Biometrics,biomathematics and the morphometric synthesis

At the core of contemporarymorphometrics—the quantitative study of biological shape variation—is a synthesis of two originally divergent methodological styles. One contributory tradition is the multivariate analysis of covariance matrices originally developed as biometrics and now dominant across a broad expanse of applied statistics. This approach, couched solely in the linear geometry of covariance structures, ignores biomathematical aspects of the original measurements. The other tributary emphasizes the direct visualization of changes in biological form. However, making objective the biological meaning of the features seen in those diagrams was always problematical; also, the representation of variation, as distinct from pairwise difference, proved infeasible. To combine these two variants of biomathematical modeling into a valid praxis for quantitative studies of biological shape was a goal earnestly sought though most of this century. That goal was finally achieved in the 1980s when techniques from mathematical statistics, multivariate biometrics, non-Euclidean geometry and computer graphics were combined in a coherent new system of tools for the complete regionalized quantitative analysis oflandmark points together with the biomedical images in which they are seen. In this morphometric synthesis, correspondence of landmarks (biologically labeled geometric points, like “bridge of the nose”) across specimens is taken as a biomathematical primitive. The shapes of configurations of landmarks are defined as equivalence classes with respect to the Euclidean similarity group and then represented as single points in David Kendall'sshape space, a Riemannian manifold with Procrustes distance as metric. All conventional multivariate strategies carry over to the study of shape variation and covariation when shapes are interpreted in the tangent space to the shape manifold at an average shape. For biomathematical interpretation of such analyses, one needs a basis for the tangent space compatible with the reality of local biotheoretical processes and explanations at many different geometric scales, and one needs graphics for visualizing average shape differences and other statistical contrasts there. Both of these needs are managed by thethin-plate spline, a deformation function that has an unusually helpful linear algebra. The spline also links the biometrics of landmarks to deformation analysis of the images from which the landmarks originally arose. This article reviews the history and principal tools of this synthesis in their biomathematical and biometrical context and demonstrates their usefulness in a study of focal neuroanatomical anomalies in schizophrenia.     

On the gestalt concept

We define a gestalt as the invariants of a collection of patterns that can mutually be transformed into each other through a class of transformations encoded by, or conversely, determining that gestalt. The class of these transformations needs to satisfy structural regularities like the ones of the mathematical structure of a group. This makes an analysis of a gestalt possible in terms of relations between its representing patterns. While the gestalt concept has its origins in cognitive psychology, it has also important implications for morphology.     

On the gestalt concept

We define a gestalt as the invariants of a collection of patterns that can mutually be transformed into each other through a class of transformations encoded by, or conversely, determining that gestalt. The class of these transformations needs to satisfy structural regularities like the ones of the mathematical structure of a group. This makes an analysis of a gestalt possible in terms of relations between its representing patterns. While the gestalt concept has its origins in cognitive psychology, it has also important implications for morphology.     

On the concept of chorotype

Recent reviews of the meaning of the word ‘chorotype’ in biogeography have led to contrasting definitions and a confusion of concepts. This is because ‘chorotype’ has been used by different authors to express two different concepts: (1) groups of species with overlapping ranges (overall distributions) and (2) groups of species with a similar distribution within a certain area. To avoid confusion, I suggest the term ‘global chorotype’ be used to indicate a group into which species with similar ranges can be classified; and ‘regional chorotype’ be used for a group of species with similar distributions within a certain region. Although the global chorotype represents the world‐wide spatial responses of species to historical and environmental pressures, and does not vary with the area under consideration, a particular species might be classified into different regional chorotypes in different study areas.     

试论生物多样性的概念

随着人口的迅速增长,人类经济活动的不断加剧,作为人类生存最为重要的基础的生物多样性受到了严重的威胁。“无法再现的基因、物种和生态系统正以人类历史上前所未有的速度消失”。如果不立即采取有效措施,人类将面临着能否继续以其固有的方式生活的挑战。     

On the concept of effective number.

A generalized conceptual basis for Wright's notion of effective size is presented. The concept is applied to the calculation of effective numbers based on the rate of change of genetic variability. With particular reference to the inbreeding, the eigen value, and the newly introduced "diversity" effective size, the use of the concept as a means for discrimination between and identification of various effective sizes is demonstrated.     

On the concept of the effective population size

The concept of the effective population size is discussed. It is shown that the “eigenvalue” and the “inbreeding” effective population sizes are in principle different, even though they have been sometimes identified in the literature. On the other hand the “eigenvalue” and “variance” effective sizes are usually both close when the latter exists. Since, however, there are many models for which a variance effective size cannot in principle exist, it seems useful to introduce the eigenvalue effective size and to examine some of its properties.