逻辑回归：简介
R 中的信用风险建模
Lore Dirick
Manager of Data Science Curriculum at Flatiron School
最终数据结构
str(training_set)
'data.frame':\t19394 行，8 列：
$ loan_status : 因子，2 水平 "0","1": 1 1 1 1 1 1 1 1 1 1 ...
$ loan_amnt : int 25000 16000 8500 9800 3600 6600 3000 7500 6000 22750 ...
$ grade : 因子，7 水平 "A","B","C","D",..: 2 4 1 2 1 1 1 2 1 1 ...
$ home_ownership: 因子，4 水平 "MORTGAGE","OTHER",..: 4 4 1 1 1 3 4 3 4 1 ...
$ annual_inc : num 91000 45000 110000 102000 40000 ...
$ age : int 34 25 29 24 59 35 24 24 26 25 ...
$ emp_cat : 因子，5 水平 "0-15","15-30",..: 1 1 1 1 1 2 1 1 1 1 ...
$ ir_cat : 因子，5 水平 "0-8","11-13.5",..: 2 3 1 4 1 1 1 4 1 1 ...
什么是逻辑回归？
- 输出介于 0 到 1 的回归模型
$$P({\text{loan status}}=1|x_1,...,x_m) = \frac{1}{1+e^{-(\beta_0 + \beta_1 x_1 + ... + \beta_m x_m)}}$$
- $x_1,...,x_m$：
loan_amnt grade age annual_inc home_ownership emp_cat ir_cat
$\beta_0,...\beta_m$： 待估参数
$\beta_0 + \beta_1 x_1 + ... + \beta_m x_m$： 线性预测子
在 R 中拟合逻辑模型
log_model <- glm(loan_status ~ age ,
family= "binomial", data = training_set)
log_model
Call: glm(formula = loan_status ~ age,
family = "binomial", data = training_set)
Coefficients:
(Intercept) age
-1.793566 -0.009726
Degrees of Freedom: 19393 Total (i.e. Null); 19392 Residual
Null Deviance:\t 13680
Residual Deviance: 13670 \tAIC: 13670
$$P({\text{loan status}}=1|\text{age}) = \frac{1}{1+e^{-(\hat{\beta_0} + \hat{\beta_1} \text{age})}}$$
违约概率
$$P({\text{loan status}}=1|x_1,...,x_m) = \frac{1}{1+e^{-(\beta_0 + \beta_1 x_1 + ... + \beta_m x_m)}} = \frac{e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}}{1 + e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}}$$
$$
$$P({\text{loan status}}=0|x_1,...,x_m) = 1- \frac{e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}}{1 + e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}} = \frac{1}{1+e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}}$$
$$
$$\frac{P({\text{loan status}}=1|x_1,...,x_m)}{P({loan \space status}=0|x_1,...,x_m)} = e^{\beta_0 + \beta_1 x_1 + ... + \beta_m x_m}$$
loan_status = 1的胜算（odds）
系数的解释
- 若变量 $x_j$ 增加 1
- 胜算被乘以 $e^{\beta j}$
- $\beta_j < 0$
- $e^{\beta j} < 1$
- 随 $x_j$ 增加，胜算下降
- $\beta_j > 0$
- $e^{\beta j} > 1$
- 随 $x_j$ 增加，胜算上升
应用到我们的模型：
- 若变量
age增加 1- 胜算被乘以 $e^{-0.009726}$
- 胜算被乘以 0.991
让我们来练习！
R 中的信用风险建模
