**L** - The crucial matrix for computing the thin-plate spline
interpolant between two landmark configurations. In this entry, *k*
stands for the number of landmarks, for historical reasons. The equation of the
thin-plate spline has coefficients
$$**L**^{-1}**h**, where
**h** is a vector of the *x*- or *y*-coordinates of
the landmarks in a target form, followed by three 0’s (for two dimensional data,
four 0’s for three-dimensional data). The entries in the matrix
**L** are wholly functions of the starting or reference form for
the spline. Bending energy is the upper *k*-by-*k* square of
$$**L**^{-1}. For the complete formula for
**L**, see the Orange
Book or Rohlf (Black
Book).

**landmark** - A specific point on a biological form or image of a form
located according to some rule. Landmarks with the same name, homologues in the
purely semantic sense, are presumed to correspond in some sensible way over the
forms of a data set. See Type
I, Type
II, and Type
III landmarks.

**least-squares estimates** - Parameter estimates that minimize the sum of
squared differences between observed and predicted sample values.

**likelihood ratio test** - A test based on the ratio of the likelihood
(the probability or density of the data given the parameters) under a general
model, to the likelihood when another, specified hypothesis is true. Many of the
commonly used statistical tests are likelihood ratio tests, e.g., the
*t*-test for comparisons of means, Hotelling's $T2$,
and the analysis of variance *F*-test.

**linear combination** - A sum of values each multiplied by some
coefficient. A linear combination can be expressed as the inner product of two
vectors, one representing the data and the other a vector of coefficients.

**linear transformation** - In multivariate statistics, a linear
transformation is the construction of a new set of variables that are all linear
combinations of the original set. In geometric morphometrics, one linear
transformation takes Procrustes-fit coordinates to partial warp scores; another
takes them to relativewarp scores. A linear transformation of a matrix **A**
can be written in the form **y** = **Ax**, where **y** is the resulting
linear combination of **x**, a column vector, with the rows of **A**.

**linear vector space** - In morphometrics, the most common
*k*-dimensional linear vector space is the set of all real
*k*-dimensional vectors, including all sums of these vectors and their
scalar multiples. More generally, but informally, a linear vector space is a set
of elements, usually bits of geometry or whole functions, that can be added
together and can be multiplied by real numbers in an intuitive way. The points
of a plane don't form a linear vector space (what is "five times a point"?), but
lines segments connecting all the points to the origin do form such a space.

**loading** - The correlation or covariance of a measured variable with a
linear combination of variables. A loading is not the same as a coefficient. In
general, coefficients supply formulas for the computation of scores whereas
loadings are used for the biological interpretation of the linear combination.

**Mahalanobis distance** - Also *D* or Mahalanobis *D*. See generalized
distance.

**MANOVA** - See multivariate
analysis of variance.

**maximum likelihood estimates** - A likelihood function is a probability
or density function for a set of data and given estimates of its parameters. A
maximum likelihood estimate is the set of parameter values that maximize this
function. In some cases, as with the arithmetic mean of a sample used as an
estimate of the parameter mu for a normally distributed population, the maximum
likelihood estimate may be identical to the least-squares estimate.

**median size** - A size measure based on the repeated median of
interlandmark distances. Used in resistant-fit methods.

**metric** - A nonnegative function,
*d _{ij}*, of two points,

**metric space** - A space and a distance function defined on every pair
of points that meets the requirements of the definition of "metric" above.

**morphometrics** - From the Greek: "morph," meaning "shape," and
"metron," meaning "measurement." Schools of morphometrics are characterized by
what aspects of biological "form" they are concerned with, what they choose to
measure, and what kinds of biostatistical questions they ask of the measurements
once they are made. The methods of this glossary emphasize configurations of
landmarks from whole organs or organisms analyzed by appropriately invariant
biometric methods (covariances of taxon, size, cause or effect with position in
Kendall's shape space) in order to answer biological questions. Another sort of
morphometrics studies tissue sections, measures the densities of points and
curves, and uses these patterns to answer questions about the random processes
that may be controlling the placement of cellular structures. A third, the
method of "allometry," measures sizes of separate organs and asks questions
about their correlations with each other and with measures of total size. There
are many others.

