前言
本节学习逻辑回归
- 解决分类问题
- 样本特征和发生概率联系在一起
最后涉及多分类的OvR和OvO
1、逻辑回归
将样本的特征和样本发生的概率联系在一起
用到了sigmoid函数
通过训练得到θ
然后可以进行预测
损失函数是
进行演化和代入如下
由于这个损失函数没有解析解
所以用梯度下降法来求解
梯度是
实现如下
import numpy as np
from sklearn.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"
# 损失函数
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')
# 梯度
def dJ(theta, X_b, y):
return X_b.T.dot(self._sigmoid(X_b.dot(theta)) - y) / len(y)
# 梯度下降
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') #布尔向量
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()"
2、决策边界
通过scikit库来实现逻辑回归
并将决策边界可视化
import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
"""用scikit库实现逻辑回归"""
# 数据
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 #噪音
plt.scatter(X[y == 0, 0], X[y == 0, 1])
plt.scatter(X[y == 1, 0], X[y == 1, 1])
plt.show()
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
# 逻辑回归
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
print(log_reg.score(X_train, y_train))
print(log_reg.score(X_test, y_test))
# 决策边界可视化
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()
# 添加多项式特征
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)
print(poly_log_reg.score(X_train, y_train))
print(poly_log_reg.score(X_test, y_test))
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个特征
poly_log_reg2 = PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X_train, y_train)
print(poly_log_reg2.score(X_train, y_train))
print(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()
# 调整分类准确度
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)
print(poly_log_reg3.score(X_train, y_train))
print(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()
得到的几个图依次是
数据
逻辑回归
添加2个多项式特征
添加20个多项式特征
调整分类准确度
3、多分类
多分类有两种办法
- OvO:两两捉对比较特征,资源消耗大,准确
- OvR:提取一种特征,将其他所有座位第二种特征,资源消耗小,但准确性有问题
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
""" 多分类 OvO和OvR"""
# 数据
iris = datasets.load_iris()
X = iris.data[:,:2]
y = iris.target
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
# 决策边界
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)
# OvR
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
print(log_reg.score(X_test, y_test))
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")
log_reg2.fit(X_train, y_train)
print(log_reg2.score(X_test, y_test))
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()
得到的结果如下
OvR
OvO
结语
逻辑回归应该算是机器学习里用的很多的一种算法
对多分类问题做了一定了解