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)