire approach does.
The sample of specimens to be studied by geometric morphometric
methods is construed under the following conditions.
The morphological structures are strictly comparable if they are
projected onto the same shape space. This means that all the specimens,
even if they belong to different groups of interest, are to be included
into the same sample (excluding some special cases, see below) for which a
common reference is to be defined, be it calculated average (consensus) or
a real/hypothetical object. For instance, if differences among sexes or
local populations will be studied by means of dispersion or discriminant
analyses, all relevant specimens are to be included into the same dataset
which undergone geometric morphometric procedure. If the purposes of a
research project presumes analysis of shape variables separately in each of
such groups, for the latters to be commensurable the next method should be
followed. All the specimens are first united in the common dataset for
which a common reference configuration is defined. Thereafter, each group
is studied separately but relative to the same reference: this presumably
means that they all are dirstributed in the same shape space.
The methods of geometric morphometrics are based on simultaneous
analysis of configuration of the landmarks located on the morphological
structure under investigation. Therefore the most «appropriate» are solid
or rigidly articulated structures with least degrees of freedom (Adams,
1999): an insect wing, a plant leaf, an axial skull projection are
examples. Unlike these structures, the bat wing is less «suitable» for
geometric morphometrics: its elements are very mobile relative to each
other, so defining «standard» position provides certain problems (Birch,
1997).
As to the disarticulated structures, landmark coordinates of only
those can be combined into the same dataset for which such a combination
has a biological meaning. For instance, it is possible to unite in a
dataset the landmarks placed on the common projection of the axial skull
and the jaw, preferably with occluded toothraws corresponding to a standard
position. Contrary to this, there is no sense to unite in the same dataset
configurations of the skull and, say, the scapula. A separate dataset
should be created for each of such structures which should be analyzed
separately: they can be compared subsequently by canonical or correlation
analysis (see the next chapter).
It is possible to calculate consesnsus configuration for any set of
specimens in a dataset. For instance, if «averaged» differences among sexes
or species are of interest, a separate consensus is to be calculated for
each of the group in question, and then all these configurations are to be
united in a new dataset. It allows to make the differences more clear-cut.
Analysis of the gnathosoma shape in several tick species of the genus
Ixodes (Voltzit, Pavlinov, 1994) indicates that the differences among adult
males are more conspicuous when consensus configurations, and not separate
individuals, are compared (Fig. 4).
As far as geometric morphometrics is based on pairwise comparisons of
the objects and does not take into consideration their distributions, the
sample size - number of both specimens and landmarks - is irrelevant in
most study cases. However, if standard statistical routines (such as
dispersion analysis) are supposed to apply, the sample size is to be large
enough. Moreover, if these routines include covariance matrix inversion,
the number of specimens should no less than 4 times exceeds the number of
landmarks (Bookstein, 1996).
The above «reasonability» of landmark placing means, before all, that
the landmarks on one object should correspond unambiguously to the
landmarks on another object. This means that the corresponding landmarks
should be fixed at the points on the object surface that are in a certain
sense «the same». There are several bases on which landmark correspondences
can be established, according to which several types of landmarks can be
recognized (Bookstein, 1990a; Slice et al., 1996; MacLeod, 2001). The
landmarks of type I are fixed in accordance to the classical criteria of
homology (special quality, befor all). The landmarks positioned at place of
certain tendon attachment to the bone or of fusion of certain veins on the
insect wing are examples. For the landmarks of type II not only strictly
biological but also geometrical criteria are taken into consideration: for
instance, certain points of maximal curvature of a contour line of a vole
dental crown or an oak leaf. Landmarks of type III are defined exclusively
geometrically: they are placed at extreme points of a curve (for instance,
depicted by the ends of diameter). Only corresponding landmarks of type I
and, in part, of type II could be treated as homologous; however, the
landmarks of type III are hardly homologous, they are simply equivalent
(MacLeod, 2001).
If exact placing of landmarks is impossible due to absence of any
strict «bindings», outline points, or semilandmarks could be applied
(Bookstein, 1997; Pavlinov, 2000a; MacLeod, 2001). They are distributed
evenly according to certain algorithm along a contour lacking any
unambiguously corresponding («homologous») inflection points. In such a
case, the specimens are compared not by particular landmarks but by the
entire sequence of semilandmarks. Respectively, equivalency is established
not between particular landmarks but between the sequence of semilandmarks
corresponding to entire contour curvature (Fig. 5). This equivalency is
warranted by two factors: number of semilandmarks is to be the same for all
the contours compared, and terminal semilandmarks of a sequence should be
fixed at strictly defined positions (they should correspond to true
landmarks of types I or II).
Efficiency of semilandmarks in analyses of shapes is still
questionable (Sampson et al., 1996; N. MacLeod, in litt.). The problem is
that the semilandmarks which sequence was generated by a single algorithm
are mutually correlated; besides, they should be placed close to each other
in order to describe outline adequately. This puts certain limitations on
applicability of methods of geometric morphometrics. First, use of
semilandmarks provides somewhat biased estimate of shape diversity (it
diminishes variability). Second, their mutual closeness makes it possible
to apply Procrustes fit only, while resistant fit method cannot be probably
used (see on the fitting methods below) (J. Rohlf, in litt.). (By the way,
standard landmarks are also being preferably placed evenly and without
local concentration for the same reason). Thus, the semilandmarks are of
subsidiary significance only.
In a real situation, a combination of both landmarks of various types
and semilandmarks is often applied. For instance, in description of lateral
projection of muroid rodent skull (Fig. 6a), the points 9, 11, 17
correspond to landmarks of type I, the points 6, 7 - to landmarks of type
II, the points 8, 13 - to landmarks of type III, and the points 1-5 - to
semilandmarks.
