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.
Remark : Let us refer to Table 2.1 . In that particular measurement
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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