# 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)