Linear regression: introduction and method
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Sep 07, 2024
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About This Presentation
Foundation Theory of Linear regression for engineers and scientists
Slide Content
7/17/2017 http://numericalmethods.eng.usf.edu 1
Linear Regression
Major: All Engineering Majors
Authors: Autar Kaw, Luke Snyder
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
Linear Regression
Major: All Engineering Majors
Authors: Autar Kaw, Luke Snyder
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
Linear Regression
http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu 3
What is Regression?
What is regression?Given ndata points
best fit )(xfy= to the data.
Residual at each point is
)(xfy=
Figure.Basic model for regression
),(),......,,(),,(
2211 nn
yxyxyx
)(
iii
xfyE −=
y
x
),(
11
yx
),(
nn
yx
),(
ii
yx
)(
iii xfyE −=
What is Regression?
What is regression?Given ndata points
best fit )(xfy= to the data.
Residual at each point is
)(xfy=
Figure.Basic model for regression
),(),......,,(),,(
2211 nn
yxyxyx
)(
iii
xfyE −=
y
x
),(
11
yx
),(
nn
yx
),(
ii
yx
)(
iii xfyE −=
http://numericalmethods.eng.usf.edu 4
Linear Regression-Criterion#1
Given ndata points best fit xaay
10
+= to the data.
Does minimizing∑
=
n
i
iE
1
work as a criterion?
x
xaay
10+=
),(
11
yx
),(
22
yx
),(
33yx
),(
nn
yx
),(
ii
yx
iii
xaayE
10−−=
y
Figure.Linear regression of yvs xdata showing residuals at a typical point, x
i
.
),(),......,,(),,(
2211 nn
yxyxyx
Linear Regression-Criterion#1
Given ndata points best fit xaay
10
+= to the data.
Does minimizing∑
=
n
i
iE
1
work as a criterion?
x
xaay
10+=
),(
11
yx
),(
22
yx
),(
33yx
),(
nn
yx
),(
ii
yx
iii
xaayE
10−−=
y
Figure.Linear regression of yvs xdata showing residuals at a typical point, x
i
.
),(),......,,(),,(
2211 nn
yxyxyx
http://numericalmethods.eng.usf.edu 5
Example for Criterion#1
x y
2.0 4.0
3.0 6.0
2.0 6.0
3.0 8.0
Example: Given the data points (2,4), (3,6), (2,6) and (3,8), best fit
the data to a straight line using Criterion#1
Figure. Data points for yvs xdata.
Table. Data Points
0
2
4
6
8
10
0 1 2 3 4
y
x
Minimize∑
=
n
i
i
E
1
Example for Criterion#1
x y
2.0 4.0
3.0 6.0
2.0 6.0
3.0 8.0
Example: Given the data points (2,4), (3,6), (2,6) and (3,8), best fit
the data to a straight line using Criterion#1
Figure. Data points for yvs xdata.
Table. Data Points
0
2
4
6
8
10
0 1 2 3 4
y
x
Minimize∑
=
n
i
i
E
1
http://numericalmethods.eng.usf.edu 6
Linear Regression-Criteria#1
0
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y=4x − 4 and yvs xdata
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
Linear Regression-Criteria#1
0
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y=4x − 4 and yvs xdata
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
http://numericalmethods.eng.usf.edu 7
Linear Regression-Criterion#1
x y y
predictedE = y -y
predicted
2.0 4.0 6.0 -2.0
3.0 6.0 6.0 0.0
2.0 6.0 6.0 0.0
3.0 8.0 6.0 2.0
0
4
1
=∑
=i
i
E
0
2
4
6
8
10
0 1 2 3 4
y
x
Table.Residuals at each point
for regression model y=6
Figure. Regression curve y=6 and y vs xdata
Using y=6 as a regression curve
Linear Regression-Criterion#1
x y y
predictedE = y -y
predicted
2.0 4.0 6.0 -2.0
3.0 6.0 6.0 0.0
2.0 6.0 6.0 0.0
3.0 8.0 6.0 2.0
0
4
1
=∑
=i
i
E
0
2
4
6
8
10
0 1 2 3 4
y
x
Table.Residuals at each point
for regression model y=6
Figure. Regression curve y=6 and y vs xdata
Using y=6 as a regression curve
http://numericalmethods.eng.usf.edu 8
Linear Regression –Criterion #1
0
4
1
=∑
=i
i
E
for both regression models of y=4x-4 and y=6
The sum of the residuals is minimized, in this case it is zero,
but the regression model is not unique.
