Biology Reference
In-Depth Information
In
Chapter 4
we presented three possible ways to express form dif-
ference using the form matrices. We have found that relative change is
a useful way of describing differences that occur due to growth and
therefore present that approach exclusively. The other approaches
described in
Chapter 4
could be used to study growth, but appropriate
precautions may be necessary when interpreting the results.
5.5 Growth measured as relative form difference
As described previously, the relative difference between two forms
FM
()
()
A
ij
2
FDM A
(,
A
)
can be written as a form difference matrix,
21
FM
A
ij
1
where the elements of the matrix correspond to the ratio of like-linear
distances and the division is done element-by-element. In this applica-
tion, the forms being compared are members of a growth series, so we
FM
()
()
,
A
ij
2
define the growth matrix (GM) as
GM A
(,
A
)
. Note the
21
FM
A
ij
1
similarity in notation and organization of the
FDM
and the
GM
. Note
also that the older (i.e., larger) form occupies the numerator position.
We adopt this convention because it is intuitive to think of values
greater than 1 corresponding with a form getting larger through time.
Adopting this convention means that when a linear distance increases
in size with advancing age, the ratio corresponding to growth of that
linear distance is greater than 1. This convention will be followed here
but does not need to be followed by users.
0
3
2 136
.
.
GM A
(,
A
)
3
0
2 13
21
2 136
.
2 13
.
0
Continuing with the example data sets given previously:
Each entry represents the ratio of like linear distances in
FM (A
1
)
and
FM (A
2
)
. Because growth matrices are square-symmetric matrices with
zeros along the diagonal, only the above-diagonal elements are needed
to describe form difference due to growth. We write the growth matrix as
a vector where only the above-diagonal elements are reported:
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