رگرسیون خطی - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle y_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}X_{i}+e_{i}}$

${\displaystyle \mathrm {SSE} =\sum _{i=1}^{N}e_{i}^{2}}$

${\displaystyle {\vec {\hat {\beta }}}={\underset {\vec {\beta }}{\mbox{arg min}}}\,L(D,{\vec {\beta }})={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}({\vec {\beta }}\,.\,{\vec {x_{i}}}-y_{i})^{2}}$

${\displaystyle {\begin{aligned}L(D,{\vec {\beta }})&=||X{\vec {\beta }}-Y||^{2}\\&=(X{\vec {\beta }}-Y)^{\top }(X{\vec {\beta }}-Y)\\&=Y^{\top }Y-Y^{\top }X{\vec {\beta }}-{\vec {\beta }}^{\top }X^{\top }Y+{\vec {\beta }}^{\top }X^{\top }X{\vec {\beta }}\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial L(D,{\vec {\beta }})}{\partial {\vec {\beta }}}}&={\frac {\partial \left(Y^{\top }Y-Y^{\top }X{\vec {\beta }}-{\vec {\beta }}^{\top }X^{\top }Y+{\vec {\beta }}^{\top }X^{\top }X{\vec {\beta }}\right)}{\partial {\vec {\beta }}}}\\&=-2X^{\top }Y+2X^{\top }X{\vec {\beta }}\end{aligned}}}$

${\displaystyle {\begin{aligned}-2X^{\top }Y&+2X^{\top }X{\vec {\beta }}=0\\&\Rightarrow X^{\top }Y=X^{\top }X{\vec {\beta }}\\&\Rightarrow {\vec {\hat {\beta }}}=(X^{\top }X)^{-1}X^{\top }Y\end{aligned}}}$

${\displaystyle {\begin{cases}\,\,{\mbox{ for }}\,\,j\,\,=\,\,1,\cdots ,m:\\\,\,\,\,\,\,\,\,\,\,{X}_{j}^{(0)}={X}_{j}\\\,\,{\mbox{ for }}\,\,i=\,\,1,\cdots ,k:\\\,\,\,\,\,\,\,\,\,\,{Z}_{i}=\sum _{j=1}^{m}\left\langle {X}_{j}^{(i-1)},{Y}\right\rangle {X}_{j}^{(i-1)}\\\,\,\,\,\,\,\,\,\,{\mbox{ for }}\,\,j\,\,=\,\,1,\cdots ,m:\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X}_{j}^{(i)}={X}_{j}^{(i)}-{\frac {\langle {Z}_{i}\,,\,{Y}\rangle }{\langle {Z}_{i}\,,\,{Z}_{i}\rangle }}{Z}_{i}\\\,\,{\hat {Y}}={\bar {Y}}\mathbf {1} +\sum _{i=1}^{k}{\frac {\langle {Z}_{i}\,,\,{Y}\rangle }{\langle {Z}_{i}\,,\,{Z}_{i}\rangle }}{Z}_{i}\\\end{cases}}}$

${\displaystyle {\begin{cases}{\mbox{Initialize}}\,\,{\vec {\beta ^{\,old}}}\,\,{\mbox{randomly}}\\{\mbox{loop until convergence :}}\\\,\,{\mbox{for}}\,\,\,\,i=0\,\,\,\,{\mbox{to}}\,\,\,\,n:\\\,\,\,\,\,\,{\mbox{for}}\,\,\,\,j=0\,\,\,\,{\mbox{to}}\,\,\,\,m:\\\,\,\,\,\,\,\,\,\,\,\,\,{\vec {\beta _{j}^{\,new}}}={\vec {\beta _{j}^{\,old}}}+\alpha \left(y_{i}-{\vec {\beta ^{\,old}}}\,.\,{\vec {x_{i}}}\right){\vec {x_{i,j}}}\\\,\,\,\,\,\,\beta ^{\,old}=\beta ^{\,new}\end{cases}}}$

${\displaystyle {\begin{aligned}H(D,{\vec {\beta }})&=\prod _{i=1}^{n}Pr(y_{i}|{\vec {x_{i}}}\,\,;{\vec {\beta }},\sigma )\\&=\prod _{i=1}^{n}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}}{2\sigma ^{2}}}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}I(D,{\vec {\beta }})&=\log \prod _{i=1}^{n}Pr(y_{i}|{\vec {x_{i}}}\,\,;{\vec {\beta }},\sigma )\\&=\log \prod _{i=1}^{n}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}}{2\sigma ^{2}}}\right)\\&=n\log {\frac {1}{{\sqrt {2\pi }}\sigma }}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}\end{aligned}}}$

