1. Transforming Relationships
Chapter 4.1: Exponential Growth and Power Law Models
Part A: Day 1: Exponential Growth

2. Beyond Linear
What if your data is clearly curved in some manner? Are there models we can use, and prediction equations we can develop?
Of course … and we will examine two basic types …
Exponential Growth & The Power Law
But ARE we really beyond LINEAR?

3. The Exponential Model Y = abx
“a” is the initial value when x = 0, it is the y-intercept.
“a” is often unrealistically small depending on the manner in which the data is entered.
Each subsequent Y value is obtained by multiplying by a factor “b”.
Taking a Logarithm (LOG) of the y-value, and using the old x-value will linearize the data.

4. Transforming Data
Enter the following data and observe the curved pattern.
Do a LinReg on L1, L2
ŷ = (x); r =
Check out the residuals, just to reinforce the non-linearity.
Cell Phone Subscribers in the U.S.,
Year
1990
1993
1994
1995
1996
1997
1998
1999
Subscribers
(1000’s)
5283
16,009
24,134
33,784
44,043
55,312
69,209
86,047

5. Logarithmic Transformation
Now, obtain the data from the Y-value list (L2), and “take the LOG” of each value.
Place the resulting LOGGED DATA into L3
Re-plot the L1/L3 data.
Are things perfectly linear?
Explore with LinReg, r-value. Comment.
Log (ŷ) = (x); r =
Year
1990
1993
1994
1995
1996
1997
1998
1999
Log of Subscribers
(1000’s)
3.7229
4.2044
4.3826
4.5287
4.6439
4.7428
4.8402
4.9347

6. Still not perfect – Is it?
Eliminate all data except the last 4 years
Dump this data into L4/L5
Do a LinReg on L4/L5 … better?
Log (ŷ) = (x); r =
How about that Residual Plot still? Grrrrrr.
Year
1996
1997
1998
1999
Log of Subscribers
(1000’s)
4.6439
4.7428
4.8402
4.9347

7. Predictions?
OK, regardless of the suspicious Residual Plot … we move on.
Let’s use the last Prediction Equation to Predict for the year 2000.
Log (ŷ) = (2000)
Log (ŷ) =
ŷ =10^ = 107, 864.5

8. Conclusions?
If a variable grows exponentially, then its logarithm grows linearly
High r and R-square values are not the total picture.
Near perfect (.99999) r values and R-Square values still are incomplete.
Residuals tell a big tale.
But the magnitude of the error can still warrant usage, if we are simply trying to predict!
Plug into the “Log Equation”, then raise answer to the 10th power.

9. Another Problem!
The combined American Indian, Eskimo, Aleut, Asian and Pacific Islander population grew in the US from 1950 to 1990 …as shown below …
When entering the year, enter 50 for 1950, 60 for 1960, etc.
Perform the Regression after transforming the data.
Make a prediction of this combined population in the year 2000 …
Year
1950
1960
1970
1980
1990
Population(1000’s)
1131
1620
2557
5150
9534

10. So how’d ya do? Log (ŷ) = 1.8246 + 0.023539(x); r = .992025
Log (ŷ) = (100) … note we are using “100” to represent the year 2000.
Log (ŷ) =
ŷ = 10 ^ =
So … the population would be predicted to be : 15,084,584 people.
Do you think your prediction (Extrapolation) is too high or too low as compared to the actual population in 2000? Why?

11. Gypsy Moths
Enter the data into L1 and L2 for the year (x–List 1) and the Acres of land defoliated by the Gypsy Moth (y-List 2).
Year
Acres
1978
63,042
1979
226,260
1980
907,075
1981
2,826,095

12. Gypsy Moths
P.212/#4.6 –
A) Plot the number of acres defoliated (y) against the year (x).
B) Check out the three consecutive ratios of the Acreage … to verify the approximate exponential growth. What is that approximate growth RATIO (to the nearest integer)?
C) “Linearize” the data – i.e. Transform the y-values, and plot the results.
D) Calculate the LSRL for the transformed data.
Log (ŷ) = (x); r =
E) Construct and interpret the residual plot.
F) Perform an inverse transformation to express ŷ as an exponential function of year.
G) Predict the number of acres defoliated in 1982.
Log (ŷ) = (1982) =
ŷ = 10 ^ = 10,719, acres.