A Glossary for Geometric Morphometrics

Dennis E. Slice
Department of Ecology and Evolution
State University of New York at Stony Brook, Stony Brook, NY 11794-5245

Fred L. Bookstein
Institute of Gerontology
University of Michigan. Ann Arbor, MI 48109-2007

Leslie F. Marcus†

F. James Rohlf
Department of Ecology and Evolution and Department of Anthropology.
State University of New York at Stony Brook, Stony Brook, NY 11794-5245

This glossary provides definitions for terms, concepts, and methods frequently encountered in morphometric literature and discussions. It includes entries for technical terms with more-or-less special meaning in shape analysis and biological morphometrics (e.g., preshape, warps, anisotropy) and some of the casual jargon that may be completely foreign to newcomers to the field (e.g., books of various color - Red, Blue, Orange, and Black). Many definitions provide the general idea behind each entry instead of a technically or mathematically rigorous treatment. As such, they are intended to give readers an intuitive understanding of a particular entry that will allow them to follow the main ideas in the literature without becoming unduly distracted, at first, with technical details. Unless otherwise indicated, the following general notation has been used: n - number of specimens, p - number of points/landmarks, k - number of dimensions, a superscript t will refer to the transpose of a matrix (e.g., At, but that may not be displayed properly by all WWW browsers). Members of the morphometrics community, especially the subscribers to the MORPHMET electronic mailing list, have helped greatly in the selection of terms to be included in the glossary.

Note: many of the mathematical symbols and equations are patched into this file as images since HTML (the language used to prepare these WWW pages) does not support mathematical symbols. For this reason, many symbols will not appear on text-only WWW browsers and may not line-up well with the rest of the text. Superscripts and subscripts will not display properly on all WWW browsers.

† Deceased.

- In a relative warps analysis, this is the exponent used to rescale partial warps before computing their principal components, the relative warps (see Rohlf's chapter in the Black Book)). Scale invariant multivariate analyses using rescaled principal warp scores, such as canonical variates analysis, are not affected by the choice of (see Rohlf, 1996, NATO volume "white book").

- The Kronecker tensor product or direct product. The Kronecker tensor product of matrices X and Y, written as X Y, results in a large matrix formed by taking all possible products of the elements of X and those of Y. For example, if X and Y are 2x2 then X Y results in a 4x4 matrix:

accuracy - The closeness of a measurement or estimate to its true value. See precision.

affine superimposition - A superimposition for which the associated transformations are all affine. See affine transformation.

affine transformation - A transformation for which parallel lines remain parallel. Affine transformations of the plane take squares into parallelograms and take circles into ellipses of the same shape. Affine transformations of a 3-dimensional space take cubes into parallelopipeds (sheared bricks) and spheres into ellipsoids all of the same shape. Similar results are produced in higher dimensional spaces. Equivalent to "uniform transformation".

As far as form is concerned (that is, ignoring translation and rotation), any affine transformation can be diagrammed as a pure strain taking a square to a rectangle on the same axes. In studies of shape, where scale is ignored as well, the picture is the same but now the sum of the squares of the axes is unchanging. Still ignoring scale (that is, as far as shape is concerned), any affine transformation can be also diagrammed as a pure shear taking a square into a parallelogram of unchanged base segment and height. This diagram of shear came into morphometrics via an application to principal components analysis somewhat before it was applied to landmark-based shape (see shear, Kendall's shape space, and tangent space).

allometry - Any change of shape with size. It describes any deviation of the bivariate relation from the simple functional form y/x = c, where c is a constant and x and y are size measures in units of the same dimension. See Klingenberg, 1996, NATO volume "white book".

anisotropy - Anisotropy is a descriptor of one aspect of an affine transformation. In two dimensions, this is the ratio of the axes of the ellipse into which a circle is transformed by an affine transformation. In general, it is the maximum ratio of extension of length in one direction to extension in a perpendicular direction.

asymptotically unbiased estimator - An estimator, , with an expected value that converges in probability on the parametric value it is estimating, , as sample size goes to infinity: as . See unbiased estimator and consistent estimator.

baseline - For a system of two-point shape coordinates for landmarks in a plane, the baseline is the line connecting the pair of landmarks that are assigned to fixed locations (0,0) and (1,0) in the construction. In general, baselines work better if they are closely aligned with the long axis of the mean landmark shape and pass near the centroid of that mean shape (see the Orange Book).

bending energy - Bending energy is a metaphor borrowed for use in morphometrics from the mechanics of thin metal plates. Imagine a configuration of landmarks that has been printed on an infinite, infinitely thin, flat metal plate, and suppose that the differences in coordinates of these same landmarks in another picture are taken as vertical displacements of this plate perpendicular to itself, one Cartesian coordinate at a time. The bending energy of one of these out-of-plane "shape changes" is the (idealized) energy that would be required to bend the metal plate so that the landmarks were lifted or lowered appropriately.

