Transcript Chapter 4 - 4.1 Exponential - ppt

Transforming
Relationships
Chapter 4.1: Exponential Growth
and Power Law Models
Part A: Day 1: Exponential Growth
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?
The Exponential Model
x
 Y = ab
 “a” is the initial value when x = 0, it is the yintercept.
 “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.
Transforming Data
 Enter the following data and observe the
curved pattern.
 Do a LinReg on L1, L2
 ŷ = -18081823.7 + 9083.3487(x); r = .96501
 Check out the residuals, just to reinforce
the non-linearity.
Cell Phone Subscribers in the U.S., 1990-1999
Year
1990
1993
1994
1995
1996
1997
1998
1999
Subscribers 5283 16,009 24,134 33,784 44,043 55,312 69,209 86,047
(1000’s)
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




Year
Re-plot the L1/L3 data.
Are things perfectly linear?
Explore with LinReg, r-value. Comment.
Log (ŷ) = -263.203 + 0.13417(x); r = .99116
1990
1993
1994
1995
1996
1997
1998
1999
Log of
3.7229 4.2044 4.3826 4.5287 4.6439 4.7428 4.8402 4.9347
Subscribers
(1000’s)
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) = -188.951 + 0.09699(x); r = .99995
How about that Residual Plot still? Grrrrrr.
Year
1996
1997
1998
1999
Log of
4.6439 4.7428 4.8402 4.9347
Subscribers
(1000’s)
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) = -188.951 + 0.09699(2000)
 Log (ŷ) = 5.032878574
 ŷ =10^ 5.032878574 = 107, 864.5
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.
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 …
1950
1960
1970
1980
1990
Population 1131
(1000’s)
1620
2557
5150
9534
Year
So how’d ya do?
 Log (ŷ) = 1.8246 + 0.023539(x); r = .992025
 Log (ŷ) = 1.8246 + 0.023539(100) … note we are using “100” to represent
the year 2000.
 Log (ŷ) = 4.17853333477
 ŷ = 10 ^ 4.17853333477 = 15084.5839
 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?
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
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) = -1094.51 + 0.5558(x); r = .999293
 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) = -1094.51 + 0.5558(1982) = 7.0302
 ŷ = 10 ^ 7.0302 = 10,719,964.92 acres.