Biology Reference
In-Depth Information
Example 8-5
.......................
We now use the Gauss-Newton algorithm to estimate parameters r and c
in the model G(r, c; t)
ce
rt
¼
from the U.S. population data in Table 8-3.
Now
@
G
@
cte
rt
and
@
G
@
e
rt
, and the matrices P and Y* become
r
¼
c
¼
2
4
3
5
2
4
3
5
ce
rt
e
r
:
ce
r
7
2
2
4
3
5
¼
2
4
3
5
¼
2ce
2r
e
2r
9
:
6
ce
2r
ct
1
e
rt
1
e
rt
1
ce
rt
1
P
1
3ce
3r
e
3r
12
:
6
ce
3r
.
.
.
and Y
¼
P
¼
:
ce
4r
4ce
4r
e
4r
17
:
1
P
6
ce
rt
61
ct
6
e
rt
6
e
rt
6
ce
5r
5ce
5r
e
5r
23
:
2
ce
6r
6ce
6r
e
6r
31
:
4
Choosing initial guesses of c
0
¼
5 and r
0
¼
0.3 in the matrices above gives
P
T
Y
Þ¼
:
0
0009
Þ
1
P
T
P
e ¼ð
ð
;
0
:
2053
r
r
0
where now
e
is the vector
e ¼
:
c
c
0
Thus, the next guesses for the parameters are
r
1
¼
r
0
0
:
0009
¼
0
:
2991 and c
1
¼
c
0
þ
0
:
2053
¼
5
:
2053
:
Substituting these values for r and c gives
0
:
0000157
0
P
T
P
Þ
1
P
T
Y
Þ¼
e ¼ð
ð
:
:
00016
With three digits of accuracy, we can terminate the process at this step
and use the values r
1
and c
1
as the least-squares estimates for the
parameters based on the data in Table 8-3.
When models involve more than two parameters, the notation becomes
quite cumbersome. Describing the steps of the computational process
becomes a bit easier if we think of a set of guesses, one for each
parameter, being produced at each step in the search for the set of
answers that minimize the SSR. Eq. (8-34), for instance, can be written as
G
ð
answers
X
i
Þ
;
Þþ
@
G
ð
guesses
X
i
Þ
;
G
ð
guesses
X
i
ð
answer
1
guess
1
Þ
;
@
guess
1
þ
@
G
ð
guesses
X
i
Þ
;
ð
answer
2
guess
2
Þ:
@
guess
2
Here, answer
1
and guess
1
refer to the answer and guess for the first
parameter (r), and answer
2
and guess
2
refer to the answer and guess for
the second parameter (c). Using
S
-notation, we write
