Biology Reference
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is a
K
K
symmetric matrix.
STEP 8: Calculate the eigenvalues and eigenvectors of
B
(
M
)
.
Let us denote its eigenvalues arranged in a decreasing order
by
1
,
2
,…,
K
and the corresponding eigenvectors by
h
1
,
h
2
,…,
h
n
.
STEP 9: If the original data are from two-dimensional objects,
the estimate of the mean form matrix is given by
, and if the original data are from three-
dimensional objects, the estimate of the mean form matrix is
given by
STEP 10: Calculate the form matrix (the matrix of all pair-
wise distances) for the above landmark coordinate matrix.
This is an improved estimator of the mean form matrix.
Note:
The
M
obtained in STEP 3 can be used to graphically repre-
sent the mean form of the sample. It should be remembered that this
estimator is a representation of the mean form
M
only up to rotation,
reflection, and translation. In practice, to obtain a graphical represen-
tation that is biologically meaningful, one may need to reflect the
above estimator by multiplying one or more of the axes by -1.
Having obtained the estimator of the mean form matrix
M
described
above, it is fairly simple to obtain the estimator of the covariance
matrix,
K
*
. The following steps provide the details.
b) Algorithm for estimation of
K
*
STEP 1: Calculate the centered matrices
X
1
c
,
X
2
c
,…,
X
n
c
corre-
sponding to the observations
X
1
,
X
2
,…,
X
n
.
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