View source for Parabolic regression

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''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|linear regression]]. By adding new components to the vector  $  X $, 
it is always possible to reduce parabolic regression to linear regression. See [[Regression|Regression]]; [[Regression analysis|Regression analysis]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.A.F. Seber,   "Linear regression analysis" , Wiley  (1977)</TD></TR></table>

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