Biology Reference

In-Depth Information

Relative form difference

The relative difference between forms is another useful way of describ-

ing form difference in biological objects. When written as a matrix, the

collection of relative differences is called the form difference matrix.

Our emphasis on this particular expression of form difference has

resulted in the following notation,

, where

FDM (B,A)

elements of the form difference matrix correspond to the ratios of like-

linear distances, and the division is done element-wise (where
0

0

by definition). Continuing with the data sets described above, the form

difference matrix for forms A and B is written:

0

02 1

20

FDM B A

(, )

.

9

1

1 59

.

0

Because form difference matrices are square, symmetric matrices

with zero along the diagonal, only the above diagonal elements are

needed to describe the form difference. As a matter of convention, we

write the form difference matrix in vector form, where only the above-

diagonal elements are reported:

2

1

159

FDM B A

(, )

.

Scaling factors and differences in scaled forms

In many biological studies, it is helpful to adjust for size differences

according to some scaling factor before comparison. Within the context

of landmark coordinate data, one can use various measures as scaling

factors. For example, a specific biological distance might be used, or the

geometric mean of all distances could be used as a scaling factor.

Mosimann (1979) provided a general mathematical definition of scal-

ing factor and referred to scaling factors as a measure of 'size.'

Following these ideas, the scaled form is referred to as 'size-corrected.'

Comparison of size-corrected forms is often considered a comparison of

'shapes.'

Following Mosimann (1979), we consider any function of the dis-

tances that always takes a positive value as a scaling factor. However,

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