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the green quadrangle in an attempt to minimize the sum of the squared
distances between corresponding landmarks in the two quadrangles.
Note that if we disregard landmarks 1 and 3, a rather nice fit is
accomplished between landmarks 2 and 4 of the red and green trans-
parencies. As the green quadrangle is rotated to match landmarks 1
and 3 of the green and red quadrangles, we tend to lose the fit between
landmarks 2 and 4. On the other hand, as landmarks 2 and 4 align
closely, landmarks 1 and 3 are not well matched.
This simple example demonstrates a basic tendency of the
Procrustes fitting criterion: corresponding landmarks farthest from
the centroid are matched closely at the cost of mismatching those that
are closer to the centroid. A consequence is that estimates of variabili-
ty for those landmarks lying farther from the centroid are reduced,
while estimates of variability for landmarks lying close to the centroid
are amplified. This effect will be most pronounced when the object is
not symmetric around the centroid and when the landmarks are not
uniformly spread on the object. This means that the Procrustes method
tends to estimate variability according to a rule that has little to do
with the natural variability of the specimens, but is instead driven by
the distance of landmarks from the centroid.
3.10 Summary
In Part 1 of this chapter, we provided the reader with working models
for landmark data and introduced concepts that should be considered
when analyzing landmark data. The terms model and method were
defined and related to one another. We defined nuisance parameters as
they exist in the study of form. We introduced the concept of invariance
and how it relates to elimination of the nuisance parameters, and pre-
sented a coordinate system-free representation of form. We presented
statistical models for landmark data and explained the estimators for
the one-sample case using Euclidean Distance Matrix Analysis.
Finally, we commented upon aspects of our approach to the analysis of
landmark data that were not essential to the central presentation of
our ideas, but that place our approach within the context of the field of
morphometrics. Part 2 of this chapter deals with the mathematical and
statistical details of the one-sample case.
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