All shape transformation in both Kendall's and tangent spaces are
continuous. Therefore all qualitative modifications causing change in
topology of a morphological structure are at the moment inaccessible to
geometric morphometrics. In particular, it is impossible to describe and
investigate analytically by its methods any differences caused by
appearance/disappearance of structural elements, such as perforations,
accessory bones on the skull or tubercles on a tooth, veins on insect wing
or on a leaf. There is a promising idea opening a possibility to apply some
apparatus similar to Tom's catastrophe theory: it implies description of
quantitative shape transformations as breaks («creases») in the surface of
the Kendall's space (Bookstein, 2000, 2001).
As long as such methods are just being elaborated, some «roundabout
maneuvers» could be applied. For example, for a large sample, an absent
structure could be indicated by a kind of «virtual landmark» which
coordinates are calculated as averaged coordinates of respective real
landmarks for entire sample (Marcus et al., 2000; Pavlinov, 2001). If
fusions and divisions of some structure are involved in shape
transformations, they could be described at least in come cases by fusions
and divisions of respective landmarks, their total number remaining
unchanged (Pavlinov, 2001).
A specific methodological problem is provided by bilateral symmetrical
structures (Bookstein, 1996). When the objects with various magnitude of
asymmetry are compared, their symmetry axes would also be rotated to reach
the best fit by the least square method. Therefore, if asymmetry is not an
immanent property of the morphological structure (as in a dolphin skull) or
a specific subject of investigation (as a fluctuating asymmetry), it is
possible to work with one of the bilateral object's sides. In such a case,
the landmarks are initially placed on one side only, or the same set of
landmarks is first defined for both sides and coordinates of each
equivalent pair are then averaged.
Methods of digitizing morphological structures depends are diverse
enough and depend on size (large - small) and configuration (3D - 2D) of
the objects themselves, on the purposes of investigation (whether it allows
to «reduce» a 3D object to 2D one), as well as on availability of
respective hardware (see a little bit outdated review: Becerra et al.,
1993). It is recommended to take coordinates on the same object more than
once and then to use their average values. For many purposes, use of screen
digitizers is most efficient; for semilandmarks it is probably the only
possible. Among most popular are softwares written for geometric
morphometrics directly: these are Morpheus et al., WinDig and especially
TPSdig.
Cartesian coordinates of landmarks or semilandmarks are most often
used in geometric morphometrics. However, if it is desirable for any reason
to fix some preferred «starting position» in the shape description, so
called Bookstein (or two-point) shape coordinates are calculated
(Bookstein, 1991). For this, two landmarks are first fixed to define a
baseline with respective coordinates (0,0) and (1,0). Other landmarks are
re-defined as vertices of triangles with the common base defined by the
baseline. Their coordinates are re-calculated respectively which leads by
implication to alignment of the objects.
The least square based methods are suggested to use to replace missing
coordinate data by respective averages (Slice, 1993c). Due to this, many
paleontological data could be studied by geometric morphometrics.

Quantitative shape comparisons

As the morphological objects are generally incomparable by original
coordinates, the numerical geometric morphometric analyses begins with
their replacement by some shape variables which values are landmark
coordinates in the shape space.
For this, the specimens (more exactly, their digitized images) are
first aligned: this action eliminates their differences due to
translations, scaling and rotations but not reflections. The most usually
applied are the standard least square methods (in particular, Procrustes
fit method) and specific for geometric morphometrics resistant fit method.
They can be imagined as rigid rotation of one object relative to another so
that their differences become minimized, either by all landmarks or by
those which differences are not exceptionally large (Rohlf, 1990; Rohlf,
Slice, 1990; Bookstein, 1991).
Some differences remain among specimens after their alignment
indicated by dispersion of landmarks around reference configuration (see
Fig. 6b). They correspond, by definition, to differences of landmark
configurations in the tangent space, that is to dissimilarity of the
shapes. This dispersion around each of the landmarks is expressed by so
called Procrustes residuals which are considered as special case of shape
variables, Procrustes coordinates (Rohlf, 1999). Their sum gives overall
Procrustes distance among the specimens.
There are two methods of comparison of shapes in geometric
morphometrics (Rohlf, 1996). One of them is based on the above least square
method, it is most efficient if overall similarity depends largely on few
landmarks. Another one is the thin-plate spline analysis, it works best if
the similarity depends on many of landmarks and so just weakly localizable.
The least square method in geometric morphometrics is represented by
the superimposition method which is, as a matter of fact, extension of
Procrustes or resistant fit rigid alignment. It is resulted in a Procrustes
distance matrix to which other appropriate multivariate routines can be
applied directly (see chapter «Application of standard routines» below). In
particular, this matrix can be decomposed into eigenvectors by principal
component analysis: this gives a new set of shape variables which are now
coordinates in the space of principal components.
Significantly different is thin-plate spline analysis based on analogy
of a 2D morphological object to a thin homogenous deformable metallic plate
(Bookstein, 1989b, 1991). Accordingly to its methodology, one specimens is
fit to another by its non-rigid «stretching/shearing», and numerical
estimate of degree of such a smooth deformation is bending energy
coefficient. Unlike Procrustes distance, the latter cannot be interpreted
as a overall dissimilarity estimate: it characterizes «localizability» of
transformation which turns the reference to (fit) into particular specimens
under investigation. It is fulfilled by softwares Morpheus et al. and
especially TPSrelw. Its application produces several kinds of shape
variables called the warps.
