什么是邏輯回歸
邏輯回歸雖然名字有回歸，但解決的是分類問題，
邏輯回歸既可以看做回歸演算法，也可以看做是分類演算法，通常作為分類演算法用，只可以解決二分類問題，
Sigmoid函式：
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()
邏輯回歸的損失函式
推導程序這里就不贅述了，高等數學基本知識，
向量化：
邏輯回歸的向量化梯度：
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):
"""根據訓練資料集X_train, y_train, 使用梯度下降法訓練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):
"""給定待預測資料集X_predict，回傳表示X_predict的結果概率向量"""
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):
"""給定待預測資料集X_predict，回傳表示X_predict的結果向量"""
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):
"""根據測驗資料集 X_test 和 y_test 確定當前模型的準確度"""
y_predict = self.predict(X_test)
return accuracy_score(y_test, y_predict)
def __repr__(self):
return "LogisticRegression()"
使用鳶尾花資料集，因為有三個特征，而邏輯回歸只適合二分類問題，所以我們取前2個特征實驗：
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玩轉機器學習\邏輯回歸\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)
準確率100%，
決策邊界
繪制決策邊界：
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()
其中那個分類錯誤的紅點是訓練資料集中的點，
不規則的決策邊界繪制方法：
如圖，遍歷每個點，看它屬于哪個類，
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的決策邊界：
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()
發現黃藍的決策邊界很陡峭，這是因為KNN的k越小，那么模型越復雜，可能會過擬合，
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()
在邏輯回歸中使用多項式特征
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玩轉機器學習\邏輯回歸\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()
發現準確率很低，這是因為邏輯回歸默認是用一條直線分類的，我們用多項式試一下：
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()
用二次多項式準確率就比較高了，我們再試一下20次多項式：
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()
形狀及其不規則，明顯是過擬合了，我們可以降低多項式的級數，當然使用正則化是更好的選擇，
scikit-learn中的邏輯回歸
邏輯回歸的正則化：
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()
用線性邏輯回歸：
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)
發現準確率較低，因為我們造的資料是拋物線，繪制一下：
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()
用二次多項式邏輯回歸：
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)
回傳的penalty就是正則化方式，默認是l2正則，即嶺回歸，
poly_log_reg.score(X_train,y_train)
poly_log_reg.score(X_test,y_test)
發現準確率比較高了，
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次多項式邏輯回歸：
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()
發現準確率下降了，根據圖就可以看出過擬合了，圖形很復雜，但因資料比較弱，所以準確率降低的比較少，
令C=0.1，l2正則：
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()
圖形比上面的要規則一點，但準確率較低，
令C=0.1，換成l1正則：
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()
雖然準確率降低了，但決策邊界比較符合我們創造的拋物線了，這是因為l1正則（lasso回歸）會盡可能使一些theta為0，起到特征選擇，
當然，C這個超引數也可以通過網格搜索來尋找，
OvR與OvO
解決多分類問題：OvR、OvO
OvR（One vs Rest）：
OvO（One vs One）:
雖然OvO更費時，但準確率要高，
使用鳶尾花資料集來測驗：
先取前兩個特征：
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必須換求解方法
log_reg2.fit(X_train,y_train)
log_reg2.score(X_test,y_test)
可見ovo準確率是比ovr高的，
我們再用所有特征測驗一下：
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準確率達到了1，
其實scikit-learn中有OVR和OVO這兩個類，以便所有二分類分類器都可以使用：
ovr：
from sklearn.multiclass import OneVsRestClassifier
ovr=OneVsRestClassifier(log_reg)
ovr.fit(X_train,y_train)
ovr.score(X_test,y_test)
from sklearn.multiclass import OneVsOneClassifier
ovo=OneVsOneClassifier(log_reg)
ovo.fit(X_train,y_train)
ovo.score(X_test,y_test)
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