Biology Reference
In-Depth Information
Combining Eqs. (8-23) and (8-24) yields:
dG
ð
r 0 ;
X i Þ
Y i
G
ð
r 0 ;
X i Þþ
ð
r
r 0 Þ:
(8-25)
dr
We use this approximation to begin an iterative procedure and produce
a better guess, r
r 1, for the parameter. As in Newton's method, we
expect using this better guess in place of r 0 will produce an even better
guess, and so on. The process terminates when two consecutive
iterations return the same value, which is the ''answer.'' Generalizing,
we can write:
¼
dG
ð
guess
X i
Þ
;
G
ð
answer
;
X i
Þ
G
ð
guess
;
X i
Þþ
ð
answer
guess
Þ:
ð
Þ
d
guess
We now have one equation of the form (8-25) for each data point. For
illustration, assume we have only three data points:
ð
X 1
;
Y 1
Þ; ð
X 2
;
Y 2
Þ; ð
X 3
;
Y 3
Þ
. The formula would yield three equations for r:
dG
ð
r 0 ;
X 1 Þ
Y 1
¼
G
ð
r 0
X 1
Þþ
ð
r
r 0
Þ
;
dr
ð
Þ
dG
r 0
;
X 2
Y 2 ¼
G
ð
r 0 ;
X 2 Þþ
ð
r
r 0 Þ
dr
dG
ð
r 0
;
X 3
Þ
Y 3 ¼
G
ð
r 0 ;
X 3 Þþ
ð
r
r 0 Þ;
dr
which can be rewritten as:
ð
Þ
dG
r 0
;
X 1
ð
r
r 0 Þ¼
Y 1
G
ð
r 0 ;
X 1 Þ
dr
dG
ð
r 0
;
X 2
Þ
ð
r
r 0 Þ¼
Y 2
G
ð
r 0 ;
X 2 Þ
(8-26)
dr
dG
ð
r 0
;
X 3
Þ
ð
r
r 0
Þ¼
Y 3
G
ð
r 0
X 3
Þ:
;
dr
We can use matrix notation to rewrite Eq. (8-26) more compactly. If we
denote
2
4
3
5
dG
ð
r 0 ;
X 1 Þ
dr
2
3
Y 1
G
ð
r 0 ;
X 1 Þ
dG
ð
r 0 ;
X 2 Þ
4
5 ;
Y ¼
P
¼
;
Y 2
G
ð
r 0 ;
X 2 Þ
and
e ¼
r
r 0 ;
(8-27)
dr
Y 3
G
ð
r 0 ;
X 3 Þ
dG
ð
r 0 ;
X 3 Þ
dr
Eq. (8-26) can be reduced to the single matrix equation:
Y :
P
e ¼
(8-28)
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