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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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