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Nonlinear modeling

Nonlinear modeling. Problem 4.6 Gypsy moths. Since gypsy moths were introduced to North America, they have proliferated unchecked and resulted in deforestation of large areas starting in the northeastern US.

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Nonlinear modeling

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  1. Nonlinear modeling Problem 4.6 Gypsy moths

  2. Since gypsy moths were introduced to North America, they have proliferated unchecked and resulted in deforestation of large areas starting in the northeastern US.

  3. Note the apparent exponential pattern the scatterplot. Let’s make a graph using some special graph paper.

  4. It is possible to use this different kind of graph paper to help us out. For this problem I am selecting what is known as semilog graph paper. The x-axis is linear and the y-axis is logarithmic. This is 3 cycle semilog paper meaning that the logarithmic scale is repeated 3 times.

  5. We can choose any scale we like for the x-axis. The y-axis scale allows for a greater spread of numbers. The first cycle will be from 1 and 10, then the second cycle will be from 10 to 100, and the third will be from 100 to 1000. Since the y values vary from 63,000 to 2,800,000, we use 10,000s of acres for the y-axis.

  6. The plot of these four points is shown. As you can see, the plot is linear. If we now go to the calculator, we can take the log of the acres defoliated and plot that by year. First enter the data into lists.

  7. Make a scatterplot of the data, then return to lists. With the cursor on the header for L3, then press <LOG> <2nd> <L2> <ENTER>. Make a scatterplot of L3 by L1. Run linear regression of L1 and L3.

  8. We make a residual plot, using our regression information. Go to the STAT lists and move the cursor to the header for L4. Now press <2nd> <LIST> and scroll down to RESID and select it. This calculates the residuals for the last linear regression performed. (n.b. Make sure that you have just performed the linear regression, otherwise your calculations may be based on earlier, and unrelated work.) Now make a residual plot. With such a small number of points it is difficult to say that there is no pattern, but this looks reasonably scattered.

  9. Now that we’ve established that our model is reasonable, we write the equation. When we write the equation we must remember to write log y, instead of just y. The calculator will report the equation in the form of y=a+bx. So our model for the gypsy moth defoliation and year is where y is acres of defoliation and x is year. While this equation is perfectly correct, it is customary to solve for y. This is accomplished by raising each side to a power of 10. The left side simplifies to . It is convenient to do this in the calculator.

  10. Go to the Y= register. (Press <Y=>) Deselect Y1 by pressing <ENTER> with the cursor on the = symbol in Y1. Then enter We can now check this model against the original data. Make a scatterplot of 1000s of Acres Defoliated (L2) by Year (L1). This model looks very good and is written Note that this is an exponential model.

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