前言
逻辑回归是统计学习中的经典分类算法，如：可用于二分类
逻辑回归有以下几个特点：
优点：计算代价不高，易于理解和实现
缺点：容易欠拟合，分类精度可能不高
适用数据类型：数值型和标称型数据
二项逻辑回归模型的数学推导
设{x, y}是输入样本，y = 1表示正类，y = 0表示负类。那么y = 1的概率和y = 0的概率可以表示为：
image
w是权值，b称为偏置
模型参数估计
逻辑回归常用的方法是极大似然估计，从而得到回归模型。
2.PNG
这样问题就变成了对对数似然函数为目标函数的最优化问题，常用的方法是梯度下降法以及拟牛顿法，本章采用梯度上升法和随机梯度上升法来求该模型的最优化问题。
3.PNG
但是这样我们无法直接求出L(w)的最大值对应的w，那么就可以采用梯度上升法求解这个问题。
4.PNG
沿着梯度的方向，每次移动一个布长，直到达到最大值。
公式exp(x) / (1 + exp(x))python代码的实现：
def sigmod(inx):
return exp(inx) / (1 + exp(inx))
梯度上升法代码的实现：
#梯度上升法
def grad_ascent(data_matin, class_label):
data_matrix = mat(data_matin) #将data_matin转为100 * 3矩阵
label_matrix = mat(class_label).transpose() #将class_label转为100 * 1的矩阵
m,n = shape(data_matrix) #m = 100, n = 100
alpha = 0.001
max_cycles = 500
weights = ones((n, 1)) #生成100 * 1权值的单位矩阵
for k in range(max_cycles):
h = sigmod(data_matrix * weights)
error = (label_matrix - h)
weights = weights + alpha * data_matrix.transpose() * error
return weights
随机梯度上升法代码的实现
#随机梯度上升法
def stoc_grad_ascent0(data_matrix, class_label):
m,n = shape(data_matrix)
alpha = 0.01
weights = ones(n) #创建权值一维数组
#print(weights)
for i in range(m):
h = sigmod(sum(data_matrix[i] * weights))
error = class_label[i] - h
temp = []
for k in data_matrix[i]:
temp.append(alpha * error * k)
#print(temp)
weights = weights + temp
return weights
改进型随机梯度上升法代码的实现：
#改进型随机梯度上升法
def stoc_grad_ascent1(data_matrix, class_label, num_iter = 150):
m,n = shape(data_matrix)
weights = ones(n)
data_index = range(m)
for j in range(num_iter):
for i in range(m):
alpha = 4 / (1 + j + i) + 0.01
rand_index = int(random.uniform(0, len(data_index)))
h = sigmod(sum(data_matrix[rand_index] * weights))
error = class_label[rand_index] - h
temp = []
for k in data_matrix[rand_index]:
temp.append(alpha * error * k)
weights = weights + temp
#del(data_index[rand_index])
return weights
完整python实现代码如下：
from numpy import *
import matplotlib.pyplot as plt
import random
def load_data_set():
data_mat = []
label_mat = []
fr = open("test_set.txt")
for lines in fr.readlines(): #读取每一行数据
line_arr = lines.strip().split() #将每一行分隔数据作为一个列表
data_mat.append([1.0, float(line_arr[0]), float(line_arr[1])]) #x0 = 1 x1 = line_arr[0] x2 = line_arr[1]
label_mat.append(int(line_arr[2])) #标签
return data_mat, label_mat #返回训练数据和标签
def sigmod(inx):
return exp(inx) / (1 + exp(inx))
#梯度上升法
def grad_ascent(data_matin, class_label):
data_matrix = mat(data_matin) #将data_matin转为100 * 3矩阵
label_matrix = mat(class_label).transpose() #将class_label转为100 * 1的矩阵
m,n = shape(data_matrix) #m = 100, n = 100
alpha = 0.001
max_cycles = 500
weights = ones((n, 1)) #生成100 * 1权值的单位矩阵
for k in range(max_cycles):
h = sigmod(data_matrix * weights)
error = (label_matrix - h)
weights = weights + alpha * data_matrix.transpose() * error
return weights
#随机梯度上升法
def stoc_grad_ascent0(data_matrix, class_label):
m,n = shape(data_matrix)
alpha = 0.01
weights = ones(n) #创建权值一维数组
#print(weights)
for i in range(m):
h = sigmod(sum(data_matrix[i] * weights))
error = class_label[i] - h
temp = []
for k in data_matrix[i]:
temp.append(alpha * error * k)
#print(temp)
weights = weights + temp
return weights
#改进型随机梯度上升法
def stoc_grad_ascent1(data_matrix, class_label, num_iter = 150):
m,n = shape(data_matrix)
weights = ones(n)
data_index = range(m)
for j in range(num_iter):
for i in range(m):
alpha = 4 / (1 + j + i) + 0.01
rand_index = int(random.uniform(0, len(data_index)))
h = sigmod(sum(data_matrix[rand_index] * weights))
error = class_label[rand_index] - h