Hence the criterion of minimizing the sum of the residuals is a
bad criterion.
0
2
4
6
8
10
0 1 2 3 4
y
x
Linear Regression –Criterion #1
0
4
1
=∑
=i
i
E
for both regression models of y=4x-4 and y=6
The sum of the residuals is minimized, in this case it is zero,
but the regression model is not unique.
Hence the criterion of minimizing the sum of the residuals is a
bad criterion.
0
2
4
6
8
10
0 1 2 3 4
y
x
http://numericalmethods.eng.usf.edu 9
Linear Regression-Criterion#1
0
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y=4x-4 and yvs xdata
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
Linear Regression-Criterion#1
0
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y=4x-4 and yvs xdata
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
http://numericalmethods.eng.usf.edu 10
Linear Regression-Criterion#2
x
),(
11
yx
),(
22yx
),(
33yx
),(
nn
yx
),(
ii
yx
iii xaayE
10−−=
y
Figure.Linear regression of yvs. xdata showing residuals at a typical point, x
i
.
Will minimizing ||
1
∑
=
n
i
i
E
work any better?
xaay
10+=
Linear Regression-Criterion#2
x
),(
11
yx
),(
22yx
),(
33yx
),(
nn
yx
),(
ii
yx
iii xaayE
10−−=
y
Figure.Linear regression of yvs. xdata showing residuals at a typical point, x
i
.
Will minimizing ||
1
∑
=
n
i
i
E
work any better?
xaay
10+=
http://numericalmethods.eng.usf.edu 11
Example for Criterion#2
x y
2.0 4.0
3.0 6.0
2.0 6.0
3.0 8.0
Example: Given the data points (2,4), (3,6), (2,6) and (3,8), best fit
the data to a straight line using Criterion#2
Figure. Data points for yvs. xdata.
Table. Data Points
0
2
4
6
8
10
0 1 2 3 4
y
x
Minimize∑
=
n
i
i
E
1
||
Example for Criterion#2
x y
2.0 4.0
3.0 6.0
2.0 6.0
3.0 8.0
Example: Given the data points (2,4), (3,6), (2,6) and (3,8), best fit
the data to a straight line using Criterion#2
Figure. Data points for yvs. xdata.
Table. Data Points
0
2
4
6
8
10
0 1 2 3 4
y
x
Minimize∑
=
n
i
i
E
1
||
http://numericalmethods.eng.usf.edu 12
Linear Regression-Criterion#2
4||
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y= y=4x − 4 and yvs. x
data
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
Linear Regression-Criterion#2
4||
4
1
=∑
=i
i
E
x y y
predicted
E = y -y
predicted
2.0 4.0 4.0 0.0
3.0 6.0 8.0 -2.0
2.0 6.0 4.0 2.0
3.0 8.0 8.0 0.0
Table. Residuals at each point
for regression model y=4x − 4
Figure.Regression curve y= y=4x − 4 and yvs. x
data
0
2
4
6
8
10
0 1 2 3 4
y
x
Using y=4x − 4 as the regression curve
http://numericalmethods.eng.usf.edu 13
Linear Regression-Criterion#2
x y y
predictedE = y -y
predicted
2.0 4.0 6.0 -2.0
3.0 6.0 6.0 0.0
2.0 6.0 6.0 0.0
3.0 8.0 6.0 2.0
4||
4
1
=∑
=i
i
E
0
2
4
6
8
10
0 1 2 3 4
y
x
Table.Residuals at each point
for regression model
y=6
Figure. Regression curve y=6 and y vs xdata
Using y=6 as a regression curve
Linear Regression-Criterion#2
x y y
predictedE = y -y
predicted
2.0 4.0 6.0 -2.0
3.0 6.0 6.0 0.0
2.0 6.0 6.0 0.0
3.0 8.0 6.0 2.0
4||
4
1
=∑
=i
i
E
0
2
4
6
8
10
0 1 2 3 4
y
x
Table.Residuals at each point
for regression model
y=6
Figure. Regression curve y=6 and y vs xdata
Using y=6 as a regression curve
http://numericalmethods.eng.usf.edu 14
Linear Regression-Criterion#2
for both regression models of y=4x − 4 and y=6.
The sum of the absolute residuals has been made as small as
possible, that is 4, but the regression model is not unique.
Hence the criterion of minimizing the sum of the absolute value
of the residuals is also a bad criterion.
