逻辑回归简介
logistic回归是一种广义的线性回归分析模型,以胃癌病情分析为例，选择两组人群，一组是胃癌组，一组是非胃癌组，两组人群必定具有不同的体征与生活方式等。因此因变量就为是否胃癌，值为“是”或“否”，自变量就可以包括很多了，如年龄、性别、饮食习惯、幽门螺杆菌感染等。
绘制sigmoid曲线
import numpy as np
import matplotlib.pyplot as plt
def sigmoid(t):
return 1. / (1. + np.exp(-t))
x = np.linspace(-10, 10, 500)
plt.plot(x, sigmoid(x))
plt.show()
逻辑回归的损失函数
y=log(x)的函数图像如下图所示：
y=-log(x)的函数图像如下图所示：
由于p的取值只能是在0到1之间，故：
当p取值为0时，按之前的sigmoid函数定义，预测分类y应该为0，但此时真实分类y=1，而-log(p)趋于无穷大（损失函数大），很好的表达了预测得不准；当p取值为1时，按之前的sigmoid函数定义，预测分类y应该为1，此时真实分类y也等于1，-log(p)趋于0（损失函数小），很好的表达了预测得准确。
另外一种情况同理。
y=-log(-x)的函数图像如下图所示（与y=-log(x)关于y轴对称）：
y=-log(1-x)的函数图像如下图所示（沿着x轴向右平移一个单位）：
合成一个式子：
考虑多个样本：
损失函数求导
推导过程
向量化：
封装Logistics模型
在之前实现的线性回归的类的基础上改写：
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()"
实现逻辑回归
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]
X.shape
(100, 2)
y.shape
(100,)
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()
使用逻辑回归
from playML.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from playML.LogisticRegression import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression()
log_reg.score(X_test, y_test)
1.0
log_reg.predict_proba(X_test)
array([ 0.92972035, 0.98664939, 0.14852024, 0.17601199, 0.0369836 ,
0.0186637 , 0.04936918, 0.99669244, 0.97993941, 0.74524655,
0.04473194, 0.00339285, 0.26131273, 0.0369836 , 0.84192923,
0.79892262, 0.82890209, 0.32358166, 0.06535323, 0.20735334])
log_reg.predict(X_test)
array([1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0])
y_test
array([1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0])
决策边界
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()
from playML.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from playML.LogisticRegression import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression()
log_reg.coef_
array([ 3.01796521, -5.04447145])
log_reg.intercept_
-0.6937719272911228
在逻辑回归中，易得决策边界为 theta*X_b=0 的直线
如果只有两个特征值，则很容易通过公式画出逻辑回归的决策边界
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()
plt.scatter(X_test[y_test==0,0], X_test[y_test==0,1], color="red")
plt.scatter(X_test[y_test==1,0], X_test[y_test==1,1], color="blue")
plt.plot(x1_plot, x2_plot)
plt.show()
不规则的决策边界的绘制
def plot_decision_boundary(model, axis):
# meshgrid网格矩阵
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),
)
# np.c_ 行拼接， ravel()高维矩阵平铺降成一维
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)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
metric_params=None, n_jobs=1, n_neighbors=5, p=2,
weights='uniform')
knn_clf.score(X_test, y_test)
1.0
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)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
metric_params=None, n_jobs=1, n_neighbors=5, p=2,
weights='uniform')
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()
image.png
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()
多项式特征应用于逻辑回归
逻辑回归中添加多项式特征
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()
使用逻辑回归
from playML.LogisticRegression import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X, y)
LogisticRegression()
log_reg.score(X, y)
0.60499999999999998
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()
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=2, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression())])
poly_log_reg.score(X, y)
0.94999999999999996
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()
poly_log_reg2 = PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X, y)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression())])
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()
逻辑回归的模型正则化
scikit-learn中的逻辑回归
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):
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()
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
使用scikit-learn中的逻辑回归
from sklearn.linear_model import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False)
log_reg.score(X_train, y_train)
0.79333333333333333
log_reg.score(X_test, y_test)
0.85999999999999999
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()
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=2, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False))])
poly_log_reg.score(X_train, y_train)
0.91333333333333333
poly_log_reg.score(X_test, y_test)
0.93999999999999995
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()
poly_log_reg2 = PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X_train, y_train)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False))])
poly_log_reg2.score(X_train, y_train)
0.93999999999999995
poly_log_reg2.score(X_test, y_test)
0.92000000000000004
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()
def PolynomialLogisticRegression(degree, C):
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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=0.1, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False))])
poly_log_reg3.score(X_train, y_train)
0.85333333333333339
poly_log_reg3.score(X_test, y_test)
0.92000000000000004
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()
def PolynomialLogisticRegression(degree, C, penalty='l2'):
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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=0.1, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l1', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False))])
poly_log_reg4.score(X_train, y_train)
0.82666666666666666
poly_log_reg4.score(X_test, y_test)
0.90000000000000002
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()
使用逻辑回归解决多分类问题
OvR 和 OvO
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()
log_reg.fit(X_train, y_train)
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False)
log_reg.score(X_test, y_test)
0.65789473684210531
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()
log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)
0.78947368421052633
plot_decision_boundary(log_reg2, 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()
使用所有的数据
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)
0.94736842105263153
log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)
1.0
OvO and OvR
from sklearn.multiclass import OneVsRestClassifier
ovr = OneVsRestClassifier(log_reg)
ovr.fit(X_train, y_train)
ovr.score(X_test, y_test)
0.94736842105263153
from sklearn.multiclass import OneVsOneClassifier
ovo = OneVsOneClassifier(log_reg)
ovo.fit(X_train, y_train)
ovo.score(X_test, y_test)
1.0
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