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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