4.1: Linearizing Data

Logs of both sides: Linearized

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.

Example: Linearizing Curved Data“Playing” with the data in order to linearize it

Linearized data/Residuals

Final Model

#4.1, 4.2: Classwork

4.1 Hint

L1 = length, L2 = weight, L3 = cube-root of L2

4.2 Hint

L1=length, L2 = period, L3 = square root of L1

Example 4.8 (Transformed Exponential)

Steps to transform data

1) Transform (linearize) the data (exponential? Log the y’s; Power? Log both the x’s and the y’s)

2) Perform regression (check r, r-squared) and write new equation (with a and b)

3) Make residual plot (check that it’s a good model)

4) Perform an inverse transformation to get the model for the original data.

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 >1) or decreasing (quotient <1) exponentially.

Example 4.5Gordon Moore

predicted that the number of transistors on an integrated circuit chip would double every 18 months. Here’s the actual data:

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

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 L4=L3/L2. These are the ratios you want.

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.

The table shows the federal debt (in trillions) for the years 1980 through 1991.

1) Construct a scatter plot. Perform an appropriate test to decide whether the data is exponential or not. Show the common ratios.

2) 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.

3) What is the correlation coefficient?4) Is this correlation between YEAR and

FEDERAL DEBT? Explain briefly.5) Now transform your linear equation back to

obtain a model for the original federal debt data. Write the equation of this model.

6) Compare and comment on your models prediction for 1990 and 1991 to the actual federal debt.

7) Use your model to predict the national debt in the year 2000.

8) 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