Nonlinear modeling

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Nonlinear modeling

• Problem 4.6
• Gypsy moths

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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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Note the apparent exponential pattern the
scatterplot. Lets make a graph using some
special graph paper.
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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.
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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.
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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.
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Make a scatterplot of the data, then return to
lists.
With the cursor on the header for L3, then press
ltLOGgt lt2ndgt ltL2gt ltENTERgt.
Make a scatterplot of L3 by L1. Run linear
regression of L1 and L3.
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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
lt2ndgt ltLISTgt 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.
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Now that weve 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 yabx. 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.
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Go to the Y register. (Press ltYgt)
Deselect Y1 by pressing ltENTERgt 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.