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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