While in physics bending energy is a real quantity, measured in appropriate units (g cm2 sec-2), there is an alternate formula that remains meaningful in morphometrics: bending energy is proportional to the integral of the summed squared second derivatives of the "vertical" displacement - the extent to which it varies from a uniform tilt. The bending energy of a shape change is the sum of the bending energies that apply to any two perpendicular coordinates in which the metaphor is evaluated. 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. Such quantities should not be interpreted as measures of dissimilarity (e.g., taxonomic or evolutionary distance) between two forms.

bending energy matrix - The formula for bending energy (see above) - the formula whose value is proportional to that integral of those summed squared second derivatives - is a quadratic form (usually written Lk-1) determined by the coordinates of the landmarks of the reference form. That is, if h is a vector describing the heights of a plate above a set of landmarks, then bending energy is htLk-1h. In morphometrics, the bending energy of a general transformation is the sum xtLk-1x +ytLk-1y of the bending energy of its horizontal x-component, modeled as a "vertical" plate, plus the bending energy of its vertical y-component, modeled similarly as a "vertical" plate.

biplot - A single diagram that represents two separate scatterplots on the same pair of axes. One scatter is of some pair of columns of the matrix U of the singular value decomposition of a matrix S, and the other scatter is of the matching pair of columns of V. When S is a centered data matrix, the effect is to plot principal component loadings and scores on the same diagram. See Marcus (Black Book) for an in depth discussion.

Black Book - Marcus, L. F., E. Bello, A. García-Valdecasas (eds.). 1993. Contributions to Morphometrics. Museo Nacional de Ciencias Naturales Monografias: Madrid.

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

Blue Book - Rohlf, F. J. and F. L. Bookstein (eds.). 1990. Proceedings of the Michigan Morphometrics Workshop. Special Publication No. 2, University of Michigan Museum of Zoology: Ann Arbor.

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

Bookstein coordinates - See two-point shape coordinates.

canonical - A canonical description of any statistical situation is a description in terms of extracted vectors that have especially simple ordered relationships. For instance, a canonical correlations analysis describes the relation between two lists of variables in terms of two lists of linear combinations that show a remarkable pattern of zero correlations. Each score (linear combination) from either list is correlated with no other combination from its list and with only one score from the other list.

canonical correlation analysis - A multivariate method for assessing the associations between two sets of variables within a data set. The analysis focuses on pairs of linear combinations of variables (one for each set) ordered by the magnitude of their correlations with each other. The first such pair is determined so as to have the maximal correlation of any such linear combinations. Subsequent pairs have maximal correlation subject to the constraint of being orthogonal to those previously determined.

canonical variates analysis - A method of multivariate analysis in which the variation among groups is expressed relative to the pooled within-group covariance matrix. Canonical variates analysis finds linear transformations of the data which maximize the among group variation relative to the pooled within-group variation. The canonical variates then may be displayed as an ordination to show the group centroids and scatter within groups. This may be thought of as a "data reduction" method in the sense that one wants to describe among group differences in few dimensions. The canonical variates are uncorrelated, however the vectors of coefficients are not orthogonal as in Principal Component Analysis. The method is closely related to multivariate analysis of variance (MANOVA), multiple discriminant analysis, and canonical correlation analysis. A critical assumption is that the within-group variance-covariance structure is similar, otherwise the pooling of the data over groups is not very sensible.

Centroid Size - Centroid Size 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. Centroid Size is used in geometric morphometrics because it is approximately uncorrelated with every shape variable when landmarks are distributed around mean positions by independent noise of the same small variance at every landmark and in every direction. Centroid Size is the size measure used to scale a configuration of landmarks so they can be plotted as a point in Kendall's shape space. The denominator of the formula for the Procrustes distance between two sets of landmark configurations is the product of their Centroid Sizes.

cluster analysis - A method of analysis that represents multivariate variation in data as a series of sets. In biology, the sets are often constructed in a hierarchical manner and shown in the form of a tree-like diagram called a dendrogram.

coefficient - A coefficient, in general, is a number multiplying a function. In multivariate data analysis, usually the "function" is a variable measured over the cases of the analysis, and the coefficients multiply these variable values before we add them up to form a score. A coefficient is not the same as a loading.