temp = []
for k in data_matrix[rand_index]:
temp.append(alpha * error * k)
weights = weights + temp
#del(data_index[rand_index])
return weights
def plot_best_fit(wei):
#weights = wei.getA()
weights = wei
data_mat, label_mat = load_data_set() #读取原始数据
data_arr = array(data_mat)
n = shape(data_arr)[0]
xcord1 = []
xcord2 = []
ycord1 = []
ycord2 = []
for i in range(n):
if int(label_mat[i]) == 1:
xcord1.append(data_arr[i, 1])
ycord1.append(data_arr[i, 2])
else:
xcord2.append(data_arr[i, 1])
ycord2.append(data_arr[i, 2])
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.scatter(xcord1, ycord1, s = 30, c = "red", marker = "s")
ax.scatter(xcord2, ycord2, s = 30, c = "green")
x = arange(-5.0, 5.0, 0.1)
y = (-weights[0] - weights[1] * x) / weights[2]
ax.plot(x, y)
plt.xlabel("X1")
plt.ylabel("X2")
plt.show()
def main():
data_mat, label_mat = load_data_set()
#weights = grad_ascent(data_mat, label_mat)
weights = stoc_grad_ascent0(data_mat, label_mat)
#weights = stoc_grad_ascent1(data_mat, label_mat)
print(weights)
plot_best_fit(weights)
main()
有几点需要注意：在画图函数中，若算法选择梯度上升法则将weights = wei注释，取消weights = wei.getA()的注释。若算法选择随机梯度上升法和改进型随机梯度上升法，则将weights = wei.getA()注释，取消weights = wei的注释。
输入数据：
-0.017612 14.053064 0
-1.395634 4.662541 1
-0.752157 6.538620 0
-1.322371 7.152853 0
0.423363 11.054677 0
0.406704 7.067335 1
0.667394 12.741452 0
-2.460150 6.866805 1
0.569411 9.548755 0
-0.026632 10.427743 0
0.850433 6.920334 1
1.347183 13.175500 0
1.176813 3.167020 1
-1.781871 9.097953 0
-0.566606 5.749003 1
0.931635 1.589505 1
-0.024205 6.151823 1
-0.036453 2.690988 1
-0.196949 0.444165 1
1.014459 5.754399 1
1.985298 3.230619 1
-1.693453 -0.557540 1
-0.576525 11.778922 0
-0.346811 -1.678730 1
-2.124484 2.672471 1
1.217916 9.597015 0
-0.733928 9.098687 0
-3.642001 -1.618087 1
0.315985 3.523953 1
1.416614 9.619232 0
-0.386323 3.989286 1
0.556921 8.294984 1
1.224863 11.587360 0
-1.347803 -2.406051 1
1.196604 4.951851 1
0.275221 9.543647 0
0.470575 9.332488 0
-1.889567 9.542662 0
-1.527893 12.150579 0
-1.185247 11.309318 0
-0.445678 3.297303 1
1.042222 6.105155 1
-0.618787 10.320986 0
1.152083 0.548467 1
0.828534 2.676045 1
-1.237728 10.549033 0
-0.683565 -2.166125 1
0.229456 5.921938 1
-0.959885 11.555336 0
0.492911 10.993324 0
0.184992 8.721488 0
-0.355715 10.325976 0
-0.397822 8.058397 0
0.824839 13.730343 0
1.507278 5.027866 1
0.099671 6.835839 1
-0.344008 10.717485 0
1.785928 7.718645 1
-0.918801 11.560217 0
-0.364009 4.747300 1
-0.841722 4.119083 1
0.490426 1.960539 1
-0.007194 9.075792 0
0.356107 12.447863 0
0.342578 12.281162 0
-0.810823 -1.466018 1
2.530777 6.476801 1
1.296683 11.607559 0
0.475487 12.040035 0
-0.783277 11.009725 0
0.074798 11.023650 0
-1.337472 0.468339 1
-0.102781 13.763651 0
-0.147324 2.874846 1
0.518389 9.887035 0
1.015399 7.571882 0
-1.658086 -0.027255 1
1.319944 2.171228 1
2.056216 5.019981 1
-0.851633 4.375691 1
-1.510047 6.061992 0
-1.076637 -3.181888 1
1.821096 10.283990 0
3.010150 8.401766 1
-1.099458 1.688274 1
-0.834872 -1.733869 1
-0.846637 3.849075 1
1.400102 12.628781 0
1.752842 5.468166 1
0.078557 0.059736 1
0.089392 -0.715300 1
1.825662 12.693808 0
0.197445 9.744638 0
0.126117 0.922311 1
-0.679797 1.220530 1
0.677983 2.556666 1
0.761349 10.693862 0
-2.168791 0.143632 1
1.388610 9.341997 0
0.317029 14.739025 0
实验结果如下所示：
梯度上升法：
5.PNG
随机梯度上升法：
6.PNG
改进型随机梯度上升法：
7.PNG
由实验结果可知，改进型随机梯度上升法和梯度上升法的效果差不多，随机梯度上升法的效果则差一些。
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