Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.     

Optimized 3D coordinate reconstruction from paired stereographs using a calibrated phantom

A refinement to a previously described three-dimensional reconstruction algorithm based on point identification in calibrated non-orthogonal radiograms (stereo-pairs) is described. The modification involves a computation of the focal point magnitude of the point in three dimensions, analogous to focusing in two dimensions, as well as the most likely location of the target point in 3-space; the focal point magnitude may be thought of as the precision of the point identification. Multiple observer studies of the same stereopair can be used to estimate three-dimensional reconstruction accuracy by providing an average location and a mean distance from average. Both measures are useful parameters for initial selection of bone landmark references and for error propagation studies.     

Morphology of the species Hyphessobrycon, heterorhabdus group (Characiformes: Characidae) in Colombia

Hyphessobrycon is the most numerous and morphologically complex genus of Characidae, and includes 18 species reported for Colombia, from which thirteen belong to the heterorhabdus-group different methods have been proposed for species identification within this genus. This study used these species to undertake a morphogeometric analysis by the Box Truss and Thin Plate Splin (TPS) methods; 13 homologous landmarks type I and three type II were used. The result of cluster analysis indicated that these species are represented by two big groups: robust and thin. The uniform and non uniform components, and the principal warps (WP) and partial warps (Wparc), described the shape changes related to body depth, involving relative displacements of the appendicular skeleton and the cephalic region. All species were characterized by allometric growth with the exception of Hyphessobrycon mavro, who presented an isometry between the landmarks 6-7. Two new measures are proposed for species discrimination in the group (snout-supraoccipital spine distance and supraoccipital spine-pelvic fin).     

TNT version 1.5, including a full implementation of phylogenetic morphometrics

Version 1.5 of the computer program TNT completely integrates landmark data into phylogenetic analysis. Landmark data consist of coordinates (in two or three dimensions) for the terminal taxa; TNT reconstructs shapes for the internal nodes such that the difference between ancestor and descendant shapes for all tree branches sums up to a minimum; this sum is used as tree score. Landmark data can be analysed alone or in combination with standard characters; all the applicable commands and options in TNT can be used transparently after reading a landmark data set. The program continues implementing all the types of analyses in former versions, including discrete and continuous characters (which can now be read at any scale, and automatically rescaled by TNT). Using algorithms described in this paper, searches for landmark data can be made tens to hundreds of times faster than it was possible before (from T to 3T times faster, where T is the number of taxa), thus making phylogenetic analysis of landmarks feasible even on standard personal computers.     

A New Method for the Analysis of Soft Tissues with Data Acquired under Field Conditions

Analyzing soft-tissue structures is particularly challenging due to the lack of homologous landmarks that can be reliably identified across time and specimens. This is particularly true when data are to be collected under field conditions. Here, we present a method that combines photogrammetric techniques and geometric morphometrics methods (GMM) to quantify soft tissues for their subsequent volumetric analysis. We combine previously developed methods for landmark data acquisition and processing with a custom program for volumetric computations. Photogrammetric methods are a particularly powerful tool for field studies as they allow for image acquisition with minimal equipment requirements and for the acquisition of the spatial coordinates of points (anatomical landmarks or others) from these images. For our method, a limited number of homologous landmarks, i.e., points that can be found on any specimen independent of space and time, and further distinctive points, which may vary over time, space and subject, are identified on two-dimensional photographs and their three-dimensional coordinates estimated using photogrammetric methods. The three-dimensional configurations are oriented by the spatial principal components (PCs) of the homologous points. Crucially, this last step orients the configuration such that x and y-information (PC1 and PC2 coordinates) constitute an anatomically-defined plane with the z-values (PC3 coordinate) in the direction of interest for volume computation. The z-coordinates are then used to estimate the volume of the tissue. We validate our method using a physical, geometric model of known dimensions and physical (wax) models designed to approximate perineal swellings in female macaques. To demonstrate the usefulness and potential of our method, we use it to estimate the volumes of Barbary macaque sexual swellings recorded in the field with video images. By analyzing both the artificial data and real monkey swellings, we validate our method''s accuracy and illustrate its potential for application in important areas of biological research.     

Principal components for allometric analysis

Logarithmic bivariate regression slopes and logarithmic principal component coefficient ratios are two methods for estimating allometry coefficients corresponding to a in the classic power formula Y = BXa. Both techniques depend on high correlation between variables. Interpretation is logically limited to the variables included in analysis. Principal components analysis depends also on relatively uniform intercorrelations; given this, it serves satisfactorily as a method for summarizing many bivariate combinations. Unmodified major principal component coefficients cannot represent scaling to body weight; rather, they represent scaling to a composite size vector which usually is highly correlated with body size or weight but has an unspecified allometry. Thus, the concepts of proportionality and of isometry must be kept distinct.     