**multiple discriminant analysis** - Discriminant analysis involving three
or more a priori-defined groups. See discriminant
analysis.

**multiple regression** - The prediction
of a dependent variable by a linear combination of two or more independent
variables using least-squares methods for parameter estimation. See multivariate
regression and multivariate
multiple regression.

**multivariate analysis of variance** - MANOVA. An
analysis of variance of two or more dependent variables considered
simultaneously.

**multivariate morphometrics** - A term
historically used for the application of standard multivariate techniques to
measurement data for the purposes of morphometric analysis. Somewhat confusing
now as any morphometric technique must be multivariate in nature. See traditional
morphometrics.

**multivariate multiple
regression** - The prediction of two or more dependent variables using two or
more independent variables. See multiple
regression and multivariate
regression.

**multivariate regression** - The
prediction of two or more dependent variables using one independent variable.
See multiple
regression and multivariate
multiple regression.

**normalize **- To normalize a geometric object is to transform it so that
some function of its coordinates or other parameters has a prespecified value.
For example, vectors are often normalized by transformation into unit vectors,
which have length one.

**nuisance parameters** - Parameters of a model
that must be fit but that are not of interest to the investigator. In
morphometrics, the parameters for translation and rotation are usually nuisance
parameters.

**null model -** The simplest model under consideration. The null model
for shape is the distribution in Kendall's shape space that arises from
landmarks that are distributed by independent circular normal noise of the same
variance in the original digitizing plane or space and drawn from a single,
homogeneous population. It is exactly analogous to the usual assumption of
"independent identically distributed error terms" in conventional linear models
(regression, ANOVA).

**oblique **- At an angle that is not a multiple of 90 degrees.

**Orange Book **- Bookstein, F. L. 1991.
*Morphometric Tools for Landmark Data. Geometry and Biology*. Cambridge
University Press: New York.

See also Black Book, Blue Book, Red Book, and Reyment's Black Book.

**ordination** - A representation of objects with respect to one or more
coordinate axes. There are many kinds of ordinations depending upon the goals of
the ordination and criteria used. For example, plotting objects according to
their scores on the first two principal component axes provides the
two-dimensional ordination best summarizing the total variability of the objects
in the original sample space. Biplots combine an ordination of specimens and an
ordination of variables.

**orthogonal **- At right angles. In linear algebra, being "at right
angles" is defined relative to a symmetric matrix **P**, such as the
bending-energy matrix; two vectors **x** and **y** are orthogonal with
respect to **P** if **x**^{t}**Py**=0. Principal warps are
orthogonal with respect to bending energy, and relative warps are orthogonal
with respect to both bending
energy and the sample covariance matrix.

**orthogonal superimposition** - A superimposition using only
transformations that are all Euclidean similarities, i. e., involve only
translation, rotation, scaling, and, possibly, reflection.

**orthonormal **- A set of vectors is orthonormal if each has length unity
and all pairs are orthogonal with respect to some relevant matrix,
**P**, such as the identity matrix. A matrix is *orthogonal*
if its rows (columns) are orthonormal as a set of vectors.

**outline** - A mathematical curve that stands for the two-dimensional
image of a physical boundary. Outline data can be archived as a sequence of
point coordinates, but such points do not share the notion of homology
associated with landmarks (but see Sampson, 1996, NATO volume "white book").

**parameter** - In general, a parameter is a number (an integer, a
decimal) indexing a function. For instance, the *F*-distribution used to
test decompositions of variance has two parameters, both integers: the counts of
the degrees of freedom for the two variances whose ratio is being tested. In
morphometrics, there are four main kinds of parameters: nuisance
parameters, which must be estimated to account for differences not of
particular scientific interest; the geometric parameters, such as shape
coordinates, in which landmark shape is expressed; statistical parameters, such
as mean differences or correlations, by which biological interpretation is
confronted with that data; and another set of geometric parameters, such as partial
warp scores or Procrustes
residuals, in which the findings of the statistical analysis are expressed.

**Partial Least Squares** - Partial Least Squares is a multivariate
statistical method for assessing relationships among two or more sets of
variables measured on the same entities. Partial Least Squares analyses the
covariances between the sets of variables rather than optimizing linear
combinations of variables in the various sets. Their computations usually do not
involve the inversion of matrices (see the Orange
Book).

