普通最小二乘法

普通最小二乘法（粵拼：pou2 tung1 zeoi3 siu2 ji6 sing4 faat3；英文：ordinary least squares）係迴歸分析當中最常用最普通嗰種估計 ${\displaystyle \beta }$（迴歸系數）嘅方法。

做法

睇埋：迴歸分析

普通最小二乘法就係建基於誤差值之上嘅。用呢種做法估計 ${\displaystyle \beta }$，首先要計個殘差平方和（residual sum of squares；RSS）出嚟，RSS 係指將所有誤差值嘅平方加埋得出嘅數：

${\displaystyle RSS=\sum _{i=1}^{n}e_{i}^{2}\,}$

${\displaystyle \beta _{0}}$ 同 ${\displaystyle \beta _{1}}$ 嘅數值可以用以下嘅算式計出嚟：

${\displaystyle {\widehat {\beta }}_{1}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}$

${\displaystyle {\widehat {\beta }}_{0}={\bar {y}}-{\widehat {\beta }}_{1}{\bar {x}}}$

當中 ${\displaystyle {\bar {x}}}$ 係 ${\displaystyle x}$ 嘅平均值，而 ${\displaystyle {\bar {y}}}$ 係 ${\displaystyle y}$ 嘅平均值。

假設個總體嘅誤差值有一個固定嘅變異數，呢個變異數可以用以下呢條式估計：

${\displaystyle {\hat {\sigma }}_{\varepsilon }^{2}={\frac {RSS}{n-2}}.\,}$

呢個數係所謂嘅均方誤差（mean square error），個分母係樣本大細減個模型要估計嘅參數嘅量。呢個迴歸模型當中有兩個未知嘅參數（${\displaystyle \beta _{0}}$ 同 ${\displaystyle \beta _{1}}$）^[1]。

而啲參數估計嘅標準誤差（standard error）係：

${\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}}$

${\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}={\hat {\sigma }}_{\beta _{1}}{\sqrt {\frac {\sum x_{i}^{2}}{n}}}}$

有咗上面呢個模型，個研究者手上有會有 ${\displaystyle \beta _{0}}$ 同 ${\displaystyle \beta _{1}}$ 嘅估計值，將來可以用呢條式嚟預測 ${\displaystyle Y}$ 嘅數值。

參考

1. Steel, R.G.D, and Torrie, J. H., Principles and Procedures of Statistics with Special Reference to the Biological Sciences., McGraw Hill, 1960, page 288.