4
4
1
=∑
=i
i
E
Linear Regression-Criterion#2
for both regression models of y=4x − 4 and y=6.
The sum of the absolute residuals has been made as small as
possible, that is 4, but the regression model is not unique.
Hence the criterion of minimizing the sum of the absolute value
of the residuals is also a bad criterion.
4
4
1
=∑
=i
i
E
http://numericalmethods.eng.usf.edu 15
Least Squares Criterion
The least squares criterion minimizes the sum of the square of the
residuals in the model, and also produces a unique line.
( )
2
1
10
1
2
∑ −−=∑=
==
n
i
ii
n
i
ir
xaayES
x
11
,yx
22
,yx
33
,yx
nn
yx,
ii
yx,
y
Figure.Linear regression of yvs xdata showing residuals at a typical point, x
i
.
xaay
10+=
iii
xaayE
10
−−=
Least Squares Criterion
The least squares criterion minimizes the sum of the square of the
residuals in the model, and also produces a unique line.
( )
2
1
10
1
2
∑ −−=∑=
==
n
i
ii
n
i
ir
xaayES
x
11
,yx
22
,yx
33
,yx
nn
yx,
ii
yx,
y
Figure.Linear regression of yvs xdata showing residuals at a typical point, x
i
.
xaay
10+=
iii
xaayE
10
−−=
http://numericalmethods.eng.usf.edu16
Finding Constants of Linear Model
( )
2
1
10
1
2
∑ −−=∑=
==
n
i
ii
n
i
ir xaayESMinimize the sum of the square of the residuals:
To find
( )() 012
1
10
0
=−−−−=
∂
∂∑
=
n
i
ii
r
xaay
a
S
( )() 02
1
10
1
=−−−−=
∂
∂∑
=
n
i
iii
r
xxaay
a
S
giving
i
n
i
ii
n
i
i
n
i
xyxaxa∑∑∑
===
=+
1
2
1
1
1
0
0aand1awe minimize with respect to1a 0aandrS .
∑∑∑
===
=+
n
i
ii
n
i
n
i
yxaa
11
1
1
0
Finding Constants of Linear Model
( )
2
1
10
1
2
∑ −−=∑=
==
n
i
ii
n
i
ir xaayESMinimize the sum of the square of the residuals:
To find
( )() 012
1
10
0
=−−−−=
∂
∂∑
=
n
i
ii
r
xaay
a
S
( )() 02
1
10
1
=−−−−=
∂
∂∑
=
n
i
iii
r
xxaay
a
S
giving
i
n
i
ii
n
i
i
n
i
xyxaxa∑∑∑
===
=+
1
2
1
1
1
0
0aand1awe minimize with respect to1a 0aandrS .
∑∑∑
===
=+
n
i
ii
n
i
n
i
yxaa
11
1
1
0
http://numericalmethods.eng.usf.edu17
Finding Constants of Linear Model
0aSolving for
2
11
2
111
1
−
−
=∑∑
∑∑∑
==
===
n
i
i
n
i
i
n
i
i
n
i
i
n
i
ii
xxn
yxyxn
a
and
2
11
2
1111
2
0
−
−
=∑∑
∑∑∑∑
==
====
n
i
i
n
i
i
n
i
ii
n
i
i
n
i
i
n
i
i
xxn
yxxyx
a
1aanddirectly yields,
xaya
10−=
Finding Constants of Linear Model
0aSolving for
2
11
2
111
1
−
−
=∑∑
∑∑∑
==
===
n
i
i
n
i
i
n
i
i
n
i
i
n
i
ii
xxn
yxyxn
a
and
2
11
2
1111
2
0
−
−
=∑∑
∑∑∑∑
==
====
n
i
i
n
i
i
n
i
ii
n
i
i
n
i
i
n
i
i
xxn
yxxyx
a
1aanddirectly yields,
xaya
10−=
http://numericalmethods.eng.usf.edu18
Example 1
The torque, Tneeded to turn the torsion spring of a mousetrap through
an angle, is given below.
Angle, θ Torque, T
Radians N-m
0.698132 0.188224
0.959931 0.209138
1.134464 0.230052
1.570796 0.250965
1.919862 0.313707
Table: Torque vs Angle for a
torsional spring
Find the constants for the model given by
θ
21kkT +=
Figure.Data points for Torque vs Angle data
0.1
0.2
0.3
0.4
0.5 1 1.5 2
θ (radians)
Torque (N-m)
Example 1
The torque, Tneeded to turn the torsion spring of a mousetrap through
an angle, is given below.