The thin-plate spline analysis begins with decomposing the bending
energy matrix into orthogonal eigenvectors which are called principal warps
(Bookstein, 1989, 1990). Their geometry is defined by reference
configuration exclusively and describes possible directions of
transformations of the latter in the tangent space. The eigenvalues
decreasing sequentially from the first to the last are determined by scale
(localizability) of reference transformations: the higher is a particular
eigenvalue, the more localizable is respective transformation. Three last
(zeroth) principal warps correspond to uniform shape component
transformations. If an investigation is aimed at non-affine (localizable)
transformations, these zeroth warps could be excluded from subsequent
analyses. The principal warps cannot be regarded each as bearing some
specific biological meaning. In a sense, they could be considered as an
analogy of Fourier harmonics which refer to different spatial scales
(Rohlf, 1998).
In order to express each specimen in term of thin-plate spline
parameters (that is, to fit it to the tangent space), a new set of shape
variables is yielded called partial warps (Bookstein, 1989). They are
calculated as coordinates of projections of aligned specimens on each of
the eigenvectors. They indicate for each specimen how much of each
principal warp is needed to transform the reference into that specimen
(Rohlf, 1993). From this it follows that geometry of partial warps is
determined exclusively by geometry of principal warps and subsequently does
not depend on landmark covariation in the sample studied. Therefore this
geometry is unstable: change of reference configuration implies change of
both the geometry itself and partial warp values. The partial warps are
intercorrelated and thus each one of them cannot be used as a separate
biological trait (Rohlf, 1998).
Distribution of specimens along the axes of partial warps illustrates
the spatial scale at which the main differences among them are observed.
For instance, analysis of distribution of sex and age groups of Ixodes
ticks (see Fig. 4) in the space of 12 partial warps indicates the
following. Adult males in total are most separable by 11th and 5th warps
and by uniform component (Fig. 7): it means that they differ from
remainders mainly by total shape of their gnathosoma. In addition, the male
group ? 8 takes quite isolated position by 3d partial warp, so its
specifics is expressed significantly by small-scaled differences.
To eliminate limitations intrinsic to both principal and partial
warps, relative warps are calculated by principal component analysis as
orthogonal eigenvectors of covariance matrix of partial warps (Bookstein,
1991; Rohlf, 1993). Dimensionality of a new shape space defined by relative
warps is n?min(n-1, p(or p-3)), that is it depends basically on the number
of objects and not of landmarks. Distribution of specimens in a newly
established shape space is defined by their projections onto relative
warps. Their eigenvalues have the same meaning as in standard PCA: they
reflect amount of total variance «explained» by respective relative warps
and decrease from the first to the last one. Subsequently, analysis of
relative warps implies reduction of shape space dimensionality and their
orthogonality allows to treat them as separate traits.
The parameter a is used in calculations of relative warps which allows
to assign different or equal «weights» to differently scaled shape
transformations (that is, to different principal warps) (Rohlf, 1993). If
a>0 then «zeroth» principal warps are excluded from the analysis and
ordination of specimens is made on the basis of bending energy matrix. The
value of a=1 yeilds the relative warp analysis in which the principal warps
are weighted inversely by the square root of their eigenvalues. As a
consequence, the large-scaled variation is given more weight than small-
scaled one. If a=0, then all principal warps are included in the analisys
and all are given the same weight, thus relative warp analysis is actually
carried out on the basis of the Procrustes fitting and not on the thin-
plate spline analysis. This value of a=0 is to be accepted if landmarks
(and especially semilandmarks) are situated close to each other.
It is essential that if a=0 then distances among specimens in the
relative warp space are the same as the distances by original landmark
coordinates (Rohlf, 1993). It means that distribution of the specimens in
that space reflects directly the structure of dissimilarity relationes
defined by Euclidian metrics. Therefore, zero value of a is generaly
preferable. However, if the research is aimed at analysis of allometric
growth gradients, then a=1 should be adopted, because allometric relations
involve usually large-scaled shape transformations.
In order to «localize» particular relative warps they have to be
«tied» with particular landmarks. For this, a matrix of loadings
(decomposed in Cartezian axes) of each landmark onto each relative warp is
calculated (Table 1). As in PCA, the loadings indicate how much is input of
the given landmark into transformations of the reference along relative
warps as the axes of respective shape space. Besides, because of
orthogonality of relative warps, distributions of landmark loadings among
these warps make it possible to discuss which components of shape
transformations are mutualy correlated or, contrary, independent.
It is seen from the Table 1 that in case of the axial skull of muroid
rodents (see Fig. 6), the highest loading in the first relative warp is
provided by landmarks 11, 17, 18, while in the second relative warp it is
provided by landmarks 14, 15, 19. This means that changes in position of
proximal part of the toothraw and the masseteric plate are most correlated
with each other, on one hand, and least correlated with displacement of
the middle portion of the zygoma and condylus, on another hand. Input of
the unform shape component is estimated similar way. It is clear from the
same Table that explaned variance increases after elimination of this
component from the analysis. Thus, one may conclude that small-scaled shape
transformations prevail over large-scaled ones.
Localizability of these transformations can also be deduced from
distributions of explaned variances among relative warps. As it is seen
from the same Table, proportion of variances explaned by respective 1st
relative warps is noticibly higher for the axial skull than for the
mandible. Together with distributions of landmarks loadings within each
relative warps, it means that non-affine shape transformations are more
promimnent and more localizable on the axial skull as comapred to the
mandible.
In verifying hypothesis of non-random differences of samples or
correlation of structures several statistical tests can be applied that are
adapted to specifics of shape variables, before all to mutual correlation
of partial warps. Overall differences among samples are tested by Goodall's
F criterium (based on analysis of Procrustes distances) or Hotelling's T2
criterium (based on analysis of Procrustes residuals) (Goodall, 1991;
Dryden, Mardia, 1998). Some randomization techniques are also employed -
permutation, bootstreping, etc. (Dryden, Mardia, 1998).
