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. Why is a linear model not the best choice?
Pattern, large residuals, no balance

8. Most popular transformations
Exponential:
Logarithmic:
Power:

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

10. 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

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

12. 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

14. Power model
Predict the heart rate for humans who way 60 kg.

15. Straightening the data
The ONLY tools available for making models are linear ones.
The only way to deal with non linear data is to make it linear
Determine what type of model would be best to model the data.
The data appears exponential
Undo the operation that made the data
The inverse of exponential functions are log functions

16. Transformed model
Use the new model to predict for x = 5. Does it match the original data?
Transformed model means transformed predictions.

17. How the exponential regression works
Linear model: y = a + bx Exponential model: y = abx
Determine that an exponential model is appropriate
Transform the data
Obtain a new linear model of the transformed data
Transform the model back to the original units

18. Transforming the model
New model: y = a + bx
Which variable was transformed? y
How was it transformed?
Log(y)
New model: log(y) = a + bx
Solve for y in the new model

19. Algebra

20. Savings Accounts Sketch a graph of the original data
Sketch a graph of the years and log(money)
Is an exponential model appropriate?
Sketch the residual plot to justify
Transform the new linear model into an exponential model
Predict for year 60

21. Harley-Davidson stock prices
Sketch a graph of the original data
Sketch a graph of the data and log(price)
Is an exponential model appropriate?
Sketch the residual plot to justify
Transform the new linear model into an exponential model
Predict for year 60

22. Other Models Open the file on heart beats
Determine if a linear model is appropriate
Determine if an exponential model is appropriate

23. Linear Model
Exponential Model

24. Power Model Transforms both x and y by logarithms
Take the log of the x’s and examine the graph for a linear relationship

26. Transforming the model
New model: y = a + bx
Which variable was transformed?
y and x
How was it transformed?
Log(y) and log(x)
New model: log(y) = a + blog(x)
Eliminate the logs in the equation

27. Algebra

28. Airplane speed and flight lengths
Sketch a graph of the original data
Sketch a graph of the mph and log(miles) and examine the residuals.
Sketch a graph of the log(mph) and log(miles) and examine the residuals.
Determine which model is best
Transform the new linear model into an exponential model.