【入門】平均分散共分散で最小二乗法【数値計算】

\(
\begin{eqnarray}
\displaystyle a\sum 1 &=& an \dots 総和の変形 \\
\displaystyle n\bar{y}&=&\sum y_i \dots 平均の変形 \\
\displaystyle \sigma_x^2 &=&\bar{x_i^2}-(\bar{x})^2 \dots 分散の変形 \\
\displaystyle \sigma_{xy}&=&\frac{1}{n}\sum x_i y_i – \bar{x}\bar{y} \dots 共分散の変形
\end{eqnarray}
\)

\(
\begin{eqnarray}
\displaystyle a\sum x_i^2 + b\sum x_i &=& \sum x_i y_i \dots (1)\\
\displaystyle a\sum x_i + b\sum 1 &=& \sum y_i \dots (2)
\end{eqnarray}
\)

\(
\begin{eqnarray}
\displaystyle a\sum x_i + b\sum 1 &=& \sum y_i \\
a\sum x_i + nb&=&\sum y_i \\
nb&=&\sum y_i – a\sum x_i \\
b&=&\frac{1}{n}\sum y_i-\frac{a}{n}\sum x_i \\
b&=&\bar{y}-a\bar{x} \qquad//
\end{eqnarray}
\)

\(
\begin{eqnarray}
\displaystyle a\sum x_i^2+b\sum x_i &=&\sum x_i y_i \\
-\sum x_i y_i+a\sum x_i^2+b\sum x_i &=& 0 \\
\end{eqnarray}
\)

ここで\(b\)を代入

\(
\begin{eqnarray}
\displaystyle -\sum x_i y_i+a\sum x_i^2+(\bar{y}-a\bar{x})\sum x_i &=& 0 \\
-\sum x_i y_i+a\sum x_i^2+\bar{y}\sum x_i-a\bar{x}\sum x_i &=& 0
\end{eqnarray}
\)

\(\displaystyle\sum x_i\)を\(n\bar{x}\)に置き換え、両辺をマイナスする。

\(
\begin{eqnarray}
\displaystyle \sum x_i y_i-n\bar{y}\bar{x}-a(\sum x_i^2-n\bar{x}\bar{x})&=&0 \\
n(\frac{1}{n}\sum x_i y_i-\bar{y}\bar{x})-an(\frac{1}{n}\sum x_i^2-(\bar{x})^2)&=&0 \\
n(\frac{1}{n}\sum x_i y_i-\bar{y}\bar{x})-an(\bar{x_i^2}-(\bar{x})^2)&=&0 \\
n\sigma_{xy}-an\sigma_x^2&=&0 \\
a&=&\frac{\sigma_{xy}}{\sigma_x^2} \qquad //
\end{eqnarray}
\)

\(
\displaystyle a=\frac{\sigma_{xy}}{\sigma_x^2}
\)

係数\(b\)を求める式

\(
b=\bar{y}-a\bar{x}
\)