# 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.

How to Cite This Entry:
Parabolic regression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_regression&oldid=48109
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article