In comparing several groups by entire set of shape variables a
multivariate dispersion analysis could be applied (MANOVA, available in
TPSregr and APS.). For example, in analysis of effects of taxonomic
allocation and trophic specialization on the skull shape in muroid rodents,
the following results was obtained. Respective Goodall's F values are equal
to, respectively: for the axial skull - 1.86 and 5.43 (df 38, 608; p=0,002
è ð<0.001), and for the mandible - 3.12 è 2.79 (df 22, 352; ð<0.001). It is
seen that for the axial skull, taxonomic allocation is less (the least,
actually) significant than trophic specialization, while for the mandible
the ratio is opposite. If particular shape variables are of interest then
standard statistical routines are employed (see chapter «Application of
standard routines» below).
Covariation of the shapes which landmark coordinates cannot be united
into the same dataset (see previous chapter) can be investigated in several
ways. One of them is based on a method similar to canonical correlation
analysis (available in the TPSpls) which allows also to explore effect of
any quantitative variable (size, climate factor etc) on the shape
transformation. For instance, correlation of the first vector for axial
skull shape with that for mandible is high, while its correlation with the
body size is low (Fig. 8): correlation coefficients are 0.96 and 0.62,
respectively.
Another method is to anaylize correlation between relative warps
calculated for different shapes (or between relative warps and other
variables). This approach is of interest, as it permits to «localize»
correlations of those shapes. For examples, in the above muroid rodents
most correlated appeared to be first relative warps of both axial skull and
mandible. Looking at the figures in the Table 1 and at landmark locations
on Fig. 6, one may conclude that changes in toothraw and masseteric plate
(landmarks 11, 17, 18) are most correlated with changes in configuration of
the base of proximal part of the mandible (landmarks 4, 6, 11).
Among most problematic in geometric morphometrics remains assessment
of diveristy (dispersion) of shape variables. At present, this methodology
lacks any statistics similar to standard coefficient of variation. It is
theoreticaly possible to evaluate overall shape variability using
Procrustes distance; however, two unclear points appear. On the one hand,
developers of geometric morphometrics used to stress that most interesting
are «local» and not «total» shape differences, so for them it does not make
much sense to apply any generalized indices of similarity/dissimilarity
like «taxonomic distances» (Bookstein, 1996; Rohlf, 2001a; N. MacLeod, in
litt.). On the other hand, statistical properties of Procrustes distance
are not properly studied yet: it is established empiricaly (personal
unpublished data) that its values positively correlated with the number of
landmarks, but the nature of this correlation is not clarified and,
consequently, no corrections factor is figured out.

graphic methods

As it was noticed in the introductory chapter, geometric morphometric
was born mainly as a «technical device» for analytical resolution of
geometric tasks exposed by D'Arcy Thompson. And developers of this
methodology used to stress that its most prominent edvantage is not the
«figures» but the geometric images. At the same time, it is advised to keep
in mind that these images are deduced from approximate algorithms and would
not be taken to literally: they are rather visual «metaphors» of shape
transformations than exact mappings of them (Bookstein, 1996).
The simplest way to visualize variabilty of shapes is decomposing
residuals obtained by resistant fit method into eigenvectors for each
landamrk separately. They characterize direction and amplitude of variation
at each landmark not taking into account landamrk covariation. The most
obvious and most easily interpretable representation is a set of ellipsoids
depicting boundaries of confidence intervals for dispersion of landmark
coordinates for a given dataset (see Fig. 6b). This approach provided by
the software GRF is actually univariant and therefore is not especially
popular among recent researchers.
More sofisticated are methods based on the thin-plate spline analysis:
they embrace correlation between lanmarks and thus are multivariate. The
resulting graphic images illustrate partial and relative warps in two ways.
One of them is transformation grid, another is a set of vectors (Fig. 9,
10). The both are available in the programs TPSsplin, TPSpls, TPSrelw,
TPSregr, Morpheus et al.
Transformation grid is initially orthogonal (Fig. 9a). The more are
differences among shapes in the vicinities of a landmark, the more is
deformation of respective fragment of the grid (Fig. 9b). Smoothness of the
function corresponding to bending energy coefficient allows to define grid
configuration between any pair of landmarks, so it might be of arbitrary
density. Contrary to this, vectors are tightly bounded to the landmarks
(Fig. 9c). They have to be analyzed in their totality, as any one vector
has no independent meaning. As dissimilarity relation defined by bending
energy is not symmetric, graphic representation of superimposition of A
onto B is not the same as B onto A (fig. 10).
In relative warp analysis, the graphic representation allows to
localize most expressed shape changes corresponding to a particular warp.
Generally, degree of grid transformation arownd a landmark (or vector
length) corresponds to loading of this landmark into respective relative
warp.
Graphic representation of shape differences avails both for a pairwise
comparison (see Fig. 10) and for a multi-object sample. In the latter case,
of importance is a possibilty to visualize potential transformations of
reference configuration in the space of relative warps. This approach
allows to demonstrate graphically, how the shape of the specimens is
transformed along particular relative warps in either «positive» or
«negative» directions from the zeroth point corresponding to consensus
configuration (Fig. 11).
Graphic approach can serves as an important addition to numerical
multivariate methods (like MANOVA) of analisys of among-group
dissimilarities. For this, consensus configurations are calculated for each
of the groups under comparison (see chapter «Data» above) and superimposed
upon each other. Configurations of respective transformation grids show
where the differences revealed statistically are localized.
An interesting possibility to trace shape transformations among taxa
is provided by program TPStree, given that there is a dendrogram with a
priory defined topology that reflects certain (for instance, phylogenetic)
relations among those taxa. For each fragment of this dendrogram, it is
possible to get a grid which transformations correspond to differences
among average (consensus, not ancestral!) configuration of respective group
of taxa and a hypothetical one placed at the base of dendrogram.