Three-dimensional space: locomotory style explains memory differences in rats and hummingbirds

While most animals live in a three-dimensional world, they move through it to different extents depending on their mode of locomotion: terrestrial animals move vertically less than do swimming and flying animals. As nearly everything we know about how animals learn and remember locations in space comes from two-dimensional experiments in the horizontal plane, here we determined whether the use of three-dimensional space by a terrestrial and a flying animal was correlated with memory for a rewarded location. In the cubic mazes in which we trained and tested rats and hummingbirds, rats moved more vertically than horizontally, whereas hummingbirds moved equally in the three dimensions. Consistent with their movement preferences, rats were more accurate in relocating the horizontal component of a rewarded location than they were in the vertical component. Hummingbirds, however, were more accurate in the vertical dimension than they were in the horizontal, a result that cannot be explained by their use of space. Either as a result of evolution or ontogeny, it appears that birds and rats prioritize horizontal versus vertical components differently when they remember three-dimensional space.     

An approach to the multiple-minima problem in protein folding by relaxing dimensionality. Tests on enkephalin

An algorithm for locating the region in conformational space containing the global energy minimum of a polypeptide is described. Distances are used as the primary variables in the minimization of an objective function that incorporates both energetic and distance-geometric terms. The latter are obtained from geometry and energy functions, rather than nuclear magnetic resonance experiments, although the algorithm can incorporate distances from nuclear magnetic resonance data if desired. The polypeptide is generated originally in a space of high dimensionality. This has two important consequences. First, all interatomic distances are initially at their energetically most favorable values; i.e. the polypeptide is initially at a global minimum-energy conformation, albeit a high-dimensional one. Second, the relaxation of dimensionality constraints in the early stages of the minimization removes many potential energy barriers that exist in three dimensions, thereby allowing a means of escaping from three-dimensional local minima. These features are used in an algorithm that produces short trajectories of three-dimensional minimum-energy conformations. A conformation in the trajectory is generated by allowing the previous conformation in the trajectory to evolve in a high-dimensional space before returning to three dimensions. The resulting three-dimensional structure is taken to be the next conformation in the trajectory, and the process is iterated. This sequence of conformations results in a limited but efficient sampling of conformational space. Results for test calculations on Met-enkephalin, a pentapeptide with the amino acid sequence H-Tyr-Gly-Gly-Phe-Met-OH, are presented. A tight cluster of conformations (in three-dimensional space) is found with ECEPP energies (Empirical Conformational Energy Program for Peptides) lower than any previously reported. This cluster of conformations defines a region in conformational space in which the global-minimum-energy conformation of enkephalin appears to lie.     

Three-dimensional technology for linear morphological studies: a re-examination of cranial variation in four southern African indigenous populations.

In order to compare linear dimensions made by traditional anthropometric techniques, and those obtained from three-dimensional coordinates, samples of four indigenous southern African populations were analysed. Linear measurements were obtained using mathematically transformed, three-dimensional landmark data on 207 male crania of Cape Nguni, Natal Nguni, Sotho and Shangaan. Univariate comparisons for accuracy of the transformed linear data were made with those in a traditional linear study by de Villiers (The Skull of the South African Negro: A Biometrical and Morphological Study. Witwatersrand University Press, Johannesburg) on similar samples and equivalent landmarks. Comparisons were not made with her Penrose (Ann Eugenics 18 (1954) 337) analysis as an apparently anomalous 'shape'-'size' statistic was found. The univariate comparisons demonstrated that accurate linear measurements could be derived from three-dimensional data, showing that it is possible to simultaneously obtain data for three-dimensional geometric 'shape' and linear interlandmark analyses. Using Penrose and canonical variates analyses of the transformed three-dimensional interlandmark measurements, similar population distances were found for the four indigenous southern African populations. The inter-population distance relationships took the form of three separated pairs of distances, with the within-pair distances very similar in size. The cranial features of the four populations were found to be overall very similar morphometrically. However the populations were each shown by CVA to have population specific features, and using discriminant analyses 50% or more of the individual crania (with the exception of the Sotho) could be referred to their correct populations.     

A comparison of random forest and its Gini importance with standard chemometric methods for the feature selection and classification of spectral data

Background  

Regularized regression methods such as principal component or partial least squares regression perform well in learning tasks on high dimensional spectral data, but cannot explicitly eliminate irrelevant features. The random forest classifier with its associated Gini feature importance, on the other hand, allows for an explicit feature elimination, but may not be optimally adapted to spectral data due to the topology of its constituent classification trees which are based on orthogonal splits in feature space.     

