Integration, Disintegration, and Self-Similarity: Characterizing the Scales of Shape Variation in Landmark Data.
Bottom Line:
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.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.
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Affiliation: Faculty of Life Sciences, University of Vienna, Vienna, Austria ; Department of Statistics, University of Washington, Seattle, WA USA.
ABSTRACT
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 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 . 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 , 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 [Formula: see text] 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. No MeSH data available. Related in: MedlinePlus |
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Mentions: I mentioned in the Prologue that the customary approaches to morphological integration based on correlations among multiple dimensions of descriptors do not suit our formalisms of Procrustes shape coordinates; it is time that I justified that claim. Figure 7 conveys two easily summarized paradoxes in this covariance-based morphometrics of distance data in order to conclude that no covariance pattern can be interpreted unambiguously unless the mean landmark configuration is an explicit component of the pattern analysis. The two triangles shown in the upper row are characterized by the same covariance structure\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma ^2{\left( \begin{array}{ccc} 0&{}0&{}0\\ 0&{}1&{}1\\ 0&{}1&{}1\\ \end{array} \right) } \end{aligned}$$\end{document}σ2000011011for the full set of three pairwise distances, where is any sufficiently small quantity and the distances are taken in the order 12, 13, 23. (That points 1 and 2 are at an invariant distance suggests that all three points might have been represented by their Bookstein coordinates at the outset of the example.) But the two descriptions of the “same” pattern are nevertheless remarkably different when considered as evidence of biological processes. On the left, landmark 3 is restricted to the line through landmarks 1 and 2. On the right, landmark 3 is restricted to their perpendicular bisector, which makes as large an angle (90°) with the collinearity constraint as it possibly could. If we add a parameter for the failure of this canalization—the signed variation of point 3 away from the line along which it was supposed to be canalized—then the rate at which the variance of the difference of distances rises, and hence cov falls, is at least eightfold greater as a function of var for the second configuration than it is for the first.Fig. 7 |
View Article: PubMed Central - PubMed
Affiliation: Faculty of Life Sciences, University of Vienna, Vienna, Austria ; Department of Statistics, University of Washington, Seattle, WA USA.
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 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 . 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 , 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 [Formula: see text] 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.
No MeSH data available.