**partial warp scores - **Partial warp
scores are the quantities that characterize the location of each specimen in the
space of the partial warps. They are a rotation of the Procrustes
residuals around the Procrustes mean configuration. For the nonuniform
partial warps, the coefficients for the rotation are the principal
warps, applied first to the *x*-coordinates of the Procrustes
residuals, then to the *y*-coordinates and, for three-dimensional data,
the *z*-coordinates. Coefficients for the uniform partial warps are
produced by special formulas (see Bookstein's "Uniform" chapter, NATO volume
"white book").

**partial warps **- Partial warps are an auxiliary structure for the
interpretation of shape changes and shape variation in sets of landmarks.
Geometrically, partial warps are an orthonormal basis for a space tangent to
Kendall's shape space. Algebraically, the partial warps are eigenvectors of the
bending
energy matrix that describes the net local information in a deformation
along each coordinate axis. Except for the very largest-scale partial warp, the
one for uniform shape change, they have an approximate location and an
approximate scale.

**precision **- The closeness of repeated
measurements to the same value. See accuracy.

**preform space** - The space corresponding to centered objects, i. e.,
differences in location have been removed. It is of *k*(*p*-1)
dimensions.

**preshape space** - The space corresponding to figures that have been
centered and scaled but not rotated to alignment. It is of
*k*(*p*-1)-1 dimensions.

**principal axes and strains** - A change of one triangle into another, or
of one tetrahedron into another, can be modelled as an affine transformation
which can be parameterized by its effect on a circle or sphere. An affine
transformation takes circles into ellipses. The principal axes of the shape
change are the directions of the diameters of the circle that are mapped into
the major and minor axes of the ellipse. The principal strains of the change are
the ratios of the lengths of the axes of the ellipse to the diameter of the
circle. In the case of the tetrahedron, there are three principal axes, the axes
of the ellipsoid into which a sphere is deformed. One has the greatest principal
strain (ratio of axis length to diameter of sphere), one the least, and there is
a third perpendicular to both, having an intermediate principal strain.

**principal components analysis **- The eigenanalysis of
the sample covariance matrix. Principal components (PC's) can be defined as the
set of vectors that are orthogonal both with respect to the identity matrix and
the sample covariance matrix. They can also be defined sequentially: the first
is the linear combination with the largest variance of all those with
coefficients summing in square to 1; the second has the largest variance (when
normalized that way) of all that are uncorrelated with the first one; etc. One
way to compute principal components is to use a singular value decomposition.
Relative warps are principal components of partial warp scores. There is a lot
to be said about PC's; see any of the colored books.

**principal warps** - Principal warps are
eigenfunctions of the bending-energy matrix interpreted as actual warped
surfaces (thin-plate splines) over the picture of the original landmark
configuration. Principal warps are like the harmonics in a Fourier
analysis (for circular shape) or Legendre polynomials (for linear shape) in
that together they decompose the relation of any sample shape to the sample
average shape as a unique summation of multiples of eigenfunctions of bending
energy. They differ from these more familiar analogues in that there are only
*p*-3 of them for a set of *p* 2D landmarks (*p*-4 for 3D
data) - they form a finite series. Together with the uniform terms, the partial
warps, which are projections (shadows) of the principal warps, supply an
orthonormal basis for a space that is tangent to Kendall's shape space in the
vicinity of a mean form.

**Procrustes distance** - Approximately (see
Bookstein's "Combining" chapter, NATO volume "white book"), the square root of
the sum of squared differences between the positions of the landmarks in two
optimally (by least-squares) superimposed configurations at centroid size. This
is the distance that defines the metric
for Kendall's
shape space.

**Procrustes mean -** The shape that has the least summed squared
Procrustes distance to all the configurations of a sample; the best choice of
consensus configuration for most subsequent morphometric analyses (see
Bookstein's "Combining" chapter, NATO volume "white book").

