## A Glossary for Geometric Morphometrics (part 2)

Continued from part 1.

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-1h, 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, dij, of two points, i and j, satisfying the following conditions: , if then , , and . The last condition is known as the "triangle inequality." The value of the function is referred to as the "distance" between the two points. A space is called "semimetric" if the distance between two points i and j can be equal to zero when .

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 xtPy=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.

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 2p-4 dimensions for 2-dimensional coordinate data and 3p-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 mxn matrix X may be decomposed into three matrices U, D, V (with dimensions mxm, mxn, and nxn, respectively) in the form: X=UDVt, 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=ELEt, 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.

T2 statistic - A multivariate generalization of the univariate t2 statistic. It is the square of the ratio of the group mean difference to the standard error of that difference. Used in the T2-test.

T2-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 T2-test may be used to test null hypotheses using the F-distribution. T2 is also closely related to Mahalanobis D2. See Marcus (Black Book).

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.

## Acknowledgments

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.

## References

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.

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