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線性模型(壹)

1 最小二乘法(Least Square)

一種數學方法,來直接求解最優解。
j = 1 n X

i j β j = y i
, ( i = 1 , 2 , . .
. , m ) , X β = y \sum_{j=1}^nX_{ij}\beta_j=y_i,(i=1,2,...,m),X\beta=y

[ X 11 X 12 . . . X 1 n X 21 X 22 . . . X 2 n X 31 X 32 . . . X 3 n . . . . . . . . . . . . X m 1 X m 2 . . . X m n ] , β = [ β 1 β 2 β 3 . . . β n ] , y = [ y 1 y 2 y 3 . . . y m ] \left[ \begin{matrix} X_{11} & X_{12} &...& X_{1n} \\ X_{21} & X_{22} &... & X_{2n}\\ X_{31} & X_{32} & ...&X_{3n}\\ ...&...&...&...\\ X_{m1} & X_{m2} & ...&X_{mn} \end{matrix} \right],\beta=\left[ \begin{matrix} \beta_{1}\\ \beta_{2} \\ \beta_{3} \\ ...\\ \beta_{n} \end{matrix} \right],y=\left[ \begin{matrix} y_{1}\\ y_{2} \\ y_{3} \\ ...\\ y_{m} \end{matrix} \right]
β ^ = a r g m i n β S ( β ) , S ( β ) = i = 1 m y i j = 1 n X i j β j 2 = y X β 2 \hat \beta =argmin_\beta S(\beta),S(\beta)=\sum_{i=1}^m|y_i-\sum_{j=1}^nX_{ij}\beta_j|^2=||y-X\beta||^2

推導:
y X β 2 = ( y X β ) T ( y X β ) = ( y T β T X T ) ( y X β ) , y T y y T X β β T X T y + β T X T X β ||y-X\beta||^2=(y-X\beta)^T(y-X\beta)=(y^T-\beta^TX^T)(y-X\beta),y^Ty-y^TX\beta-\beta^TX^Ty+\beta^TX^TX\beta