Graphics Programs Reference
In-Depth Information
This discrepancy can belargely eliminatedbyweighting the logarithmic fit.We
note fromEq. (3.29b)thatln(
r
i
−
ln(
ae
bx
i
)
y
i
)
=
=
ln
a
+
bx
i
,sothat Eq. (3.29a)can
be writtenas
ln
1
r
i
y
i
R
i
=
ln
y
i
−
ln(
r
i
−
y
i
)
=
−
If the residuals
r
i
aresufficiently small (
r
i
<<
y
i
), wecanuse the approximation ln(1
−
r
i
/
y
i
)
r
i
/
y
i
,sothat
≈
y
i
Wecan now see that by minimizing
R
i
, we inadvertentlyintroduced the weights
1
R
i
≈
r
i
/
y
i
. This effect can be negatedif we apply the weights
y
i
whenfitting
F
(
x
)to
(ln
y
i
,
/
x
i
); that is, by minimizing
n
y
i
R
i
S
=
(3.30)
i
=
1
Other examples that also benefitfrom the weights
W
i
=
y
i
are giveninTable 3.3.
f
(
x
)
F
(
x
)
Data to be fittedby
F
(
x
)
x
i
,
x
i
)
axe
bx
ln
[
f
(
x
)
/
x
]
=
ln
a
+
bx
ln(
y
i
/
ln
x
i
,
ln
y
i
ax
b
ln
f
(
x
)
=
ln
a
+
b
ln(
x
)
Table 3.3
EXAMPLE 3.8
Fit a straight linetothedata shown and compute the standard deviation.
x
0
.
0
1
.
0
2
.
0
2
.
5
3
.
0
y
2
.
9
3
.
7
4
.
1
4
.
4
5
.
0
Solution
The averages of the dataare
5
x
i
=
1
0
.
0
+
1
.
0
+
2
.
0
+
2
.
5
+
3
.
0
x
=
=
1
.
7
5
5
y
i
=
1
2
.
9
+
3
.
7
+
4
.
1
+
4
.
4
+
5
.
0
y
=
=
4
.
02
5
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