Biology Reference
In-Depth Information
number of landmarks is large and/or the objects are three-dimension-
al. It is not uncommon in medical research to need at least 10, 15, or
more landmarks to represent an object reasonably. In such a situation,
the exact shape distribution is extremely complicated as can be seen
below. Notice that it involves telescoping sums whose number of com-
ponents increase geometrically with the number of landmarks.
In the following, we briefly discuss the exact distribution of the
shape coordinates, sometimes referred to as the Mardia-Dryden distri-
bution. For the details of the derivation, the reader should refer to
Dryden and Mardia (1991,1992). Their paper deals only with two-
dimensional objects and we restrict our attention to that case. For
generalizations and details, the reader may refer to Dryden and
Mardia (1998).
a) Definition of Kendall shape variables
Let
X
be a
K
2 landmark coordinate matrix and let
X
c
=
C
1
X
be the
centered landmark coordinate matrix where
C
1
=
diag
(
H
1
,
H
1
) and
H
1
T
is the usual Helmert matrix without the first column. Notice that
this centering matrix is different than the centering matrix L intro-
duced earlier in this chapter. Define new variables as follows:
for
i
2)
variables. These are
called Kendall Shape variables. Let us denote the shape variables by a
1,2,…,
K
2.
Notice that there are
2(
K
vector
S
, where
b) The probability density function for the Kendall Shape vari-
ables
Assume that
vec
(
X
)~
N
(
vec
(
M
),
)
.
First define new vectors
U
+
and
V
+
as follows:
U
+
V
+
=
(1,
(1,
U
1
,
U
2
,…,
U
K
-2
,0,
V
1
,
V
2
,…,
V
K
-2
)
and
V
1
,
V
2
,…,
V
K
-2
,0,
U
1
,
U
2
,…,
U
K
-2
)
. Let
M
*
*
C
T
.
C
{
vec
(
M
)},
C
Define new matrices:
é
ù
é
ù
C
=
U
+
T
*
-
1
U
+
U
+
T
*
-
1
˙
,
b=
M
*
T
*
-
1
W
W
V
+
W
U
+
ê
ê
ú
˙
V
+
T
*
-
1
V
+
T
*
-
1
M
*
T
*
-
1
W
U
+
W
V
+
W
V
+
ë
û
ë
û
M
*T
*
-1
U
+
T
C
-1
and
g
.
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