机器学习（6）——逻辑回归

什么是逻辑回归

逻辑回归虽然名字有回归，但解决的是分类问题。

逻辑回归既可以看做回归算法，也可以看做是分类算法，通常作为分类算法用，只可以解决二分类问题。

Sigmoid函数：

import numpy as np
import matplotlib.pyplot as plt

def sigmoid(t):
    return 1 / (1+np.exp(-t))
    
x=np.linspace(-10,10,500)
y=sigmoid(x)
plt.plot(x,y)
plt.show()

逻辑回归的损失函数

推导过程这里就不赘述了，高等数学基本知识。

向量化：

逻辑回归的向量化梯度：

LogisticRegression.py：

import numpy as np
from .metrics import accuracy_score

class LogisticRegression:

    def __init__(self):
        """初始化Logistic Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def _sigmoid(self, t):
        return 1. / (1. + np.exp(-t))

    def fit(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Logistic Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            y_hat = self._sigmoid(X_b.dot(theta))
            try:
                return - np.sum(y*np.log(y_hat) + (1-y)*np.log(1-y_hat)) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            return X_b.T.dot(self._sigmoid(X_b.dot(theta)) - y) / len(y)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def predict_proba(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果概率向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return self._sigmoid(X_b.dot(self._theta))

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        proba = self.predict_proba(X_predict)
        return np.array(proba >= 0.5, dtype='int')

    def score(self, X_test, y_test):
        """根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return accuracy_score(y_test, y_predict)

    def __repr__(self):
        return "LogisticRegression()"

使用鸢尾花数据集，因为有三个特征，而逻辑回归只适合二分类问题，所以我们取前2个特征实验：

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets

iris=datasets.load_iris()

X=iris.data
y=iris.target

X=X[y<2,:2]
y=y[y<2]

plt.scatter(X[y==0,0],X[y==0,1],color="red")
plt.scatter(X[y==1,0],X[y==1,1],color="blue")
plt.show()

%run f:\python3玩转机器学习\逻辑回归\LogisticRegression.py

from sklearn.model_selection import train_test_split
X_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.2,random_state=666)

log_reg=LogisticRegression()
log_reg.fit(X_train,y_train)

log_reg.score(X_test,y_test)
log_reg.predict_proba(X_test)
log_reg.predict(X_test)

准确率100%。

决策边界

绘制决策边界：

def x2(x1):
    return (-log_reg.coef_[0] * x1 - log_reg.intercept_)/log_reg.coef_[1]
    
x1_plot=np.linspace(4,8,1000)
x2_plot=x2(x1_plot)

plt.scatter(X[y==0,0],X[y==0,1],color="red")
plt.scatter(X[y==1,0],X[y==1,1],color="blue")
plt.plot(x1_plot,x2_plot)
plt.show()

其中那个分类错误的红点是训练数据集中的点。

不规则的决策边界绘制方法：

如图，遍历每个点，看它属于哪个类。

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_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
    
plot_decision_boundary(log_reg, axis=[4, 7.5, 1.5, 4.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

KNN的决策边界：

from sklearn.neighbors import KNeighborsClassifier
knn_clf=KNeighborsClassifier()
knn_clf.fit(X_train,y_train)

knn_clf.score(X_test,y_test)

plot_decision_boundary(knn_clf,axis=[4,7.5,1.5,4.5])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

knn_clf_all=KNeighborsClassifier()
knn_clf_all.fit(iris.data[:,:2],iris.target)

plot_decision_boundary(knn_clf_all,axis=[4,8,1.5,4.5])
plt.scatter(iris.data[iris.target==0,0],iris.data[iris.target==0,1])
plt.scatter(iris.data[iris.target==1,0],iris.data[iris.target==1,1])
plt.scatter(iris.data[iris.target==2,0],iris.data[iris.target==2,1])

plt.show()

发现黄蓝的决策边界很陡峭，这是因为KNN的k越小，那么模型越复杂，可能会过拟合。

取k=50：

knn_clf_all=KNeighborsClassifier(n_neighbors=50)
knn_clf_all.fit(iris.data[:,:2],iris.target)

plot_decision_boundary(knn_clf_all,axis=[4,8,1.5,4.5])
plt.scatter(iris.data[iris.target==0,0],iris.data[iris.target==0,1])
plt.scatter(iris.data[iris.target==1,0],iris.data[iris.target==1,1])
plt.scatter(iris.data[iris.target==2,0],iris.data[iris.target==2,1])

plt.show()

在逻辑回归中使用多项式特征

import numpy as np


import matplotlib.pyplot as plt

np.random.seed(666)
X=np.random.normal(0,1,size=(200,2))
y=np.array(X[:,0]**2+X[:,1]**2<1.5,dtype='int')

plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

%run f:\python3玩转机器学习\逻辑回归\LogisticRegression.py

log_reg=LogisticRegression()
log_reg.fit(X,y)

log_reg.score(X,y)

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_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
    
plot_decision_boundary(log_reg,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

发现准确率很低，这是因为逻辑回归默认是用一条直线分类的，我们用多项式试一下：

from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
from sklearn.preprocessing import StandardScaler
def PolynomialLogisticRegression(degree):
    return Pipeline([
        ('poly',PolynomialFeatures(degree=degree)),
        ('std_scaler',StandardScaler()),
        ('log_reg',LogisticRegression())
    ])

poly_log_reg=PolynomialLogisticRegression(degree=2)
poly_log_reg.fit(X,y)

poly_log_reg.score(X,y)

plot_decision_boundary(poly_log_reg,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

