Graphics Programs Reference
In-Depth Information
EXAMPLE 3.9
Determine the parameters a and b so that f ( x )
ae bx fits the following datainthe
=
least-squares sense.
x
1
.
2
2
.
8
4
.
3
5
.
4
6
.
8
7
.
9
.
.
.
.
.
.
y
7
5
16
1
38
9
67
0
146
6
266
2
Use two different methods: (1) fitln y i ; and (2) fitln y i with weights W i =
y i .Compute
the standard deviationineach case.
Solution of Part (1) The problemistofit the function ln( ae bx )
=
ln a
+
bx to the data
x
1
.
2
2
.
8
4
.
3
5
.
4
6
.
8
7
.
9
z
=
ln y
2
.
015
2
.
779
3
.
661
4
.
205
4
.
988
5
.
584
We are nowdealing with linear regression, where the parameterstobefound are
A
ln a and b .Following the steps in Example 3.8, we get (skipping some of the
arithmetic details)
=
6 x i =
6 z i =
1
1
x
=
4
.
733
z
=
3
.
872
z i ( x i
x )
16
.
716
b
=
x i ( x i
=
153 =
0
.
5366
A
=
z
xb
=
1
.
3323
x )
31
.
e A
790 e 0 . 5366 . The
Therefore, a
=
=
3
.
790 and the fitting functionbecomes f ( x )
=
3
.
plots of f ( x ) and the datapoints are shown in the figure.
300
250
200
150
y
100
50
0
1
2
3
4
5
6
7
8
x
Here is the computation of standard deviation:
y
7
.
50
16
.
10
38
.
90
67
.
00
146
.
60
266
.
20
f ( x )
7
.
21
17
.
02
38
.
07
68
.
69
145
.
60
262
.
72
.
.
.
.
.
.
y
f ( x )
0
29
0
92
0
83
1
69
1
00
3
48
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