Parabolic regression

polynomial regression

A regression model in which the regression functions are polynomials. More precisely, let $ X = ( X _ {1} \dots X _ {m} ) ^ {T} $ and $ Y = ( Y _ {1} \dots Y _ {n} ) ^ {T} $ be random vectors taking values $ x = ( x _ {1} \dots x _ {m} ) ^ {T} \in \mathbf R ^ {m} $ and $ y = ( y _ {1} \dots y _ {n} ) ^ {T} \in \mathbf R ^ {n} $, and suppose that

$$ {\mathsf E} \{ Y \mid X \} = \ f( X) = ( f _ {1} ( X) \dots f _ {n} ( X)) ^ {T} $$

exists (i.e. suppose that $ {\mathsf E} \{ Y _ {1} \mid X \} = f _ {1} ( X) \dots $ $ {\mathsf E} \{ Y _ {n} \mid X \} = f _ {n} ( X) $ exist). The regression is called parabolic (polynomial) if the components of the vector $ {\mathsf E} \{ Y \mid X \} = f( x) $ are polynomial functions in the components of the vector $ X $. For example, in the elementary case where $ Y $ and $ X $ are ordinary random variables, a polynomial regression equation is of the form

$$ y = \beta _ {0} + \beta _ {1} X + \dots + \beta _ {p} X ^ {p} , $$

where $ \beta _ {0} \dots \beta _ {p} $ are the regression coefficients. A special case of parabolic regression is linear regression. By adding new components to the vector $ X $, it is always possible to reduce parabolic regression to linear regression. See Regression; Regression analysis.

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2]	G.A.F. Seber, "Linear regression analysis" , Wiley (1977)

Comments

The phrase "parabolic regression" is seldom used in the Western literature; one uses "polynomial regression" almost exclusively.