**Procrustes methods** - A term for least-squares methods for estimating
nuisance parameters of the Euclidean similarity transformations. The adjective
"Procrustes" refers to the Greek giant who would stretch or shorten victims to
fit a bed and was first used in the context of superimposition methods by Hurley
and Cattell, 1962, The Procrustes program: producing a direct rotation to test
an hypothesized factor structure, *Behav. Sci. *7:258-262. Modern workers
have often cited Mosier (1939), a psychometrician, as the earliest known
developer of these methods. However, Cole (1996) reports that Franz Boas in 1905
suggested the "method of least differences" (ordinary Procrustes analysis) as a
means of comparing homologous points to address obvious problems with the
standard point-line registrations (Boas, 1905). Cole further points out that one
of Boas' students extended the method to the construction of mean configurations
from the superimposition of multiple specimens using either the standard
registrations of Boas' method (Phelps, 1932). The latter being essentially a
Generalized Procrustes Analysis. *References*:

Cole, T. M. 1996.
Historical note: early anthropological contributions to "geometric
morphometrics." Amer. J. Phys. Anthropol. 101:291-296.

Boas, F. 1905. The
horizontal plane of the skull and the general problem of the comparision of
variable forms. Science, 21:862-863.

Phelps, E. M. 1932. A critique of the
principle of the horizontal plane of the skull. Amer. J. Phys. Anthropol.,
17:71-98.

Mosier, 1939, Determining a simple structure when loadings for
certain tests are known, *Psychometrika *4:149-162.

**Procrustes residuals -** The set of vectors
connecting the landmarks of a specimen to corresponding landmarks in the
consensus configuration after a Procrustes fit. The sum of squared lengths of
these vectors is approximately the squared Procrustes distance between the
specimen and the consensus in Kendall's
shape space. The partial warp scores are an orthogonal rotation of the full
set of these residuals.

**Procrustes scatter -** A collection of forms all superimposed by
ordinary orthogonal Procrustes fit over one single consensus configuration that
is their Procrustes mean; a scatter of all the Procrustes residuals each
centered at the corresponding landmark of the Procrustes mean shape.

**Procrustes superimposition -** The construction of a two-form
superimposition by least squares using orthogonal or affine transformations.

**Red Book **- Bookstein, F. L., B. Chernoff, R.
Elder, J. Humphries, G. Smith, and R. Strauss. 1985. *Morphometrics in
Evolutionary Biology*. Special Publication No. 15, Academy of Natural
Sciences: Philadelphia.

See also Black Book, Blue Book, Orange Book, and Reyment's Black Book.

**reference configuration** - In the context of superimposition methods,
this is the configuration to which data are fit. It may be another specimen in
the sample but usually it will be the average (consensus) configuration for a
sample. The construction of two-point
shape coordinates does not involve a reference specimen, though the
intelligent choice of baseline for the construction usually does. The reference
configuration corresponds to the point of tangency of the linear tangent space
used to approximate Kendall's shape space. The mean configuration is usually
used as the reference in order to minimize distortions caused by this
approximation. When splines and warps are part of the analysis, the bending
energy that goes with them is computed using the geometry of the grand mean
shape, and the orthogonality that characterizes the partial warps is with
respect to this particular formula for bending energy.

There has been some controversery regarding the choice of reference. See the
following papers.

Rohlf, F. James. 1998. On applications of geometric
morphometrics to studies of ontogeny and phylogeny. Systematic Biology,
47:147-158.

Adams, D. C. and M. S. Rosenberg. 1998. Partial warps,
phylogeny, and ontogeny: a comment on Fink and Zelditch (1995). Systematic
Biology, 47:168-173.

Zelditch, M. L., W. L. Fink, D. L. Swiderski, and B. L.
Lundrigan. 1998. On applications of geometric morphometrics to studies of
ontogeny and phylogeny: a reply to Rohlf. Systematic Biology, 47:159-167.

Zelditch, M. L. and W. L. Fink. 1998. Partial warps, phylogeny and ontogeny:
a reply to Adams and Rosenberg. Systematic Biology, 47:345-348.

**regression **- A model for predicting one variable from another. Due to
Francis Galton, the word comes from the fact that when measurements of
offspring, whether peas or people, were plotted against the same measurements of
their parents, the offspring measurements "went back" or regressed towards the
mean.

