Transforming to Achieve Linearity

1
Lesson 4 - 1

• Transforming to Achieve Linearity

2
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.

3
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

4
Vocabulary

• Exponential Growth
• Hierarchy of Power Transformations
• Ladder of Power Transformations
• Linear Growth
• Logarithmic Transformation
• Power Model
• Transformation

5
Brain wt vs Body Wt
Direction positive Form ? linear
? Strength moderate Outliers y Human
Dolphin x Hippo Elelphant Clusters
maybe near 600
6
Mammals - Outliers Removed
Direction positive Form curved Strength
moderate Outliers y 2 upper dots Clusters
maybe ?
7
Scatter plot and LS Regression
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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

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Transforming with Powers

• Form xn where n is a number
• For n 1 we have a line
• For n gt 1 we have curves that bend upward
• For 0 lt n lt 1 we have curves that bend downward
• For n lt 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,
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Hierarchy of Power Functions
n 0 corresponds to the logarithm function
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Trial and Error is not Recommended
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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

13
Laws of Logs
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Summary and Homework

• Summary
• Homework
• pg