Biology Reference
In-Depth Information
Eq. (8-21) are presented in Table 8-5(A). Thus, the least-squares value for
r is r*
A:
Iteration i
Guess x
i
0.3 was rather close
to the root. If we start with a less accurate guess, more iterations may
be required before finding the root, as illustrated in Table 8-5(B) with the
guess x
0
¼
¼
0.29591448299395. Our initial guess of x
0
¼
0
0.3
1
0.29604731216119
0.5. In addition, because nonlinear equations may have more
than one solution, if the initial guess is chosen at random, the method
may converge to a false root, finding a minimum for the SSR that results
in a value for r that is meaningless in the context of the problem.
2
0.29591462767617
3
0.29591448299412
4
0.29591448299395
5
0.29591448299395
In principle, Newton's method for one variable can be generalized to
two or more variables and used to solve systems of nonlinear equations,
such as the system defined by Eq. (8-14). However, for several reasons,
such methods are not computationally optimal for determining the
least-squares parameter estimates. First, minimizing the SSR function
requires that its partial derivatives with respect to the model parameters
be calculated and set to zero. This process may lead to complicated
systems of equations, where the lengths of the algebraic expressions
grow with the number of experimental data points. Second, to use
Newton's method to solve these equations would require yet another
differentiation. That is, the cumbersome expressions defining the
systems of equations for the parameters will need to be differentiated
again, and their derivatives used to calculate the iterations
approximating the solutions. Instead, improved versions of the
procedures, such as the following, are usually applied.
6
0.29591448299395
B:
Iteration i
Guess x
i
0
0.5
1
0.42811073165450
2
0.36676718238807
3
0.32253692486329
4
0.30074060321933
5
0.29609881338742
6
0.29591476150979
7
0.29591448299458
8
0.29591448299395
9
0.29591448299395
TABLE 8-5.
Role of the initial guess on outcome of Newton's
method.
B.TheGauss-NewtonMethodforOneVariable
We consider a one-parameter model of the general form Y
¼
G(r;X),
experimental data
ð
X
1
;
Y
1
Þ; ð
X
2
;
Y
2
Þ;
...
; ð
X
n
;
Y
n
Þ
and the SSR, which in
this case is defined by
X
2
SSR
ð
r
Þ¼
1
½
Y
i
G
ð
r
X
i
Þ
:
(8-22)
;
¼
i
As with the Newton's method, the process is iterative. We begin by
making a guess r
r
0
of the parameter's value. Next, we use the Taylor
series approximation for G(r; X), as in Eq. (8-16). The variable of
interest is r, so we obtain
¼
dG
ð
r
0
X
i
Þ
;
G
ð
r
X
i
Þ
G
ð
r
0
;
X
i
Þþ
ð
r
r
0
Þ:
(8-23)
;
dr
¼
Because we assume the model Y
G(r;X) is correct, we seek the value
for the parameter r for which:
Y
i
¼
G
ð
r
;
X
i
Þþ
experimental uncertainties
:
Ignoring the experimental uncertainties, we write:
Y
i
G
ð
r
;
X
i
Þ:
(8-24)
