适用类型：解决二分类问题
逻辑回归的出现：线性回归可以预测连续值，但是不能解决分类问题，我们需要根据预测的结果判定其属于正类还是负类。所以逻辑回归就是将线性回归的结果，通过Sigmoid函数映射到（0，1）之间
线性回归的决策函数：数据与θ的乘法，数据的矩阵格式(样本数×列数)，θ的矩阵格式(列数×1)
将其通过Sigmoid函数，获得逻辑回归的决策函数
可以对(-∞, +∞)的结果，映射到(0, 1)之间作为概率
可以将1/2作为决策边界
数学特性好，求导容易
线性回归的损失函数维平方损失函数，如果将其用于逻辑回归的损失函数，则其数学特性不好，有很多局部极小值，难以用梯度下降法求解最优
这里使用对数损失函数
解释：如果一个样本为正样本，那么我们希望将其预测为正样本的概率p越大越好，也就是决策函数的值越大越好，则logp越大越好，逻辑回归的决策函数值就是样本为正的概率；如果一个样本为负样本，那么我们希望将其预测为负样本的概率越大越好，也就是（1-p）越大越好，即log(1-p)越大越好
为什么使用对数函数：样本集中有很多样本，要求其概率连乘，概率为0-1之间的数，连乘越来越小，利用log变换将其变为连加，不会溢出，不会超出计算精度
损失函数:： y(1->m)表示Sigmoid值(样本数×1)，hθx(1->m)表示决策函数值(样本数×1)，所以中括号的值(1×1)
import numpy as np
from matplotlib import pyplot as plt
from scipy.optimize import minimize
from sklearn.preprocessing import PolynomialFeatures
class MyLogisticRegression:
def __init__(self):
plt.rcParams["font.sans-serif"] = ["SimHei"]
# 包含数据和标签的数据集
self.data = np.loadtxt("./data2.txt", delimiter=",")
self.data_mat = self.data[:, 0:2]
self.label_mat = self.data[:, 2]
self.thetas = np.zeros((self.data_mat.shape[1]))
# 生成多项式特征，最高6次项
self.poly = PolynomialFeatures(6)
self.p_data_mat = self.poly.fit_transform(self.data_mat)
def cost_func_reg(self, theta, reg):
"""
损失函数具体实现
:param theta: 逻辑回归系数
:param data_mat: 带有截距项的数据集
:param label_mat: 标签数据集
:param reg:
:return:
"""
m = self.label_mat.size
label_mat = self.label_mat.reshape(-1, 1)
h = self.sigmoid(self.p_data_mat.dot(theta))
J = -1 * (1/m)*(np.log(h).T.dot(label_mat) + np.log(1-h).T.dot(1-label_mat))\
+ (reg / (2*m)) * np.sum(np.square(theta[1:]))
if np.isnan(J[0]):
return np.inf
return J[0]
def gradient_reg(self, theta, reg):
m = self.label_mat.size
h = self.sigmoid(self.p_data_mat.dot(theta.reshape(-1, 1)))
label_mat = self.label_mat.reshape(-1, 1)
grad = (1 / m)*self.p_data_mat.T.dot(h-label_mat) + (reg/m)*np.r_[[[0]], theta[1:].reshape(-1, 1)]
return grad
def gradient_descent_reg(self, alpha=0.01, reg=0, iterations=200):
"""
逻辑回归梯度下降收敛函数
:param alpha: 学习率
:param reg:
:param iterations: 最大迭代次数
:return: 逻辑回归系数组
"""
m, n = self.p_data_mat.shape
theta = np.zeros((n, 1))
theta_set = []
for i in range(iterations):
grad = self.gradient_reg(theta, reg)
theta = theta - alpha*grad.reshape(-1, 1)
theta_set.append(theta)
return theta, theta_set
def plot_data_reg(self, x_label=None, y_label=None, neg_text="negative", pos_text="positive", thetas=None):
neg = self.label_mat == 0
pos = self.label_mat == 1
fig1 = plt.figure(figsize=(12, 8))
ax1 = fig1.add_subplot(111)
ax1.scatter(self.p_data_mat[neg][:, 1], self.p_data_mat[neg][:, 2], marker="o", s=100, label=neg_text)
ax1.scatter(self.p_data_mat[pos][:, 1], self.p_data_mat[pos][:, 2], marker="+", s=100, label=pos_text)
ax1.set_xlabel(x_label, fontsize=14)
# 描绘逻辑回归直线（曲线）
if isinstance(thetas, type(np.array([]))):
x1_min, x1_max = self.p_data_mat[:, 1].min(), self.p_data_mat[:, 1].max()
x2_min, x2_max = self.p_data_mat[:, 2].min(), self.p_data_mat[:, 2].max()
xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
h = self.sigmoid(self.poly.fit_transform(np.c_[xx1.ravel(), xx2.ravel()]).dot(thetas))
h = h.reshape(xx1.shape)
ax1.contour(xx1, xx2, h, [0.5], linewidths=3)
ax1.legend(fontsize=14)
plt.show()
@staticmethod
def sigmoid(z):
return 1.0 / (1 + np.exp(-z))
if __name__ == '__main__':
my_logistic_regression = MyLogisticRegression()
# my_logistic_regression.plot_data(x_label="线性不可分数据集")
thetas, theta_set = my_logistic_regression.gradient_descent_reg(alpha=0.5, reg=0, iterations=500)
my_logistic_regression.plot_data_reg(thetas=thetas, x_label="$\\lambda$ = {}".format(0))
