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## Lesson 4 - 4

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**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**• 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 Exponential Power**Exponential Data**• We would like to fit an exponentialmodel 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(abx) = 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 = abx • 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) 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 • 1990 1.1609 • 1991 2.6988 • 1992 4.5381 • 1993 5.3379 • 1994 6.8032 • 1995 7.0328 • 1996 11.5585**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 powermodel y = axb • 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 + BX • 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) 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 • 1990 1.1609 • 1991 2.6988 • 1992 4.5381 • 1993 5.3379 • 1994 6.8032 • 1995 7.0328 • 1996 11.5585**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