Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 64

cies in front of you. Assume that these transparencies are not the
familiar rectangular ones, but instead are circular, each with a differ-
ent radius. The importance of the circular transparencies will become
evident as you work through the experiment. On one transparency,
draw a triangle (using a red pen) and label each of the three vertices
as a landmark using the numbers 1, 2, and 3. Take a second trans-
parency and put it on top of the first. Using a black pen, place a single
point in the proximity of the original (red) landmark 1 at a random dis-
tance and direction from landmark 1. Next, on the same transparency,
place a point in the proximity of landmark 2 at a random (though
small) distance and direction from the original landmark 2. Do the
same for landmark 3. You now have a three-point configuration on the
second transparency that was created using the original red triangle
as a template. Repeat this procedure of forming a novel three point
configuration in black ink nine more times, using a different trans-
parency each time. At completion, you will have a stack of eleven
transparencies. Ten of these contain a random three-point configura-
tion drawn in black that are related to the original three point
configuration (drawn in red). The spread of the black points around
each of the red points can be thought of as the variability around an
average location.
Biological organisms that constitute a group resemble each other to
such a degree that we have an intuitive understanding of a typical or
“average” form (the red triangle) that represents all members of the
group. Individuals within a group correspond to the ten black triangles
that were just created on the circular transparencies. Some members
are very similar to the average, while others are less like the average.
Since all forms differ from each other in various ways, a scheme for
characterizing these differences is needed. It is convenient to organize
and specify these differences as divergences from an average form.
In biology, genetic and environmental influences combine to affect
structures represented by each of the landmarks, thereby creating ran-
domly perturbed forms. When observing a group of forms, we think of
these individuals and the variation among them by relating them to a
typical form that does not exist but that we are able to envision. This
is formally done in statistics by calculating an average form and mea-
suring variation in the sample with reference to an average.
To quantify variability within a sample, the researcher needs to
make some decisions about how the individuals in a sample vary.
These decisions constitute the choice of a perturbation model , the
mathematical construct that characterizes the way in which individu-