complex numbers Complex numbers are an algebraic way of coding points in the ordinary Euclidean plane so that translation (shift of position) corresponds to the addition of complex numbers and both rescaling (enlargement or shrinking) and rotation correspond to multiplication of complex numbers. In this system of notation, invented by Gauss, the x-axis is identified with the "real numbers" (ordinary decimals numbers) and the y-axis is identified with "imaginary numbers" (the square roots of negative numbers). When you multiply points on this axis by themselves according to the rules, you get negative points on the "real" axis just defined. Many operations on data in two dimensions can be proved valid more directly if they are written out as operations on complex numbers.

consensus configuration - A single set of landmarks intended to represent the central tendency of an observed sample for the production of superimpositions, of a weight matrix, or some other morphometric purpose. Often a consensus configuration is computed to optimize some measure of fit to the full sample: in particular, the Procrustes mean shape is computed to minimize the sum of squared Procrustes distances from the the consensus landmarks to those of the sample.

consistent estimator - An estimator, , that converges in probability on the parametric value it is estimating, , as sample size goes to infinity: for any positive . Asymptotically unbiased estimators are consistent estimators if their variance goes to zero as sample size goes to infinity. See unbiased estimator.

coordinates - A set of parameters that locate a point in some geometrical space. Cartesian coordinates, for instance, locate a point on a plane or in physical space by projection onto perpendicular lines through one single point, the origin. The elements of any vector may be thought of as coordinates in a geometric sense.

correlation - Relation between two or more variables. Frequently the word is used for Pearson's product-moment correlation which is the covariance divided by the product of the standard deviations, . This correlation coefficient is +1 or -1 when all values fall on a straight line, not parallel to either axis. However, there are also Kendall, Spearman, tetrachoric, etc. correlations which measure other aspects of the relation between two variables.

covariance - A measure of the degree to which two variables vary together. Computed as for two variables X and Y in a sample of size n. See correlation.

covariant - A covariant of a particular shape change is a shape variable whose gradient vector as a function of changes in any complete set of shape coordinates lies precisely along the change in question.

For transformations of triangles, the relation between invariants and covariants is a rotation by 90 degrees in the shape-coordinate plane. For more than three landmarks, a given transformation has only one direction of covariants, but a full plane (four landmarks) or hyperplane (five or more landmarks) of invariants (see the Orange Book). See invariant.

curved space - A space with coordinates and a distance function such that the area of circles, volume of spheres, etc. are not proportional to the appropriate power of the radius, e. g., Kendall's shape space. In curved spaces, the usual intuitions about what "straight lines" can be expected to do will be faulty. For instance, corresponding to every triangular shape in Kendall's shape space, there is another that is "as far from it as possible," just like there is a point on the surface of the earth as far as possible from where you now sit.

D - See 1) generalized distance or 2) fractal dimension.

D2 - Squared Mahalanobis, or generalized, distance.

deficient coordinate - In addition to landmark locations, a digitizer can be used to supply information of other sorts. For example, a point can be used to encode part of the information about a curving arc by identifying the spot at which the arc lies farthest from some other image structure (perhaps another such curving arc). The null model of independent Gaussian noise does not apply to position along the tangent direction of the curve that is digitized in this way, and so that Cartesian coordinate is "deficient." The usual model of independent Gaussian noise is inapplicable in principle for such points. See Type III landmark.

degrees of freedom - Given a set of parameters estimated from the data, the "degrees of freedom" of some statistic is the number of independent observations required to compute the statistic. For example, the variance has n-1 degrees of freedom because only n-1 of the observations are needed for its computation given the sample mean. The missing observation can be computed as .

dilation - Increase of length in a particular direction, or along a particular interlandmark segment.

discriminant analysis - A broad class of methods concerned with the development of rules for assigning unclassified objects/specimens to previously defined groups. See discriminant function.

discriminant function - A discriminant function is used to assign an observation to one of a set of groups. Linear discriminant functions take a vector of observations from a specimen and multiplies it by a vector of coefficients to produce a score which can be used to classify the specimen as belonging to one or another predefined group. See discriminant analysis.

distance - This term has several meanings in morphometrics; it should never be used without a prefixed adjective to qualify it, e.g., Euclidean distance, Mahalanobis distance, Procrustes distance, taxonomic distance.

edgel - An extension of the notion of landmark to include partial information about a curve through the landmark. An edgel specifies rotation of a direction through a landmark, extension along a direction through a landmark, or both. The formula for thin-plate splines on landmarks can be extended to encompass data about edgels as well. They are intended eventually to circumvent any need for deficient coordinates in multivariate morphometric analysis. See Little (1996, NATO volume "white book") and Bookstein and Green, 1993, A feature space for edgels in images with landmarks, Journal of Mathematical Imaging and Vision 3: 231-261.