Angle, θ Torque, T
Radians N-m
0.698132 0.188224
0.959931 0.209138
1.134464 0.230052
1.570796 0.250965
1.919862 0.313707
Table: Torque vs Angle for a
torsional spring
Find the constants for the model given by
θ
21kkT +=
Figure.Data points for Torque vs Angle data
0.1
0.2
0.3
0.4
0.5 1 1.5 2
θ (radians)
Torque (N-m)
http://numericalmethods.eng.usf.edu19
Example 1 cont.
1a
The following table shows the summations needed for the calculations of
the constants in the regression model.
θ
2
θ θT
Radians N-m Radians
2
N-m-Radians
0.698132 0.188224 0.487388 0.131405
0.959931 0.209138 0.921468 0.200758
1.134464 0.230052 1.2870 0.260986
1.570796 0.250965 2.4674 0.394215
1.919862 0.313707 3.6859 0.602274
6.2831 1.1921 8.8491 1.5896
Table.Tabulation of data for calculation of important
∑
=
=
5
1i
5=n
Using equations described for
2
5
1
5
1
2
5
1
5
1
5
1
2
−
−
=∑∑
∑∑∑
==
===
i
i
i
i
i
i
i
i
i
ii
n
TTn
k θθ
θθ
( )( )( )
( )( )
2
28316849185
1921128316589615
..
...
−
−
=
2
1060919
−
×=. N-m/rad
summations
0a
T
andwith
Example 1 cont.
1a
The following table shows the summations needed for the calculations of
the constants in the regression model.
θ
2
θ θT
Radians N-m Radians
2
N-m-Radians
0.698132 0.188224 0.487388 0.131405
0.959931 0.209138 0.921468 0.200758
1.134464 0.230052 1.2870 0.260986
1.570796 0.250965 2.4674 0.394215
1.919862 0.313707 3.6859 0.602274
6.2831 1.1921 8.8491 1.5896
Table.Tabulation of data for calculation of important
∑
=
=
5
1i
5=n
Using equations described for
2
5
1
5
1
2
5
1
5
1
5
1
2
−
−
=∑∑
∑∑∑
==
===
i
i
i
i
i
i
i
i
i
ii
n
TTn
k θθ
θθ
( )( )( )
( )( )
2
28316849185
1921128316589615
..
...
−
−
=
2
1060919
−
×=. N-m/rad
summations
0a
T
andwith
http://numericalmethods.eng.usf.edu20
Example 1 cont.
n
T
T
i
i
∑
=
=
5
1
_
Use the average torque and average angle to calculate1k
_
2
_
1
θkTk −=
n
i
i∑
=
=
5
1
_
θ
θ
5
1921.1
=
1
103842.2
−
×=
5
2831.6
=
2566.1=
Using,
)2566.1)(106091.9(103842.2
21 −−
×−×=
1
101767.1
−
×= N-m
Example 1 cont.
n
T
T
i
i
∑
=
=
5
1
_
Use the average torque and average angle to calculate1k
_
2
_
1
θkTk −=
n
i
i∑
=
=
5
1
_
θ
θ
5
1921.1
=
1
103842.2
−
×=
5
2831.6
=
2566.1=
Using,
)2566.1)(106091.9(103842.2
21 −−
×−×=
1
101767.1
−
×= N-m
http://numericalmethods.eng.usf.edu21
Example 1 Results
Figure.Linear regression of Torque versus Angle data
Using linear regression, a trend line is found from the data
Can you find the energy in the spring if it is twisted from 0 to 180 degrees?
Example 1 Results
Figure.Linear regression of Torque versus Angle data
Using linear regression, a trend line is found from the data
Can you find the energy in the spring if it is twisted from 0 to 180 degrees?
http://numericalmethods.eng.usf.edu22
Linear Regression (special case)
Given
best fit
to the data.
),( , ... ),,(),,(
2211
nn
yxyx yx
xay
1
=
Linear Regression (special case)
Given
best fit
to the data.
),( , ... ),,(),,(
2211
nn
yxyx yx
xay
1
=
http://numericalmethods.eng.usf.edu23
Linear Regression (special case cont.)
Is this correct?
2
11
2
111
1
−
−
=
∑∑
∑∑∑
==
===
n
i
i
n
i
i
n
i
i
n
i
i
n
i
ii
xxn
yxyxn
a
xay
1=
Linear Regression (special case cont.)
