, y x. X Y (x 1, x 2,, xn), (y 1, y 2,, yn) , n , Mi (i =1,2,, n) (xi, yi), xy. y = f (x). , X Y, : y = kx + b, (1) k b . k b , , (). k b (1), k b . S (k, b) k b, , a b: (2) (2), k b, S (k, b) : (3) , . k b (1), . |
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R2=1−V(y|x)V(y)=1−σ2σy2,{\displaystyle R^{2}=1-{\frac {V(y|x)}{V(y)}}=1-{\frac {\sigma ^{2}}{\sigma _{y}^{2}}},}
V(y|x)=σ2{\displaystyle V(y|x)=\sigma ^{2}} ( x) ( ).
, . , ( ):
R2=1−σ^2σ^y2=1−SSres/nSStot/n=1−SSresSStot,{\displaystyle R^{2}=1-{\frac {{\hat {\sigma }}^{2}}{{\hat {\sigma }}_{y}^{2}}}=1-{\frac {SS_{res}/n}{SS_{tot}/n}}=1-{\frac {SS_{res}}{SS_{tot}}},}
SSres=∑t=1net2=∑t=1n(yt−y^t)2{\displaystyle SS_{res}=\sum _{t=1}^{n}e_{t}^{2}=\sum _{t=1}^{n}(y_{t}-{\hat {y}}_{t})^{2}} , yt,y^t{\displaystyle y_{t},{\hat {y}}_{t}} .
SStot=∑t=1n(yt−y¯)2=nσ^y2{\displaystyle SS_{tot}=\sum _{t=1}^{n}(y_{t}-{\overline {y}})^{2}=n{\hat {\sigma }}_{y}^{2}} .
y¯=1n∑i=1nyi{\displaystyle {\bar {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}
SStot=SSreg+SSres{\displaystyle SS_{tot}=SS_{reg}+SS_{res}} , SSreg=∑t=1n(y^t−y¯)2{\displaystyle SS_{reg}=\sum _{t=1}^{n}({\hat {y}}_{t}-{\overline {y}})^{2}} , :
R2=SSregSStot{\displaystyle R^{2}={\frac {SS_{reg}}{SS_{tot}}}}
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