用二次多项式准确率就比较高了，我们再试一下20次多项式：

poly_log_reg20=PolynomialLogisticRegression(degree=20)
poly_log_reg20.fit(X,y)

poly_log_reg20.score(X,y)

plot_decision_boundary(poly_log_reg20,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

形状及其不规则，明显是过拟合了，我们可以降低多项式的级数，当然使用正则化是更好的选择。

scikit-learn中的逻辑回归

逻辑回归的正则化：

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(666)
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): #添加噪音
    y[np.random.randint(200)]=1
    
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

用线性逻辑回归：

from sklearn.model_selection import train_test_split
X_train,X_test,y_train,y_test=train_test_split(X,y,random_state=666)

from sklearn.linear_model import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X_train,y_train)

log_reg.score(X_train,y_train)

log_reg.score(X_test,y_test)

发现准确率较低，因为我们造的数据是抛物线。绘制一下：

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_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
    
plot_decision_boundary(log_reg,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

用二次多项式逻辑回归：

from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
from sklearn.preprocessing import StandardScaler
def PolynomialLogisticRegression(degree):
    return Pipeline([
        ('poly',PolynomialFeatures(degree=degree)),
        ('std_scaler',StandardScaler()),
        ('log_reg',LogisticRegression())
    ])  
    
poly_log_reg=PolynomialLogisticRegression(degree=2)
poly_log_reg.fit(X_train,y_train)

返回的penalty就是正则化方式，默认是l2正则，即岭回归。

poly_log_reg.score(X_train,y_train)

poly_log_reg.score(X_test,y_test)

发现准确率比较高了。

plot_decision_boundary(poly_log_reg,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

20次多项式逻辑回归：

poly_log_reg2=PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X_train,y_train)

poly_log_reg2.score(X_train,y_train)

poly_log_reg2.score(X_test,y_test)

plot_decision_boundary(poly_log_reg2,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

发现准确率下降了，根据图就可以看出过拟合了，图形很复杂，但因数据比较弱，所以准确率降低的比较少。

令C=0.1，l2正则：

def PolynomialLogisticRegression(degree,C):#C是比重
    return Pipeline([
        ('poly',PolynomialFeatures(degree=degree)),
        ('std_scaler',StandardScaler()),
        ('log_reg',LogisticRegression(C=C))
    ])
    
poly_log_reg3=PolynomialLogisticRegression(degree=20,C=0.1)
poly_log_reg3.fit(X_train,y_train)

poly_log_reg3.score(X_train,y_train)

poly_log_reg3.score(X_test,y_test)

plot_decision_boundary(poly_log_reg3,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

图形比上面的要规则一点，但准确率较低。

令C=0.1，换成l1正则：

def PolynomialLogisticRegression(degree,C,penalty='l2'):#C是比重
    return Pipeline([
        ('poly',PolynomialFeatures(degree=degree)),
        ('std_scaler',StandardScaler()),
        ('log_reg',LogisticRegression(C=C,penalty=penalty))
    ])
    
poly_log_reg4=PolynomialLogisticRegression(degree=20,C=0.1,penalty='l1')
poly_log_reg4.fit(X_train,y_train)

poly_log_reg4.score(X_train,y_train)

poly_log_reg4.score(X_test,y_test)

plot_decision_boundary(poly_log_reg4,axis=[-4,4,-4,4])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.show()

虽然准确率降低了，但决策边界比较符合我们创造的抛物线了，这是因为l1正则（lasso回归）会尽可能使一些theta为0，起到特征选择。

当然，C这个超参数也可以通过网格搜索来寻找。

OvR与OvO

解决多分类问题：OvR、OvO

OvR（One vs Rest）：

OvO（One vs One）:

虽然OvO更费时，但准确率要高。

使用鸢尾花数据集来测试：

先取前两个特征：

ovr：

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets

iris=datasets.load_iris()
X=iris.data[:,:2]
y=iris.target

from sklearn.model_selection import train_test_split
X_train,X_test,y_train,y_test=train_test_split(X,y,random_state=666)

from sklearn.linear_model import LogisticRegression

log_reg=LogisticRegression(multi_class='ovr')
log_reg.fit(X_train,y_train)

log_reg.score(X_test,y_test)

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_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
    
plot_decision_boundary(log_reg,axis=[4,8.5,1.5,4.5])
plt.scatter(X[y==0,0],X[y==0,1])
plt.scatter(X[y==1,0],X[y==1,1])
plt.scatter(X[y==2,0],X[y==2,1])
plt.show()

ovo：

log_reg2=LogisticRegression(multi_class='multinomial',solver="newton-cg")#ovo必须换求解方法

log_reg2.fit(X_train,y_train)
log_reg2.score(X_test,y_test)

可见ovo准确率是比ovr高的。

我们再用所有特征测试一下：

X=iris.data
y=iris.target
X_train,X_test,y_train,y_test=train_test_split(X,y,random_state=666)

log_reg=LogisticRegression()
log_reg.fit(X_train,y_train)
log_reg.score(X_test,y_test)

log_reg2=LogisticRegression(multi_class='multinomial',solver="newton-cg")
log_reg2.fit(X_train,y_train)
log_reg2.score(X_test,y_test)

ovo准确率达到了1。

其实scikit-learn中有OVR和OVO这两个类，以便所有二分类分类器都可以使用：