Is this correct?
2
11
2
111
1
−
−
=
∑∑
∑∑∑
==
===
n
i
i
n
i
i
n
i
i
n
i
i
n
i
ii
xxn
yxyxn
a
xay
1=
http://numericalmethods.eng.usf.edu24
x
11
,yx
iixax
1
,
nn
yx,
ii
yx,
iii xay
1−=ε
y
Linear Regression (special case cont.)
x
11
,yx
iixax
1
,
nn
yx,
ii
yx,
iii xay
1−=ε
y
Linear Regression (special case cont.)
http://numericalmethods.eng.usf.edu25
Linear Regression (special case cont.)
iii
xay
1−=ε
∑
=
=
n
i
irS
1
2
ε
( )
2
1
1∑
=
−=
n
i
ii
xay
Residual at each data point
Sum of square of residuals
Linear Regression (special case cont.)
iii
xay
1−=ε
∑
=
=
n
i
irS
1
2
ε
( )
2
1
1∑
=
−=
n
i
ii
xay
Residual at each data point
Sum of square of residuals
http://numericalmethods.eng.usf.edu26
Linear Regression (special case cont.)
Differentiate with respect to
gives
( )()∑
=
−−=
n
i
iii
r
xxay
da
dS
1
1
1
2
( )∑
=
+−=
n
i
iii
xaxy
1
2
1
22
0
1
=
da
dS
r
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
1
a
Linear Regression (special case cont.)
Differentiate with respect to
gives
( )()∑
=
−−=
n
i
iii
r
xxay
da
dS
1
1
1
2
( )∑
=
+−=
n
i
iii
xaxy
1
2
1
22
0
1
=
da
dS
r
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
1
a
http://numericalmethods.eng.usf.edu27
Linear Regression (special case cont.)
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
( )∑
=
+−=
n
i
iii
r
xaxy
da
dS
1
2
1
1
22
02
1
2
2
1
2
>=∑
=
n
i
i
r
x
da
Sd
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
Does this value of a
1correspond to a local minima or local
maxima?
Yes, it corresponds to a local minima.
Linear Regression (special case cont.)
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
( )∑
=
+−=
n
i
iii
r
xaxy
da
dS
1
2
1
1
22
02
1
2
2
1
2
>=∑
=
n
i
i
r
x
da
Sd
∑
∑
=
=
=
n
i
i
n
i
ii
x
yx
a
1
2
1
1
Does this value of a
1correspond to a local minima or local
maxima?
Yes, it corresponds to a local minima.
http://numericalmethods.eng.usf.edu28
Linear Regression (special case cont.)
Is this local minima of an absolute minimum of ?
rS
rS
1a
rS
Linear Regression (special case cont.)
Is this local minima of an absolute minimum of ?
rS
rS
1a
rS
http://numericalmethods.eng.usf.edu29
Example 2
Strain Stress
(%) (MPa)
0 0
0.183 306
0.36 612
0.5324 917
0.702 1223
0.867 1529
1.0244 1835
1.1774 2140
1.329 2446
1.479 2752
1.5 2767
1.56 2896
To find the longitudinal modulus of composite, the following data is
collected. Find the longitudinal modulus,
Table.Stress vs. Strain data
Eusing the regression model
εσE= and the sum of the square of the
0.0E+00
1.0E+09
2.0E+09
3.0E+09
0 0.005 0.01 0.015 0.02
Strain, ε (m/m)
Stress, σ (Pa)
residuals.
Figure.Data points for Stress vs. Strain data
Example 2
Strain Stress
(%) (MPa)
0 0
0.183 306
0.36 612
0.5324 917
0.702 1223
0.867 1529
1.0244 1835
1.1774 2140
1.329 2446
1.479 2752
1.5 2767
1.56 2896
To find the longitudinal modulus of composite, the following data is
collected. Find the longitudinal modulus,
Table.Stress vs. Strain data
Eusing the regression model
εσE= and the sum of the square of the
0.0E+00
1.0E+09
2.0E+09
3.0E+09
0 0.005 0.01 0.015 0.02
Strain, ε (m/m)
Stress, σ (Pa)
residuals.
