Lesson 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 positive Form ? linear ? Strength moderate Outliers y: Human Dolphin x: Hippo Elelphant Clusters maybe near 600

Mammals - Outliers Removed Direction positive Form curved Strength moderate Outliers y: 2 upper dots Clusters 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 = ax b –Logistic Regression –Sinusoidal Regression Only these 3 do we need be concerned with

Transforming with Powers Form: x n 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, x ½, x, x 2, x 3, …

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