Biology Reference
In-Depth Information
Example 8-4
.......................
We use the Gauss-Newton algorithm to estimate parameter r in the
model G(r;t)
5.3 e
rt
¼
from the U.S. population data in Table 8-3.
The matrices are
2
3
2
3
2
3
2
3
3e
rt
0
3e
rt
0
t
0
5
:
0
P
0
5
:
0
4
5
4
5
4
5
4
5
3e
rt
1
e
r
3e
rt
1
e
r
t
1
5
:
1
ð
5
:
3
Þ
P
1
5
:
7
:
2
ð
5
:
3
Þ
3e
rt
2
e
2r
3e
rt
2
e
2r
t
2
5
:
2
ð
5
:
3
Þ
P
2
5
:
9
:
6
ð
5
:
3
Þ
and Y
¼
3e
rt
3
e
3r
3e
rt
3
e
3r
P
¼
t
3
5
:
¼
3
ð
5
:
3
Þ
P
3
5
:
¼
12
:
9
ð
5
:
3
Þ
:
3e
rt
4
e
4r
3e
rt
4
e
4r
t
4
5
:
4
ð
5
:
3
Þ
P
4
5
:
17
:
1
ð
5
:
3
Þ
3e
rt
5
e
5r
3e
rt
5
e
5r
t
5
5
:
5
ð
5
:
3
Þ
P
5
5
:
23
:
2
ð
5
:
3
Þ
3e
rt
6
e
6r
3e
rt
6
e
6r
t
6
5
:
6
ð
5
:
3
Þ
P
6
5
:
31
:
4
ð
5
:
3
Þ
(8-32)
We begin with an initial guess of r
0
¼
0.3 and compute
Þ
1
P
T
P
P
T
Y
Þ¼
e¼ð
ð
0
:
0040379399. The improved guess, r
1
, is then
calculated to be
r
1
¼
r
0
þe ¼
0
:
3
0
:
0040379399
¼
0
:
295962601
:
Using this value for r in Eq. (8-31), we compute
e¼ð
Þ
1
P
T
P
P
T
Y
Þ¼
ð
0
:
000048084. The value for the next guess will
now be
r
2
¼
r
1
þe ¼
0
:
295962601
0
:
000048084
¼
0
:
295914517
:
The process continues until the desired accuracy is achieved. Assume
we only want to calculate an answer accurate to at least three decimal
places. The value
e ¼
0.000048084 calculated above is then
e ¼
0.000, and,
therefore, r
2
is the answer.
3
C. The Gauss-Newton Method for Two and More Variables
For models involving two or more parameters, the idea behind the
Gauss-Newton method is the same, but the matrices P and Y* need
to be changed appropriately. We outline the process for two parameters
and then discuss how it generalizes for an arbitrary number of
parameters.
¼
¼
Consider the model Y
G(r,c;X), where the goal is to
find the least-squares values of the parameters c and r estimated from
the data
G(parameters; X)
ð
X
1
;
Y
1
Þ; ð
X
2
;
Y
2
Þ;
...
; ð
X
n
;
Y
n
Þ
.
3. Compare this value with the least-squares value for r we obtained for the same
model and data set earlier (Table 8-5) using Newton's method.