Application of standard routines

Geometric morphometrics deals with shape deformations proper, so its
«ultimate aim», strictly speaking, is limited to Procrustes distances and
partial warp scores that indicate to what extent and how one shape differs
from other(s). However, if the research is managed to go further, other
numerical techniques are to be applied borrowed from the standard
statistics.
Some of these techniques are built in the geometric morphometric
toolkit: among them are PCA, canonical correlation, regression and
dispersion analyses (considered in short above). At the same time,
geometric morphometrics still lacks some commonly used standard methods -
for instance, cluster analysis, multidimensional scaling, stepwise
discriminant analysis etc. Their applications are in part limited by
properties of shape variables distribution that should certainly be taken
into consideration when a research project is planned (Bookstein, 1996;
Rohlf, 1998, 2001a). Most of these properties were mentioned above; to sum
up, the following points should be stressed.
Before all, it should be reminded that any comparison of different
groups of specimens by shape variables is valid only if the specimens
belonging to those groups are all included in a common dataset. Numerical
results obtained independently for different datasets by means of geometric
morphometric tools are not strictly commensurable.
Procrustes distance is a metric and may be dealt with as the Euclidean
distance - for instance, it may be used in multidimensional scaling (Gower,
1975) or in cluster analysis. In contrast to it, bending energy coefficient
is neither metrics nor ultrametrics, so the bending energy matrix should
not be studied by standard methods.
Partial warps are mutually correlated and therefore reflect shape
changes in their total only. Any one of them taken in isolation does not
bear any sensible biological information and thus should not be analyzed
separately from others. Besides, this shape variable, as a matter of fact,
is not a vector but a tensor as it is defined by a pair (or a triplet) of
x,y(z) coordinates. Therefore it is not recommended to explore n?p?k matrix
of partial warp scores by any of univariance methods or by the
multivariance methods assuming orthonormalitry (such as standard factor
analysis). To this matrix, applied could be such methods as multiple
regression, multiple analyses of variance and covariance (MANOVA, MANCOVA),
complete discriminant analysis (without selection of variables); cluster
analysis of Euclidean distances calculated for the entire set of partial
warps; also applicable are such multivariate estimates as Hotelling's Ò2,
Mahalanobis' D2 , Wilkson's ( etc.
Unlike partial warps, relative warps are not correlated with each
other, so it is possible to apply to them not only all the above methods
but univariance ones and a stepwise discriminant analysis, as well. The
only reservation should be made that relative warps (as principal
components in general) are a kind of «mathematical artifacts», therefore
analysis of differences among samples by each warp using, say, the
Student's T criterium makes no special sense.
As in the case of partial warps, differences by relative warps can be
evaluated numerically by Euclidean distance, which produces the same
estimates as the Procrustes distance. This technique is of special meaning,
as there some tasks exist that require standardization of the data, and
such a standardization meets least objections just in the case of relative
warps. For instance, it is possible to study structure of shape diversity
by means of multidimensional scaling. For this, distributions of stresses
calculated at each step of iterative procedure of dimensionality reduction
are compared for the distance matrices calculated for real and random data
(Puzachenko, 2001). But because the stress values are directly proportional
to the distance values, the both datasets are to be equally standardized,
and this could most easy be done for the original variables rather then for
calculated distances.
Stress distributions obtained for two Euclidean distance matrices are
shown on Fig. 12. One of these matrices was calculated for relative warps
scores of 60 third upper molars of the rock vole genus Alticola (some are
drawn on Fig. 5); another one was calculated for a set of variables
generated by a random number calculus; in both cases the data were
standardized in the interval [0,1] (SYSTAT package was used for
standardization and randomization and Statistica for Windows package was
used for multidimensional scaling). As it is seen, two distributions
appeared to be nearly the same which proves basically random distribution
of the tooth shapes in a shape space. One can conclude from this that there
is no sign of discreteness in the shape differences thus estimated and to
argue (at least for this particular case) against the vary possibility of
recognizing «discrete» morphotypes.
Analysis of effect of taxonomic allocation and trophic specialization
on the skull shape in muroid rodents using Fisher's F estimation of
relative warps (ANOVA in Statistica for Windows) gave the following
results. The highest correlation with both factors was obtained, as it had
to be anticipated, for 1st relative warp which explained the largest
portion of a total variance. As to the ratio of the two factor effects, the
present results differ to some degree from those obtained using Goodall's
test (see chapter «Numerical shape comparisons» above). The both
morphological structures now appeared to be most dependent on taxonomic
allocation rather than on trophic preferences: Fisher's F is equal to,
respectively, 30.38 and 12.81 for axial skull and 16.73 and 4.01 for
mandible (trophic specialization was more significant for the axial skull
by Goodall's F). It is probable that 1st relative warp extracted by thin-
plate spline analysis does not yet incorporate some details of shape
differences which is contained in a complete set of Procrustes residuals.
A possibility of incorporating numerical results of geometric
morphometric analyses in cladistics, which methods became now quite
«routinal», constitutes a special problem. Several attempts based on
partial warps (Zelditch et al., 1995; Naylor, 1996; Zelditch, Fink, 1998)
met rather sharp criticism (Adams, Rosenberg, 1998; Rohlf, 1998, 2001a;
MacLeod, 2001). The causes are evident enough. Partial warps are
«mathematical artifacts»: they are linear combinations of original landmark
coordinates calculated according to certain algorithm. Therefore, unlike
the original landmarks, they could not be treated as «homologous» traits.
In this respect, relative warps have some benefits over partial ones, as
they could be «tied» to particular landmarks (through their «weights», see
above) and thus could be used in cladistic analysis as the traits (MacLeod,
2001). However, another problem emerges: the both shape variables are
strictly continuous, this property following not from peculiarities of
morphological structures themselves but from smoothness of the
interpolation function corresponding to bending energy coefficient. It is
evident that bringing any «discreteness» into their distribution, as it is
requested by parsimony methods in cladistics, contradicts to the nature of
these variables.