**relative warps** - Relative warps are principal components of a
distribution of shapes in a space tangent to Kendall's shape space. They are the
axes of the "ellipsoid" occupied by the sample of shapes in a geometry in which
spheres are defined by Procrustes distance. Each relative warp, as a direction
of shape change about the mean form, can be interpreted as specifying multiples
of one single transformation, a transformation that can often be usefully drawn
out as a thin-plate spline. In a relative warps analysis, the parameter can be used to weight
shape variation by the geometric scale of shape differences. Relative warps can
be computed from Procrustes residuals or from partial warps (see Bookstein's
"Combining" chapter, NATO volume "white book").

**repeated median** - A median of medians. Repeated medians are used to
estimate some superimposition parameters in the resistant-fit methods. For
example, the resistant-fit rotation estimate is the median of the estimates
obtained for each landmark, which is, in turn, the median of angular differences
between the reference configuration and the configuration being fit of the line
segments defined using that landmark and the other *n*-1 landmarks.
Repeated medians are insensitive to larger subsets of extremely deviant values
than simple medians.

**residual** - The deviations of an observed value or vector of values
from some expectation, e.g., the differences between a shape and its prediction
by an allometric regression expressed in any set of shape coordinates.

**resistant-fit superimposition** - Superimposition methods that use
median- and repeated-median-based estimates of fitting parameters rather than
least-squares estimates. Resistant-fit procedures are less sensitive to subsets
of extreme values than those of comparable least-squares methods. As such, their
results may provide a simple description of differences in shape that are due to
changes in the positions of just a few landmarks. However, resistant-fit methods
lack the well-developed distributional theory associated with the least-squares
fitting methods. See Slice, 1996, NATO volume "white book".

**resolution** - The smallest scale distinguishable by a digitizing,
imaging, or display device.

**Reyment's Black Book** - Reyment, R. A.
1991. *Multidimensional Palaeobiology*. Pergamon Press: Oxford.

See also Black Book, Blue Book, Orange Book, and Red Book.

**ridge curve **- Ridge curves are curves on a surface along which the
curvature *perpendicular to the curve* is a local maximum. For instance on
a skull, the line of the jaw or the rim of an orbit. See Dean, 1996, NATO volume
"white book".

**rigid rotation **- An orthogonal transformation of a real vector space
with respect to the Euclidean distance metric. Such transformations leave
distances between points and angles between vectors unchanged. A principal
components analysis represents a rigid rotation to new orthogonal axes. A canonical
variates analysis does not.

**score** - A linear combination of an observed set of measured variables.
The coefficients for the linear combination are usually determined by some
matrix computation. Multivariate statistical findings in the form of coefficient
vectors can usually be more easily interpreted if scores are also shown case by
case, their scatters, their loadings (correlations with the original variables),
etc.

**shape** - The geometric properties of a configuration of points that are
invariant to changes in translation, rotation, and scale. In morphometrics, we
represent the shape of an object by a point in a space of shape variables, which
are measurements of a geometric object that are unchanged under similarity
transformations. For data that are configurations of landmarks, there is also a
representation of shapes per se, without any nuisance
parameters (position, rotation, scale), as single points in a space, Kendall's
shape space, with a geometry given by Procrustes
distance. Other sorts of shapes (e.g., those of outlines, surfaces, or
functions) correspond to quite different statistical spaces.

**shape coordinates** - In the past, any system of distance-ratios and
perpendicular projections permitting the exact reconstruction of a system of
landmarks by a rigid trusswork. Now, more generally, coordinates with respect to
any basis for the tangent space to Kendall's shape space in the vicinity of a
mean form: see Procrustes
residuals, partial
warp scores, two-point
shape coordinates.

**shape space** - A space in which the shape of a figure is represented by
a single point. It is of 2*p*-4 dimensions for 2-dimensional coordinate
data and 3*p*-7 dimensions for 3-dimensional coordinate data. See Kendall's
shape space.

**shape variable** - Any measure of the geometry of a biological form, or
the image of a form, that does not change under similarity transformations:
translations, rotations, and changes of geometric scale (enlargements or
reductions). Useful shape variables include angles, ratios of distances, and any
of the sets of shape coordinates that arise in geometric morphometrics.

