直線近似の求め方-03
\( \Large \displaystyle a \displaystyle \sum_{i=1}^n x_i^2 +b \displaystyle \sum_{i=1}^n x_i = \displaystyle \sum_{i=1}^n x_i y_i \)
\( \Large \displaystyle a \displaystyle \sum_{i=1}^n x_i + nb = \displaystyle \sum_{i=1}^n y_i \)
まずは,先ほどの方程式の一式に,\( \Large \displaystyle \displaystyle \sum_{i=1}^n x_i \),を,二式に,\( \Large \displaystyle \displaystyle \sum_{i=1}^n x_i^2 \),をかけると,
\( \Large \displaystyle a \displaystyle \sum_{i=1}^n x_i^2 \cdot \displaystyle \sum_{i=1}^n x_i +b \left(\displaystyle \sum_{i=1}^n x_i \right)^2= \displaystyle \sum_{i=1}^n x_i y_i \cdot \displaystyle \sum_{i=1}^n x_i\)
\( \Large \displaystyle a \displaystyle \sum_{i=1}^n x_i \cdot \displaystyle \sum_{i=1}^n x_i^2+ nb \displaystyle \sum_{i=1}^n x_i^2= \displaystyle \sum_{i=1}^n y_i \cdot \displaystyle \sum_{i=1}^n x_i^2\)
\( \Large \displaystyle b\left\{\left(\displaystyle \sum_{i=1}^n x_i \right)^2 - nb \displaystyle \sum_{i=1}^n x_i^2 \right\} = \displaystyle \sum_{i=1}^n x_i y_i \cdot \displaystyle \sum_{i=1}^n x_i - \displaystyle \sum_{i=1}^n y_i \cdot \displaystyle \sum_{i=1}^n x_i^2 \)
\( \Large \displaystyle b
= \frac{ \displaystyle \sum_{i=1}^n x_i y_i \cdot \displaystyle \sum_{i=1}^n x_i - \displaystyle \sum_{i=1}^n y_i \cdot \displaystyle \sum_{i=1}^n x_i^2}
{\left(\displaystyle \sum_{i=1}^n x_i \right)^2 - nb \displaystyle \sum_{i=1}^n x_i^2 } \)
となり,b,を求めることができました.
次に,先ほどの方程式の一式に,n,を,二式に,\( \Large \displaystyle \displaystyle \sum_{i=1}^n x_i \),をかけると,
\( \Large \displaystyle na \displaystyle \sum_{i=1}^n x_i^2 +nb \displaystyle \sum_{i=1}^n x_i = n \displaystyle \sum_{i=1}^n x_i y_i \)
\( \Large \displaystyle a \left(\displaystyle \sum_{i=1}^n x_i \right)^2+ nb \displaystyle \sum_{i=1}^n x_i = \displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i \)
\( \Large \displaystyle a \left\{ n \displaystyle \sum_{i=1}^n x_i^2 - \left(\displaystyle \sum_{i=1}^n x_i \right)^2 \right\} = n \displaystyle \sum_{i=1}^n x_i y_i-\displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i \)
\( \Large \displaystyle a = \frac{ n \displaystyle \sum_{i=1}^n x_i y_i-\displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i}{n \displaystyle \sum_{i=1}^n x_i^2 - \left(\displaystyle \sum_{i=1}^n x_i \right)^2} \)
となって,無事,a,bを求めることができました.
もう少し,a,をいじってみると,
分子:
\( \Large \displaystyle n \displaystyle \sum_{i=1}^n x_i y_i-\displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i \)
\( \Large \displaystyle = n \displaystyle \sum_{i=1}^n x_i y_i-\displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i + \displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i - \displaystyle \sum_{i=1}^n x_i \displaystyle \sum_{i=1}^n y_i\)
\( \Large \displaystyle = n \displaystyle \sum_{i=1}^n x_i y_i-n \bar{x} \displaystyle \sum_{i=1}^n y_i + n^2 \bar{x} \bar{y} - n \displaystyle \sum_{i=1}^n x_i \bar{y}\)
\( \Large \displaystyle = n \displaystyle \sum_{i=1}^n (x_i y_i- \bar{x} y_i- x_i \bar{y} + \bar{x} \bar{y} ) \)
\( \Large \displaystyle = n \displaystyle \sum_{i=1}^n (x_i - \bar{x})( y_i - \bar{y} ) \)
分母:
\( \Large \displaystyle n \sum_{i=1}^n x_i^2 - \left(\displaystyle \sum_{i=1}^n x_i \right)^2 \)
\( \Large = \displaystyle n \sum_{i=1}^n x_i^2 - n \bar{x} \displaystyle \sum_{i=1}^n x_i \)
\( \Large = \displaystyle n \sum_{i=1}^n x_i^2 - 2n \bar{x} \displaystyle \sum_{i=1}^n x_i + n \bar{x} \displaystyle \sum_{i=1}^n x_i \)
\( \Large = \displaystyle n \sum_{i=1}^n x_i^2 - 2n \bar{x} \displaystyle \sum_{i=1}^n x_i + n^2 \bar{x}^2 \)
\( \Large = \displaystyle n \sum_{i=1}^n \left( x_i^2 - 2n \bar{x} x_i + \bar{x}^2 \right) \)
\( \Large = \displaystyle n \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 \)
したがって,
\( \Large a =\frac{ \displaystyle \sum_{i=1}^n (x_i - \bar{x})( y_i - \bar{y} )}{ \displaystyle \sum_{i=1}^n \left( x_i - \bar{x} \right)^2} \)
\( \Large \displaystyle a =\frac{ Cov(X,Y) }{ \sigma_X^2} \)
もしくは,
\( \Large \displaystyle a =\frac{ S_{xy} }{ S_x^2} \)
と記述することができます.
bは,ここ,より,
\( \Large \displaystyle b = \overline{y} - a \overline{x} \)
なので,書き換えると
\( \Large \displaystyle b = \overline{y} - \frac{ S_{xy} }{ S_x^2} \overline{x} \)
となります.
次に,全体のばらつきからの決定係数の求め方,を考えましょう.