Analysis of Ratios in Multivariate Morphometry

The analysis of ratios of body measurements is deeply ingrained in the taxonomic literature. Whether for plants or animals, certain ratios are commonly indicated in identification keys, diagnoses, and descriptions. They often provide the only means for separation of cryptic species that mostly lack distinguishing qualitative characters. Additionally, they provide an obvious way to study differences in body proportions, as ratios reflect geometric shape differences. However, when it comes to multivariate analysis of body measurements, for instance, with linear discriminant analysis (LDA) or principal component analysis (PCA), interpretation using body ratios is difficult. Both techniques are commonly applied for separating similar taxa or for exploring the structure of variation, respectively, and require standardized raw or log-transformed variables as input. Here, we develop statistical procedures for the analysis of body ratios in a consistent multivariate statistical framework. In particular, we present algorithms adapted to LDA and PCA that allow the interpretation of numerical results in terms of body proportions. We first introduce a method called the “LDA ratio extractor,” which reveals the best ratios for separation of two or more groups with the help of discriminant analysis. We also provide measures for deciding how much of the total differences between individuals or groups of individuals is due to size and how much is due to shape. The second method, a graphical tool called the “PCA ratio spectrum,” aims at the interpretation of principal components in terms of body ratios. Based on a similar idea, the “allometry ratio spectrum” is developed which can be used for studying the allometric behavior of ratios. Because size can be defined in different ways, we discuss several concepts of size. Central to this discussion is Jolicoeur's multivariate generalization of the allometry equation, a concept that was derived only with a heuristic argument. Here we present a statistical derivation of the allometric size vector using the method of least squares. The application of the above methods is extensively demonstrated using published data sets from parasitic wasps and rock crabs

Correlated evolution of multivariate traits: detecting co-divergence across multiple dimensions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75039/1/j.1420-9101.2007.01415.x.pd

Morphometry of the Cranial Base in Subjects with Class III Malocclusion

The significance of the cranial base in the development of Class III malocclusion remains uncertain. The purpose of this study was to determine whether the form of the cranial base differs between prepubertal Class I and Class III subjects. Lateral cephalographs of 73 children of European-American descent aged between 5 and 11 years with Class III malocclusion were compared with those of their counterparts with a normal, Class I molar occlusion. The cephalographs were traced, checked, and subdivided into seven age- and sex-matched groups. Average geometries, scaled to an equivalent size, were generated based on 13 craniofacial landmarks by means of Procrustes analysis, and these configurations were statistically tested for equivalence. Bivariate and multivariate analyses utilizing 5 linear and angular measurements were undertaken to corroborate the Procrustes analysis. Graphical analysis, utilizing thin-plate spline and finite element methods, was performed for localization of differences in cranial base morphology. Results indicated that cranial base morphology differed statistically for all age-wise comparisons. Graphical analysis revealed that the greatest differences in morphology occurred in the posterior cranial base region, which generally consisted of horizontal compression, vertical expansion, and size contraction. The sphenoidal region displayed expansion, while the anterior regions showed shearing and local increases in size. It is concluded that the shape of the cranial base differs in subjects with Class III malocclusion compared with the normal Class I configuration, due in part to deficient orthocephalization, or failure of the cranial base to flatten during development.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67377/2/10.1177_00220345970760021101.pd

Parameter-sparse modification of Fourier methods to analyse the shape of closed contours with application to otolith outlines

-Elliptical Fourier descriptors (EFDs) have been used extensively in shape analysis of closed contours and have a range of marine applications, such as automatic identification of fish species and discrimination between fish stocks based on EFDs of otolith contours. A recent method (the ‘MIRR’ method) transforms the two-dimensional contour to a one-dimensional function by mirroring (reflecting) the lower half of the contour around a vertical axis at the right end of the contour. MIRR then applies the fast Fourier transform (FFT) to the vertical contour points corresponding to equidistant coordinate values along the horizontal axis. MIRR has the advantage of reducing the number of Fourier coefficients to two coefficients per frequency component compared with four EFDs. However, both Fourier methods require several frequency components to reproduce a pure ellipse properly. This paper shows how the methods can be easily modified so that a virtually perfect reproduction of a pure ellipse is obtained with only one frequency component. In addition, real otolith examples for cod (Gadus morhua) and Greenland halibut (Reinhardtius hippoglossoides) are used to demonstrate that the modified methods give better approximations to the large-scale shape of the original contour with fewer coefficients than the traditional Fourier methods, with negligible additional computing time

Unsupervised model-based clustering for typological classification of Middle Bronze Age flanged axes

The classification of Western European flanged axes dating to the Middle Bronze Age (1650–1350 BC) is very complex. Many types of axe have been identified, some of which have numerous variant forms. In the current French terminology, all axes are divided into two generic groups: namely "Atlantic" (Atlantique) and "Eastern" (Orientale). Each of these generic groups, however, is highly polymorphic, so that it is often very difficult for the operator to classify individual axes with absolute confidence and certainty. In order to overcome such problems, a new shape classification is proposed, using morphometric analysis (Elliptic Fourier Analysis) followed by unsupervised model-based clustering and discriminant analysis, both based on Gaussian mixture modelling. Together, these methods produce a clearer pattern, which is independently validated by the spatial distribution of the findings, and multinomial scan statistics. This approach is fast, reproducible, and operator-independent, allowing artefacts of unknown membership to be classified rapidly. The method is designed to be amendable by the introduction of new artefacts, in the light of future discoveries. This method can be adapted to suit many other archaeological artefacts, providing information about the material, social and cultural relations of ancient populations

Dysmorphometrics: the modelling of morphological abnormalities

<p>Abstract</p> <p>Background</p> <p>The study of typical morphological variations using quantitative, morphometric descriptors has always interested biologists in general. However, unusual examples of form, such as abnormalities are often encountered in biomedical sciences. Despite the long history of morphometrics, the means to identify and quantify such unusual form differences remains limited.</p> <p>Methods</p> <p>A theoretical concept, called dysmorphometrics, is introduced augmenting current geometric morphometrics with a focus on identifying and modelling form abnormalities. Dysmorphometrics applies the paradigm of detecting form differences as outliers compared to an appropriate norm. To achieve this, the likelihood formulation of landmark superimpositions is extended with outlier processes explicitly introducing a latent variable coding for abnormalities. A tractable solution to this augmented superimposition problem is obtained using Expectation-Maximization. The topography of detected abnormalities is encoded in a dysmorphogram.</p> <p>Results</p> <p>We demonstrate the use of dysmorphometrics to measure abrupt changes in time, asymmetry and discordancy in a set of human faces presenting with facial abnormalities.</p> <p>Conclusion</p> <p>The results clearly illustrate the unique power to reveal unusual form differences given only normative data with clear applications in both biomedical practice & research.</p