Biology Reference

In-Depth Information

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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