${\displaystyle {\begin{aligned}{\underset {\vec {\beta }}{\mbox{arg max}}}\,I(D,{\vec {\beta }})&={\underset {\vec {\beta }}{\mbox{arg max}}}\left(n\log {\frac {1}{{\sqrt {2\pi }}\sigma }}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}\right)\\&={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}\\&={\underset {\vec {\beta }}{\mbox{arg min}}}\,L(D,{\vec {\beta }})\\&={\vec {\hat {\beta }}}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}{\underset {\vec {\beta }}{\mbox{arg max}}}\,Pr\left({\vec {\beta }}\,|\,D\right)&={\underset {\vec {\beta }}{\mbox{arg max}}}\,{\frac {Pr\left(D\,|\,{\vec {\beta }}\right)\times Pr\left({\vec {\beta }}\right)}{Pr\left(D\right)}}\\&={\underset {\vec {\beta }}{\mbox{arg max}}}\,Pr\left(D\,|\,{\vec {\beta }}\right)\times Pr\left({\vec {\beta }}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\underset {\vec {\beta }}{\mbox{arg max}}}\,\,\log Pr\left(D\,|\,{\vec {\beta }}\right)&={\underset {\vec {\beta }}{\mbox{arg max}}}\,\,\log \prod _{i=1}^{n}Pr(y_{i}|{\vec {x_{i}}}\,\,;{\vec {\beta }},\sigma )Pr\left({\vec {\beta }}\right)\\&={\underset {\vec {\beta }}{\mbox{arg max}}}\,\,\log \prod _{i=1}^{n}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}}{2\sigma ^{2}}}\right){\frac {1}{{\sqrt {2\pi }}\tau }}\exp \left(-{\frac {||{\vec {\beta }}||^{2}}{2\tau ^{2}}}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\underset {\vec {\beta }}{\mbox{arg max}}}\,\,Pr\left(D\,|\,{\vec {\beta }}\right)&={\underset {\vec {\beta }}{\mbox{arg max}}}\left(n\log {\frac {1}{{\sqrt {2\pi }}\sigma }}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}+\log {\frac {1}{{\sqrt {2\pi }}\tau }}-{\frac {1}{2\tau ^{2}}}||{\vec {\beta }}||^{2}\right)\\&={\underset {\vec {\beta }}{\mbox{arg min}}}\left(\sum _{i=1}^{n}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x_{i}}}\right)^{2}+\lambda ||{\vec {\beta }}||^{2}\right)\end{aligned}}}$

${\displaystyle Q_{Y}(\tau )=F_{Y}^{-1}(\tau )=\inf \left\{y:F_{Y}(y)\geq \tau \right\}}$

${\displaystyle Q_{Y}(\tau )={\underset {u}{\min }}\,E(\rho _{\tau }(Y-u))={\underset {u}{\min }}\,\left\{(\tau -1)\int _{-\infty }^{u}(y-u)dF_{Y}(y)+\tau \int _{u}^{\infty }(y-u)dF_{Y}(y)\right\}}$

${\displaystyle {\hat {Q}}_{Y}(\tau )={\underset {u\in \mathbb {R} }{\mbox{arg min}}}\sum _{i=1}^{n}\rho _{\tau }\left(y_{i}-u\right)={\underset {u\in \mathbb {R} }{\mbox{arg min}}}\left[(\tau -1)\sum _{y_{i}<u}(y_{i}-u)+\tau \sum _{y_{i}\geq u}(y_{i}-u)\right]}$

${\displaystyle {\vec {\hat {\beta }}}_{\tau }={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}\rho _{\tau }\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x}}_{i}\right)}$

${\displaystyle {\vec {\hat {\beta }}}_{0.5}={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}\rho _{0.5}\left(y_{i}-{\vec {\beta }}\,.\,{\vec {x}}_{i}\right)={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}\left|y_{i}-{\vec {\beta }}\,.\,{\vec {x}}_{i}\right|}$

${\displaystyle {\vec {\hat {\beta }}}({\vec {x}})={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}w_{i}({\vec {x}})\times ({\vec {\beta }}\,.\,{\vec {x_{i}}}-y_{i})^{2}}$

${\displaystyle w_{i}({\vec {x}})=\exp \left(-{\frac {||{\vec {x}}-{\vec {x_{i}}}||^{2}}{2\sigma ^{2}}}\right)}$

${\displaystyle f({\vec {x}})=\beta _{0}+\beta _{1}\phi ({\vec {x}})_{1}+\cdots +\beta _{h}\phi ({\vec {x}})_{h}}$

${\displaystyle f(x)=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\cdots +\beta _{h}x^{h}}$

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

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

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