Figure.Data points for Stress vs. Strain data
http://numericalmethods.eng.usf.edu30
Example 2 cont.
i ε σ ε
2
εσ
1 0.0000 0.0000 0.0000 0.0000
2 1.8300× 10
−3
3.0600× 10
8
3.3489× 10
−6
5.5998× 10
5
3 3.6000× 10
−3
6.1200× 10
8
1.2960× 10
−5
2.2032× 10
6
4 5.3240× 10
−3
9.1700× 10
8
2.8345× 10
−5
4.8821× 10
6
5 7.0200× 10
−3
1.2230× 10
9
4.9280× 10
−5
8.5855× 10
6
6 8.6700× 10
−3
1.5290× 10
9
7.5169× 10
−5
1.3256× 10
7
7 1.0244× 10
−2
1.8350× 10
9
1.0494× 10
−4
1.8798× 10
7
8 1.1774× 10
−2
2.1400× 10
9
1.3863× 10
−4
2.5196× 10
7
9 1.3290× 10
−2
2.4460× 10
9
1.7662× 10
−4
3.2507× 10
7
10 1.4790× 10
−2
2.7520× 10
9
2.1874× 10
−4
4.0702× 10
7
11 1.5000× 10
−2
2.7670× 10
9
2.2500× 10
−4
4.1505× 10
7
12 1.5600× 10
−2
2.8960× 10
9
2.4336× 10
−4
4.5178× 10
7
1.2764× 10
−3
2.3337× 10
8
Table.Summation data for regression model
∑
=
12
1i
∑
=
−
×=
12
1
32
102764.1
i
i
ε
∑
=
×=
12
1
8
103337.2
i
ii
εσ
∑
∑
=
=
=
12
1
2
12
1
i
i
i
ii
E
ε
εσ
3
8
102764.1
103337.2
−
×
×
=
GPa84.182=
∑
∑
=
=
=
n
i
i
n
i
ii
E
1
2
1
ε
εσ
Example 2 cont.
i ε σ ε
2
εσ
1 0.0000 0.0000 0.0000 0.0000
2 1.8300× 10
−3
3.0600× 10
8
3.3489× 10
−6
5.5998× 10
5
3 3.6000× 10
−3
6.1200× 10
8
1.2960× 10
−5
2.2032× 10
6
4 5.3240× 10
−3
9.1700× 10
8
2.8345× 10
−5
4.8821× 10
6
5 7.0200× 10
−3
1.2230× 10
9
4.9280× 10
−5
8.5855× 10
6
6 8.6700× 10
−3
1.5290× 10
9
7.5169× 10
−5
1.3256× 10
7
7 1.0244× 10
−2
1.8350× 10
9
1.0494× 10
−4
1.8798× 10
7
8 1.1774× 10
−2
2.1400× 10
9
1.3863× 10
−4
2.5196× 10
7
9 1.3290× 10
−2
2.4460× 10
9
1.7662× 10
−4
3.2507× 10
7
10 1.4790× 10
−2
2.7520× 10
9
2.1874× 10
−4
4.0702× 10
7
11 1.5000× 10
−2
2.7670× 10
9
2.2500× 10
−4
4.1505× 10
7
12 1.5600× 10
−2
2.8960× 10
9
2.4336× 10
−4
4.5178× 10
7
1.2764× 10
−3
2.3337× 10
8
Table.Summation data for regression model
∑
=
12
1i
∑
=
−
×=
12
1
32
102764.1
i
i
ε
∑
=
×=
12
1
8
103337.2
i
ii
εσ
∑
∑
=
=
=
12
1
2
12
1
i
i
i
ii
E
ε
εσ
3
8
102764.1
103337.2
−
×
×
=
GPa84.182=
∑
∑
=
=
=
n
i
i
n
i
ii
E
1
2
1
ε
εσ
http://numericalmethods.eng.usf.edu31
Example 2 Results
εσ
9
1084.182×=The equation
Figure.Linear regression for stress vs. strain data
describes the data.
Example 2 Results
εσ
9
1084.182×=The equation
Figure.Linear regression for stress vs. strain data
describes the data.
Additional Resources
For all resources on this topic such as digital audiovisual
lectures, primers, textbook chapters, multiple-choice
tests, worksheets in MATLAB, MATHEMATICA, MathCad
and MAPLE, blogs, related physical problems, please
visit
http://numericalmethods.eng.usf.edu/topics/linear_regr ession.html
For all resources on this topic such as digital audiovisual
lectures, primers, textbook chapters, multiple-choice
tests, worksheets in MATLAB, MATHEMATICA, MathCad
and MAPLE, blogs, related physical problems, please
visit
http://numericalmethods.eng.usf.edu/topics/linear_regr ession.html
THE END
http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu
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