Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 97

CHAPTER 3
PART 2
Statistical Theory for the
Analysis of a Single Population
In this part, we discuss the statistical theory involved in the analysis
of data obtained from a single population. In particular, we describe
the use of invariance to eliminate nuisance parameters. The distribu-
tion of the maximal invariant is used to shed some light on the
identifiability of the parameters of interest. The method of moments is
used to obtain estimators of the parameters of interest. It is argued
that these estimators are consistent and are computationally simple to
obtain. We also compare and contrast competing methods of estima-
tion: the Procrustes method, the shape coordinates method, and the
method of maximum likelihood from both the theoretical point of view
and the practical point of view. This discussion is based on Lele (1993)
and Lele and McCulloch (2000).
3.11 The perturbation model
We model the inter-individual variability by the Gaussian perturbation
model that has been previously described (Goodall, 1991 or Lele, 1993).
The perturbation model may be thought of as representing the follow-
ing process. To generate a random geometrical object or equivalently a
K point configuration in D dimensional Euclidean space, one must first
choose a mean form (represented by matrix M ) and perturbs the ele-
ments of this matrix by adding noise to this mean form according to a
matrix valued Gaussian distribution. The K point configuration so
obtained is then rotated and/or reflected by an unknown angle and
translated by an unknown amount. Such perturbed, translated, rotat-
ed, and/or reflected K point configurations constitute our data. The
above description can be put in a mathematical form as follows.