Nonlinear Regression:

1
Lesson 4 - 4

• Nonlinear Regression
• Transformations

2
Objectives

• Change exponential expressions to logarithmic
  expressions and logarithmic expressions to
  exponential expressions
• Simplify expressions containing logarithms
• Use logarithmic transformations to linearize
  exponential relations
• Use logarithmic transformations to linearize
  power relations

3
Vocabulary

• Response Variable variable whose value can be
  explained by the value of the explanatory or
  predictor variable
• Predictor Variable independent variable
  explains the response variable variability
• Lurking Variable variable that may affect the
  response variable, but is excluded from the
  analysis
• Positively Associated if predictor variable
  goes up, then the response variable goes up (or
  vice versa)
• Negatively Associated if predictor variable
  goes up, then the response variable goes down (or
  vice versa)

4
Non-linear Scatter Diagrams

• Some relationships that are nonlinear can be
  modeled with exponential or power models
• y a bx (with b gt 1)
• y a bx (with b lt 1)
• y a xb

Exponential
Exponential
Power
5
Exponential Data

• We would like to fit an exponential model
• y a bx
• We would still like to use our least-squares
  linear regression techniques on this model
• If we take the logarithms of both sides, we get
• log y log a x log b
• because
• log(a bx) log(a) log(bx) log a x log b

6
Exponential to Linear Transform

• We modify this transformed equation
• log y log a x log b
• Define these new variables
• Y log y
• A log a
• B log b
• X x
• Then this equation becomes
• Y A B X

7
Least Squares on Exponential Model

• We started with an exponential model
• y a bx
• We transformed that into a linear model
• Y A B X
• After we solve the linear model, we match up
• b 10B
• a 10A
• In this way, we are able to use the method of
  least-squares to find an exponential model

8
Harley Davidson Dataset

• Year (x) Closing Price (y)

• Year (x) Closing Price (y)

• 1990 1.1609
• 1991 2.6988
• 1992 4.5381
• 1993 5.3379
• 1994 6.8032
• 1995 7.0328
• 1996 11.5585

• 1997 13.4799
• 1998 23.5424
• 1999 31.9342
• 2000 39.7277
• 2001 54.31
• 2002 46.20
• 2003 47.53
• 2004 60.75

9
Exponential Example

• The scatter diagram below appears to be
  exponential (curved) and not linear
• A line is not an appropriate model

10
Fitting an Exponential Model

• We use Y log y and X x
• The first observation is x 1 and y 1.1609,
  thus the first observation of the transformed
  data is X 1 and Y log 1.1609 0.0648
• The second observation is x 2 and y 2.6988,
  thus the second observation of the transformed
  data is X 2 and Y log 2.6988 0.4312
• We continue and take logs of all of the y values

11
Using your Calculator

• To get the scatter plot we inputted x-values into
  L1 and the y-values into L2
• To change the y-values into logs we go to the top
  of L3 and hit LOG(L2) ENTER and then use LINREG
  to find a and b (first part of the slide after
  next)
• Or a simpler way yet, use the ExpReg calculation
  under STAT and CALC and get it directly without
  having to convert back (last line of the slide
  after next)

12
Transformed Line

• The scatter diagram of the transformed data (Y
  and X) is more linear
• We now calculate least-squares regression line
  for this data

13
Least Squares to Exponential

• The least squares line is
• Y 0.1161 X 0.2107
• B 0.1161
• A 0.2107
• This is transformed back to
• b 10B 100.1161 1.3064 and
• a 10A 100.2107 1.6244, so
• y 1.6244 (1.3064)x

14
Exponential Model

• We now plot y 1.624 (1.306)x, our exponential
  model, on the original scatter diagram
• This is a better fit to the data, but we need to
  be careful if we try to extrapolate

15
Part Two

• Power Models

16
Power Model Data

• We would now like to fit an power model
• y a xb
• We would still like to use our least-squares
  linear regression techniques on this model
• If we take the logarithms of both sides, we get
• log y log a b log x
• because
• log(a xb) log(a) log(xb) log a b log x

17
Power Function to Linear Transform

• We modify this transformed equation
• log y log a x log b
• Define these new variables
• Y log y
• A log a
• B log b
• X x
• Then this equation becomes
• Y A B X

18
Least Squares on Power Model

• We started with a power model
• y a bx
• We transformed that into a linear model
• Y A B X
• After we solve the linear model, we find that
• b B
• a 10A
• In this way, we are able to use the method of
  least-squares to find a power model

19
Harley Davidson Dataset

• Year (x) Closing Price (y)

• Year (x) Closing Price (y)

• 1990 1.1609
• 1991 2.6988
• 1992 4.5381
• 1993 5.3379
• 1994 6.8032
• 1995 7.0328
• 1996 11.5585

• 1997 13.4799
• 1998 23.5424
• 1999 31.9342
• 2000 39.7277
• 2001 54.31
• 2002 46.20
• 2003 47.53
• 2004 60.75

20
Power Function Example

• The scatter diagram below appears to be
  exponential (curved) and not linear
• A line is not an appropriate model

21
Fitting a Power Function Model

• We use Y log y and X log x
• The first observation is x 1 and y 1.1609,
  thus the first observation of the transformed
  data isX log 1 0 and Y log 1.1609 0.0648
• The second observation is x 2 and y 2.6988,
  thus the second observation of the transformed
  data is X log 2 .3010 and Y log 2.6988
  0.4312
• We continue and take logs of all of the x values
  and all the y values

22
Using your Calculator

• To get the scatter plot we inputted x-values into
  L1 and the y-values into L2
• To change the x-values into logs we go to the top
  of L3 and hit LOG(L1) ENTER and then repeat using
  L4 and the y-values (L2). Then use LINREG to
  find a and b (first part of the slide after next)
• Or a simpler way yet, use the PwrReg calculation
  under STAT and CALC and get it directly without
  having to convert back (last line of the slide
  after next)

23
Transformed Line

• The scatter diagram of the transformed data (Y
  and X) is more linear
• We now calculate least-squares regression line
  for this data

24
Least Squares to Power

• The least squares line is
• Y 1.5252 X 0.0928
• B 1.5252
• A 0.0928
• This is transformed back to
• b B 1.5252 and
• a 10A 0.8076, so
• y 0.8076 x1.5252

25
Power Model

• We now plot y 0.8076 x1.5252, our power model,
  on the original scatter diagram
• This is a better fit to the data, but we need to
  be careful if we try to extrapolate

26
Summary and Homework

• Summary
• Transformations can enable us to construct
  certain nonlinear models
• Exponential models, or y a bx, can be created
  using least-squares techniques after taking
  logarithms of both sides
• Power models, or y a xb, can also be created
  using least-squares techniques after taking
  logarithms of both sides
• Homework