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It is easy to show that the form difference matrix
D
F
obtained by the

least squares constraint will be different than the one obtained by the

edge superimposition constraint.

Similar to the discussion of the deformation approach, the principle

of invariance suggests that we should restrict ourselves to inferences

that are invariant to the choice of the external constraint. In
Part 1,

Section 5
of this chapter, we have illustrated the practical conse-

quences of the lack of invariance of the superimposition approach. In

the following, we discuss the relationship between invariance and

identifiability in a mathematically rigorous fashion and show that the

transformations considered by the deformation and superimposition

approaches are non-identifiable.

4.16 Matrix transformations, invariance,

and identifiability issues

The problem of studying form differences can be put in terms of study-

ing transformation of one
K

D
matrix. The

invariance ideas described in
Chapter 3
can then be easily extended to

the two-sample problem. We then discuss the non-identifiability of the

transformations considered by the superimposition and deformation

approaches. Our argument is that non-identifiable transformations

should not be used for statistical or scientific inferences. We then show

that identifiability is equivalent to invariance. In the next section we

discuss invariant procedures to study form difference.

Consider the class of all transformations that map the space of

D
matrix into another
K

K

D
matrices to the space of
K

D
matrices. Let
denote the space of

all
K

D
matrices. Let
H:

be the collection of all transformations

that map any given
K

D
matrix. For exam-

ple, the affine transformation discussed earlier belongs to this class, so

do the thin-plate splines and superimposition transformations belong

to this class.

Recall that two
K

D
matrix to another
K

D
matrices are equivalent to each other if they

are rotations, reflections, and/or translations of each other. We can par-

tition the space

into equivalency classes, each class consisting of

equivalent matrices. Let
O(
M
)
denote the class of matrices equivalent

to a given matrix
M
.

Let
h
1
and
h
2
be two members of H such that
h
1
(
MR
1

1
t
1
)

h
2

1
t
. That is, the resultant matrices from both transforma-

tions acting on equivalent matrices are also equivalent to each other.

(
MR
2

1
t
2
)
R

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