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TEXAS A&M UNIVERSITY
DEPARTMENT OF MECHANICAL ENGINEERING
TEXAS A&M UNIVERSITY
COLLEGE STATION, TX 778433123
979 845 1251
FAX 979 845 3081
1 of 16
R. Langari, 2/12/06
Linear and Nonlinear Regression
Objective: To find the best function that can fit a given set of data
points:
•
The choice of linear or nonlinear curvefit or regression is up to the user but must
be based on some quantifiable measure(goodness of fit.)
•
Linear regression is the simplest choice but some types of nonlinear regression
can be done using the same formulations as linear regression.
x
y
1
2
3
4
5
i
x
i
y
i
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
x
5
y
5
x
1
x
2
x
3
x
4
x
5
y
1
y
5
…
x
y
x
1
x
2
x
3
x
4
x
5
y
1
y
5
DEPARTMENT OF MECHANICAL ENGINEERING
TEXAS A&M UNIVERSITY
COLLEGE STATION, TX 778433123
979 845 1251
FAX 979 845 3081
2 of 16
Linear and Nonlinear Regression
R. Langari, 02/12/06
Linear Regression
Find the best line,
, that fits a given data set:
•
For each
we have the originally given
as well as the value of
given by
(1)
•
There is usually an error,
between these two values,
, which is the
basis for finding the best
and
discussed next.
ya
xb
+
=
12
.
1
22
.
3
33
.
5
43
.
4
54
.
3
x
i
y
i
x
y
012345
1
2
3
b
4
1
a
slope
offset
e
i
y
i
yx
i
()
–
=
x
i
y
i
y
i
ax
i
b
+
=
e
i
e
i
y
i
i
–
=
a
b
DEPARTMENT OF MECHANICAL ENGINEERING
TEXAS A&M UNIVERSITY
COLLEGE STATION, TX 778433123
979 845 1251
FAX 979 845 3081
3 of 16
Linear and Nonlinear Regression
R. Langari, 02/12/06
Linear Regression Analysis
The slope,
and the offset,
of the line are found through least squares minimization of sum
of all squared errors,
s:
(2)
Substituting for
its value from (1) we have
(3)
which expands into
(4)
Now taking the derivative with respect to
and setting it to zero, we have
(5)
Similarly for
(6)
a
b
e
i
2
e
i
2
i
1
=
N
∑
y
i
i
–
2
i
1
=
N
∑
=
i
e
i
2
i
1
=
N
∑
y
i
i
b
+
–
2
i
1
=
N
∑
=
e
i
2
i
1
=
N
∑
y
i
2
i
1
=
N
∑
a
2
x
i
2
i
1
=
N
∑
2
ab
x
i
i
1
=
N
∑
Nb
2
2
y
i
i
b
+
i
1
=
N
∑
–
++
+
=
a
2
i
2
i
1
=
N
∑
2
bx
i
i
1
=
N
∑
2
y
i
x
i
i
1
=
N
∑
–
+0
=
b
2
i
i
1
=
N
∑
2
2
y
i
i
1
=
N
∑
–
=
DEPARTMENT OF MECHANICAL ENGINEERING
TEXAS A&M UNIVERSITY
COLLEGE STATION, TX 778433123
979 845 1251
FAX 979 845 3081
4 of 16
Linear and Nonlinear Regression
R. Langari, 02/12/06
Equations of Linear Regression
The two equations just derived form a set of simultaneous equations to be solved for
and
leading to the solution given next
(7)
and
(8)
Remarks:
•
We can find
and
from the above to determine the best linear fit,
for the
given data set.
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This note was uploaded on 12/26/2010 for the course MEEN 260 taught by Professor Langari during the Fall '08 term at Texas A&M.
 Fall '08
 LANGARI

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