1
The University of Texas at Austin
PGE 310: Formulation and Solution in Geosystems Engineering
Homework #8: Curve Fitting and Interpolation
SOLUTION
By HosseinRoodi
1.
Curve Fitting:
(By HAND)
A graduate student is looking for an analytical equation that can describe his lab data the best. In his experimental
tests, he is changing parameter x, which can be something like the temperature of the test, and he is measuring
parameter y, which can be something like the stress in the specimens. After investigation of different types of
equations, he has come up with 4 final choices as follows:
a)
1
1
b
x
a
y
!
"
1
1
b
x
a
y
!
"
b)
x
b
e
a
y
2
2
"
x
b
a
y
2
2
)
ln(
)
ln(
!
"
c)
3
3
b
x
a
y
"
)
log(
)
log(
)
log(
3
3
x
b
a
y
!
"
No.
x
y
X =x
Y=y
XY
X
2
Y
2
y
pred
(y
pred
"
y)
2
1
5
2.760
5
2.760
13.800
25
7.618
2.839
0.006
2
10
3.050
10
3.050
30.500
100
9.303
3.070
0.000
3
15
3.577
15
3.577
53.655
225
12.795
3.301
0.076
4
20
3.355
20
3.355
67.100
400
11.256
3.532
0.031
#
50
12.742
165.055
750
40.971
#
12.742
0.114
Slope =(a
1
)=
0.046
a
1
=
0.046
S
r
=
0.114
Interc =(b
1
)=
2.608
b
1
=
2.608
r
2
=
0.701
Error Calc
y=a
1
x+b
1
Part a
1
1
b
x
a
y
!
"
No.
x
y
X =x
Y=ln(y)
XY
X
2
Y
2
y
pred
(y
pred
"
y)
2
1
5
2.760
5
1.015
5.076
25
1.031
2.835
0.006
2
10
3.050
10
1.115
11.151
100
1.244
3.054
0.000
3
15
3.577
15
1.275
19.118
225
1.624
3.291
0.082
4
20
3.355
20
1.210
24.209
400
1.465
3.545
0.036
#
50
4.615
59.554
750
5.364
#
12.725
0.124
Slope = (b
2
) =
0.015
a
2
=
2.632
S
r
=
0.124
Interc = ln(a
2
) =
0.968
b
2
=
0.015
r
2
=
0.721
Part b
Error Calc
ln(y)=b
2
x+ln(a
2
)
x
b
e
a
y
2
2
"
No.
x
y
X =log(x)
Y=log(y)
XY
X
2
Y
2
y
pred
(y
pred
"
y)
2
1
5
2.760
0.699
0.441
0.308
0.489
0.194
2.770
0.000
2
10
3.050
1.000
0.484
0.484
1
0.235
3.116
0.004
3
15
3.577
1.176
0.554
0.651
1.38319
0.306
3.338
0.057
4
20
3.355
1.301
0.526
0.684
1.69268
0.276
3.506
0.023
#
4.176
2.004
2.127
4.56443
1.012
#
12.730
0.084
Slope = (b
3
) =
0.170
a
3
=
2.107
S
r
=
0.084
Interc = log(a
3
) =
0.324
b
3
=
0.170
r
2
=
0.814
Part c
log(y)=b
3
log(x)+log(a
3
)
Error Calc
3
3
b
x
a
y
"
X
Y
interceptslope
X
Y
interceptslope
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2
d)
x
b
x
a
y
!
"
4
4
4
4
4
1
1
1
a
x
a
b
y
!
"
Note that although it is possible to calculate the error, (S
r
), in XY space (i.e
2
))
(
)
(
(
i
Ypredict
i
Y
$
), but as shown
in the above tables, it would be better to obtain it in original space xy (i.e.
2
))
(
)
(
(
i
ypredict
i
y
$
).
Here is a good sample of his measured data:
%
&
%
&
355
.
3
577
.
3
050
.
3
760
.
2
20
15
10
5
"
"
y
x
Basedon this sample data help him to find which equation fits better to the data. Use linearization and find the
coefficients of the fitted curve for each type of equations.
Then compare these four types of equations based on the
r
S
and
2
r
values.
Based on the obtained values for
S
r
and
r
2
it is obvious that Equation 4 can represents the best (among the four
equations) fit to the data. Because it returns larger value for
r
2
and lower value for
S
r
.
You should do this problem by HAND. You may use EXCEL to make your calculations easy and faster, but you are
not supposed to use curve fitting feature of EXCEL.
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 Spring '06
 Klaus
 Coefficient, 20 30 40, Polynomial interpolation, order polynomial

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