Lesson 4 - 1

Transforming to Achieve Linearity

Knowledge Objectives

• Explain what is meant by transforming (re-expressing) data.

• Tell where y = log(x) fits into the hierarchy of power transformations.

• Explain the ladder of power transformations.

• Explain how linear growth differs from exponential growth.

Construction Objectives• Discuss the advantages of transforming nonlinear

data

• Identify real-life situations in which a transformation can be used to linearize data from an exponential growth model

• Use a logarithmic transformation to linearize a data set that can be modeled by an exponential model

• Identify situations in which a transformation is required to linearize a power model

• Use a transformation to linearize a data set that can be modeled by a power model

Vocabulary• Exponential Growth –

• Hierarchy of Power Transformations –

• Ladder of Power Transformations –

• Linear Growth –

• Logarithmic Transformation –

• Power Model –

• Transformation –

Brain wt vs Body Wt

Direction positiveForm ? linear ?Strength moderateOutliers y: Human Dolphin x: Hippo ElelphantClusters maybe near 600

Mammals - Outliers Removed

Direction positiveForm curvedStrength moderateOutliers y: 2 upper dotsClusters maybe ?

Scatter plot and LS Regression

Data Transformations• From our calculator:

– Linear Regression y-hat = a + b x– Quadratic Regression– Cubic Regression– Quartic Regression– Natural Log Regression y-hat = a + b ln(x)– Exponential Regression– Power Regression y-hat = axb

– Logistic Regression– Sinusoidal Regression

• Only these 3 do we need be concerned with

Transforming with Powers

• Form: xn where n is a number

• For n = 1 we have a line

• For n > 1 we have curves that bend upward

• For 0 < n < 1 we have curves that bend downward

• For n < 0 we have curves that decrease as x increases (the bigger the negative the quicker the decrease)

…, x-2, x-1, x-½, x½, x, x2, x3, …

Hierarchy of Power Functions

n = 0 corresponds to the logarithm function

Trial and Error is not Recommended

Real-Life: What do you do?

• We begin with a mathematical model that we expect the data to adhere to (experience is the key!)

• Linear growth is an additive process• Exponential growth is a multiplicative process

Laws of Logs

Summary and Homework

• Summary

• Homework– pg