4.1: Linearizing Data

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4.1 Linearizing Data
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Logs of both sides Linearized
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Which model for prediction should you choose?

• Plot the data and look for patterns.
• Is there a linear pattern? Use Ch. 3.
• If no to 1 try exponential model.
• If no to 2, try power function model.
• If no to 3, check for a dimensional
  relationship.
• If no to 4, try the hierarchy of powers.

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Example Linearizing Curved DataPlaying with
the data in order to linearize it
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Linearized data/Residuals
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Final Model
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4.1, 4.2 Classwork

• 4.1 Hint
• L1 length, L2 weight, L3 cube-root of L2
• 4.2 Hint
• L1length, L2 period, L3 square root of L1

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Example 4.8 (Transformed Exponential)
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Steps to transform data

1. Transform (linearize) the data (exponential? Log
   the ys Power? Log both the xs and the ys)
2. Perform regression (check r, r-squared) and write
   new equation (with a and b)
3. Make residual plot (check that its a good model)
4. Perform an inverse transformation to get the
   model for the original data.

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Ratio Test

• To see if a set of data with a consistent
  increment in the x-values is growing
  exponentially, we calculate the ratios of
  y-values to previous y-values.
• If these quotients are approximately the same,
  then the points are increasing (quotient gt1) or
  decreasing (quotient lt1) exponentially.

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Example 4.5
Processor Years since 1970 Transistors
4004 1 2250
8008 2 2500
8080 4 5000
8086 8 29000
286 12 120000
386 15 275000
486DX 19 1180000
Pentium 23 3100000
Pent II 27 7500000
Pent III 29 24000000
Pent 4 30 42000000

• Gordon Moore predicted that the number of
  transistors on an integrated circuit chip would
  double every 18 months. Heres the actual data

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Calculator Tip for Ratio Test

1. Copy the y values (in L2) into L3.
2. Since we want to calculate y2/y1, delete the
   first number in L3.
3. Since L2 and L3 need to be the same length,
   delete the last number in L2.
4. Define L4L3/L2. These are the ratios you want.

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If our data is growing exponentially and we plot
the log of y against x, we should observe a
straight line for the transformed data.
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• The table shows the federal debt (in trillions)
  for the years 1980 through 1991.
• Construct a scatter plot. Perform an appropriate
  test to decide whether the data is exponential or
  not. Show the common ratios.
• Calculate the logarithms of the y-values and
  extend the table above to show the transformed
  data Then perform least-squares regression on the
  transformed data. Write the LSRL equation for the
  transformed data.
• What is the correlation coefficient?
• Is this correlation between YEAR and FEDERAL
  DEBT? Explain briefly.
• Now transform your linear equation back to obtain
  a model for the original federal debt data. Write
  the equation of this model.
• Compare and comment on your models prediction for
  1990 and 1991 to the actual federal debt.
• Use your model to predict the national debt in
  the year 2000.
• Would you use this model to predict future debts?
  Why/why not?

YEAR DEBT
1980 .909
1981 .994
1982 1.1
1983 1.4
1984 1.6
1985 1.8
1986 2.1
1987 2.3
1988 2.6
1989 2.9
1990 3.2
1991 3.6