Biology Reference
In-Depth Information
Let
M
be a
K
D
matrix corresponding to the mean form. Let
E
i
be
a
K
D
matrix-valued Gaussian random variable (Arnold, 1981, pages
309-323) representing the error for the
i
-th individual. We assume
E
i
to
be Gaussian with a mean matrix of 0 and covariance matrices given by
K
and
D
. The matrix
K
describes the covariances between elements
within the same column of
E
i
and
D
describes the covariances within
the rows of
E
i
. In terms of notation we have
E
i
~N
(0,
D
).
More pre-
cisely, if we stack the matrix,
E
i
, into a vector,
vec
(
E
i
)
, then we have
var
(
vec
(
E
T
))
K
,
K
D
. Let
R
i
be an orthogonal matrix corresponding
to the rotation of the
i
-th individual, and let
t
i
be a
1
D
matrix corre-
sponding to the translation of the
i
-th individual. Then the landmark
coordinate matrix corresponding to the
i
-th individual may be repre-
sented as:
X
i
1
t
i
where
1
is a
K
1
matrix of 1's. The
(
M
E
i
)
R
i
random matrices
X
i
thus follow:
X
i
~N
(
MR
i
K
,
R
i
D
R
i
T
)
.
An important thing to notice is that the matrices
R
i
and
t
i
are
unknown and unknowable. These are nuisance parameters, while the
parameters of interest are
(
M
,
1
t
i
,
D
)
. In fact, there are more unknown
parameters than there are observations and the number of parameters
grows with the sample size. This problem falls within the class
described by Neyman and Scott (1948).
K
,
3.12 Invariance and elimination of nuisance parameters
One way to eliminate the nuisance parameters is by considering a
maximal invariant under a group of transformations. We briefly review
the definition of a maximal invariant and an important consequence of
using a maximal invariant for the statistical inference to the identifi-
ability of the underlying parameters. For an excellent discussion of
invariance, we refer the reader to Lehmann (1959, Chapter 6), Berger
(1982, Chapter 6) or Arnold (1981, Chapter 1).
Definition 1:
Let
G
be a group of invertible functions from a set
C
to
itself. A function
T
(
c
)
is called a maximal invariant if it satisfies the fol-
lowing two conditions:
a)
T
(
g
(
c
))
T
(
c
)
for all g
G and c
C.
b) If
T
(
c
1
)
T
(
c
2
)
, then there exists g
G such that
c
2
g
(
c
1
)
.
For any set
C
and any group
G
of invertible functions from
C
into
itself, there exists a maximal invariant. Any one to one function of a
maximal invariant is itself a maximal invariant. Hence, maximal
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