**shear** - In two-dimensional problems, shape aspects
of any affine transformation can be diagrammed as a *pure shear*, a map
taking a square to a parallelogram of unchanged base segment and height. This is
a transformation that leaves one Cartesian coordinate, *y*, invariant and
alters the other by a translation that is a multiple of *y*: for instance,
what happens when you slide the top of a square sideways without altering its
vertical position or the length of the horizontal edges. The score for such a
translation, together with a separate score for change in the
horizontal/vertical ratio, supplies one orthonormal basis for the subspace of
uniform shape changes of two-dimensional data. Without the adjective "pure,"
geometric morphometricians usually use the word "shear" as an informal synonym
for "affine transformation," since any 2D uniform transformation can be drawn as
one if you wish.

In multivariate morphometrics, a somewhat different use of pure shear is in a
transformation of the "shape principal components" of an allometric analysis of
distances to be uncorrelated with within-group size (see refs). See Bookstein et
al. (1985) for a description of the method of shearing and the critique by Rohlf
& Bookstein (1987) of the technique as a method of size correction.
*References*:

Bookstein, F. L., B. Chernoff, R. Elder, J.
Humphries, G. Smith, and R. Strauss. 1985. *Morphometrics in Evolutionary
Biology*. Special Publication No. 15, Academy of Natural Sciences:
Philadelphia.

Rohlf, F. James and Bookstein, F. L. 1987. A comment on
shearing as a method for "size correction". Systematic Zoology, 36:356-367.

**similarity transformation** - A change of Cartesian coordinate system
that leaves all ratios of distances unchanged. The term proper or special
similarity group of similarities is sometimes used when the transformations do
not involve reflection. Similarities are arbitrary combinations of translations,
rotations, and changes of scale. Compare affine
transformation.

**singular value decomposition** - Any *m*x*n*
matrix **X** may be decomposed into three matrices **U**, **D**,
**V** (with dimensions *m*x*m*, *m*x*n*, and
*n*x*n*, respectively) in the form:
**X**=**UDV**^{t}, where the columns of **U** are orthogonal,
**D** is a diagonal matrix of singular values, and the columns of **V**
are orthogonal. The singular value decomposition of a variance-covariance matrix
**S** is written as **S**=**ELE**^{t}, where **L** is the
diagonal matrix of eigenvalues and **E** the matrix of eigenvectors.

**size measure** - In general, some measure of a form (i. e., an invariant
under the group of isometries) that scales as a positive power of the geometric
scale of the form. Interlandmark lengths are size measures of dimension one,
areas are size measures of dimension two, etc.

**space** - In statistics, a collection of objects or measurements of
objects, treated as if they were points in a plane, a volume, on the surface of
a sphere, or on any higher-dimensional generalization of these intuitive
structures. Examples are: Euclidean spaces, sample spaces, shape spaces, linear
vector spaces, etc.

**superimposition** - The transformation of one or more figures to achieve
some geometric relationship to another figure. The transformations are usually
affine transformations or similarities. They can be computed by matching two or
three landmarks, by least-squares optimization of squared residuals at all
landmarks, or in other ways. Sometimes informally referred to as a "fit" or
"fitting," e.g., a resistant fit.

**SVD -** See singular value
decomposition.

**T**** ^{2} statistic **- A multivariate generalization
of the univariate

**T**** ^{2}-test** - A test due to Hotelling for
comparing an observed mean vector to a parametric mean; or comparing the
difference between two mean vectors to a parametric difference (usually the zero
vector). If the observations are independently multivariate normal, then the

**tangent space** - Informally, if S is a
curving space and P a point in it, the tangent space to S at P is a linear space
T having points with the same "names" as the points in S and in which the metric
on S "in the vicinity of P" is very nearly the ordinary Euclidean metric on T.
One can visualize T as the projection of S onto a "tangent plane" "touching" at
P just like a map is a projection of the surface of the earth onto flat paper.

In geometric morphometrics, the most relevant tangent space is a linear vector space that is tangent to Kendall's shape space at a point corresponding to the shape of a reference configuration (usually taken as the mean of a sample of shapes). If variation in shape is small then Euclidean distances in the tangent space can be used to approximate Procrustes distances in Kendall's shape space. Since the tangent space is linear, it is possible to apply conventional statistical methods to study variation in shape. See Rohlf, 1996, NATO volume "white book", and Bookstein’s 1996 "Combining" chapter (NATO volume "white book").

