Biology Reference
In-Depth Information
E
XERCISE
8-8
(a) Show that the set of equations from Eq. (8-40) can be written in
matrix form as
Y
;
P
e ¼
(8-41)
where the matrices P, Y*, and
e
are :
0
@
1
A
SEM
1
@
1
G
ð
guesses
;
X
1
Þ
SEM
1
@
1
G
ð
guesses
;
X
1
Þ
...
@
guess
1
@
guess
2
P
¼
(8-42)
SEM
2
@
1
G
ð
guesses
;
X
2
Þ
SEM
2
@
1
G
ð
guesses
;
X
2
Þ
...
@
guess
1
@
guess
2
...
...
...
0
1
Y
1
G
ð
guesses
;
X
1
Þ
0
@
1
A
@
A
answer
1
guess
1
SEM
1
Y
¼
e ¼
answer
2
guess
2
and
:
Y
2
G
ð
guesses
;
X
2
Þ
...
SEM
2
...
(8-43)
The number of rows for the matrices P and Y* are determined by
the number of data points. The number of parameters determines
the number of columns for P. The same is true for the number
of rows for the matrix
e
.
(b) Show that the representation in Eq. (8-41) leads to
P
T
P
P
T
Y
Þ;
ð
Þe ¼ð
(8-44)
Þ
1
P
T
P
P
T
Y
Þ:
e ¼ð
ð
(8-45)
The iterative scheme
e þ
guess
)
better guess
;
(8-46)
terminates when
e
is found.
E
XERCISE
8-9
Show that when it converges, the Gauss-Newton algorithm from
Eq. (8-40) produces parameter estimates that minimize the function
WSSR defined in Eq. (8-37).
