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.