Logistic回归和梯度上升算法
一. Logistic回归原理
Logistic回归是一种广义线性回归，常用的分类器函数是Sigmoid函数，其公式如下：
σ(z)=11+e−z
其中， z 可由下面公式得出：
z=w0x0+w1x1+w2x2+⋅⋅⋅+wnxn
如果采用向量的写法，上面的公式可以写成：
z=wTx
我们的主要任务是找到最佳参数 w 使得分类器尽可能准确。
二. 梯度上升算法
1. 梯度
函数 f(x,y) 的梯度的公式如下：
∇f(x,y)=⎡⎣⎢⎢⎢∂f(x,y)∂x∂f(x,y)∂y⎤⎦⎥⎥⎥
2. 梯度上升算法的原理
梯度上升算法的迭代公式如下：
w=w+α∇wf(w)
上面的公式一直被迭代执行，直到符合某个条件为止。
三. 关键点
1. 目标函数是啥？
我们考虑二分类问题，其中的包含的类别为类别1和类别0。
可以得到预测函数，其公式如下：
hw(x)=σ(wTx)=11+e−wTx
hw(x) 的值表示 y=1 的概率，因此分类结果为类别1和类别0的概率分别为：
P(y=1|x;w)=hw(x)
P(y=0|x;w)=1−hw(x)
上面的公式可以合起来写成：
P(y|x;w)=(hw(x))y(1−hw(x))1−y
其似然函数为：
L(w)=∏i=1mP(y(i)|x(i);w)=∏i=1m(hw(x(i)))y(i)(1−hw(x(i)))1−y(i)
对数似然函数为：
l(w)=logL(w)=∑i=1m(y(i)loghw(x(i))+(1−y(i))log(1−hw(x(i))))
最大似然估计就是要求使得 l(w) 取值最大时的 w ，所以我们的目标函数就是 l(w) 。
2. 梯度怎么求？
∂l(w)∂wj=∑i=1m(y(i)1hw(x(i))∂hw(x(i))∂wj−(1−y(i))11−hw(x(i))∂hw(x(i))∂wj)=∑i=1m(y(i)1σ(wTx(i))−(1−y(i))11−σ(wTx(i)))∂σ(wTx(i))∂wj=∑i=1m(y(i)1σ(wTx(i))−(1−y(i))11−σ(wTx(i)))σ(wTx(i))(1−σ(wTx(i)))∂wTx(i)∂wj=∑i=1m(y(i)(1−σ(wTx(i)))−(1−y(i))σ(wTx(i)))x(i)j=∑i=1m(y(i)−σ(wTx(i)))x(i)j=∑i=1m(y(i)−hw(x(i)))x(i)j
四. 实验数据
-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 0五. 实验代码
#encoding: utf-8
import numpy
import matplotlib.pyplot as plt
def load_data():
data = []
label = []
file_object = open('data.txt')
for line in file_object.readlines():
arr = line.strip().split()
data.append([1.0, float(arr[0]), float(arr[1])])
label.append(int(arr[2]))
return data, label
def sigmoid(x):
return 1.0 / (1 + numpy.exp(-x))
def grad_ascent(data, label):
data_mat = numpy.mat(data)
label_mat = numpy.mat(label).transpose()
n, m = data_mat.shape
alpha = 0.001
max_step = 500
weights = numpy.ones((m, 1))
for i in range(max_step):
h = sigmoid(data_mat * weights)
err = (label_mat - h)
weights = weights + alpha * data_mat.transpose() * err
return weights
#随机梯度上升
def stoc_grad_ascent0(data, label):
data = numpy.array(data)
n, m = data.shape
alpha = 0.01
weights = numpy.ones(m)
for i in range(n):
h = sigmoid(numpy.sum(data[i] * weights))
err = label[i] - h
weights = weights + data[i] * alpha * err
return weights
#优化后的随机梯度上升
def stoc_grad_ascent1(data, label, max_step):
data = numpy.array(data)
n, m = data.shape
weights = numpy.ones(m)
for i in range(max_step):
data_index = range(n)
for j in range(n):
alpha = 4 / (1.0 + i + j) + 0.01
rand_index = int(numpy.random.uniform(0, len(data_index)))
h = sigmoid(numpy.sum(data[rand_index] * weights))
error = label[rand_index] - h
weights = weights + alpha * error * data[rand_index]
del(data_index[rand_index])
return weights
def plot(weights):
data, label = load_data()
n = len(data)
x1 = []
y1 = []
x2 = []
y2 = []
for i in range(n):
if label[i] == 1:
x1.append(data[i][1])
y1.append(data[i][2])
else:
x2.append(data[i][1])
y2.append(data[i][2])
fig = plt.figure()
ax = fig.add_subplot(111)
ax.scatter(x1, y1, s = 30, c = 'red', marker = 's')
ax.scatter(x2, y2, s = 30, c = 'green')
x = numpy.arange(-3.0, 3.0, 0.1)
y = (-weights[0] - weights[1] * x) / weights[2]
ax.plot(x, y)
plt.xlabel('X1')
plt.ylabel('X2')
plt.show()
if __name__=="__main__":
data, label = load_data()
weights = grad_ascent(data, label)
# print weights
plot(weights.getA())
# weights = stoc_grad_ascent0(data, label)
# # print weights
# plot(weights)
# weights = stoc_grad_ascent1(data, label, 500)
# # print weights
# plot(weights)六. 实验结果
如有错误，请指正