fitting polynomial data

© Bart Lauwers LSSBB, [email protected] Polynomial Data

Fitting Polynomial Data

with Linear Regression using Minitab


FITTING QUADRATIC DATA


Visualizing the Data

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YScatterplot of Y vs X

Graph>Scatterplot… Quadratic.MPJ


Fitting the Data using Linear Regression Model

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S 7.54763R-Sq 56.9%R-Sq(adj) 56.8%

Fitted Line PlotY = 7.324 + 2.998 X

Stat>Regression>Fitted Line Plot…


Regression Analysis: Y versus X

The regression equation isY = 7.32 + 3.00 X

Predictor Coef SE Coef T PConstant 7.3238 0.3375 21.70 0.000X 2.9984 0.1169 25.64 0.000

S = 7.54763 R-Sq = 56.9% R-Sq(adj) = 56.8%

Stat>Regression>Regression…


Calculating the Quadratic Term

Calc>Calculator…


The regression equation isY = - 0.998 + 3.02 X + 0.999 X^2

Predictor Coef SE Coef T PConstant -0.99830 0.07765 -12.86 0.000X 3.01840 0.01793 168.31 0.000X^2 0.998671 0.006945 143.79 0.000

S = 1.15755 R-Sq = 99.0% R-Sq(adj) = 99.0%


Regression Analysis: Y versus X, X^2



Polynomial Regression Analysis: Y versus X

The regression equation isY = - 0.9983 + 3.018 X + 0.9987 X**2

S = 1.15755 R-Sq = 99.0% R-Sq(adj) = 99.0%

Analysis of Variance

Source DF SS MS F PRegression 2 65163.9 32582.0 24316.13 0.000Error 497 665.9 1.3Total 499 65829.9

Sequential Analysis of Variance

Source DF SS F PLinear 1 37460.5 657.59 0.000Quadratic 1 27703.5 20675.26 0.000


Fitting the Data using Quadratic Regression Model


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S 1.15755R-Sq 99.0%R-Sq(adj) 99.0%

Fitted Line PlotY = - 0.9983 + 3.018 X

+ 0.9987 X**2


FITTING POLYNOMIAL DATA



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Graph>Scatterplot… Polynomial.MPJ



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S 59.0593R-Sq 0.9%R-Sq(adj) 0.7%

Fitted Line PlotY = 50.89 - 1.931 X

Stat>Regression>Fitted Line Plot… (linear regression model)


Calculating the 2nd through 5th Degree Polynomial Terms

Calc>Calculator…



Regression Analysis: Y versus X - X^5

The regression equation isY = - 7.48 + 3.01 X + 7.09 X^2 - 2.98 X^3 - 0.00355 X^4 + 0.149 X^5

Predictor Coef SE Coef T PConstant -7.4753 0.1852 -40.35 0.000X 3.0137 0.1497 20.13 0.000X^2 7.08514 0.04640 152.70 0.000X^3 -2.97756 0.02353 -126.56 0.000X^4 -0.003554 0.002075 -1.71 0.087X^5 0.148801 0.000826 180.19 0.000

S = 2.20915 R-Sq = 99.9% R-Sq(adj) = 99.9%

Evaluate the P-values.• We notice that for X^4 the

P-value > 0.05. Therefore it must be removed from the regression!



Regression Analysis: Y versus X, X^2, X^3, X^5

The regression equation isY = - 7.28 + 3.02 X + 7.01 X^2 - 2.98 X^3 + 0.149 X^5

Predictor Coef SE Coef T PConstant -7.2850 0.1485 -49.06 0.000X 3.0164 0.1500 20.11 0.000X^2 7.00898 0.01328 527.66 0.000X^3 -2.97820 0.02357 -126.36 0.000X^5 0.148829 0.000827 179.91 0.000

S = 2.21346 R-Sq = 99.9% R-Sq(adj) = 99.9%


FITTING A PURE POLYNOMIAL



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Graph>Scatterplot… Pure.MPJ


Graph>Regression>Fitted Line Plot… (linear regression model)

Regression Analysis: Y versus X

The regression equation isY = 55.10 - 1.961 X

S = 59.0735 R-Sq = 0.9% R-Sq(adj) = 0.7%


Source DF SS MS F PRegression 1 16017 16016.8 4.59 0.033Error 498 1737857 3489.7Total 499 1753874



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S 59.0735R-Sq 0.9%R-Sq(adj) 0.7%

Fitted Line PlotY = 55.10 - 1.961 X

Graph>Regression>Fitted Line Plot… (linear regression model)


Graph>Regression>Fitted Line Plot… (quadratic regression model)


The regression equation isY = - 2.646 - 1.822 X + 6.929 X**2

S = 28.5157 R-Sq = 77.0% R-Sq(adj) = 76.9%


Source DF SS MS F PRegression 2 1349741 674870 829.95 0.000Error 497 404133 813Total 499 1753874

Sequential Analysis of Variance

Source DF SS F PLinear 1 16017 4.59 0.033Quadratic 1 1333724 1640.20 0.000

Quadratic.MPJ


Fitting the Data using Quadratic Regression Model

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S 28.5157R-Sq 77.0%R-Sq(adj) 76.9%

Fitted Line PlotY = - 2.646 - 1.822 X

+ 6.929 X**2

Graph>Regression>Fitted Line Plot… (quadratic regression model)


Graph>Regression>Fitted Line Plot… (cubic regression model)


The regression equation isY = - 2.821 - 19.32 X + 6.964 X**2 + 1.167 X**3

S = 18.0209 R-Sq = 90.8% R-Sq(adj) = 90.8%


Source DF SS MS F PRegression 3 1592796 530932 1634.88 0.000Error 496 161078 325Total 499 1753874


Fitting the Data using Cubic Regression Model

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S 18.0209R-Sq 90.8%R-Sq(adj) 90.8%

Fitted Line PlotY = - 2.821 - 19.32 X

+ 6.964 X**2 + 1.167 X**3

Graph>Regression>Fitted Line Plot… (cubic regression model)


Calculating the 2nd through 5th Degree Polynomial Terms

Calc>Calculator…


The regression equation isY = - 3.00 + 3.00 X + 7.00 X^2 - 3.00 X^3 + 0.000000 X^4 + 0.150 X^5

Predictor Coef SE Coef T PConstant -3.00000 0.00000 * *X 3.00000 0.00000 * *X^2 7.00000 0.00000 * *X^3 -3.00000 0.00000 * *X^4 0.00000000 0.00000000 * *X^5 0.150000 0.000000 * *

S = 0 R-Sq = 100.0% R-Sq(adj) = 100.0%


Regression Analysis: Y versus X - X^5

The actual polynomial equation was:


EXERCISE


Exercise Data Visualized

-4 -3 -2 -1 0 1 2 3 4

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Exercise.MPJ


Exercise Objectives

• Visualize the Data using Minitab.• Will any of the built-in regression models match this data?• Use the technique explained above to find the coefficients for the

polynomial terms.


Solution

• The equation was:

• What were your results? Explain the variance.

fitting polynomial data

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