Defining shape in non-mathematical terms has always been a challenge. We define shape as what remains after non-shape features are ignored when comparing objects, which immediately suggests that shape attributes are not easily encapsulated in one or a few variables. An object's shape, at least when captured using methods that preserve spatial information, is characterized as a vector of differences relative to a reference object. These differences jointly describe a transformation that maps every point in the reference into a point in the target object. Shape transformations are in general based on incomplete information---not every spot in either the reference or the target object is or can be measured, and so we often rely on some kind of spatial sampling, such as landmarks. The upshot of this compromise is that interpreting shape, and shape variation, necessarily entails some amount of guesswork. In practice, "guessing" assumes that sampled landmarks reflect shape changes in the unsampled spaces in between landmarks, which we usually describe as local stretching and shrinking, which often lead to hypotheses about tissue growth and differentiation.
Interpreting shape differences is often aided by visualizations computed via interpolation, which is a form of estimation that uses "local" information to build a function that allows us to infer unseen states of variables. But even when explicit functions are not used, interpretation of shape differences is always based on some subjective form of interpolation. This highlights two potential issues: first, it suggests that it is preferable to base interpretations on a known set of simplifying assumptions (i.e., functional interpolation) than on an unknowable set of subjective ones; and second, that the customary way of performing statistical analysis of shape, i.e., using landmark coordinates as the raw data, is disconnected from the traditional basis for interpretation of shape differences, i.e., interpolation, and the consequences of this are hardly discussed.
In Márquez et al. (2012), we discuss an approach that deals with both problems. First, although we agree that the assumptions of oft-used interpolation functions are not rooted on biology, we see that a solution for this is not to dismiss interpolated data, but to shift the focus of research in morphometrics toward improving on these functions. We favor making explicit assumptions as it forces us to bind these assumptions to the domain of reality. Shifting toward modeling of inter-landmark shape differences entails integration of morphometric analysis and biological theory, and emphasizes explanation over description, particularly as model proliferation will demand testing among hypotheses competing to explain observed patterns of variation.
Second, using landmarks as raw data for statistical analysis may be granted when interpretation of shape differences is irrelevant. Whenever we wish to make inferences about local variation, however, interpretation plays a tacit role because the deformations implied by a subset of landmarks are informed by the behavior of all other landmarks. In these situations, which include, among others, tests of variational modularity, an interpolating function-based approach integrates global information continuously over the entire object, restoring our ability to make local inferences about shape variation.
Quantifying shape using interpolation functions immediately receives all of the benefits of defining any kind of trait as functions. As mentioned above, function-based approaches can easily become theory-building endeavors, where sets of explicit assumptions become falsifiable hypotheses that can be re-applied to other data sets. Function-valued traits are infinitely comparable across studies regardless of whether the original data inhabit different spaces (e.g., configurations need not have the same landmarks to be comparable). Finally, functions can be re-sampled at any resolution to adjust to limitations imposed by statistical power, computational resources, or just healthy skepticism to the quality of the original data.
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