Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 65

als within a population vary with respect to an average form. The
degree to which a specific model correctly approximates the perturbed
copies of the average form depends upon the appropriateness, or the
“fit” of the model to the data. The better we are able to anticipate these
perturbations, the better the model will fit the data. Again, knowledge
of the study specimens puts the researcher in a better position to
choose the appropriate model.
Suppose we assume that a normal distribution will model the per-
turbation of the transparency data appropriately. Let M denote the
mean form. M can be likened to the notion we have of a form that has
the typical features of individuals within the group and that fairly rep-
resents members of a particular group. In the transparency
experiment M corresponds to the red triangle. Since we are dealing
with landmark data, M is a K

D matrix where K corresponds to the
number of landmarks and D corresponds to the dimension of the form.
Each of the perturbed observations (black triangles) are obtained by
adding some noise, namely E i , to the mean form, where E i 's are
assumed to be matrix-valued normal random variables with a mean of
0 and a covariance matrix. Each observation is written as M

E i .In

our transparency experiment, each M

E i corresponds to one of the
ten configurations drawn using the black pen.
In a simple world, these would be our only concerns in choosing a
model. However, there is an additional complication. This complication
involves the relationship between the coordinate system local to the
object and the coordinate system used to collect the landmark coordi-
nate data. No information is available regarding how these coordinate
systems relate to one another. To clarify this problem, imagine that
after creating the ten transparencies, they are dropped onto the floor.
When we pick the transparencies up and put them onto the table, their
original orientations are changed. Because they are circular, we have
no outside reference to inform us about how they were related to one
another before they fell to the floor. In order to study biological vari-
ability, knowledge of the relationship between the transparencies
before they fell to the floor is essential.
Let us now go back to the original, undisturbed stack of trans-
parencies and consider the effect of lifting a single transparency from
its original position and putting it elsewhere on the table. Movement
of a single transparency introduces a particular translation and rota-
tion to the definition of this transparency in relation to its original
location. Lifting the next transparency and moving it to another loca-