Biology Reference
In-Depth Information
Hint: Notice that
0
1
X
SEM
i
@
1
G
ð
guesses
;
X
i
Þ
½
Y
i
G
ð
guesses
;
X
i
Þ
@
A
@
guess
1
i
X
SEM
i
@
1
G
ð
guesses
;
X
i
Þ
P
T
Y
¼
;
(8-47)
½
Y
i
G
ð
guesses
;
X
i
Þ
@
guess
2
i
.
and that the individual elements of the P
T
Y* are proportional to the
derivatives of the WSSR with respect to each of the parameters being
estimated.
We need to stress that the Gauss-Newton approach is not guaranteed to
converge. If the higher-order terms (
in Eq. [8-38]) do not converge
sufficiently rapidly to zero, then this algorithm might actually diverge,
because if higher-order terms cannot be ignored, then their omission in
Eq. (8-40) might cause irreparable error. The Gauss-Newton approach
will converge rapidly in most cases, and, when it does not, there are
many adaptations, such as the Marquardt-Levenberg and damped
Gauss-Newton algorithms, which specifically correct for the failure to
converge (see Johnson and Frasier [1985]). The damped Gauss-Newton
algorithm simply checks that the new value of the WSSR, Eq. (8-37), is
lower for the guesses
...
þ e
than it was for the previous guesses.Ifitisnot
lower, then
e
was too big—so it is divided by 2, and this new value
of
e
is used in Eq. (8-46). This process of dividing
e
by 2 if the WSSR has
increased is repeated until it decreases.
There are many weighted nonlinear least-squares algorithms in addition
to the Gauss-Newton. Some converge faster, and some require more
computer memory; but when correctly implemented, they all provide
equivalent results.
V. OBJECTIVES OF THE DATA-FITTING PROCEDURES
We have explained why the data-fitting procedure provides parameter
values affording the best description of a data set and have described
some computational methods for finding the best least-squares fit. To
obtain a complete analysis of the experimental data, any data-fitting
procedure will have multiple objectives, which include estimating:
1. Optimal model parameters with respect to the desired criteria;
2. Cross-correlation of the estimated model parameters;
3. Precision of the model parameters;
