Biology Reference
In-Depth Information
Various choices are possible for W i , and what values are used depends
upon the design of the experiment. If each point is measured only once,
and there is no known distribution of the errors, it is reasonable to
assume all weights are equal to one, which makes the measure WSSR
from Eq. (8-37) equivalent to the measure SSR from Eq. (8-1). Thus,
WSSR in Eq. (8-37) is a generalization of SSR introduced in Eq. (8-1).
Similarly, the two formulas are equivalent if the measurement errors of
all data points have the same distribution, with a certain known
standard deviation. When each data point is measured only once and
there are different known errors in different measurements, it is
reasonable to assign each data point a weight that is reciprocal to its
measurement error. Thus, more uncertain data points contribute less to
the least-squares estimates. Finally, if each data point is a result of
several (
15) measurements, then we can calculate the standard error of
measurement SEM as an empirical estimate of the measurement errors,
and the weights W i can be computed as W i ¼
>
1/SEM i (see, for example,
Johnson and Frasier [1985]).
The weighted least-squares estimates for the model parameters are those
that minimize the function WSSR from Eq. (8-37). Different values of the
model parameters correspond to different values of the WSSR. For
example, Figure 8-6 is a two-dimensional contour map of the WSSR as a
function of the two-parameters of the fitting equation for dimeric
hemoglobin given in Eq. (8-7). Each of the contours represents a constant
value for the WSSR, with contours nearer the center denoting lower
WSSR values. The objective of the fitting procedure is to find the optimal
parameter values corresponding to the dot in the center, which
represents the lowest WSSR value. In the case of normally distributed
errors, these values are also called maximum likelihood solutions.
As before, the minimization procedure for determining those solutions
is based on series expansions and is not much different from the Gauss-
Newton methods described earlier for data points with equal weights.
Specifically, the Gauss-Newton method is based on the Taylor
expansion:
K 21
FIGURE 8-6.
Contour map of WSSR. The contours represent
constant values of WSSR, with the values getting
smaller as we approach the center contour.
X
@
G
ð
guesses
;
X i
Þ
G
ð
answers
;
X i Þ¼
G
ð
guesses
;
X i Þþ
ð
answer j
guess j Þþ ... ;
@
guess j
j
(8-38)
using the same notation as in Eq. (8-37). Again, Eq. (8-38) consists of
one equation for each data point, hence the subscript i. The objective
of the least-squares fitting is to determine the answers for which
Y i ¼
G
ð
answers
;
X i Þþ
experimental uncertainties
:
(8-39)
By neglecting the experimental uncertainties in Eq. (8-39), ignoring the
higher derivatives terms (i.e., the
) in Eq. (8-38), dividing Eq. (8-38) by
SEM i , and combining the results, we obtain
...
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