EDMA - See euclidean distance matrix analysis.

eigenshapes - Principal components for outline data. An eigenshape analysis begins with the selection of a distance function between pairs of outlines. At the end one gets "eigenshapes," which have the properties of principal component vectors (uncorrelated, describing the sample in decreasing order of variance) and also are outline shapes themselves, so that the scores for each specimen of the sample can be combined to produce a new outline shape that approximates it in some possibly useful way. Eigenshapes apply to curves as relative warps apply to landmark shape. See the chapter by Lohmann and Schweitzer in the Blue Book and that by Sampson, 1996, NATO volume "white book".

eigenvalues - Eigenvalues, , are the diagonal elements of the diagonal matrix in the equation: . In the common data analysis case, S is a symmetrical variance-covariance matrix, E is a matrix of eigenvectors, , and . The order of the columns of E and is arbitrary, but by convention they are usually sorted from largest to smallest eigenvalue. See eigenvectors and singular value decomposition.

eigenvectors - In the equation given to define eigenvalues, E contains the eigenvectors. In the common data analysis case, E is an orthonormal matrix (i. e., EtE=I and EEt=I). When sorted by descending eigenvalues, the first eigenvector is that linear combination of variables that has the greatest variance. The second eigenvector is the linear combination of variables that has the greatest variance of such combinations orthogonal to the first, and so on. See eigenvalues and singular value decomposition.

elliptic Fourier analysis - A type of outline analysis in which differences in x and y (and possibly z) coordinates of an outline are fit separately as a function of arc length by Fourier analysis. The chapter by Rohlf in the blue book provides an overview of various methods of fitting curves to outline data.

Euclidean distance - Defined as: for coordinates of points xl and xm on the axes of a k-dimensional space. This can be expressed in matrix notation as , where xl and xm are the 1xk row vectors of the coordinates of points l and m in some coordinate system.

euclidean distance matrix analysis --EDMA. A method for the statistical analysis of full matrices of all interlandmark distances, averaging elementwise within samples, and then comparing those averages between samples by computing the ratios of corresponding mean distances. See 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.

Euclidean space - A space where distances between two points are defined as Euclidean distances in some system of coordinates.

factor analysis - Factor analysis is a multivariate technique for describing a set of measured variables in terms of a set of causal or underlying variables. A factor model can be characterized in terms of path diagrams to show relations between measured variables and factors. See the chapter by Marcus in the Blue Book and Reyment and Joreskog, 1993, Applied Factor Analysis in the Natural Sciences, Cambridge University Press: Cambridge, United Kingdom.

FESA - See finite element scaling analysis.

fiber - The set of preshapes (configurations that have been centered at the origin and scaled to unit centroid size) that differ only by a rotation. It is the path, through preshape space, followed by a centered and scaled configuration under all possible rotations.

figure - A representation of an object by the coordinates of a specified set of points, the landmarks.

figure space - The 2p- or 3p-space of figures, i. e., the original coordinate data vectors.

finite element scaling analysis - Without the word "scaling," finite element analysis is a computational system for continuum mechanics that estimates the deformation (fully detailed changes of position of all component particles) that are expected to result from a specified pattern of stresses (forces) upon a mechanical system. As applied in morphometrics, FESA solves the inverse problem of estimating the strains representing the hypothetical forces that deformed one specimen into another. These results are a function of the "finite elements" into which the space between the landmarks is subdivided. FESA can be compared with the thin-plate spline, which interpolates a set of landmark coordinates under an entirely different set of assumptions.

form - In morphometrics, we represent the form of an object by a point in a space of form variables, which are measurements of a geometric object that are unchanged by translations and rotations. If you allow for reflections, forms stand for all the figures that have all the same interlandmark distances. A form is usually represented by one of its figures at some specified location and in some specified orientation. When represented in this way, location and orientation are said to have been "removed."

form space - The space of figures with differences due to location and orientation removed. It is of 2p-3 dimensions for two-dimensional coordinate data and 3p-6 dimensions for three-dimensional coordinate data.

