Biology Reference
In-Depth Information
CHAPTER 4
PART 2
Statistical Theory for the
Comparison of Two Forms
In
Chapter 3
, we studied the effects of the nuisance parameters of
translation, rotation, and reflection on the identifiability and estima-
tion of the mean form and the covariance parameters. In particular, it
was shown that mean forms are identifiable only up to the orbit to
which they belong and not their precise location on the orbit. In this
chapter, we study the effect of nuisance parameters on the identifia-
bility and estimation of the differences in the mean forms of two
populations.
The statistical setting is as follows. We have two populations under
study. Following the convention of the last chapter we assume that
they follow matrix valued Gaussian distributions. Let
X
1
, X
2
,…, X
n
be
the landmark coordinate matrices for the sample of
n
individuals from
the first population. Let us assume that
X
i
~ N
(
M
1
R
i
K,
1
, R
i
T
1
t
i
,
D,
1
R
i
)
for
i
1, 2,…,
n
. Similarly, let
Y
1
,
Y
2
,…,
Y
m
be the landmark coordinate
matrices for the sample of
m
individuals from the second population.
Let us assume that
Y
j
~ N
(
M
2
R
j
1, 2,…,
m
.
When comparing two forms, the interest usually lies in studying
the relationship between the two mean forms
M
1
and
M
2
. Recall that
mean forms are identifiable only up to the orbit to which they belong
and the precise location on this orbit cannot be specified. An implica-
tion of this result is that we can estimate those differences between the
mean forms that depend only on the specification of the orbits and that
we cannot estimate differences that require information as to a specif-
ic location on the orbit. Hence, we make the following definition.
Definition 1:
Let
Diff
(
M
1
,
M
2
)
be a measure of the difference
between two mean forms
M
1
and
M
2
. Then
Diff
(
M
1
,
M
2
)
is an invariant
measure of form difference if and only if
Diff
(
M
1
,
M
2
)
K,
2
,
R
j
T
D,
2
R
j
)
for
j
1
t
j
,
Diff
(
M
1
R
1
1
t
1
T
,
M
2
R
2
1
t
2
T
)
for all orthogonal matrices
R
i
and translation vectors
t
i
.
In other words, the notion of form difference should not depend on
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