Business Analytics Predictive Modeling using Linear Regression
Business Analytics Predictive Modeling using Linear Regression
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4.b.Explained and Unexplained Variation
y yi
•SSE = Sum of squared errors
SSE = (yi - yi )2
SST = Total Sum of Squares
y
_
y
_ y
SST = (yi - y)2 _ 2 RSS = (yi - y) •RSS = Regression sum of squares
Xi © Pristine
_ y
x 1
4.b.Explained and Unexplained Variation (Cont…) SST = Total sum of squares • Measures the variation of the yi values around their mean y
(continued)
SSE = Sum of squared errors • Variation attributable to factors other than the relationship between x and y SSR = Regression sum of squares • Explained variation attributable to the relationship between x and y
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2
4.b.Explained and Unexplained Variation (Cont…) Total variation is made up of two parts:
SST SSE Total sum of Squares
Sum of Squared Errors
SST ( y y )
2
RSS Regression Sum of Squares Also known as Square Sum of Regression SSR
SSE ( y yˆ ) 2 SSR ( yˆ y ) 2
Where:
y = Average value of the dependent variable
y = Observed values of the dependent variable
yˆ = Estimated value of y for the given x value
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3
4.b.Coefficient of Determination, R2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable
The coefficient of determination is also called R-squared and is denoted as R2
SSR R SST 2
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where
0 R 1 2
4
4.b.Coefficient of Determination, R2 (Cont…) Coefficient of determination
(continued)
SSR sum of squaresexplained by regression R SST total sum of squares 2
Note: In the single independent variable case, the coefficient of determination is
Where:
R r 2
2
• R2 = Coefficient of determination
• r = Simple correlation coefficient
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5
4.b.Examples of Approximate R2 Values
y R2 = 1 Perfect linear relationship between x and y:
R2
=1
x
100% of the variation in y is explained by variation in x
y
R2 © Pristine
= +1
x 6
4.b.Examples of Approximate R2 Values (Cont…)
y
0 < R2 < 1 Weaker linear relationship between x and y:
x
Some but not all of the variation in y is explained by variation in x
y
x
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7
4.b.Examples of Approximate R2 Values (Cont…)
R2 = 0
y
No linear relationship between x and y: The value of Y does not depend on x. (None of the variation in y is explained by variation in x)
R2 = 0
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x
8
4.b.Limitations of Regression Analysis Parameter Instability - This happens in situations where correlations change over a period of time. This is very common in financial markets where economic, tax, regulatory, and political factors change frequently.
Public knowledge of a specific regression relation may cause a large number of people to react in a similar fashion towards the variables, negating its future usefulness. If any regression assumptions are violated, predicted dependent variables and hypothesis tests will not hold valid.
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9
Thank you!
Pristine 702, Raaj Chambers, Old Nagardas Road, Andheri (E), Mumbai-400 069. INDIA www.edupristine.com Ph. +91 22 3215 6191
© Pristine
© Pristine – www.edupristine.com
© Pristine
© Pristine – www.edupristine.com
4.b.Explained and Unexplained Variation
y yi
•SSE = Sum of squared errors
SSE = (yi - yi )2
SST = Total Sum of Squares
y
_
y
_ y
SST = (yi - y)2 _ 2 RSS = (yi - y) •RSS = Regression sum of squares
Xi © Pristine
_ y
x 1
4.b.Explained and Unexplained Variation (Cont…) SST = Total sum of squares • Measures the variation of the yi values around their mean y
(continued)
SSE = Sum of squared errors • Variation attributable to factors other than the relationship between x and y SSR = Regression sum of squares • Explained variation attributable to the relationship between x and y
© Pristine
2
4.b.Explained and Unexplained Variation (Cont…) Total variation is made up of two parts:
SST SSE Total sum of Squares
Sum of Squared Errors
SST ( y y )
2
RSS Regression Sum of Squares Also known as Square Sum of Regression SSR
SSE ( y yˆ ) 2 SSR ( yˆ y ) 2
Where:
y = Average value of the dependent variable
y = Observed values of the dependent variable
yˆ = Estimated value of y for the given x value
© Pristine
3
4.b.Coefficient of Determination, R2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable
The coefficient of determination is also called R-squared and is denoted as R2
SSR R SST 2
© Pristine
where
0 R 1 2
4
4.b.Coefficient of Determination, R2 (Cont…) Coefficient of determination
(continued)
SSR sum of squaresexplained by regression R SST total sum of squares 2
Note: In the single independent variable case, the coefficient of determination is
Where:
R r 2
2
• R2 = Coefficient of determination
• r = Simple correlation coefficient
© Pristine
5
4.b.Examples of Approximate R2 Values
y R2 = 1 Perfect linear relationship between x and y:
R2
=1
x
100% of the variation in y is explained by variation in x
y
R2 © Pristine
= +1
x 6
4.b.Examples of Approximate R2 Values (Cont…)
y
0 < R2 < 1 Weaker linear relationship between x and y:
x
Some but not all of the variation in y is explained by variation in x
y
x
© Pristine
7
4.b.Examples of Approximate R2 Values (Cont…)
R2 = 0
y
No linear relationship between x and y: The value of Y does not depend on x. (None of the variation in y is explained by variation in x)
R2 = 0
© Pristine
x
8
4.b.Limitations of Regression Analysis Parameter Instability - This happens in situations where correlations change over a period of time. This is very common in financial markets where economic, tax, regulatory, and political factors change frequently.
Public knowledge of a specific regression relation may cause a large number of people to react in a similar fashion towards the variables, negating its future usefulness. If any regression assumptions are violated, predicted dependent variables and hypothesis tests will not hold valid.
© Pristine
9
Thank you!
Pristine 702, Raaj Chambers, Old Nagardas Road, Andheri (E), Mumbai-400 069. INDIA www.edupristine.com Ph. +91 22 3215 6191
© Pristine
© Pristine – www.edupristine.com
