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Curvilinear Regression

Simple linear regression has been developed to fit straight lines to data points. However, sometimes the relationship between two variables may be represented by a curve instead of a straight line. Such "non-linear" relationships need not be non-linear in a mathematical sense. For example, a parabolic relationship may be well-modeled by a (modified) linear regression, since a parabola is a linear equation, as far as its parameters are concerned. Sometimes, such relationships are called "curvilinear".
 


There are several ways to fit a curve other than a line (or, generally speaking, an n-dimensional hyperplane) to the data:
 


The first two approaches require the type of functional relationship to be known. In many standard cases, the second approach may be appropriate:
 


Below is a table of the transformations for linearizing some common relationships.
 
 

Non-Linear Model Step 1: Linearized Model Step 2: Calculate Linear Model Step 3: Back Transformation
y = abx lg y = a* + b*x a = 10a* b = 10b*
y = axb lg y = a* + b* lg x a = 10a* b = 10b*
y = aebx ln y = a* + b*x a = ea* b = b*
y = ae(b / x) ln y = a* + b* (1/x) a = ea* b = b*
y = a + b/x y = a* + b* (1/x) -- y is plotted against (1/x) a = a* b = b*
y = a / (b + x) (1/y) = a* + b*x a = b/a* a = 1/b*
y = a + bxn y = a* + b*xn a = a* b = b*

Last Update: 2006-Jšn-17