簡介

邏輯迴歸： 使用了邏輯迴歸函式對資料進行了擬合就叫邏輯迴歸？？

\[P(x)=\frac{1}{1+e^{-x}}（sigmoid function） \] \[y= \begin{cases}1, & P(x) \geq 0.5 \\ \hline 0, & P(x)<0.5\end{cases} \]

其中y為分類結果，P為概率分佈，x為特徵值。

分類問題的核心就是尋找決策邊界。

損失函式

\[J_{i}=\left\{\begin{array}{l} -\log \left(P\left(x_{i}\right)\right), \text { if } y_{i}=1 \\ -\log \left(1-P\left(x_{i}\right)\right), \text { if } y_{i}=0 \end{array}\right. \]

聯合表示

\[J=\frac{1}{m} \sum_{i=1}^{m} J_{i}=-\frac{1}{m}\left[\sum_{i=1}^{m}\left(\left(y_{i} \log \left(P\left(x_{i}\right)\right)\right.+\left(1-y_{i}\right) \log \left(1-P\left(x_{i}\right)\right)\right)\right] \]

重複計算直至收斂

\[\left\{\begin{aligned} t e m p_{\theta_{j}} &=\theta_{j}-\alpha \frac{\partial}{\partial \theta_{j}} J(\theta) \\ \theta_{j} &=t e m p_{\theta_{j}} \end{aligned}\right\} \]

評估模型表現
使用準確率

\[\text { Accuracy }=\frac{\text { 正確預測樣本數量 }}{\text { 總樣本數量 }} \]

參考連結

https://blog.csdn.net/weixin_46344368/article/details/105904589?spm=1001.2014.3001.5502

code

學生考試成績已知兩門成績判定第三門成績是否會合格

import pandas as pd
import numpy as np
data = pd.read_csv('examdata.csv')
data.head()
#visalize the data
from matplotlib import pyplot as plt
fig1=plt.figure()
plt.scatter(data.loc[:,'Exam1'],data.loc[:,'Exam2'])
plt.title('Exam1-Exam2')
plt.xlabel('Exam1')
plt.ylabel('Exam2')
plt.show()
# add label mask
mask=data.loc[:,'Pass']==1
print(mask) # print(~mask)
fig2=plt.figure()
passed=plt.scatter(data.loc[:,'Exam1'][mask],data.loc[:,'Exam2'][mask])
failed=plt.scatter(data.loc[:,'Exam1'][~mask],data.loc[:,'Exam2'][~mask])
plt.title('Exam1-Exam2')
plt.xlabel('Exam1')
plt.ylabel('Exam2')
plt.legend((passed,failed),('passed','failed'))
plt.show()
#define X,y
X = data.drop(['Pass'],axis=1)
y = data.loc[:,'Pass']
#X.head()
X1 = data.loc[:,'Exam1']
X2 = data.loc[:, 'Exam2']
y.head()
# eatablish the model and train it
from sklearn.linear_model import LogisticRegression
LR = LogisticRegression()
LR.fit(X,y)
# show the predicted result and its accuracy
y_predict = LR.predict(X)
print(y_predict)
from sklearn.metrics import accuracy_score
accuracy = accuracy_score(y,y_predict)
print(accuracy)
#exam1 = 70, exam2=65
y_test = LR.predict([[70,65]])
print('passed' if y_test==1 else 'failed')
print(LR.coef_,LR.intercept_)
theta0 = LR.intercept_
theta1,theta2 = LR.coef_[0][0],LR.coef_[0][1]
print(theta0,theta1,theta2)
X2_new = -(theta0+theta1*X1)/theta2
print(X2_new)
# 邊界函式
fig3 = plt.figure()
passed=plt.scatter(data.loc[:,'Exam1'][mask],data.loc[:,'Exam2'][mask])
failed=plt.scatter(data.loc[:,'Exam1'][~mask],data.loc[:,'Exam2'][~mask])
plt.plot(X1,X2_new) #畫出決策邊界
plt.title('Exam1-Exam2')
plt.xlabel('Exam1')
plt.ylabel('Exam2')
plt.legend((passed,failed),('passed','failed'))
plt.show()  
# create new data
X1_2 = X1*X1
X2_2 = X2*X2
X1_X2 = X1*X2
print(X1,X1_2)
X_new = {'X1':X1,'X2':X2,'X1_2':X1_2,'X2_2':X2_2,'X1_X2':X1_X2}
X_new = pd.DataFrame(X_new)
print(X_new)
LR2 = LogisticRegression()
LR2.fit(X_new,y)
y2_predict = LR2.predict(X_new)
accuracy2 = accuracy_score(y,y2_predict)
print(accuracy2)
X1_new = X1.sort_values()
print(X1,X1_new)
theta0=LR2.intercept_
theta1,theta2,theta3,theta4,theta5=LR2.coef_[0][0],LR2.coef_[0][1],LR2.coef_[0][2],LR2.coef_[0][3],LR2.coef_[0][4]
a = theta4
b = theta5*X1_new+theta2
c = theta0+theta1*X1_new+theta3*X1_new*X1_new
X2_new_boundary = (-b+np.sqrt(b*b-4*a*c))/(2*a)
print(X2_new_boundary)
fig5 = plt.figure()
passed=plt.scatter(data.loc[:,'Exam1'][mask],data.loc[:,'Exam2'][mask])
failed=plt.scatter(data.loc[:,'Exam1'][~mask],data.loc[:,'Exam2'][~mask])
plt.plot(X1_new,X2_new_boundary) #畫出決策邊界
plt.title('Exam1-Exam2')
plt.xlabel('Exam1')
plt.ylabel('Exam2')
plt.legend((passed,failed),('passed','failed'))
plt.show()