View source for Parabolic regression

''polynomial regression''

A regression model in which the regression functions are polynomials. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712402.png" /> be random vectors taking values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712404.png" />, and suppose that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712405.png" /></td> </tr></table>

exists (i.e. suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712406.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712407.png" /> exist). The regression is called parabolic (polynomial) if the components of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712408.png" /> are polynomial functions in the components of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p0712409.png" />. For example, in the elementary case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p07124010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p07124011.png" /> are ordinary random variables, a polynomial regression equation is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p07124012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p07124013.png" /> are the regression coefficients. A special case of parabolic regression is [[Linear regression|linear regression]]. By adding new components to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071240/p07124014.png" />, 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.