Biology Reference

In-Depth Information

In the following, we provide detailed computational algorithms for

estimation of the mean form matrix and variance-covariance matrix.

In order to program these algorithms, the reader will need to under-

stand the following concepts and related computation procedures:

Form matrix, Centered landmark coordinate matrix, Centered inner

product matrix, Eigenvalues, and Eigenvectors. Except for the concepts

of eigenvalues and eigenvectors, all other concepts are described below.

For details on eigenvalues and eigenvectors of a square, symmetric

matrix, refer to any standard matrix algebra textbook such as Barnett

(1990). For computational algorithms, see Press et al., (1986).

Definition of a centered landmark coordinate matrix:
Let
A
be a

landmark coordinate matrix. First, calculate the mean of the

first column and subtract it from all the elements of the first col-

umn; calculate the mean of the second column and subtract it

from all the elements of the second column, and calculate the

mean of the third column and subtract it from all the elements

of the third column. The resultant matrix, denoted by
A
c
,is

called the centered landmark coordinate matrix.

For example, let the landmark coordinate matrix be

Then the centered landmark coordinate matrix is given by subtracting

from each of the entry in a column by the average of the corresponding

column in the landmark coordinate matrix
A.
Thus

is the centered landmark coordinate matrix.

Notice that the column sum of
A
c
is always 0. Notice that this matrix

is obtained when one shifts the original triangle such that its centroid,

instead of matching landmark 1, matches with (0,0).

Centered inner product matrix:
Let
A
c
be a centered landmark

coordinate matrix. Define a new matrix
B

A
c
(
A
c
)
T
. This matrix

is called the centered inner product matrix corresponding to
A
.

The centered inner product matrix corresponding to the above

example is given by

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