James Rohlf (1998, 2001a) suggests that most accurate, in respect to
phylogenetic interpretation of numerical data resulted from geometric
morphometrics, is a maximum likelihood approach (in sense of Felsenstein,
1973, 1988, 2001) applied to Procrustes distance. His opinion is based on
an idea that, for the both, the initial statistical model is a Brownian
motion around central momentum. However, at least as far as complex
morphological structures are concerned, such a model has probably no
biological meaning. At best, it could be used to formulate a kind of a null-
hypothesis, not a working one.
On the other hand, if there is a cladogram deduced from some other
data source, it is possible to «fit» to it the shape transformations
resulted from geometric morphometrics (see chapter «Graphic comparisons»
above). Besides, one could wish to apply contrasts method to determine a
«phylogenetic load» into the shape diversity and thus to discuss more
thoroughly possible causes of historical transformations of that shape
(e.g. David, Laurin, 1996; Schaefer, Lauder, 1996).

The Software

Numerical methods of geometric morphometrics are not yet built in
standard statistical packages such as SYSTAT, SPSS, SAS, Statistica and
others. However, they are available in a number of more special programs
many of which were already mentioned above. Most of them are designed for
PC platform (with operational systems DOS, Windows, less frequently OS/2
and UNIX), and few are for Apple Mac. PC-designed programs work under DOS
in earlier versions and under Windows (or occasionally under other systems)
in their more recent upgrades. The landmark coordinate data are kept in
standard text files in ASCII codes, sometimes in electronic table format.
Nearly all these programs are free-ware (besides Morphologika and the last
version of NTSYSpc which are commercial products) available through the
Internet site by address http://life.bio.sunysb.edu/morph/ (supported by
F.J. Rohlf).
Below is given a brief compendium of these programs (listed
alphabetically). The outdated program products, for which more recent
upgrades or more advanced softs are available, are not reviewed.
APS (Penin, 2000) explores a multi-specimen sample, extracts principal
components (relative warps) based on procrustes residuals; compares two
subsamples by discriminant analysis of procrustes residuals; fulfils
regression of each of the relative warps against centroid size. Graphical
representation of results is a scatter-plot of specimens in the space of
those principal components.
GRF (Rohlf, Slice, 1991) and GRF-nd (Slice, 1993b) are DOS
applications working with 2D (GRF) or nD (GRF-nd) objects. GRF-nd supports
datasets with partially missing landmark coordinates. The specimens are
fitted by least square or resistant-fit methods, Procrustes residuals are
decomposed by PCA into eigenvectors, dispersion limits and direction are
displaced graphically as ellipsoids or vectors.
Morpheus et al. (Slice, 1993c) is an advanced version of GRF-nd
designed for PC (under Windows, OS/2, UNIX) and Apple Mac. Undertakes
Procrustes analysis (pairwise comparison of specimens by all landmarks),
transformation grid provides graphical representation of results. Fulfils
Fourier analysis, as well. Allows to save coordinates of a consensus
configuration, Fourier harmonic coefficients.
Morphologika (O'Higgins, Jones, 1998) is a beautifully designed
Windows application supporting both 2D and 3D coordinate data; allows to
select specimens and landmarks. Analytical facilities are limited, however:
it makes Procrustes fit, calculates principal components (relative warps)
of Procrustes residuals, displays both scatter-plot of specimens and
transformation grid configurations in the relative warps space.
NTSYS-pc (Rohlf, 2000a) of 2.0 and higher versions is a Windows
application in which some geometric morphometric tools are built-in. Allows
to analyze both 2D and 3D objects (unlike TPS series of the same author),
calculates principal and partial warps.
Shape (Cavalcanti, 1996) calculates and saves in a data file the
Bookstein coordinates relative to a priory user-defined baseline.
TPS is a series of programs issued by F.J. Rohlf. Their earlier
versions were designed for DOS (are not considered here), more recent are
Windows applications designed in both 16 and 32 bit versions. They allow to
work with 2D objects only. Many of them have built-in screen graphic editor
(manipulations with colours, size, labeling) and save the images in graphic
format files. The latest versions constitute a most powerful toolkit for
geometric morphometrics, all with rather detailed helpers.
TPSdig (Rohlf, 2001b) is a screen digitizer, operates with the files
of most standard raster formats; also permits loading some multimedia video
files and capturing images from them; simple enhancement operations with
the images are possible. Allows to locate, maintain and edit on the screen
landmark and outlines coordinates from digitized images and to save them in
a text file. When the original image file is also saved in a directory
indicated in the data file, loading the latter yields appearance of the
image with the landmarks previously collected.
TPSpls (Rohlf, 1998a) computes canonical correlations among two shapes
or among a shape and a non-shape variable. Most of the facilities are
similar to those in TPSregr (see below).
TPSpower (Rohlf, 1999d) is an auxiliary program. Given an estimate of
the means of two populations, the expected amount of variability at the
landmarks, and a sample size, it computes the statistical power expected
using various statistical methods.
TPSregr (Rohlf, 1998b) undertakes multiple regression and dispersion
analyses of partial warps using linear models. In the first case, one or
several continuous independent variables (such as size, temperature etc)
are used, in the second case they are discrete (group belonging). Displaces
shape changes predicted by regression model in form of transformation grid
changes. Provides statistical estimates of congruence of independent and
shape variables using standard criteria or permutation approach. Displays
empirical distribution of specimens along regression line; permits saving
this picture in a graphic file, as well as regression coefficients and
Procrustes residuals.
