رگرسیون خطی

${\displaystyle y_i={\hat{\beta }}_0+{\hat{\beta }}_1{X}_i+e_i}$

${\displaystyle \mathrm{S}\mathrm{S}\mathrm{E}=\sum _{i=1}^N{e}_i^2}$

${\displaystyle \overrightarrow{\hat{\beta }}=\underset{\overrightarrow{\beta }}{\text{arg min}}L(D,\overrightarrow{\beta })=\underset{\overrightarrow{\beta }}{\text{arg min}}\sum _{i=1}^n(\overrightarrow{\beta }.\overrightarrow{x_i}-y_i{)}^2}$

${\displaystyle \{\begin{array}{l}\text{ for }j=1,\cdots ,m:\\ X_j^{(0)}=X_j\\ \text{ for }i=1,\cdots ,k:\\ Z_i=\sum _{j=1}^m⟨X_j^{(i-1)},Y⟩X_j^{(i-1)}\\ \text{ for }j=1,\cdots ,m:\\ X_j^{(i)}=X_j^{(i)}-\frac{⟨Z_i,Y⟩}{⟨Z_i,Z_i⟩}{Z}_i\\ \hat{Y}=\overline{Y}\mathbf{1}+\sum _{i=1}^k\frac{⟨Z_i,Y⟩}{⟨Z_i,Z_i⟩}{Z}_i\end{array}}$

${\displaystyle \{\begin{array}{l}\text{Initialize}\overrightarrow{{\beta }^{old}}\text{randomly}\\ \text{loop until convergence :}\\ \text{for}i=0\text{to}n:\\ \text{for}j=0\text{to}m:\\ \overrightarrow{{\beta }_j^{new}}=\overrightarrow{{\beta }_j^{old}}+\alpha \left(y_i-\overrightarrow{{\beta }^{old}}.\overrightarrow{x_i}\right)\overrightarrow{x_{i,j}}\\ {\beta }^{old}={\beta }^{new}\end{array}}$

${\displaystyle Pr\left(\overrightarrow{\beta }|D\right)=\frac{Pr\left(D|\overrightarrow{\beta }\right)\times Pr\left(\overrightarrow{\beta }\right)}{Pr\left(D\right)}}$

${\displaystyle Q_Y(\tau )=F_Y^{-1}(\tau )=inf\left\{y:F_Y(y)\ge \tau \right\}}$

${\displaystyle Q_Y(\tau )=\underset{u}{min}E({\rho }_{\tau }(Y-u))=\underset{u}{min}\left\{(\tau -1){\int }_{-\mathrm{\infty }}^u(y-u)d{F}_Y(y)+\tau {\int }_u^{\mathrm{\infty }}(y-u)d{F}_Y(y)\right\}}$

${\displaystyle {\overrightarrow{\hat{\beta }}}_{\tau }=\underset{\overrightarrow{\beta }}{\text{arg min}}\sum _{i=1}^n{\rho }_{\tau }\left(y_i-\overrightarrow{\beta }.{\overrightarrow{x}}_i\right)}$

${\displaystyle {\overrightarrow{\hat{\beta }}}_{0.5}=\underset{\overrightarrow{\beta }}{\text{arg min}}\sum _{i=1}^n{\rho }_{0.5}\left(y_i-\overrightarrow{\beta }.{\overrightarrow{x}}_i\right)=\underset{\overrightarrow{\beta }}{\text{arg min}}\sum _{i=1}^n\left|y_i-\overrightarrow{\beta }.{\overrightarrow{x}}_i\right|}$

${\displaystyle \overrightarrow{\hat{\beta }}(\overrightarrow{x})=\underset{\overrightarrow{\beta }}{\text{arg min}}\sum _{i=1}^n{w}_i(\overrightarrow{x})\times (\overrightarrow{\beta }.\overrightarrow{x_i}-y_i{)}^2}$

${\displaystyle w_i(\overrightarrow{x})=\exp \left(-\frac{||\overrightarrow{x}-\overrightarrow{x_i}|{|}^2}{2{\sigma }^2}\right)}$

${\displaystyle f(\overrightarrow{x})={\beta }_0+{\beta }_1\varphi (\overrightarrow{x}{)}_1+\cdots +{\beta }_h\varphi (\overrightarrow{x}{)}_h}$

${\displaystyle f(x)={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+\cdots +{\beta }_h{x}^h}$

${\displaystyle Y_{ij}={\beta }_{0j}+{\beta }_{1j}{X}_{ij}+e_{ij}}$

${\displaystyle {\beta }_{0j}={\gamma }_{00}+{\gamma }_{01}{W}_j+u_{0j}}$

${\displaystyle {\beta }_{1j}={\gamma }_{10}+{\gamma }_{11}{W}_j+u_{1j}}$