**tensor **- An example of a tensor in morphometrics is
the representation of a uniform component of shape change as a transformation
matrix. The transformation matrix assigns to each vector in a starting (or
average) form a vector in a second form. A rigorous, general definition of a
tensor would be beyond the scope of this glossary, but a reasonably intuitive
characterization comes from Misner, Thorne, and Wheeler, *Gravitation*
(Freeman, 1973): a tensor is a "geometric machine" that is fed one or more
vectors in an arbitrary Cartesian coordinate system and that produces scalar
values (ordinary decimal numbers) that are independent of that coordinate
system. In morphometrics, these "numbers" will be ordinary geometric entities
like lengths, areas, or angles: anything that doesn't change when the coordinate
system changes. For the representation of a uniform component as a
transformation matrix, the "scalars" of the Misner-Thorne-Wheeler metaphor are
the lengths of the resulting vectors and the angles among them.

A different tensor representing the same uniform transformation is the
*relative metric tensor*, which you probably know as the ellipse of
principal axes and principal strains. This tensor produces the necessary
numerical invariants (distances in the second form as a function of coordinates
on the first form) directly. Other tensors include the *metric tensor* of a
curving surface which expresses distance on the surface as a function of the
parameters in which surface points are expressed and the *curvature tensor*
of the same surface which expresses the way in which the surface "falls away"
from its tangent plane at any point.

**thin-plate spline** - In continuum mechanics, a thin-plate spline models
the form taken by a metal plate that is constrained at some combination of
points and lines and otherwise free to adopt the form that minimizes bending
energy. (The extent of bending is taken as so small that elastic energy -
stretches and shrinks in the plane of the original plate - can be neglected.)
One particular version of this problem - an infinite, uniform plate constrained
only by displacements at a set of discrete points - can be solved algebraically
by a simple matrix inversion. In that form, the technique is a convenient
general approach to the problem of surface interpolation for computer graphics
and computer-aided design. In morphometrics, the same interpolation (applied
once for each Cartesian coordinate) provides a unique solution to the
construction of D'Arcy Thompson-type deformation grids for data in the form of
two landmark configurations.

**traditional morphometrics** - Application
of multivariate statistical methods to arbitrary collections of size or shape
variables such as distances and angles. "Traditional morphometrics" differs from
the geometric morphometrics discussed here in that even though the distances or
measurements are defined to record biologically meaningful aspects of the
organism, but the geometrical relationships between these measurements are not
taken into account. Traditional morphometrics makes no reference to Procrustes
distance or any other aspect of Kendall's
shape space. See multivariate
morphometrics and geometric
morphometrics

**transformation **- In general, a replacement of landmark coordinates by
another set purporting to pertain to the same landmarks. For example, a matrix
of landmark coordinates might be transformed by multiplication by another matrix
to produce a new set of coordinates that have been scaled, rotated, and
translated with respect to the original data.

**two-point shape coordinates -** A
convenient system of shape coordinates, originally Francis Galton's,
rediscovered by Bookstein, consisting (for two-dimensional data) of the
coordinates of landmarks 3, 4, ... after forms are rescaled and repositioned so
that landmark 1 is fixed at (0,0) and landmark 2 is fixed at (1,0) in a
Cartesian coordinate system. Also referred to as Bookstein coordinates or
Bookstein's shape coordinates.

**Type I landmark** - A mathematical point whose
claimed homology from case to case is supported by the strongest evidence, such
as a local pattern of juxtaposition of tissue types or a small patch of some
unusual histology.

**Type II landmark** - A mathematical point
whose claimed homology from case to case is supported only by geometric, not
histological, evidence: for instance, the sharpest curvature of a tooth.

**Type III landmark** - A landmark having at
least one deficient coordinate, for instance, either end of a longest diameter,
or the bottom of a concavity. Type III landmarks characterize more than one
region of the form. The multivariate machinery of geometric morphometrics
permits them to be treated as landmark points in some analyses, but the
deficiency they embody must be kept in mind in the course of any geometric or
biological interpretation.

**unbiased estimator** - An estimator, , that has as its
expected value the parametric value, *q*, it is intended to estimate: . See consistent
estimator and asymptotically
unbiased estimator.

