Numpy实现逻辑回归(Logistic Regression)算法
前言:本文旨在对如何使用 numpy 实现逻辑回归拟合的过程做具体分析,有关逻辑回归原理部分不做过多论述。
数据集准备
准备用于二分类的数据集,可直接复制到对应的 txt 文件-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
读取数据
构造出[[1,data,data],]
与[result,]
两种矩阵
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from numpy import *
filename = 'Resources/LogisticRegressionSet.txt'
def loadDataSet():
dataMat = []
labelMat = [] # 构造两个空列表
fr = open(filename)
for line in fr.readlines():
lineArr = line.strip().split() # 切割元素
dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
# 前面的1,表示方程的常量。比如两个特征X1,X2,共需要三个参数,W1+W2*X1+W3*X2
labelMat.append(int(lineArr[2]))
return dataMat, labelMat
构造 Logistic(Sigmoid)函数
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def sigmoid(inX): # sigmoid函数
return 1.0 / (1 + exp(-inX))
构造梯度上升函数
普通梯度上升
通过矩阵乘法后算出差值反复迭代,利用梯度上升法求出结果,注意矩阵转置的意义
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def gradAscent(dataMat, labelMat): # 梯度上升求最优参数
dataMatrix = mat(dataMat) # 将读取的数据转换为矩阵
classLabels = mat(labelMat).transpose() # 将读取的数据转换为矩阵
m, n = shape(dataMatrix)
alpha = 0.001 # 设置梯度的阀值,该值越大梯度上升幅度越大
maxCycles = 500 # 设置迭代的次数,一般看实际数据进行设定,有些可能200次就够了
weights = ones((n, 1)) # 设置初始的参数,并都赋默认值为1。注意这里权重以矩阵形式表示三个参数。
for k in range(maxCycles):
h = sigmoid(dataMatrix * weights)
error = (classLabels - h) # 求导后差值
weights = weights + alpha * dataMatrix.transpose() * error # 迭代更新权重
return weights
随机梯度上升
与上述算法的区别在于只选择一行数据更新权重
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def stocGradAscent0(dataMat, labelMat):
# 随机梯度上升,当数据量比较大时,每次迭代都选择全量数据进行计算,
#计算量会非常大。所以采用每次迭代中一次只选择其中的一行数据进行更新权重。
dataMatrix = mat(dataMat)
classLabels = labelMat
m, n = shape(dataMatrix)
alpha = 0.01
maxCycles = 500
weights = ones((n, 1))
for k in range(maxCycles):
for i in range(m): # 遍历计算每一行
h = sigmoid(sum(dataMatrix[i] * weights))
error = classLabels[i] - h
weights = weights + alpha * error * dataMatrix[i].transpose()
return weights
改进版随机梯度上升
该算法与上述算法的区别在于两点:
- 步长随迭代次数的增加而减小
- 采用了随机抽样的方法
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def stocGradAscent1(dataMat, labelMat):
# 改进版随机梯度上升,在每次迭代中随机选择样本来更新权重,并且随迭代次数增加,权重变化越小。
dataMatrix = mat(dataMat)
classLabels = labelMat
m, n = shape(dataMatrix)
weights = ones((n, 1))
maxCycles = 500
for j in range(maxCycles): # 迭代
dataIndex = [i for i in range(m)]
for i in range(m): # 随机遍历每一行
alpha = 4 / (1 + j + i) + 0.0001 # 随迭代次数增加,权重变化越小。
randIndex = int(random.uniform(0, len(dataIndex))) # 随机抽样
h = sigmoid(sum(dataMatrix[randIndex] * weights))
error = classLabels[randIndex] - h
weights = weights + alpha * error * dataMatrix[randIndex].transpose()
del (dataIndex[randIndex]) # 去除已经抽取的样本
return weights
画出图像
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def plotBestFit(weights): # 画出最终分类的图
import matplotlib.pyplot as plt
dataMat, labelMat = loadDataSet()
dataArr = array(dataMat)
n = shape(dataArr)[0]
xcord1 = []
ycord1 = []
xcord2 = []
ycord2 = []
for i in range(n):
if int(labelMat[i]) == 1:
xcord1.append(dataArr[i, 1])
ycord1.append(dataArr[i, 2])
else:
xcord2.append(dataArr[i, 1])
ycord2.append(dataArr[i, 2])
fig = plt.figure()
ax = fig.add_subplot(111) # 一行一列一个格子
ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
ax.scatter(xcord2, ycord2, s=30, c='green') # 画出散点图
x = arange(-3.0, 3.0, 0.1) # 生成数组
y = (-weights[0] - weights[1] * x) / weights[2] # 决策边界
ax.plot(x, y)
plt.xlabel('X1')
plt.ylabel('X2')
plt.show()
定义主函数
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def main():
dataMat, labelMat = loadDataSet()
weights = gradAscent(dataMat, labelMat).getA()
plotBestFit(weights)
调用主函数
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if __name__ == '__main__':
main()
运行结果
- 普通梯度上升
- 随机梯度上升
- 改进版随机梯度上升
本文源码参考自 逻辑回归原理(python 代码实现)