机器学习之logistic回归算法
一:Sigmoid函数和Logistic回归分类器
其函数表达式为:
其显示的图象为:
2:Logistic回归分类器
Simoid函数的输入记为:z=w0x0 + w1x1 + w2x2 .... + wnxn
如果采用向量的写法,上述公式可以写成z=w^t * x(w^t表示系数w的转置矩阵)
代入到Sigmoid函数可得:
其输出分大于0.5和小于0.5,表示两个类别,也就实现了分类,确定了分类器的函数形式,接下来问题就是求最佳回归系数
二:基于最优化方法的最佳回归系数确定
2.1:梯度上升法
主要思想:要找到某函数的最大值,最好的办法是沿着该函数的梯度方向探寻
下边这种图片是机器学习实战对梯度的数学解释:
梯度是有方向的,总是沿着函数值上升最快的方向移动(这有点感觉想物理中的加速度),因此我们沿着梯度方向或者反方向行进时,就能达到一个函数的最大值或者最小值,因此梯度上升算法就是不断更新梯度值,直到梯度不再变化或者变化很小,即函数达到了最大值
梯度算法的迭代公式为(alpha为步长,即每一步移动量):
那么问题来了,我们如何求解函数的梯度,在 Machine Learning in Action一书中,作者没有解释,直接给出了代码
- h = sigmoid(dataMatrix*weights)
- error = (labelMat - h)
- weights = weights + alpha * dataMatrix.transpose()* error
当然在实战这本书也没有具体说明(这里有一篇博客对这个公式进行了猜想推测:http://blog.sina.com.cn/s/blog_61f1db170101k1wr.html )
求梯度上升算法的代码,并画出图形:
- #coding:utf-8
- '''''
- Created on 2016/4/24
- @author: Gamer Think
- '''
- from numpy import *
- #加载数据集
- def loadDataSet():
- dataMat = []
- labelMat = []
- fp = open("ex1.txt")
- for line in fp.readlines():
- lineArr = line.strip().split("\t") #每行按\t分割
- dataMat.append([1.0,float(lineArr[0]), float(lineArr[1])])
- labelMat.append(float(lineArr[2]))
- return dataMat,labelMat
- #定义Sigmoid函数
- def sigmoid(inX):
- return 1.0/(1+exp(-inX))
- #定义求解最佳回归系数
- def gradAscent(dataMatIn,classLabels):
- dataMatrix = mat(dataMatIn) #将数组转为矩阵
- labelMat = mat(classLabels).transpose()
- m,n = shape(dataMatrix) #返回矩阵的行和列
- alpha = 0.001 #初始化 alpha的值
- maxCycles = 500 #最大迭代次数
- weights = ones((n,1)) #初始化最佳回归系数
- for i in range(0,maxCycles):
- #引用原书的代码,求梯度
- h = sigmoid(dataMatrix*weights)
- error = labelMat - h
- weights = weights + alpha * dataMatrix.transpose() * error
- return weights
- #分析数据,画出决策边界
- def plotBestFit(wei,dataMatrix,labelMat):
- import matplotlib.pyplot as plt
- weights = wei.getA() #将矩阵wei转化为list
- dataArr = array(dataMatrix) #将矩阵转化为数组
- n = shape(dataMatrix)[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)
- 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") #X轴的标签
- plt.ylabel("x2") #Y轴的标签
- plt.show()
- if __name__=="__main__":
- dataMatrix,labelMat = loadDataSet()
- weight = gradAscent(dataMatrix, labelMat)
- plotBestFit(weight,dataMatrix,labelMat)
显示效果图:
2.2随机梯度上升算法
梯度上升算法在每次更新回归系数时都需要遍历整个数据集,该方法在处理100个左右的数据集尚可,但如果数据量增大,那该方法的计算量就太大了,有一种改进方法是一次仅用一个样本点来更新回归系数,该方法称为随机梯度上升算法,由于可以在新样本到来时对分类器进行增量式更新,因而随机梯度上升算法是一个在线学习算法。
随机梯度上升算法的代码如下:
- <span style="font-size:18px;">#随机梯度上升算法求回归系数
- def stocGradAscent0(dataMatrix,labelMat):
- dataMatrix = array(dataMatrix)
- m,n = shape(dataMatrix)
- alpha = 0.01
- weights = ones(n)
- for i in range(0,m):
- h = sigmoid(sum(dataMatrix[i]*weights))
- error = labelMat[i] - h
- weights = weights + alpha * error * dataMatrix[i]
- return weights</span><span style="font-size: 14px;">
- </span>
- #随机梯度上升算法
- weight = stocGradAscent0(dataMatrix, labelMat)
- print weight
- plotBestFit(weight,dataMatrix,labelMat)
- <span style="font-size:18px;">#改进版的随机梯度上升算法
- def stocGradAscent1(dataMatrix,labelMat,numIter=150):
- m,n = shape(dataMatrix)
- weights = ones(n)
- for i in range(0,numIter):
- dataIndex = range(m)
- for j in range(0,m):
- alpha = 4/(1.0+j+i)+0.01 #(1)
- randIndex = int(random.uniform(0,len(dataIndex))) #(2)
- h = sigmoid(sum(dataMatrix[randIndex] * weights))
- error = labelMat[randIndex] - h
- weights = weights + alpha * error * dataMatrix[randIndex]
- del(dataIndex[randIndex])
- return weights </span><span style="font-size: 14px;">
- </span>
(1):alpha在每次 迭代的时候都会调整,会缓解数据波动和高频波动,另外alpha会随着迭代次数不断减小,但永远不会减小到0,这是因为(1)中存在一个常数项,这样做的目的是保证在多次迭代后新数据仍有一定的影响力,如果处理的问题是动态的,可以适当加大上边的常数项,来保证新的书获得更大的回归系数,另外一点值得注意的是,在降低alpha的函数中,alpha每次减小1/(j+i),其中j是迭代次数,i是样本点的下标,这样当j<<max(i)时,alpha就不是严格下降的,避免参数的严格下降也是常见于模拟退火算法等其他优化算法中
(2):通过随机选择样本来更新回归系数,这样方法将减小周期性波动,每次随机从列表中选出一个值,然后从列表中删除该值。
此外增加了一个迭代次数作为第三个参数,如果不给定的话,默认是150次。
- <span style="font-size:18px;"> #改进版的随机梯度上升算法
- weight = stocGradAscent1(array(dataMatrix), labelMat)
- print weight
- plotBestFit(weight,dataMatrix,labelMat)</span>
显示效果图如下:
数据集内容如下:
-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
-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