At 06:57 PM 9/14/2010, Doug wrote:
If X is linear, and Y is linear,
how can X*Y be nonlinear?
To look from a different, but related, point of view:
The linear model
Y = a*X + b*Y + c
is independent of coordinate changes. That is, if you replace
X by X+z (where Z is a constant); or if you replace X by Y-X; you get the
same model. NOT that the coefficients will be the same (they won't), but
that it will reach the same set of predicted values.
Now, if you have
Y = a*X + b*Y + c*X*Y + d
then if you change coordinates
X'=Y-X, meaning
X=Y-X', you get
Y = a*(Y-X) + b*Y + c*(Y-X)*Y + d
The third term becomes
c*Y^2 -
c*X*Y and the new, equivalent model
does have a square
term, namely Y^2.
If you're going to include product terms (X*Y) but exclude square terms
(Y^2), then the second form of the model is "wrong" -- it
includes a squared term. But they're the same model.
If you include all second-order terms when you include any (that is, if
you include X^2 and Y^2 when you include X*Y), you do get equivalent
models after you change coordinates; that is, the model is
coordinate-independent. And that's important, because you rarely have a
unique 'right' set of coordinates.
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