TPSrelw (Rohlf, 1998c) calculates principal, partial, and relative
warps. Calculates and optionally saves in a file the landmark coordinates
of both references configuration and of each specimen in a sample after
their alignment, partial and relative warps scores, and landmark loadings.
Varying parameter a values and inclusion/exclusion of uniform shape
component is possible. Calculates SS for each of relative warp. Displays
shape changes as deformations grids or vectors, scatter plot of specimens
in the spaces of partial and relative warps, permits saving all screen
images in graphic files.
TPSsmall (Rohlf, 1998d) is used to determine whether the amount of
variation in shape in a data set is small enough to permit statistical
analyses using linear model.
TPSsplin (Rohlf, 1997) compares two shapes based on thin-plate spline
analysis, displays results as transformation grids or vectors (facilities
are the same as in TPSrelw). Also optionally calculates and saves in a file
Procrustes and geodesic distances.
TPStree (Rohlf, 2000b) makes it possible to trace shape changes on a
user-defined hierarchical tree (both ultrametric and additive). Reads tree
description from a NEXUS file. Displays the tree and transformation grid
which configuration changes accordingly to cursor position on the tree. For
each position (corresponding to a hypothetical object), grid configuration,
landmark coordinates and partial warp scores can be saved in respective
files.
TPSutil (Rohlf, 2000c) is a small program to edit data files: permits
combining several files into one, to select and deselect landmarks, and to
randomize specimen list.
WinDig (Lovi, 1996) is a rather simple screen digitizer with some
facilities permitting editing screen images.
Besides the above programs, some geometric morphometric routines are
available from standard statistic packages, such as Matlab, SAS, if
respective program languages is used (Marcus, 1993).

A brief glossary

The below glossary is a brief version of the one published by Slice et
al. (1996). Its updated version can be found in the Internet site by
already indicated address http://life.bio.sunysb.edu/morph/.
a - a «weighting» parameter used in relative warp analysis to give
different (if not equal to zero) or same (if zeroth) «weights» to different
principal warps.
Affine transformation - a linear transformation of a shape in which
parallel lines remain parallel (turns square into parallelogram).
Equivalent to uniform transformation.
Alignment - an isometric transformation of the objects by which their
centoids are scaled to unit in respect to that of reference configuration.
Baseline - an imaginary line connecting the pair of landmarks that are
assigned to fixed locations (0,0) and (1,0) in Cartesian coordinate system
to define a basis for calculating Bookstein shape coordinates.
Bending energy - a metaphor borrowed from the mechanics of thin metal
plates. It is the (idealized) energy that would be required to bend the
metal plate so that the landmarks were change their position appropriately.
The bending energy of an affine transformation is zero since it corresponds
to a tilting of the plate without any bending. The value obtained for the
bending energy corresponding to a given displacement is inversely
proportional to scale of shape transformations. Such quantity should not be
interpreted as a measure of dissimilarity (e.g., taxonomic distance)
between two forms.
Bookstein shape coordinates, two-point shape coordinates - for the 2D
data, recalculated coordinates of landmarks defined as vertices of
triangles which bases coincide with a baseline.
Centroid size - the size measure used in geometric morphometrics to
scale a configuration of landmarks so it can be plotted as a point in the
shape space. It is the square root of the sum of squared distances of a set
of landmarks from their centroid, or, equivalently, the square root of the
sum of the variances of the landmarks about that centroid in x- and y-
directions.
Consensus configuration - a set of landmarks representing the central
momentum of the sample studied. It is often computed to optimize some
measure of fit: in particular, in the Procrustes fit a mean shape is
computed to minimize the sum of squared Procrustes distances among
specimens in the sample.
Kendall's space - the basic non-linear interpretation of the shape
space in geometric morphometrics. Informally, it is represented by a sphere
on which surface the points corresponding to particular shapes are
distributed. It is defined by Procrustes metrics and provides a complete
geometric setting for analyses of the shapes. Most multivariate methods of
geometric morphometrics are linearizations of statistical analyses of
distances and directions in this underlying space.
Landmark - a specific point on a morphological object or on its
digitized image located according to certain rule. Position of landmarks is
defined based either on the classical homology criteria (landmarks of type
I) or on geometric properties of the object (landmarks of types II and III)
(see also semilandmarks).
Morphospace - any hyperspace in which the objects being compared are
distributed. The objects are characterized by any arbitrary (not obligatory
equivalent) variables that define axes of this space.
Partial warps - an auxiliary structure for the interpretation of shape
changes. Geometrically, partial warps constitute an orthonormal basis for a
tangent space. Algebraically, they are eigenvectors of the bending energy
matrix describing a shape deformation along each coordinate axis. Except
for the very largest-scale and for uniform shape component, partial warps
are (approximately) localizable and have (an approximate) scale. The
partial warp scores are projections of the shapes on principal warps
calculated from an orthogonal rotation of the full set of Procrustes
residuals.
Principal warps - eigenfunctions of the bending energy matrix
interpreted as actual warped surfaces (thin-plate splines) over the
original landmark configuration. Principal warps together describe complete
decomposition of a sample of shapes relative to the sample reference shape.
Together with the uniform shape component, supply an orthonormal basis for
a space that is tangent to Kendall's space.
Procrustes distance - a measure of dissimilarities among shapes by
landmark coordinates. Calculated as the square root of the sum of squared
differences between the positions of the landmarks in two optimally (by
least-squares) aligned and superimposed configurations at centroid size.
Procrustes fit - a set of least-squares methods for estimating
dissimilarities among shapes by landmark coordinates. The adjective
«Procrustes» refers to the Greek giant who used to stretch or shorten
victims to «fit» his bed.
