In today’s geometric morphometrics the commonest multivariate statistical procedures, such as principal component analysis or regressions of Procrustes shape coordinates on Centroid Size, embody a tacit roster of symmetries—axioms concerning the homogeneity of the multiple spatial domains or descriptor vectors involved—that do not correspond to actual biological fact. These techniques are hence inappropriate for any application regarding which we have a-priori biological knowledge to the contrary (e.g., genetic/morphogenetic processes common to multiple landmarks, the range of normal in anatomy atlases, the consequences of growth or function for form). But nearly every morphometric investigation is motivated by prior insights of this sort. We therefore need new tools that explicitly incorporate these elements of knowledge, should they be quantitative, to break the symmetries of the classic morphometric approaches. Some of these are already available in our literature but deserve to be known more widely: deflated (spatially adaptive) reference distributions of Procrustes coordinates, Sewall Wright’s century-old variant of factor analysis, the geometric algebra of importing explicit biomechanical formulas into Procrustes space. Other methods, not yet fully formulated, might involve parameterized models for strain in idealized forms under load, principled approaches to the separation of functional from Brownian aspects of shape variation over time, and, in general, a better understanding of how the formalism of landmarks interacts with the many other approaches to quantification of anatomy. To more powerfully organize inferences from the high-dimensional measurements that characterize so much of today’s organismal biology, tomorrow’s toolkit must rely neither on principal component analysis nor on the Procrustes distance formula, but instead on sound prior biological knowledge as expressed in formulas whose coefficients are not all the same. I describe the problems of the standard techniques, discuss several examples of the alternatives, and draw some conclusions.

中文翻译：

---

几何形态计量学中多元统计分析的不适当对称性。


在当今的几何形态计量学中，最常见的多元统计程序，例如主成分分析或质心尺寸上的普罗克拉斯特形状坐标回归，体现了一个默认的对称名册——涉及多个空间域或所涉及的描述符向量的同质性的公理——它们不对应实际的生物学事实。因此，这些技术不适用于我们拥有相反的先验生物学知识的任何应用（例如，多个地标共有的遗传/形态发生过程、解剖图谱中的正常范围、生长或功能对形态的影响）。但几乎每一次形态测量研究都是由此类先前的见解推动的。因此，我们需要新的工具来明确地整合这些知识要素（如果它们是定量的），以打破经典形态测量方法的对称性。其中一些已经在我们的文献中提供，但值得更广泛地了解：普罗克拉斯特坐标的紧缩（空间自适应）参考分布、Sewall Wright 的百年因子分析变体、将显式生物力学公式导入普罗克拉斯特空间的几何代数。其他尚未完全制定的方法可能涉及负载下理想形式应变的参数化模型，将形状随时间变化的功能与布朗方面分离的原则性方法，以及总体上更好地理解地标的形式主义如何与许多其他解剖学量化方法相互作用。 为了更有力地从表征当今有机生物学特征的高维测量中组织推论，明天的工具包必须既不能依赖于主成分分析，也不能依赖于 Procrustes 距离公式，而是依赖于可靠的先验生物学知识，如公式所示，其系数为不都是一样的。我描述了标准技术的问题，讨论了几个替代方案的例子，并得出了一些结论。