Logistic regression 逻辑回归

• Logistic function 逻辑函数

$$y\ =\ \frac{1}{1+e^{-x}}$$

• Equation for logistic regression
  逻辑回归公式

$$y\ =\ \frac{1}{1+e^{-\left(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p\right)}}$$

• • Relationship between logistic and linear regression
    逻辑回归与线性回归的关系

$$\log\left(\frac{y}{1-y}\right)=\beta_0+\beta_1x$$

Logistic regression with sklearn
使用 sklearn 进行逻辑回归

• Build the model 建立模型

• • [$\beta_0$] $\rightarrow$ .intercept_
    [ $\beta_0$ ] $\rightarrow$ .intercept_

  • [[$\beta_1$, ...., $\beta_n$]] $\rightarrow$ .coef_
    [[ $\beta_1$ , ...., $\beta_n$ ]] $\rightarrow$ .coef_

• Making predictions 进行预测

Classification metrics 分类指标

• Confusion matrix 混淆矩阵

• Further metrics 进一步的衡量标准

$$\text{Precision}=\frac{TP}{TP\ +\ FP} \hspace{1.5cm}\text{Recall}\ =\ \frac{TP}{TP+FN}$$

$$\text{F}_1=\frac{2\ \times\text{precision} \times \text{recall}}{\text{precision + recall}}$$

Customer types 客户类型

1. Sure things: customers that will achieve the campaign goal with certainty
   有把握的客户：能确保实现活动目标的客户

2. Persuadables: customers that will respond to the campaign if the are treated e.g. sent a message about a promotion
   可被说服的客户：如果受到对待，例如收到促销信息，就会对活动做出回应的客户

3. Lost causes: the opposite of "Sure things" in that they will not respond to the campaign regardless of whether they are treated
   失落的原因：与 "肯定的事情 "相反，无论是否对其进行治疗，它们都不会对活动做出反应

4. Sleeping dogs: customers who will not achieve the campaign goal if treated so they respond negatively to treatment
   睡狗：如果对其进行处理，他们将无法实现活动目标，因此他们会对处理做出消极反应

Note: treatment here refers to a customer being targeted with the campaign
注：此处的 "处理 "是指营销活动的目标客户

Uplift modelling 上升模型

Control models calculate the probability that a customer naturally achieves the goal of the marketing campaign.
控制模型计算客户自然实现营销活动目标的概率。

Treatment models calculate the probability that a customer achieves the goal of the marketing campaign after the customer shown the marketing material i.e. treated.
治疗模型计算的是客户在观看营销材料（即接受治疗）后实现营销活动目标的概率。

Uplift is defined as the following:
上浮的定义如下：

$$\text{Uplift} = P(\text{action based on treatment}) - P(\text{action based on no treatment})$$