Procrustes residuals - the set of vectors connecting the landmarks of
a specimen to corresponding landmarks in the reference configuration after
a Procrustes fit. The sum of squared lengths of these vectors is
(approximately) the squared Procrustes distance between the specimen and
the reference.
Reference configuration, reference - a configuration of landmarks to
which data are fit. It may be another specimen in the sample or the average
(consensus) configuration for a sample. The reference configuration
corresponds to the point of tangency of the linear tangent space to spheric
Kendall's space. The average configuration is preferable as the reference
in order to minimize distortions caused by this linear approximation.
Relative warps - principal components of a distribution of shapes in a
tangent space. Each relative warp, as a direction of shape change around
the reference, can be drawn out as a thin-plate spline transformation. In a
relative warps analysis, the parameter a is used to «weight» shape
variation by the geometric scale of shape differences. Relative warps can
be computed from partial warps or from Procrustes residuals.
Resistant fit - a superimposition methods that use median-based
estimate of fitting parameters rather than least-squares estimates. It
takes into account landmarks that provide small differences among shapes
and thus is less sensitive to extreme values than those of comparable
procrustes fit methods. However, resistant-fit methods lack the well-
developed distributional theory.
Semilandmark - an element of a single sequence of points along an
outline that is generated by a single algorithm.
Shape - in geometric morphometrics, a geometric properties of a
configuration of points that are invariant to changes in translation,
rotation, and scale. The shape of an object is represented by a
configurations of landmarks, as a single point in a shape space defined by
a set of shape variables. In morphometrics there are also other sorts of
shapes (e.g., those of outlines, surfaces, or functions) correspond to
quite different statistical spaces.
Shape space - a fundamental algebraic construction in geometric
morphometrics. Each point in this space represents a configuration of
landmarks irrespective of size, position, and orientation, which is the
shape by definition. In the shape space, scatters of points correspond to
scatters of entire landmark configurations, not merely scatters of single
landmarks.
Shape variable - any measure of the geometry of a biological form or
of its digitized image that does not change under translations, rotations,
and changes of geometric scale. Shape variables include angles, ratios, and
any of the sets of shape coordinates that arise in geometric morphometrics
(Procrustes residuals, principal, partial and relative warps etc).
Superimposition - transformation of one or more shape to achieve some
geometric relationship to another shape. These transformations can be
computed by matching two a priori defined landmarks (Bookstein
coordinates), few landmarks (resistant fit), or by least-squares
optimization of residuals at all landmarks (Procrustes fit).
Tangent space - a linear space (plane) that is tangent to Kendall's
space at a point corresponding to the shape of a reference configuration.
If variation in shape is small then Euclidean distances in the tangent
space can be used to approximate Procrustes distances in the Kendall's
space. Since the tangent space is linear, it is possible to apply
conventional statistical methods to study variation in shape.
Uniform shape component - a part of the difference in shape between a
set of configurations that can be modeled by an affine transformation.
Together with the partial warps, the uniform component supplies an
orthonormal basis for shape space.

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Table 1
Numerical estimates (loadings, explaned variance) of 1st and 2d relative
warps (RW1, RW2) calculated for axial skull and mandible of muroid rodents
(after Pavlinov, 2000b, modified)
| |Loadings |
|Landmark | |
|number | |
| |Axial skull |Mandible |
| |RW1 |RW2 |RW1 |RW2 |
| |x |y |x |y |x |y |x |y |
|1 |16.1 |-28.5|27.4 |-27.8 |-19.2|0.2 |4.0 |-1.2 |
|2 |-10.9|-0.6 |-8.0 |-6.8 |1.1 |2.3 |-15.8|5.4 |
|3 |-6.5 |11.8 |-19.0 |11.5 |23.0 |4.2 |-24.9|1.2 |
|4 |-18.7|6.1 |-9.4 |-13.0 |-33.5|3.2 |-0.2 |12.2 |
|5 |-16.7|-8.0 |3.2 |-1.5 |-12.0|36. 2|24.9 |20.4 |
|6 |-25.8|-8.9 |-8.2 |-8.6 |8.8 |-41.7|-38.8|-28.9|
|7 |11.9 |-2.7 |-0.3 |-1.7 |7.3 |-10.9|1.9 |-1.7 |
|8 |43.7 |-3.9 |47.2 |28.0 |0.2 |1.2 |25.6 |32.9 |
|9 |-9.7 |49.3 |-62.5 |58.3 |4.3 |8.0 |-29.9|-28.6|
|10 |47.4 |-3.8 |17.9 |-29.9 |-26.6|1.8 |31.8 |13.9 |
|11 |128.9|115.2|-41.3 |-4.5 |39.3 |1.8 |-21.4|-11.6|
|12 |48.4 |-55.3|0.6 |-16.5 |0.4 |-1.6 |-18.7|5.6 |
|13 |21.9 |-6.5 |-41.1 |-21.9 |6.5 |-5.1 |61.5 |-19.7|
|14 |-64.4|37.0 |192.5 |121.9 | | | | |
|15 |30.0 |-32.8|-144.6|-104.1| | | | |
|16 |-7.5 |18.0 |-7.3 |39.5 | | | | |
|17 |107.2|-21.4|-57.7 |-117.0| | | | |
|18 |119.8|41.4 |2.9 |180.3 | | | | |
|19 |44.3 |-92.5|128.3 |-60.3 | | | | |
|20 |6.3 |-74.1|0.6 |53.3 | | | | |
|21 |25.9 |27.0 |-2.3 |-9.5 | | | | |
|Explain|Uni+|43.23 |12.15 |32.88 |21.25 |
|ed | | | | | |
|varianc| | | | | |
|e, % | | | | | |
| |Uni-|46.07 |12.57 |32.61 |22.58 |


Comment. Landmark ns see on Fig. 6. Uni+ - uniform included, Uni- -
uniform excluded.