1. EQ: How can we use linear models for non linear data?
Straightening Data

2. River water velocity and Distance from shore
(cm/s) .5 22
1.5
23.18
2.5
25.48
3.5
25.25
4.5
27.15
5.5
27.83
6.5
28.49
7.5
28.18
8.5
28.50
9.5
28.63

3. Linear model analysis

4. Straighten the data

5. Writing the model
Original model:
Transformed model:

6. Using the model Predict the velocity of the river water at a
distance of 5 meters from the shore

7. Most popular transformations
Exponential:
Power:

8. Exponential X Y 1 2 3 4 5 16 7 64 8
128 9
256 10
512

9. Straighten the data X Log(y) 1 2 0.301 3 0.602 4 0.903 5 1.204 7 1.5052
0.301 3
0.602 4
0.903 5
1.204 7
1.505 8
1.806 9
2.107 10
2.709

10. Use the model
Predict the y value for an x of 2.5

11. Find the model Mammal Weight (kg) Heart Rate (BPM) Mouse 0.03 580 At
0.32
320
Rabbit
3.97
170
Monkey
6.55
150
Dog 16
120
Elephant
2500 25

12. Power Model
Exponential Model

13. Power model
Predict the heart rate for humans who weigh 60 kg.

14. Creating a Transformed data model
Determine which model is appropriate
Plot the original data
Plot x vs log y (exponential model)
Plot log(x) vs log(y) (power model)
Check residuals for each plot and determine which is best
2. Algebraically transform the model so the original units are used
Write the model using the log variables
Use inverse operations to eliminate the logs
3. Check the model
Plot the original data (x vs y)
Find the chosen model (exponential or power)
Check the calculator model with your model

15. Algebra: Transforming exponential models

16. Savings Accounts Sketch a graph of the original data
Comment on why a linear model is NOT appropriate
Sketch a graph of the years vs log(money)
Determine why an exponential model IS appropriate by sketching the residual plot and commenting on it
Find the linear model for the transformed data and algebraically transform it back into the original units
Predict the amount of savings after 60 years

17. Algebra: Transforming Power models

18. Harley stock Prices Sketch a graph of the original data
Comment on why a linear model is NOT appropriate
Sketch a graph of the years vs log(money) and then log(years) vs log(money)
Determine which model is better by examining the residuals
Find the linear model for the transformed data and algebraically transform it back into the original units
Predict the amount of savings after 13 years