Biology Reference
In-Depth Information
object is three. When the object is three dimensional, the first K diag-
onal elements of
V
quantify variability in the X direction, the next
K
elements quantify the variability in the Y direction, and the last
K
elements quantify the variability in the Z direction for landmarks 1
through
K
. In the measurement error study we concentrate on the
diagonal elements (variances) and ignore the off-diagonal elements
(covariances). The matrix
V
can be used to identify those landmarks
that are highly error prone.
In practice, the matrix
V
is unknown, and we have to use the obser-
vations from
N
data collection episodes to estimate it. Let
M
1
,
M
2
,...
M
N
denote the observations (the landmark coordinate matrices) from
N
data collection episodes.
We start with calculating the mean of these observations. Let
denote the sample mean matrix. Notice that
M
is a
K
by 2 or a
K
by 3 matrix, depending on the dimension of the object.
Let
A
be an
m
n
matrix. Then
A
<i>
denotes the
i
-th column of
A
.For
example,
M
i
<2>
denotes the second column of the observation
M
i
. With
this notation, for a two-dimensional object, the variance-covariance
matrix
V
is obtained using the following formula. The first equation
provides the variance-covariance matix in the X direction, the second
equation provides the variance-covariance matrix in the Y direction,
and the third one provides the covariances between X and Y directions.
N
1
V
(
MMMM
1
1
)(
1
1
)
T
X
i
i
N
i
1
N
1
2
2
2
2
T
V
(
MMMM
)(
)
Y
i
i
N
1
i
N
1
1
1
2
2
T
V
(
MMMM
)(
)
XY
,
i
i
N
i
1
Extension to three-dimensional objects is straightforward. The
additional terms needed are given by:
Search WWH ::
Custom Search