D.3.3.:  Curve Fitting

Find the curve of best fit to a given set of data.  This means that we need to find an equation which will model the data closely with the smallest error.  You have several options:

 

   Linear Regression                              y = ax + b

   Quadratic Regression                        y = ax2 + bx + c

   Cubic Regression                              y = ax3 + bx2 + cx + d

   Quartic Regression                            y = ax4 + bx3 + cx2 + dx + e

   Logarthmic Regression                       y = a + b ln(x)

   Exponential Regression                      y = abx

   Power Regression                              y = axb

   Logistic Regression                            y =   c / (1 + a*e(-bx))

   Sinusoidal Regression                        y = a sin (bx+c) + d

 

The most commonly used curve fitting models are  Linear, Quadratic, Cubic and Quartic, Logarithmic and Exponential Regression. The type of model you will choose depends on the type of data. Hence, it is advantageous to first graph your data using a scatter plot.

 

1.)  Linear Regression

Lets use the data Height versus Weight from the previous section. Is there a relationship between the height and weight of an individual? You can determine the line that will best fit the data by performing a linear regression using your calculator.

 

Height (in.)

Weight (lbs.)

62

105

67

124

58

120

60

90

66

130

65

120

69

162

69

134

74

122

76

185

 

Graph the data and find the best fit line using LinReg:

1.)     Use a scatter plot to graph the data

2.)     Input your data in L1 and L2

3.)     STAT

4.)     CALC

5.)     Select 4:LinReg(ax + b)

6.)     ENTER ENTER

      

Note: If you do not get r2 and r, turn your Diagnostic On.

 

Interpretation:  Your linear regression line is as follows:

 Y = 3.45 x 100.62  

 Slope a = 3.45

 Y-intercept of -100.62

 Correlation coefficient of r = .7322 

 Coefficient of determination of r2 = .5361.

 

Explain the slope a, y-intercept, r and r2. What do they mean in terms of our data?

 Slope:  _______________________________________________

 y-intercept:  ___________________________________________

 r  =  _________________________________________________

 r2 = _________________________________________________

 

 

Activity:  Airline Data

 

American Airlines Flights Departing from Chicago

Flight

Gate-to-Gate Minutes

(taxiing and flight time)

Miles

To Boston

130

868

To Dallas

140

803

To Denver

150

902

To Indianapolis

52

178

To Nashville

85

410

To New Orleans

125

838

To New York City

120

734

To Orlando

157

1006

To Toronto

86

438

To Washington, D.C.

104

613

 

  1. Graph the data. Is there a linear trend?
  2. Find the equation of the line of best fit using LinReg:  y =  ________
  3. Interpret the slope, y-intercept, r, and r2.
  4. Find the x-intercept. What are the airplanes doing during that time?

 

 

2.) Quadratic, Cubic and Quartic Regression

If the data does not follow a straight line or shows a linear trend, you may want to explore other options such as QuadReg, CubicReg and QuartReg.  The model with the highest r and r2 indicates the best possible fit.  However, curve fitting is always cumbersome and tricky. Sometimes none of these models will prove to model the data very well.

 

Example:  Gas consumption vs. Speed of car on a 250 mile trip

 

Speed of Car (miles) = X

Gas Consumption (in gal.)  = Y

20

5.2

30

5.8

40

6.2

50

6.4

60

6.4

70

6.2

80

5.8

 

Find the equation of best fit:

1.)     Graph the data.

The points show the shape of a parabola, hence use 5:QuadReg

2.)     STAT

3.)     CALC

4.)     Select 5:QuadReg L1,L2

5.)     ENTER ENTER

 

    

 

The equation of best fit is:  y = -.003x2 + .32x 1.61 with R2 = .87.

 

Graph the equation y in Y1 and compare the results.  Is it a good fit?

 

 

 

Suggestion: A quick procedure to determine which model to use is finding the differences between the  y-values.  Create a table and compute the first, second, third, etc. differences of ys.  When the differences computed are all about the same it is an indication of the model you should use.  For example, if your third differences are equal you should use a cubic model. 

 

Speed of Car (miles) = X

Gas Consumption (in gal.)  = Y

D1Y

D2Y

D3Y

20

5.2

 

 

 

30

5.8

 

 

 

40

6.2

 

 

 

50

6.4

 

 

 

60

6.4

 

 

 

70

6.2

 

 

 

80

5.8

 

 

 

 

Activity: Fit a curve to the following data.

1

-4

2

0

3

1

4

0

5

-2

6

0

7

3

8

15

 

  1. What model should you choose?
  2. Find the equation:  y = ______________________
  3. What is R2?  _____________
  4. Graph the data and the equation you found.  Is this a good model?  __________

 

Activity:  Fit a curve to the following data.

1

4

2

1

3

.5

4

.2

5

2

6

5

7

7

8

15

 

  1. What model should you choose?
  2. Find the equation:  y = ______________________
  3. What is R2?  _____________

Graph the data and the equation you found.  Is this a good model?  __________

 

copyright 2004 Elisabeth Knowlton