Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 61

CHAPTER 3
Statistical Models for
Landmark Coordinate Data
In this chapter, we present models that are used to study form and
variability among forms as represented by landmark data. In Part 1 of
this chapter, we discuss the concept of variability among organisms,
models that are used to characterize variability and the way in which
these models relate to the methods used for analysis of form. Based on
these considerations, we then discuss the landmark coordinate repre-
sentation of form and present our method for characterizing form
without reference to a coordinate system. The final portion of Part 1 of
this chapter presents analysis of the example data sets presented in
Chapter 1 using these methods and models. Part 2 of this chapter pre-
sents the formal statistical theory for characterizing a single
population of forms using landmark data and presents the necessary
computational algorithms for estimation of the mean and variance.
Before we discuss models and methods for the analysis of landmark
data, it is necessary to differentiate between a statistical model and a
statistical method. A model, as used in this monograph, is a mathe-
matical construct that attempts to characterize certain aspects of the
underlying phenomenon. This mathematical construct includes quan-
tities called parameters. For example, consider the simple linear
regression model that describes the relationship between height and
weight of an individual. Let Y denote the weight of an individual and
X denote the height of the same individual. The statistical model used
in this situation may be: Y = 0 + 1 X+ . This model states that the
weight of an individual is a linear function of the height of the indi-
vidual with the addition of some random variability. The parameters of
this model are the intercept, 0, and the slope, 1 . The term denotes
the variability around the mean response 0 + 1 X . Usually, we