**uniform shape component** - That
part of the difference in shape between a set of configurations that can be
modeled by an affine
transformation. Once a metric
is supplied for shape space one can ascertain which such transformation takes a
reference form closest to a particular target form. For the Procrustes metric
(the geometry of Kendall's shape space), that uniform transformation is computed
by a formula based in Procrustes
residuals or by another based in two-point
shape coordinates (see Bookstein's 1996 "Uniform" chapter, NATO volume
"white book"). Together with the partial warps, the uniform component defined in
this way supplies an orthonormal basis for all of shape space in the vicinity of
a mean form. In this setting, the uniform shape component may also be
interpreted as the projection of a shape difference (between two group means, or
between a mean and a particular specimen) into the plane (or hyperplane for data
of dimension greater than two) through that mean form and all nearby forms
related to it by affine transformations. For descriptive purposes, the uniform
component is parameterized not by a vector, like the partial warps, but by a
representation as a tensor,
in terms of sets of shears and dilations with respect to a fixed, orthogonal set
of Cartesian axes.

**weight matrix **, W matrix -The matrix of partial warp scores, together
with the uniform component, for a sample of shapes. The weight matrix is
computed as a rotation of the Procrustes-residual shape coordinates; like them,
they are a set of shape coordinates for which the sum of squared differences is
the squared Procrustes distance between any two specimens.

**Wright factor analysis **- A version of factor analysis, due to Sewall
Wright, in which a path model is used to describe the relation between the
measured variables and the factors of interest. It is usually exploratory, in
that one fits a simple one factor model iteratively to maximally explain the
correlations among variables, and then proceeds to find additional factors to
fit to the residuals, and so on until the data is adequately fit. See the Orange
book for examples and discussion of the application of this approach to the
analysis of size and group factors for morphometric data.

**z -** Notation for complex numbers in two-dimensional Procrustes
formulas.

Work on this glossary by Slice and Rohlf was been supported by grants BSR-89-18630 and DEB-93-17572 from the Systematic Biology Program of the National Science Foundation.

Bookstein's work in morphometrics is supported by NIH grants DA-09009 and GM-37251. The former of these is jointly supported by the National Institute on Drug Abuse, the National Institute of Mental Health, and the National Institute on Aging as part of the Human Brain Project.

This is publication number 944 from the Graduate Studies in Ecology and Evolution, State University of New York at Stony Brook.

Bookstein, F. L. 1991. Morphometric tools for landmark data. Geometry and biology. Cambridge University Press: New York.

Bookstein, F. L., B. Chernoff, R. Elder, J. Humphries, G. Smith, and R. Strauss. 1985. Morphometrics in evolutionary biology: The geometry of size and shape change, with examples from fishes. Academy of Natural Sciences of Philadelphia Special Publication 15.

Bookstein, F. L., and W. D. K. Green. 1993. A feature space for edgels in images with landmarks. Journal of Mathematical Imaging and Vision 3:231–261.

Hurley and Cattell. 1962. The Procrustes program: Producing a direct rotation to test an hypothesized factor structure. Behavior Science 7:258–262.

Lele, S., and J. T. Richtsmeier. 1991. Euclidean distance matrix analysis: A coordinate free approach for comparing biological shapes using landmark data, American Journal of Physical Anthropology 86:415-428.

Marcus, L. F., E. Bello, and A. García-Valdecasas (*eds.*). 1993.
Contributions to morphometrics. Monografias del Museo Nacional de Ciencias
Naturales 8, Madrid.

Misner, C., K. Thorne, and J. Wheeler. 1973. Gravitation. Freeman: New York.

Mosier, C. I. 1939. Determining a simple structure when loadings for certain tests are known. Psychometrika 4:149–162.

Reyment, R. A. 1991. Multidimensional palaeobiology. Pergamon Press: Oxford.

Reyment, R. A., and K. G. Jöreskog. 1993. Applied factor analysis in the natural sciences. Cambridge University Press: Cambridge.

Rohlf, F. J., and F. L. Bookstein (eds.). 1990. Proceedings of the Michigan morphometrics workshop. University of Michigan Museum of Zoology Special Publication 2.

Revised November 24, 1998 by F. James Rohlf