thetas = np.zeros((my_logistic_regression.p_data_mat.shape[1], 1))
# 未知错误，有大佬解决可留言
result = minimize(my_logistic_regression.cost_func_reg, thetas,
args=(0, ),
method=None,
jac=my_logistic_regression.gradient_reg)
my_logistic_regression.plot_data_reg(thetas=result.x, x_label="$\\lambda$ = {}".format(0))
import numpy as np
from matplotlib import pyplot as plt
from scipy.optimize import minimize
from sklearn.preprocessing import PolynomialFeatures
class MyLogisticRegression:
def __init__(self):
plt.rcParams["font.sans-serif"] = ["SimHei"]
# 包含数据和标签的数据集
self.data = np.loadtxt("./data2.txt", delimiter=",")
self.data_mat = self.data[:, 0:2]
self.label_mat = self.data[:, 2]
self.thetas = np.zeros((self.data_mat.shape[1]))
# 生成多项式特征，最高6次项
self.poly = PolynomialFeatures(6)
self.p_data_mat = self.poly.fit_transform(self.data_mat)
def cost_func_reg(self, theta, reg):
"""
损失函数具体实现
:param theta: 逻辑回归系数
:param data_mat: 带有截距项的数据集
:param label_mat: 标签数据集
:param reg:
:return:
"""
m = self.label_mat.size
label_mat = self.label_mat.reshape(-1, 1)
h = self.sigmoid(self.p_data_mat.dot(theta))
J = -1 * (1/m)*(np.log(h).T.dot(label_mat) + np.log(1-h).T.dot(1-label_mat))\
+ (reg / (2*m)) * np.sum(np.square(theta[1:]))
if np.isnan(J[0]):
return np.inf
return J[0]
def gradient_reg(self, theta, reg):
m = self.label_mat.size
h = self.sigmoid(self.p_data_mat.dot(theta.reshape(-1, 1)))
label_mat = self.label_mat.reshape(-1, 1)
grad = (1 / m)*self.p_data_mat.T.dot(h-label_mat) + (reg/m)*np.r_[[[0]], theta[1:].reshape(-1, 1)]
return grad
def gradient_descent_reg(self, alpha=0.01, reg=0, iterations=200):
"""
逻辑回归梯度下降收敛函数
:param alpha: 学习率
:param reg:
:param iterations: 最大迭代次数
:return: 逻辑回归系数组
"""
m, n = self.p_data_mat.shape
theta = np.zeros((n, 1))
theta_set = []
for i in range(iterations):
grad = self.gradient_reg(theta, reg)
theta = theta - alpha*grad.reshape(-1, 1)
theta_set.append(theta)
return theta, theta_set
def plot_data_reg(self, x_label=None, y_label=None, neg_text="negative", pos_text="positive", thetas=None):
neg = self.label_mat == 0
pos = self.label_mat == 1
fig1 = plt.figure(figsize=(12, 8))
ax1 = fig1.add_subplot(111)
ax1.scatter(self.p_data_mat[neg][:, 1], self.p_data_mat[neg][:, 2], marker="o", s=100, label=neg_text)
ax1.scatter(self.p_data_mat[pos][:, 1], self.p_data_mat[pos][:, 2], marker="+", s=100, label=pos_text)
ax1.set_xlabel(x_label, fontsize=14)
# 描绘逻辑回归直线（曲线）
if isinstance(thetas, type(np.array([]))):
x1_min, x1_max = self.p_data_mat[:, 1].min(), self.p_data_mat[:, 1].max()
x2_min, x2_max = self.p_data_mat[:, 2].min(), self.p_data_mat[:, 2].max()
xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
h = self.sigmoid(self.poly.fit_transform(np.c_[xx1.ravel(), xx2.ravel()]).dot(thetas))
h = h.reshape(xx1.shape)
ax1.contour(xx1, xx2, h, [0.5], linewidths=3)
ax1.legend(fontsize=14)
plt.show()
@staticmethod
def sigmoid(z):
return 1.0 / (1 + np.exp(-z))
if __name__ == '__main__':
my_logistic_regression = MyLogisticRegression()
# my_logistic_regression.plot_data(x_label="线性不可分数据集")
thetas, theta_set = my_logistic_regression.gradient_descent_reg(alpha=0.5, reg=0, iterations=500)
my_logistic_regression.plot_data_reg(thetas=thetas, x_label="$\\lambda$ = {}".format(0))
thetas = np.zeros((my_logistic_regression.p_data_mat.shape[1], 1))
# 未知错误，有大佬解决可留言
result = minimize(my_logistic_regression.cost_func_reg, thetas,
args=(0, ),
method=None,
jac=my_logistic_regression.gradient_reg)
my_logistic_regression.plot_data_reg(thetas=result.x, x_label="$\\lambda$ = {}".format(0))
