The biologist examining samples of multicellular organisms in anatomical detail must already have an intuitive concept of morphological integration. But quantifying that intuition has always been fraught with difficulties and paradoxes, especially for the anatomically labelled Cartesian coordinate data that drive today’s toolkits of geometric morphometrics. Covariance analyses of interpoint distances, such as the Olson–Miller factor approach of the 1950’s, cannot validly be extended to handle the spatial structure of complete morphometric descriptions; neither can analyses of shape coordinates that ignore the mean form. This paper introduces a formal parametric quantification of integration by analogy with how time series are approached in modern paleobiology. Over there, a finding of trend falls under one tail of a distribution for which stasis comprises the other tail. The null hypothesis separating these two classes of finding is the random walks, which are self-similar, meaning that they show no interpretable structure at any temporal scale. Trend and stasis are the two contrasting ways of deviating from this null. The present manuscript introduces an analogous maneuver for the spatial aspects of ontogenetic or phylogenetic organismal studies: a subspace within the space of shape covariance structures for which the standard isotropic (Procrustes) model lies at one extreme of a characteristic parameter and the strongest growth-gradient models at the other. In-between lies the suggested new construct, the spatially self-similar processes that can be generated within the standard morphometric toolkit by a startlingly simple algebraic manipulation of partial warp scores. In this view, integration and “disintegration” as in the Procrustes model are two modes of organismal variation according to which morphometric data can deviate from this common null, which, as in the temporal domain, is formally featureless, incapable of supporting any summary beyond a single parameter for amplitude. In practice the classification can proceed by examining the regression coefficient for log partial warp variance against log bending energy in the standard thin-plate spline setup. The self-similarity model, for which the regression slope is precisely \(-1,\) corresponds well to the background against which the evolutionist’s or systematist’s a-priori notion of “local shape features” can be delineated. Integration as detected by the regression slope can be visualized by the first relative intrinsic warp (first relative eigenvector of the nonaffine part of a shape coordinate configuration with respect to bending energy) and may be summarized by the corresponding quadratic growth gradient. The paper begins with a seemingly innocent toy example, uncovers an unexpected invariance as an example of the general manipulation proposed, then applies the new modeling tactic to three data sets from the existing morphometric literature. Conclusions follow regarding findings and methodology alike.

中文翻译：

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集成，分解和自相似：表征地标数据中形状变化的尺度。

生物学家在解剖学上详细地检查了多细胞生物的样本时，必须已经有了一个直观的形态学整合概念。但是量化直觉总是充满困难和悖论，尤其是对于驱动当今几何形态计量学工具包的解剖学标记的笛卡尔坐标数据而言。点间距离的协方差分析，例如1950年代的Olson-Miller因子方法，不能有效地扩展以处理完整形态描述的空间结构。形状坐标的分析也不能忽略均值形式。本文介绍了一种形式化的积分参数化量化方法，该方法通过类推法与现代古生物学中如何处理时间序列进行类比。在那边，趋势的发现落在分布的一个尾部，而停滞则包括另一尾部。将这两类发现分开的零假设是随机游走，它们是自相似，表示它们在任何时间尺度上都没有可解释的结构。趋势和停滞是两种与之相反的方式。本手稿针对个体发生或系统发生生物研究的空间方面引入了一种类似的操作：形状协方差结构空间内的一个子空间，标准的各向同性（Procrustes）模型处于该特征空间的一个极端，并且该参数是最强的增长梯度。在另一个模型。介于中间的是建议的新结构，即空间自相似过程可以在标准形态计量工具包中通过对部分翘曲分数的简单简单的代数运算来生成。在这种观点下，Procrustes模型中的积分和“崩解”是两种有机体变异模式，根据形态学，形态计量学数据可能会偏离此通用零点，而这种零点在时间域上是形式上无特征的，无法支持超出范围的任何摘要。振幅的单个参数。在实践中，可以通过检查标准薄板样条设置中对数部分翘曲方差与对数弯曲能的回归系数来进行分类。自相似模型，其回归斜率精确为\（-1，\）与可以描述进化论者或系统论者的先验概念“局部形状特征”的背景非常吻合。通过回归斜率检测到的积分可以通过第一相对固有扭曲（相对于弯曲能量的形状坐标配置的非仿射部分的第一相对特征向量）进行可视化，并可以通过相应的二次生长梯度进行总结。本文从一个看似纯真的玩具示例开始，发现了一个意想不到的不变性作为所建议的一般操作的示例，然后将新的建模策略应用于现有形态学文献中的三个数据集。关于发现和方法的结论如下。