# 定义

$\displaystyle{ 0=E(X)=AP-x(1-P); P=\frac{x}{A+x} }$

$\displaystyle{ \frac{p}{1-p}=\frac{x}{A}, \frac{p^*}{1-p^*}=\frac{2x}{A} }$，...

$\displaystyle{ \mbox{Logit}(P)=log(\frac{p}{1-p}) = f(X) = \beta_0+\sum_{i=1}^k{\beta_kX_k} + \epsilon }$

# 与线性回归的比较

 $\displaystyle{ \bar{P}\backslash X }$ 观测数 X1 X2 0.8 $\displaystyle{ N_1 }$ 0 0 0.65 $\displaystyle{ N_2 }$ 1 0 0.7 $\displaystyle{ N_3 }$ 0 1 0.55 $\displaystyle{ N_4 }$ 1 1

$\displaystyle{ \mbox{Logit}(P)=log(\frac{p}{1-p}) = f(X) = \beta_0+\beta_1X_1+\beta_2X_2 + \epsilon }$

# Logistic回归求解

Logistic回归的目标在于如何更准确的建立泛用性的Logistic线性模型，允许变量集$\displaystyle{ \{X_1,X_2,...X_k\} }$是连续型变量，如下图所示：

 Y $\displaystyle{ \backslash }$ X 观测数 X1 X2 1 1 $\displaystyle{ x_{1,1} }$ $\displaystyle{ x_{1,2} }$ 0 1 $\displaystyle{ x_{2,1} }$ $\displaystyle{ x_{2,2} }$ 0 1 $\displaystyle{ x_{3,1} }$ $\displaystyle{ x_{2,3} }$ 1 1 $\displaystyle{ x_{4,1} }$ $\displaystyle{ x_{2,4} }$

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