Biology Reference
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(a) Isotropic variance. Perhaps the simplest type of covariance
structure for the perturbation distribution is one in which all
landmarks are perturbed with the same variance irrespective of
direction. This corresponds to adding a Normal random variate
with mean 0 and variance
to each landmark coordinate along
both the X- and the Y-axes. In this model
2
is equal along the X-
and Y-axes. Figure 3.1a shows a collection of 100 objects that are
generated from the template of a simple 3 landmark, two-
dimensional object using this perturbation covariance structure.
2
This isotropic variance structure is easy to visualize, but may not
be biologically realistic in the study of certain biological structures or
certain populations. For example, we found in a sample of primate
skulls (Lele and Richtsmeier, 1990) that the variances for osseous
landmarks that mark sites of muscle attachment are larger than the
variance of those that mark foramina for neurovascular bundles.
(b) Independent and different variances. To account for observa-
tions that suggest differing variability for specific landmarks, a
model can be constructed that allows for perturbation of each
landmark by a different variance. A graphical depiction of such
a covariance structure is provided in Figure 3.1b . Differences
between Figure 3.1a and 3.1b are apparent.
(c) Correlated landmarks. Since subsets of landmarks can corre-
spond with functionally or developmentally integrated units
(Atchley, 1987; Atchley, Cowley et al., 1990; Atchley and Hall,
1991; Atchley, 1993; Cheverud 1995; Cheverud, 1996; Cheverud,
Routman et al., 1996), in some cases it may be appropriate to
assume correlation among landmark perturbations. For exam-
ple, if landmark data were available from a long bone, we might
expect that the landmarks on the shaft would share certain
aspects of variability, while landmarks on a joint surface might
share other aspects of variability. Variability of landmarks on
female pelves may vary depending on whether the landmark is
located at a site associated with locomotion or parturition. A
covariance structure can be made somewhat more realistic by
allowing correlation among the perturbations local to specific
subsets of landmarks. For mathematical convenience, we
assume independence between X- and Y-axes. Figure 3.1c
depicts this covariance structure.
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