線形回帰

$$ \begin{aligned} &j \in \{ 1, \ldots , m \} \subset \mathbb{N} \\ &x_{j} : \text{explanatory variable} \\ &y : \text{objective variable} \end{aligned} $$

$$ \begin{aligned} y &= \beta_{0} + \beta_{1} x_{1} + \cdots + \beta_{j} x_{j} + \cdots + \beta_{m} x_{m} + \epsilon \\ &= \beta_{0} + \sum_{j=1}^{m} \beta_{j} x_{j} + \epsilon : \text{linear regression} \end{aligned} $$

$$ \begin{aligned} y &= \beta_{0} + \sum_{j=1}^{m} \beta_{j} x_{j} + \epsilon \\ &= \beta_{0} + \left( \begin{array}{c} x_{1} \\ \vdots \\ x_{j} \\ \vdots \\ x_{m} \end{array} \right) ^{\mathrm{T}} \left( \begin{array}{c} \beta_{1} \\ \vdots \\ \beta_{j} \\ \vdots \\ \beta_{m} \end{array} \right) + \epsilon \\ \\ &= \left( \begin{array}{c} 1 \\ x_{1} \\ \vdots \\ x_{j} \\ \vdots \\ x_{m} \end{array} \right) ^{\mathrm{T}} \left( \begin{array}{c} \beta_{0} \\ \beta_{1} \\ \vdots \\ \beta_{j} \\ \vdots \\ \beta_{m} \end{array} \right) + \epsilon \\ \end{aligned} $$

$$ \begin{aligned} &E[ \epsilon ] = 0 \\ &\text{Var}( \epsilon ) = \sigma ^ 2 \end{aligned} $$

$$ \begin{aligned} y = \beta_{0} + \beta_{1} x + \beta_{2} x^{2} + \epsilon \end{aligned} $$