Download - Lesson 4 - 4
Lesson 4 - 4
Nonlinear Regression:
Transformations
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
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)
Non-linear Scatter Diagrams
Exponential Exponential Power
Some relationships that are nonlinear can be modeled with exponential or power models
y = a bx (with b > 1) y = a bx (with b < 1) y = a xb
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
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
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
Harley Davidson Dataset
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
Year (x) Closing Price (y)
• 1997 13.4799• 1998 23.5424• 1999 31.9342• 2000 39.7277• 2001 54.31• 2002 46.20• 2003 47.53• 2004 60.75
Exponential Example
• The scatter diagram below appears to be exponential (curved) and not linear
• A line is not an appropriate model
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
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)
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
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
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
Part Two
Power Models
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
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
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
Harley Davidson Dataset
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
Year (x) Closing Price (y)
• 1997 13.4799• 1998 23.5424• 1999 31.9342• 2000 39.7277• 2001 54.31• 2002 46.20• 2003 47.53• 2004 60.75
Power Function Example
• The scatter diagram below appears to be exponential (curved) and not linear
• A line is not an appropriate model
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
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)
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
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
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
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