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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