Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 144

final location of the hiker on a given contour. Although we would like
to describe the exact path that the climber followed to gain this eleva-
tion, without further information we cannot. We can assign a “sensible”
starting point for the hiker, but this information may be wrong and
eventually misleading. The description of the exact path can be pro-
vided if, and only if, we know exactly the point where the hiker started
and every intermediate point that he traversed as he attained the
higher elevation. If our information is limited to elevation, we simply
do not have the information to obtain a description of his path.
To take this analogy a bit further, let us say that we know the exact
position of a specific path between the elevations of 2000ft. and 2560ft.
The most parsimonious conclusion in this case is that the hiker used
this path to gain 560ft in elevation. But, perhaps the path was
designed for beginners and this particular hiker prefers a challenge.
The hiker forges trails up embankments and scales walls to reach
2560ft. We do not have this information, however, and parsimony
forces us to assume that the hiker used the well-known path. In this
case, the assumption of parsimony provides us with incorrect informa-
tion regarding how the hiker got from his starting point to his ending
point. As scientists, our choice is between using only the information
that is known, or including information or assumptions that are not
testable but that provide a more “complete” answer.
The complete mathematical description of these concepts is pre-
sented in Part 2 of this chapter. A simple and brief mathematical
description is presented here. Let M 1 be any point on the first orbit in
the space of the landmark coordinate matrices. The orbit of M 1 contains
all translated, rotated, and reflected versions of M 1 . Let M 2 , which is
also a two-dimensional three-landmark object, be represented by any
point on the second orbit in the space of the landmark coordinate
matrices. Let Diff (M 1 M 2 ) denote a description of the difference between
the two forms. Because we only know the orbits on which the two forms
lie but not the exact locations of the forms on the orbits, the descrip-
tion of the difference between the two forms should not require
information pertaining to the exact locations on the orbits. Instead this
description should be invariant to the exact location of the forms on
their respective orbits (Lele and McCulloch, 2000). Mathematically,
this is written as Diff (M 1 , M 2 ) 1 t 2 ).

Inclusion of information other than what is unambiguously known
may provide a seemingly more complete, but potentially misleading
answer. The information that is unambiguously known consists of
identification of the orbit on which the form lies. An invariant descrip-

Diff (M 1 , R 1 1 t 1 , M 2 R 2