Linear Regression

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Linear Regression

• Industrial Engineering Majors
• Authors Autar Kaw, Luke Snyder
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Linear Regression http//numericalmethods.e
ng.usf.edu
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What is Regression?
What is regression? Given n data points
best fit
to the data. The best fit is generally based on
minimizing the sum of the square of the
residuals,
.

Residual at a point is
Sum of the square of the residuals
Figure. Basic model for regression
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Linear Regression-Criterion1
Given n data points
best fit
to the data.

Figure. Linear regression of y vs. x data showing
residuals at a typical point, xi .
Does minimizing
work as a criterion, where
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Example for Criterion1
Example Given the data points (2,4), (3,6),
(2,6) and (3,8), best fit the data to a straight
line using Criterion1
Table. Data Points
Figure. Data points for y vs. x data.
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Linear Regression-Criteria1
Using y4x-4 as the regression curve
Table. Residuals at each point for regression
model y 4x 4.
Figure. Regression curve for y4x-4, y vs. x data
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Linear Regression-Criteria1
Using y6 as a regression curve
Table. Residuals at each point for y6
Figure. Regression curve for y6, y vs. x data
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Linear Regression Criterion 1
for both regression models of y4x-4 and y6.
The sum of the residuals is as small as possible,
that is zero, but the regression model is not
unique. Hence the above criterion of minimizing
the sum of the residuals is a bad criterion.
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Linear Regression-Criterion2
Will minimizing
work any better?

Figure. Linear regression of y vs. x data showing
residuals at a typical point, xi .
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Linear Regression-Criteria 2
Using y4x-4 as the regression curve
Table. The absolute residuals employing the
y4x-4 regression model


Figure. Regression curve for y4x-4, y vs. x data
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Linear Regression-Criteria2
Using y6 as a regression curve
Table. Absolute residuals employing the y6 model
Figure. Regression curve for y6, y vs. x data
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Linear Regression-Criterion2


for both regression models of y4x-4 and y6.



The sum of the errors has been made as small as
possible, that is 4, but the regression model is
not unique. Hence the above criterion of
minimizing the sum of the absolute value of the
residuals is also a bad criterion.




Can you find a regression line for which
and has unique
regression coefficients?
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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.






Figure. Linear regression of y vs. x data showing
residuals at a typical point, xi .
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Finding Constants of Linear Model
Minimize the sum of the square of the residuals
To find
and
we minimize
with respect to
and
.




giving
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Finding Constants of Linear Model


Solving for
and
directly yields,






and



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Example 1
As machines are used over long periods of time,
the output product can get off target. Below is
the average value of how much off target a
product is getting manufactured as a function of
machine use.
Table. Off target value as a function of machine
use.
Figure. Data points for h vs. t data
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Example 1 cont.
Regress the data to and
find when the product will be 2mm off target.
Table. Summation data for linear regression
With
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Example 1 cont.
The value for
can then be found using
where
mm
hours
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Example 1 cont.
The linear regression model is now given by
Figure. Linear regression of hours of use vs.
millimeters off target.
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Example 1 cont.
Solving for when
yields
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Example 2
To find the longitudinal modulus of composite,
the following data is collected. Find the
longitudinal modulus,
using the regression model
Table. Stress vs. Strain data
and the sum of the square of the
residuals.
Figure. Data points for Stress vs. Strain data
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Example 2 cont.
Residual at each point is given by
The sum of the square of the residuals then is
Differentiate with respect to
Therefore
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Example 2 cont.
Table. Summation data for regression model
With
and
Using
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Example 2 Results
The equation
describes the data.
Figure. Linear regression for Stress vs. Strain
data
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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
  _regression.html

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• http//numericalmethods.eng.usf.edu