The ontogenetic complexity of developmental constraints

Developmental constraint is a theoretically important construct bridging ontogenetic and evolutionary studies. We propose a new operationalization of this notion that exploits the unusually rich measurement structure of landmark data. We represent landmark configurations by their partial warps, a basis for morphospace that represents a set of localized features of form. A finding of developmental constraint arises from the interplay between age-varying means and age-specific variances in these subspaces of morphospace. Examination of variances and means in 16 ventral skull landmarks in the cotton rat S. fulviventer at ages 1, 10, 20, and 30 days yielded three types of developmental constraint: canalization (constraint to relatively constant form age by age); chreods (reduction of variance orthogonal to the mean trajectory over ages); and opposition (reduction of age-specific variance along the mean trajectory over ages). While canalization and chreodic constraints have been noted previously, the oppositional type of constraint appears novel. Only one of our characters, relative length and orientation of the incisive foramen, appears to be canalized. Although skull growth becomes increasingly integrated through ontogeny, our characters display a remarkable spatiotemporal complexity in patterns of variance reduction. The specific assortment of constraints observed may be related to the precociality of Sigmodon. We suggest that Waddington's diagrammatic presentation of the “epigenetic landscape” may be misleading in quantitative studies of developmental regulation.     

Stability and Control of Human Trunk Movement During Walking

A mathematical model has been developed to study the control mechanisms of human trunk movement during walking. The trunk is modeled as a base-excited inverted pendulum with two-degrees of rotational freedom. The base point, corresponding to the bony landmark of the sacrum, can move in three-dimensional space in a general way. Since the stability of upright posture is essential for human walking, a controller has been designed such that the stability of the pendulum about the upright position is guaranteed. The control laws are developed based on Lyapunov' stability theory and include feedforward and linear feedback components. It is found that the feedforward component plays a critical role in keeping postural stability, and the linear feedback component, (resulting from viscoelastic function of the musculoskeletal system) can effectively duplicate the pattern of trunk movement. The mathematical model is validated by comparing the simulation results with those based on gait measurements performed in the Biomechanics Laboratory at the University of Manitoba.     

Stability and Control of Human Trunk Movement During Walking

A mathematical model has been developed to study the control mechanisms of human trunk movement during walking. The trunk is modeled as a base-excited inverted pendulum with two-degrees of rotational freedom. The base point, corresponding to the bony landmark of the sacrum, can move in three-dimensional space in a general way. Since the stability of upright posture is essential for human walking, a controller has been designed such that the stability of the pendulum about the upright position is guaranteed. The control laws are developed based on Lyapunov's stability theory and include feedforward and linear feedback components. It is found that the feedforward component plays a critical role in keeping postural stability, and the linear feedback component, (resulting from viscoelastic function of the musculoskeletal system) can effectively duplicate the pattern of trunk movement. The mathematical model is validated by comparing the simulation results with those based on gait measurements performed in the Biomechanics Laboratory at the University of Manitoba.     

PCA 和PLS 应用于胃癌亚型分类研究

文章研究了基于微阵列基因表达数据的胃癌亚型分类。微阵列基因表达数据样本少、纬度高、噪声大的特点,使得数据降维成为分类成功的关键。作者将主成分分析(PCA) 和偏最小二乘（PLS）两种降维方法应用于胃癌亚型分类研究,以支持向量机（SVM）、K- 近邻法（KNN）为分类器对两套胃癌数据进行亚型分类。分类效果相比传统的医理诊断略高,最高准确率可达100%。研究结果表明,主成分分析和偏最小二乘方法能够有效地提取分类特征信息,并能在保持较高的分类准确率的前提下大幅度地降低基因表达数据的维数。     

A model to predict canine pelvic limb musculoskeletal geometry

We developed a model to predict the three-dimensional canine pelvic limb muscular geometry (i.e., all muscle moment arms during any instant in gait). Forty-one muscle origins and insertions, as well as external landmarks (to obtain anthropometric dimensions) were marked on both pelvic limbs of five dogs and digitized on biplanar radiographs. Reference frames in the pelvis, femur, and tibia established the three-dimensional coordinates of each origin, insertion, and landmark. A set of dimensionless 'scaled coordinates' was created by dividing the actual origin and insertion coordinates by selected anthropometric dimensions of each animal. Combining scaled coordinates from all ten limbs produced an averaged 'template' of scaled coordinates. To provide limited validation of the scaling procedure, we measured the anthropometric dimensions between externally palpable landmarks of two additional pelvic limbs. The anthropometric dimensions were multiplied by the averaged template coordinates to calculate two new sets of hindlimb muscle coordinates within the three bony reference frames. The two limbs then were dissected, muscle endpoints were marked, and biplanar radiographs of each of the limb segments were digitized. The actual coordinates so obtained were similar to those predicted by the template and anthropometric measures.     

Functional Uniform Priors for Nonlinear Modeling

Summary This article considers the topic of finding prior distributions when a major component of the statistical model depends on a nonlinear function. Using results on how to construct uniform distributions in general metric spaces, we propose a prior distribution that is uniform in the space of functional shapes of the underlying nonlinear function and then back‐transform to obtain a prior distribution for the original model parameters. The primary application considered in this article is nonlinear regression, but the idea might be of interest beyond this case. For nonlinear regression the so constructed priors have the advantage that they are parametrization invariant and do not violate the likelihood principle, as opposed to uniform distributions on the parameters or the Jeffrey’s prior, respectively. The utility of the proposed priors is demonstrated in the context of design and analysis of nonlinear regression modeling in clinical dose‐finding trials, through a real data example and simulation.