线性回归
1 一元线性回归重要公式
一元线性回归的均方误差：
E ( w , b ) = ∑ i = 1 m ( y i − w x i − b ) 2 {
{\rm{E}}_{(w,b)}} = {\sum\limits_{i = 1}^m {({y_i} - w{x_i} - b)} ^2} E(w,b)=i=1∑m(yi−wxi−b)2
对w和b分别求导，得
∂ E ( w , b ) ∂ w = 2 ( w ∑ i = 1 m x i 2 − ∑ i = 1 m ( y i − b ) x i ) \frac{
{\partial {E_{(w,b)}}}}{
{\partial w}} = 2(w\sum\limits_{i = 1}^m {x_i^2 - \sum\limits_{i = 1}^m {({y_i} - b){x_i}} } ) ∂w∂E(w,b)=2(wi=1∑mxi2−i=1∑m(yi−b)xi)
∂ E ( w , b ) ∂ b = 2 ( m b − ∑ i = 1 m y i − w x i ) \frac{
{\partial {E_{(w,b)}}}}{
{\partial b}} = 2(mb - \sum\limits_{i = 1}^m {
{y_i} - w{x_i}} ) ∂b∂E(w,b)=2(mb−i=1∑myi−wxi)
令以上式子分别等于0，得
w = ∑ i = 1 m y i ( x i − x ˉ ) ∑ i = 1 m x i 2 − 1 m ( ∑ i = 1 m x i ) 2 w = \frac{
{\sum\limits_{i = 1}^m {
{y_i}({x_i} - \bar x)} }}{
{\sum\limits_{i = 1}^m {x_i^2 - \frac{1}{m}{
{(\sum\limits_{i = 1}^m {
{x_i}} )}^2}} }} w=i=1∑mxi2−m1(i=1∑mxi)2i=1∑myi(xi−
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