对于特征向量，我们设置参数
，通过计算：
获得一个处于0到1之间的值，当这个值大于0.5时，我们把它判定为类别A，当这个值小于0.5时，我们把它判定为类别B，等于说这个值代表了它是类别A的概率。这里的是一个偏置项，我们可以把它放入到参数矩阵中，此时
，其中
，这时候只需要计算
就可以了。
当然这里输出只是一个标量，只能解决二分类的问题，这时候只需要将参数向量变成参数矩阵
，输出就是一个多维的向量了，每个维度代表着一个分类并给出对这个分类的概率，我们选择概率最大的作为我们的分类。这种逻辑回归的方式，可以看做一个全连接的前馈神经网络。
首先构造一个数据集testSet.txt：
-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
前两个数字是两个特征的值，最后一个数字是分类，属于0或者1。
然后载入数据集：
def loadDataSet():
dataMat, labelMat = [], []
with open("./testSet.txt") as fr:
for line in fr:
lineArr = line.strip().split()
dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
labelMat.append(int(lineArr[2]))
return dataMat, labelMat
dataMat, labelMat = loadDataSet()
print(dataMat)
# [[1.0, -0.017612, 14.053064], [1.0, -1.395634, 4.662541], [1.0, -0.752157, 6.53862], [1.0, -1.322371, 7.152853],...]
print(labelMat)
# [0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1,...]
参数通过梯度下降/上升进行更新：
def sigmoid(inX):
return 1.0 / (1+exp(-inX))
def gradAscent(dataMatIn, classLabels):
dataMatrix = mat(dataMatIn) # [size, 3]
labelMat = mat(classLabels).transpose() #
m, n = shape(dataMatrix) # size, 3
alpha = 0.001
maxCycles = 500
weights = ones((n, 1)) # [3, 1]
for k in range(maxCycles):
h = sigmoid(dataMatrix * weights)
error = (labelMat - h)
weights = weights + alpha * dataMatrix.transpose() * error
return weights
weights = gradAscent(dataMat,labelMat)
print(weights)
# [[ 4.12414349]
# [ 0.48007329]
# [-0.6168482 ]]
最后看一下划分的结果：
def plotBestFit(weights):
import matplotlib.pyplot as plt
dataMat, labelMat = loadDataSet()
dataArr = array(dataMat)
n = shape(dataArr)[0] # size
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)
ax.scatter(xcord1, ycord1, s=30, c="red", marker="s")
ax.scatter(xcord2, ycord2, s=30, c="green")
x = 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()
plotBestFit(weights.getA())
logRegres.png
更新参数的方式还有随机梯度下降等等和前馈神经网络类似，所以逻辑回归这里就不详细写了。
