Biology Reference
In-Depth Information
E
XERCISE
8-1
Derive Eq. (8-5) for the solution (a,b) of the system of linear Eq. (8-4).
E
XERCISE
8-2
Derive a formula for the least-squares value of the parameter a in the
linear model Y
¼
G
ð
a
X
Þ¼
aX
:
;
E
XERCISE
8-3
X
0.25
0.5
0.75
1.0
Y
1.3
2.7
3.3
5.1
Consider the data in Table 8-1, collected to fit the model Y
¼
aX
þ
b.
TABLE 8-1.
Data for exercise 8-3.
Determine the least-square estimates for parameters a and b.
Thus far, we have considered the specific linear model
Y
The principle definitions of the least-squares
measure remain the same, however, for any model given by a function y
of the form Y
¼
aX
þ
b
¼
G
ð
a
;
b
X
Þ:
;
G (parameters; X). Consider, for example, Eq. (8-6),
a rewritten form of Eq. (7-24) from Chapter 7:
¼
2K
22
X
2
1
2
K
21
X
þ
Y
¼
K
22
X
2
¼
G
ð
K
21
;
K
22
;
X
Þ:
(8-6)
1
þ
K
21
X
þ
In this case, the measured quantity is the fractional saturation Y, the
dependent variable. The experimentally manipulated quantity, the
oxygen concentration X, is the independent variable. The model G
defined by Eq. (8-6) has two parameters, K
21
and K
22,
to be adjusted by
the data-fitting procedure.
Using the model from Eq. (8-6), we can write
2K
22
X
i
1
2
K
21
X
i
þ
ð
;
Þ¼
Y
i
G
K
21
K
22
;
X
i
(8-7)
K
22
X
i
1
þ
K
21
X
i
þ
where the best possible fit between the model and the data is determined
by minimizing the sum of squared residuals
X
2
SSR
ð
K
21
;
K
22
Þ¼
½
Y
i
G
ð
K
21
;
K
22
;
X
i
Þ
:
(8-8)
i
Equation (8-6) is assumed to hold only approximately because it ignores the
measurement errors always present in the experimental values of Y
i
.By
finding the values for K
21
and K
22
that minimize the SSR in Eq. (8-8), we
find the best possible approximation for the entire set of data points.
Notice that in this case the function SSR is not a linear function of
K
21
and K
22
.