Fourier analysis - In morphometrics, the decomposition of an outline into a weighted sum of sine and cosine functions. The chapter by Rohlf in the Blue Book provides an overview of this and other methods of analyzing outline data.

fractal dimension - D. A measure of the complexity of a structure assuming a consistent pattern of self-similarity (structural complexity at smaller scales is mathematically indistiguishable from that at larger-scales) over all scales considered. See the chapter by Slice in the Black Book.

generalized distance - D. A synonym for Mahalanobis distance. Defined by the equation for two row vectors xi and xj for two individuals, and p variables as: , where S is the pxp variance-covariance matrix. It takes into consideration the variance and correlation of the variables in measuring distances between points, i. e., differences in directions in which there is less variation within groups are given greater weight than are differences in directions in which there is more variation.

generalized superimposition - The superimposition of a set of configurations onto their consensus configuration. The fitting may involve least-squares, resistant-fit, or other algorithms and may be strictly orthogonal or allow affine transformations.

geodesic distance - The length of the shortest path between two points in a suitable geometric space (one for which curving paths have lengths). On a sphere, it is the distance between two points as measured along a great circle.

geometric morphometrics - Geometric morphometrics is a collection of approaches for the multivariate statistical analysis of Cartesian coordinate data, usually (but not always) limited to landmark point locations. The "geometry" referred to by the word "geometric" is the geometry of Kendall's shape space: the estimation of mean shapes and the description of sample variation of shape using the geometry of Procrustes distance. The multivariate part of geometric morphometrics is usually carried out in a linear tangent space to the non-Euclidean shape space in the vicinity of the mean shape.

More generally, it is the class of morphometric methods that preserve complete information about the relative spatial arrangements of the data throughout an analysis. As such, these methods allow for the visualization of group and individual differences, sample variation, and other results in the space of the original specimens.

great circle - A circle on a sphere with a diameter equal to that of the sphere. The shortest path connecting two points on the surface of a sphere lies along the great circle passing through the points. See geodesic distance.

homology - The notion of homology bridges the language of geometric morphometrics and the language of its biological or biomathematical applications. In theoretical biology, only the explicit entities of evolution or development, such as molecules, organs or tissues, can be "homologous." Following D'Arcy Thompson, morphometricians often apply the concept instead to discrete geometric structures, such as points or curves, and, by a further extension, to the multivariate descriptors (e.g., partial warp scores) that arise as part of most multivariate analyses. In this context, the term "homologous" has no meaning other than that the same name is used for corresponding parts in different species or developmental stages. To declare something "homologous" is simply to assert that we want to talk about processes affecting such structures as if they had a consistent biological or biomechanical meaning. Similarly, to declare an interpolation (such as a thin-plate spline) a "homology map" means that one intends to refer to its features as if they had something to do with valid biological explanations pertaining to the regions between the landmarks, about which we have no data.

Hotelling's T2 - See T2 statistic.

hyperplane - A k-1 dimensional subspace of a k-dimensional space. A hyperplane is typically characterized by the vector to which it is orthogonal.

hyperspace - A space of more than three dimensions.

hypersphere - A generalization of the idea of a sphere to a space of greater than three dimensions.

hypervolume - A generalization of the idea of volume to a space of more than three dimensions.

invariant - An invariant, generally speaking, is a quantity that is unchanged (even though its formula may have changed) when one changes some inessential aspect of a measurement. For instance, Euclidean distance is an invariant under translation or rotation of one's coordinate system, and ratio of distances in the same direction is an invariant under affine transformations. In the morphometrics of triangles, the invariants of a particular transformation are the shape variables that do not change under that transformation (see the Orange Book). See covariant.

isometry - An isometry is a transformation of a geometric space that leaves distances between points unchanged. If the space is the Euclidean space of a picture or an organism, and the distances are distances between landmarks, the isometries are the Euclidean translations, rotations, and reflections. If the distances are Procrustes distances between shapes, the isometries (for the simplest case, landmarks in two dimensions) are the rotations of Kendall's shape space. For triangles, these can be visualized as ordinary rotations of Kendall's "spherical blackboard."

isotropic - Invariant with respect to direction. Isotropic errors have the same statistical distribution in all directions implying equal variance and zero correlation between the original variables (e.g., axis coordinates).

Kendall's shape space - The fundamental geometric construction, due to David Kendall, underlying geometric morphometrics. Kendall's shape space provides a complete geometric setting for analyses of Procrustes distances among arbitrary sets of landmarks. Each point in this shape space represents the shape of a configuration of points in some Euclidean space, irrespective of size, position, and orientation. In shape space, scatters of points correspond to scatters of entire landmark configurations, not merely scatters of single landmarks. Most multivariate methods of geometric morphometrics are linearizations of statistical analyses of distances and directions in this underlying space.

Kronecker product - See .

For additional entries, see part 2.

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