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: The present paper intends just such a recentering for the complementary domain of spatial variation (and, by extension, their joint combination in the spatiotemporal processes that are of central interest in the evo-devo sciences and in phylogenetic inference). The difference between the two approaches to a model is usually more dramatic than what was demonstrated in the Prologue. For instance, from a set of (artificial) landmarks in a grid, we can generate precisely 50 different squares that vary by size, grid position, and orientation. In the isotropic Procrustes model, the nonaffine shape variance of these squares itself varies strongly by size and to some extent by position and orientation as well. After the deflation by bending energy, though, they all show exactly the same distribution of nonaffine shape. Figure 3 numbers the landmarks and displays the basic Procrustes and bending-deflated scatters. The concluding panel shows the proportionality of variance after deflation to bending energy in the form of the log-log plot with slope in order to anticipate the findings in two of the empirical examples in Sect. "Visualizing Integration: Three Examples", which extract other slopes for this same plot in realistic settings. It is this slope that stands for the actual parameter of integration when integration is actually found to be a meaningful partial description of a data set. Figure 4 collects examples of forms over a narrow range of Procrustes distances, showing how biologically uninterpretable the majority of such shape dimensions would be, and then the corresponding bending-deflated grids, which would be much more suggestive of interpretable biological patterns were they to have arisen in real data analyses. Figure 5 confirms that in the deflated version of the isotropic Procrustes distribution, the nonaffine shape variation of any square highlighted within this grid is not dependent on the size, position, or orientation of that square upon the mean landmark configuration of Fig. 3. It is quite startling that such a distribution of multiple shape coordinates should exist at all, let alone that it can be generated from the standard Procrustes shape space by such a simple manipulation.Fig. 3 |
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.