Biology Reference

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Notice that the first three diagonal elements of this matrix corre-

spond to the variance in the X direction for landmark 1, 2, and 3

respectively. Similarly the next three diagonal elements of this matrix

correspond to the variance in the Y direction for landmark 1, 2, and 3

respectively. Other elements (called the off-diagonal elements) provide

information on the covariances. Such a matrix
V
is called either a “vari-

ance-covariance matrix,” a “covariance matrix,” or a “variance matrix.”

All three names are acceptable. In our application the matrix
V
sum-

marizes the measurement errors. This matrix
V
is a 2
K

2
K
matrix

where
K
, the number of landmarks is 3 and the dimension of the object

is 2.

error study, we have 11 landmarks and the
Macaca fascicularis
skull

is a three-dimensional object. The resultant covariance matrix is a 33

by 33 symmetric matrix. This table provides only the square root

(standard deviations) of the diagonal elements of the covariance

matrix. For example, the first entry in the f 0.249, is equal to

V
X,11
, the standard deviation of the measurement error for landmark

1 along the X direction. The remaining entries in the column include

the values of

V
X,33
, and so on.

Exercise
1) Verify that if the object under study is two-dimension-

al and has
K
landmarks, the variance-covariance matrix
V
is a 2
K

V
X,22
,

2
K

square, symmetric matrix.

Exercise
2) Verify that if the object under study is three-dimen-

sional and has
K
landmarks the variance-covariance matrix
V
is a 3
K

3
K
square, symmetric matrix. Notice that for three dimensional

objects,

.

The observations just presented, that the true landmark coordinate

matrix is
M
, that measurement errors
E
i
are Normally distributed with

variance covariance matrix
V
, and that
M
i

E
i
, are summarized

by the notation:
M
i
~ N
(
M,V
). We read this as “
M
i
follows a matrix

Normal distribution with mean matrix
M
and the variance-covariance

matrix (or simply variance)
V.
”

Remember that for a two dimensional object
M
is a
K
by 2 matrix,

and for a three-dimensional object, it is a
K
by 3 matrix. The variance

matrix
V
is a square, symmetric matrix of dimension 2
K

M

2
K
when the

dimension of the object is two, and 3
K

3
K
when the dimension of the

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