Module

for

Logistic Curve Fitting

 

Background for the Logistic Curve Fitting.  

    Fit the curve  [Graphics:Images/LogisticEquationMod_gr_1.gif]  to the data points  [Graphics:Images/LogisticEquationMod_gr_2.gif].  

Rearrange the terms  [Graphics:Images/LogisticEquationMod_gr_3.gif].  Then take the logarithm of both sides:  

        [Graphics:Images/LogisticEquationMod_gr_4.gif].  

Introduce the change of variables: [Graphics:Images/LogisticEquationMod_gr_5.gif].  The previous equation becomes  

        [Graphics:Images/LogisticEquationMod_gr_6.gif]     which is now "linearized."

Use this change of variables on the data points  ,  i.e. same abscissa's but transformed ordinates.

Now you have transformed data points:  [Graphics:Images/LogisticEquationMod_gr_8.gif].  

Use the "Fit" procedure get  Y = A X + B, which must match the form  [Graphics:Images/LogisticEquationMod_gr_9.gif],  hence we must have  [Graphics:Images/LogisticEquationMod_gr_10.gif]  and  a = A.  

 

Proof  Logistic Curve Fitting  Logistic Curve Fitting  

 

Remark.  For the method of "data linearization" we must know the constant  L in advance.  Since  L  is the "limiting population" for the  "S"  shaped logistic curve, a value of  L  that is appropriate to the problem at hand can usually be obtained by guessing.   

 

Computer Programs  Logistic Curve Fitting  Logistic Curve Fitting  

 

Example 1.  Use the method of "data linearization" to find the logistic curve that fits the data for the population of the U.S. for the years 1900-1990.  Fit the curve  [Graphics:Images/LogisticEquationMod_gr_11.gif]  to the census data for the population of the U.S.
        

Date

Populatlion

[Graphics:Images/LogisticEquationMod_gr_12.gif]

76094000

[Graphics:Images/LogisticEquationMod_gr_13.gif]

92407000

[Graphics:Images/LogisticEquationMod_gr_14.gif]

106461000

[Graphics:Images/LogisticEquationMod_gr_15.gif]

123076741

[Graphics:Images/LogisticEquationMod_gr_16.gif]

132122446

[Graphics:Images/LogisticEquationMod_gr_17.gif]

152271417

[Graphics:Images/LogisticEquationMod_gr_18.gif]

180671158

[Graphics:Images/LogisticEquationMod_gr_19.gif]

205052174

[Graphics:Images/LogisticEquationMod_gr_20.gif]

227224681

[Graphics:Images/LogisticEquationMod_gr_21.gif]

249464396

Solution 1.

 

Example 2.  Use the mathematical model  [Graphics:Images/LogisticEquationMod_gr_61.gif]  in Example 1 to estimate the population in 2000.
Solution 2.

 

Example 3.  Follow one of the hyperlinks to a U.S.government computer database of population census values.
        Your Gateway to Census 2000   or   Introduction to Census 2000 Data Products
Solution 3.

 

Example 4.  Use the data in Example 3 for the 2000 census value.
4 (a).  How close is the predicted value  [Graphics:Images/LogisticEquationMod_gr_79.gif] in Example 2 ?
4 (b).  What is the percentage error for the predicted value  [Graphics:Images/LogisticEquationMod_gr_80.gif]?  
Solution 4.

 

Caveat.  Various curves can be fit, but they all depend on the value of  L.  No one knows this value in advance and it must be estimated.

 

Old Lab Project (Logistic Curve Fitting  Logistic Curve Fitting).  Internet hyperlinks to an old lab project.  

 

Reference 

This module is based on the following article.

Bounded Population Growth: a Curve Fitting Lesson, J. Mathews, Math. and Computer Education J., Vol. 26, No. 2, Spring 1992, pp. 169-176.

 

Research Experience for Undergraduates

The Logistic Curve  The Logistic Curve  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook The Logistic Curve

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005