Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 77

targeted at the scientist who is interested in understanding the bio-
logical significance of the mathematical results presented in Part 2 .

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.
An algorithm for the estimation of the mean FM is given in Part 2
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-