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

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

Linear model analysis

Straighten the data

Writing the model Original model: Transformed model:

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

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

Most popular transformations Exponential: Logarithmic: Power:

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

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

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

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

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

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

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

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

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

Algebra

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

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

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

Linear Model Exponential Model

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

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

Algebra

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.