逻辑回归模型算法原理
逻辑回归模型的数学原理
知识点：Sigmoid函数绘制
import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-6, 6) # 通过linspace()函数生成-6到6的等差数列，默认50个数
y = 1.0 / (1.0 + np.exp(-x)) # Sigmoid函数计算公式，exp()函数表示指数函数
print(x)
print(y)
plt.plot(x,y) # 画图
plt.show() # 展示
[-6. -5.75510204 -5.51020408 -5.26530612 -5.02040816 -4.7755102
-4.53061224 -4.28571429 -4.04081633 -3.79591837 -3.55102041 -3.30612245
-3.06122449 -2.81632653 -2.57142857 -2.32653061 -2.08163265 -1.83673469
-1.59183673 -1.34693878 -1.10204082 -0.85714286 -0.6122449 -0.36734694
-0.12244898 0.12244898 0.36734694 0.6122449 0.85714286 1.10204082
1.34693878 1.59183673 1.83673469 2.08163265 2.32653061 2.57142857
2.81632653 3.06122449 3.30612245 3.55102041 3.79591837 4.04081633
4.28571429 4.53061224 4.7755102 5.02040816 5.26530612 5.51020408
5.75510204 6. ]
[0.00247262 0.00315659 0.00402898 0.00514124 0.00655853 0.00836325
0.01065923 0.01357692 0.01727929 0.0219688 0.02789489 0.03536175
0.04473535 0.05644827 0.07100002 0.08894941 0.11089489 0.13743793
0.16912564 0.2063713 0.2493577 0.29793663 0.35154728 0.40918225
0.46942595 0.53057405 0.59081775 0.64845272 0.70206337 0.7506423
0.7936287 0.83087436 0.86256207 0.88910511 0.91105059 0.92899998
0.94355173 0.95526465 0.96463825 0.97210511 0.9780312 0.98272071
0.98642308 0.98934077 0.99163675 0.99344147 0.99485876 0.99597102
0.99684341 0.99752738]
# 演示下linespace()函数
import numpy as np
x = np.linspace(-6, 6)
x
array([-6. , -5.75510204, -5.51020408, -5.26530612, -5.02040816,
-4.7755102 , -4.53061224, -4.28571429, -4.04081633, -3.79591837,
-3.55102041, -3.30612245, -3.06122449, -2.81632653, -2.57142857,
-2.32653061, -2.08163265, -1.83673469, -1.59183673, -1.34693878,
-1.10204082, -0.85714286, -0.6122449 , -0.36734694, -0.12244898,
0.12244898, 0.36734694, 0.6122449 , 0.85714286, 1.10204082,
1.34693878, 1.59183673, 1.83673469, 2.08163265, 2.32653061,
2.57142857, 2.81632653, 3.06122449, 3.30612245, 3.55102041,
3.79591837, 4.04081633, 4.28571429, 4.53061224, 4.7755102 ,
5.02040816, 5.26530612, 5.51020408, 5.75510204, 6. ])
# 演示np.exp()函数
x = -1
np.exp(-x)
2.718281828459045
逻辑回归模型的代码实现
# 构造数据
X = [[1, 0], [5, 1], [6, 4], [4, 2], [3, 2]]
y = [0, 1, 1, 0, 0]
# 模型训练
from sklearn.linear_model import LogisticRegression
model = LogisticRegression()
model.fit(X, y) # 运行时下面FutureWarning警告，意为以后模型的官方默认参数会有所调整，不是报错
LogisticRegression()
# 如果不想看到FutureWarning这样的警告信息，可以在代码最上面加上如下内容
import warnings
warnings.filterwarnings('ignore')
# 模型预测 - 预测单个数据
print(model.predict([[2,2]]))
[0]
# 模型预测 - 预测多个数据1
print(model.predict([[1,1], [2,2], [5, 5]]))
[0 0 1]
# 模型预测 - 预测多个数据2
print(model.predict([[1, 0], [5, 1], [6, 4], [4, 2], [3, 2]])) # 因为这里演示的多个数据和X是一样的，所以也可以直接写成model.predict(X)
[0 1 1 0 0]
可以看到其预测准确度为100%。
逻辑回归模型的深入理解
# 预测概率：左列是分类为0的概率，右列是分类为1的概率
y_pred_proba = model.predict_proba(X)
y_pred_proba # 直接打印
array([[0.97344854, 0.02655146],
[0.39071972, 0.60928028],
[0.17991028, 0.82008972],
[0.63167893, 0.36832107],
[0.82424527, 0.17575473]])
# 另外一种打印概率的方式：通过DataFrame展示，更加好看些
import pandas as pd
a = pd.DataFrame(y_pred_proba, columns=['分类为0的概率', '分类为1的概率']) # 通过numpy数组创建DataFrame
a
| 分类为0的概率 | 分类为1的概率 | |
|---|---|---|
| 0 | 0.973449 | 0.026551 |
| 1 | 0.390720 | 0.609280 |
| 2 | 0.179910 | 0.820090 |
| 3 | 0.631679 | 0.368321 |
| 4 | 0.824245 | 0.175755 |
# 打印系数和截距项
print(model.coef_) # 系数k1与k2
print(model.intercept_) # 截距项k0
[[1.00595248 0.02223835]]
[-4.60771284]
model.coef_.T
array([[1.00595248],
[0.02223835]])
# 如果想批量查看预测概率
import numpy as np
for i in range(5): # 这里共有5条数据，所以循环5次
print(1 / (1 + np.exp(-(np.dot(X[i], model.coef_.T) + model.intercept_))))
[0.02655146]
[0.60928028]
[0.82008972]
[0.36832107]
[0.17575473]
补充知识点：多分类逻辑回归模型演示
# 构造数据，此时y有多个分类
X = [[1, 0], [5, 1], [6, 4], [4, 2], [3, 2]]
y = [-1, 0, 1, 1, 1] # 这里有三个分类-1、0、1
# 模型训练
from sklearn.linear_model import LogisticRegression
model = LogisticRegression()
model.fit(X, y)
LogisticRegression()
# x1=0, x2 = 0, 预测结果为-1
print(model.predict([[0, 0]]))
[-1]
# 把x里的样本全部预测一遍
model.predict(X)
array([-1, 0, 1, 1, 1])
# 查看预测值在三种分类下的概率
print(model.predict_proba([[0, 0]]))
[[0.88352311 0.02340026 0.09307662]]