Regression surface

regression hypersurface

The general geometric representation of a regression equation. If one is given random variables $X _ {1} \dots X _ {n}$ and

$$f ( x _ {2} \dots x _ {n} ) = \ {\mathsf E} ( X _ {1} \mid X _ {2} = x _ {2} \dots X _ {n} = x _ {n} )$$

is the regression of $X _ {1}$ with respect to $X _ {2} \dots X _ {n}$, then the equation $y = f ( x _ {2} \dots x _ {n} )$ describes the corresponding regression hypersurface in an $n$- dimensional space. When $n = 2$, a regression hypersurface is usually called a regression curve. These terms are sometimes used to emphasize that the corresponding regression equations are not linear. In the linear case, a regression hypersurface or curve is called a regression plane or line, respectively. See Regression.