Biology Reference

In-Depth Information

targeted at the scientist who is interested in understanding the bio-

Result 1:
The
mean
landmark coordinate matrix,
M
, consisting of

landmark coordinates cannot be estimated. However, the matrix

of all possible pair-wise distances corresponding to
M
, otherwise

known as the form matrix of
M
,or
FM
(
M
)
,
can
be estimated.

of this chapter. If we know the form matrix for a given object, we have

all the relevant information about the form of that object that can be

obtained from landmark coordinates. This result means that given the

landmark coordinate data, we can capture the essence of the mean

form by using the vector of all possible linear distances among land-

marks. This can be done even in the presence of nuisance parameters

of translation, rotation, and reflection.

However, the unfortunate effect of these nuisance parameters

becomes apparent when we attempt to estimate the variance. Suppose

the variance-covariance matrix
V
characterizes the perturbation pat-

tern, where
V

K
D
(See
Chapter 2, Part 2
for details on
K
and

D
)

Result 2:
Neither
K
nor
D
can be estimated. What we can esti-

mate is a singular version of
K
, denoted by
K
*
, and only the

eigenvalues of
D
.

A consequence of the non-estimability of
K
and
D
is that inter-

pretation of the estimators of the variances becomes complex. In fact,

this constraint means that we cannot estimate the variability local to

any particular landmark. However, the eigenvalues of
D
can be used

to determine whether or not there is a difference in the variability cal-

culated along the three major axes. We underscore that the nuisance

parameters prohibit valid estimates of the exact magnitude of vari-

ability surrounding landmarks. Consequently, biological questions that

require specific values (estimates) of local variability cannot be

addressed. Since the estimation of exact quantities of local variability

is impossible due to the presence of nuisance parameters, we need to

determine how we might use those quantities that can be estimated in

scientific analysis.

We refer to the quantities that are estimable, namely
K
*
and the

eigenvalues of
D
, as the perturbation pattern. These estimators can

be used as tools to evaluate differences in form or shape between pop-

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