Module for Logistic Curve Fitting

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Background for the Logistic Curve Fitting.  

We wish to fit the curve    to the data points  .  

Rearrange the terms  .  

Take the logarithm of both sides:  .  

Introduce the change of variables: .  

The previous equation becomes    which is now "linearized."

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

Now you have transformed data points:  .  

Use the "Fit" procedure get  Y = A X + B, which must match the form  Y = ln(c) + a X  so you see that    and  a = A.  

Remark 1.  For the first 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.   

Remark 2.  The purpose of the second example it to use true least squares techniques to find the curve.  The computer will find A, B, and L using the second method, but good estimates are needed.

Remark 3.  The data for example 1 can be obtained from the U.S. Census Bureau, Historical National Population Estimates: July 1, 1900 to July 1, 1999.

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    to the census data for the population of the U.S.
    

Date	Populatlion
	76094000
	92407000
	106461000
	123076741
	132122446
	152271417
	180671158
	205052174
	227224681
	249464396

Solution 1.

Example 2.  Use the mathematical model    in Example 1 to estimate the population in 2000.
Solution 2.

Example 3.  Follow this WWW hyperlink to a U.S.government computer database of population census figures.
Click on the Link!  Then look at the table and find the U.S. census population figure for  July 1, 2000

Population Estimates (GCT-T1) - United States and states, or state and counties  

If you are having difficulty connecting to the internet, then click on this link to obtain the desired data.

If you are curious to know today's estimate of the population follow this hyperlink to obtain the current estimates of the U.S. and world population.

U.S. Census Bureau, Population Division  

Example 4.  Go to the world wide web and verify the 2000 census figure.
4 (a).  How close is the predictions in example 2 ?
4 (b).  What is the percentage error for the predicted value ?  
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.  

Research Experience for Undergraduates

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

Downloads (The Logistic Curve The Logistic Curve).  
Download this Mathematica notebook.  

(c) John H. Mathews 2003