# 机器学习----逻辑回归

2021/8/21 14:09:11 浏览：

### 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：

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）：

###### 封装Logistics模型

``````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:
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

``````
``````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

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
``````

``````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()
``````

``````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()
``````

### 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()
``````

### 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()
``````

### OvR 和 OvO

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

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
``````