1、逻辑回归算法即可以看做是回归算法，也可以看作是分类算法，通常用来解决分类问题，主要是二分类问题，对于多分类问题并不适合，也可以通过一定的技巧变形来间接解决。
2、决策边界是指不同分类结果之间的边界线(或者边界实体)，它具体的表现形式一定程度上说明了算法训练模型的过拟合程度，我们可以通过决策边界来调整算法的超参数。
注解：左边逻辑回归拟合决策边界嘈杂冗余说明过拟合，右边决策边界分层清晰说明拟合度好
3、在逻辑回归中随着算法的复杂度不断地提高，其算法的过拟合也会越来越严重，为了避免这个现象，我们在逻辑回归中也需要进行正则化，以减小整体拟合的均方差，减少训练的过拟合现象。因此sklearn中调用逻辑回归时含有三个重要的超参数degree(多项式的最高次数)，C（正则化系数）以及penalty（正则化的方式l1/l2）
4、sklearn中逻辑回归使用的正则化方式如下：
import numpy as npimport matplotlib.pyplot as plt#定义概率转换函数sigmoid函数def sigmoid(t): return 1/(1+np.exp(-t))x=np.linspace(-10,10,100)y=sigmoid(x)plt.figure()plt.plot(x,y,"r",label="Sigmoid")plt.legend(loc=2)plt.show()
from sklearn import datasetsd=datasets.load_iris()x=d.datay=d.targetx=x[y<2,:2]y=y[y<2]
#定义机器学习算法的决策边界输出函数def plot_decision_boundary(model,axis): x0,x1=np.meshgrid( np.linspace(axis[0],axis[1],int((axis[1]-axis[0])*100)).reshape(-1,1), np.linspace(axis[2],axis[3], int((axis[3] - axis[2]) * 100)).reshape(-1,1) ) x_new=np.c_[x0.ravel(),x1.ravel()] y_pre=model.predict(x_new) zz=y_pre.reshape(x0.shape) from matplotlib.colors import ListedColormap cus=ListedColormap(["#EF9A9A","#FFF59D","#90CAF9"]) plt.contourf(x0,x1,zz,cmap=cus)
from sklearn.model_selection import train_test_splitx_train,x_test,y_train,y_test=train_test_split(x,y,random_state=666)
from sklearn.neighbors import KNeighborsClassifierknn1=KNeighborsClassifier()knn1.fit(x_train,y_train)plot_decision_boundary(knn1,axis=[4,8,1,5])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()knn2=KNeighborsClassifier(n_neighbors=50) #k越大，模型越简单，也意味着过拟合的程度越轻，决策边界越清晰knn2.fit(d.data[:,:2],d.target)x=d.datay=d.targetplot_decision_boundary(knn2,axis=[4,8,1,5])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.scatter(x[y==2,0],x[y==2,1],color="b")plt.show()#逻辑回归添加多项式回归import numpy as npimport matplotlib.pyplot as pltnp.random.seed=666x=np.random.normal(0,1,size=(100,2))y=np.array(x[:,0]**2+x[:,1]**2<1.5,dtype="int")knn2=KNeighborsClassifier() knn2.fit(x,y)plot_decision_boundary(knn2,axis=[-4,4,-3,3])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()### sklearn中调用逻辑回归算法函数import numpy as npimport matplotlib.pyplot as pltnp.random.seed=666x=np.random.normal(0,1,size=(200,2))y=np.array(x[:,0]**2+x[:,1]<1.5,dtype="int")for _ in range(20): y[np.random.randint(200)]=1plt.figure()plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()#1-1单纯的逻辑回归算法from sklearn.model_selection import train_test_splitx_train,x_test,y_train,y_test=train_test_split(x,y,random_state=666)from sklearn.linear_model import LogisticRegressionlog=LogisticRegression()log.fit(x_train,y_train)print(log.score(x_test,y_test))knn3=KNeighborsClassifier()knn3.fit(x_train,y_train)print(knn3.score(x_test,y_test))#1-2sklearn中的逻辑回归（多项式参与，并不带正则化）from sklearn.preprocessing import PolynomialFeaturesfrom sklearn.pipeline import Pipelinefrom sklearn.preprocessing import StandardScalerdef polynomiallogisticregression(degree): return Pipeline([ ("poly",PolynomialFeatures(degree=degree)), ("std_reg",StandardScaler()), ("log_reg",LogisticRegression()) ])x=np.random.normal(0,1,size=(200,2))y=np.array(x[:,0]**2+x[:,1]<1.5,dtype="int")for _ in range(20): y[np.random.randint(200)]=1from sklearn.model_selection import train_test_splitx_train,x_test,y_train,y_test=train_test_split(x,y,random_state=666)p1=polynomiallogisticregression(degree=2)p1.fit(x_train,y_train)print(p1.score(x_train,y_train))print(p1.score(x_test,y_test))plot_decision_boundary(p1,axis=[-4,4,-4,4])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()p1=polynomiallogisticregression(degree=20) #当其次数变为高次时，其训练模型已经过拟合p1.fit(x_train,y_train)print(p1.score(x_train,y_train))print(p1.score(x_test,y_test))plot_decision_boundary(p1,axis=[-4,4,-4,4])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()#1-3逻辑回归的正则化形式函数def Polynomiallogisticregression(degree,C,penalty): #逻辑回归的三大超参数 return Pipeline([ ("poly",PolynomialFeatures(degree=degree)), ("std_reg",StandardScaler()), ("log_reg",LogisticRegression(C=C,penalty=penalty)) ])p1=Polynomiallogisticregression(degree=20,C=1,penalty="l2") #当其次数变为高次时，其训练模型已经过拟合p1.fit(x_train,y_train)print(p1.score(x_train,y_train))print(p1.score(x_test,y_test))plot_decision_boundary(p1,axis=[-4,4,-4,4])plt.scatter(x[y==0,0],x[y==0,1],color="r")plt.scatter(x[y==1,0],x[y==1,1],color="g")plt.show()其输出结果对比如下所示：
注：左为拟合度比较好的决策边界，右边为高次的过拟合训练模型