第五章 Logistic回归
假设现在有一些数据点，我们利用一条直线对这些点进行拟合（该线称为最佳拟合直线），这个拟合过程就称作回归。
为了实现Logistic回归分类器，我们可以在每个特征上都乘以一个回归系数，然后把所有的结果值相加，将这个结果代入Sigmoid函数中，进而得到一个范围在0-1之间的数值。任何大于0.5的数据被分入1类，小于0.5即被归入0类。
1.Sigmoid函数的输入记为 (z),由下面的公式得出：
\[z = {\omega_0}{x_0} + {\omega _1}{x _1} + {\omega _2}{x _2} + ... + {\omega _n}{x _n}
\]
\]
2.梯度算法的迭代公式为：
\[\omega = \omega + \alpha {\nabla _\omega }f(\omega )
\]
\]
\[\theta _j = \theta _j + \alpha({y^i - h({x^i})}) * {x _j^i}
\]
\]
3.Logistic回归梯度上升优化算法
原始数据：
Logistic回归梯度上升优化算法-python代码:
def loadDataSet():
dataMat = []; labelMat = []
fr = open('testSet.txt')
for line in fr.readlines():
lineArr = line.strip().split()
dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
labelMat.append(int(lineArr[2]))
return dataMat,labelMat
def sigmoid(inX):
return 1.0/(1+exp(-inX))
def gradAscent(dataMatIn, classLabels):
dataMatrix = mat(dataMatIn) #convert to NumPy matrix
labelMat = mat(classLabels).transpose() #convert to NumPy matrix
m,n = shape(dataMatrix)
alpha = 0.001
maxCycles = 500 #迭代次数
weights = ones((n,1))
for k in range(maxCycles): #heavy on matrix operations
h = sigmoid(dataMatrix*weights) #matrix mult
error = (labelMat - h) #vector subtraction
weights = weights + alpha * dataMatrix.transpose()* error #matrix mult 这是梯度上升公式更新参数
return weights #返回值为[w0,w1,w2]
在python提示符下，敲下如下代码：
dataArr, labelMat = logRegres.loadDataSet()
logRegres.gradAscent(dataArr, labelMat)
Out[12]:
matrix([[ 4.12414349],
[ 0.48007329],
[-0.6168482 ]])
4.分析数据：画出决策边界
** 画出数据集和Logistic回归最佳拟合直线的函数 **
def plotBestFit(weights): #此处weights即为上面求出的[w0,w1,w2]
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)
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] # 0=w0x0+w1x1+w2x2 (x0=1) 此处x=x1；y=x2
ax.plot(x, y)
plt.xlabel('X1'); plt.ylabel('X2');
plt.show()
在python提示符下，敲下如下代码：
logRegres.plotBestFit(weights.getA()) #getA() 将矩阵转化为数组array
运行结果为：
5.训练算法：随机梯度上升
改进算法，一次仅用一个样本点来更新回归系数——在线学习算法
** 随机梯度上升算法 **
def stocGradAscent0(dataMatrix, classLabels):
m,n = shape(dataMatrix)
alpha = 0.01
weights = ones(n) #initialize to all ones
for i in range(m):
h = sigmoid(sum(dataMatrix[i]*weights))
error = classLabels[i] - h
weights = weights + alpha * error * dataMatrix[i]
return weights
在python提示符下，敲下如下代码：
weights = logRegres.stocGradAscent0(array(dataArr), labelMat)
运行结果为：
**改进的随机梯度上升算法 **
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
m,n = shape(dataMatrix)
weights = ones(n) #initialize to all ones
for j in range(numIter):
dataIndex = range(m)
for i in range(m):
alpha = 4/(1.0+j+i)+0.0001 #apha decreases with iteration, does not
randIndex = int(random.uniform(0,len(dataIndex)))#go to 0 because of the constant
h = sigmoid(sum(dataMatrix[randIndex]*weights))
error = classLabels[randIndex] - h
weights = weights + alpha * error * dataMatrix[randIndex]
del(dataIndex[randIndex])
return weights
- 步长alpha每次迭代时都会调整，可以缓解数据波动或者高频扰动。且alpha会随着迭代次数不断减小，但永远不会减小到0（常数项）——可以保证多次迭代后新数据仍然有影响。如果要处理的问题是动态变化的，可以适当加大常数项，确保新值获得更大的回归系数。当( j << max(i) )时，步长alpha不是严格下降的。避免参数严格下降也常见于模拟退火等其他优化算法中。
- 随机选取样本值来更新回归系数。这种方法将减少周期性的波动——具体实现方法：每次随机从样本中选出一个值，更新回归系数后将其删除，再进行下一次迭代。
改进后回归系数变化情况
收敛速度更快，波动更小
在python提示符下，敲下如下代码：
dataArr, labelMat = logRegres.loadDataSet()
weights = logRegres.stocGradAscent1(array(dataArr), labelMat)
logRegres.plotBestFit(weights)
运行结果为：
6. 示例：从疝气病症预测病马的死亡率
Logistic回归分类函数
def classifyVector(inX, weights):
prob = sigmoid(sum(inX*weights))
if prob > 0.5: return 1.0
else: return 0.0
def colicTest():
frTrain = open('horseColicTraining.txt'); frTest = open('horseColicTest.txt')
trainingSet = []; trainingLabels = []
for line in frTrain.readlines():
currLine = line.strip().split('\t')
lineArr =[]
for i in range(21):
lineArr.append(float(currLine[i]))
trainingSet.append(lineArr)
trainingLabels.append(float(currLine[21]))
trainWeights = stocGradAscent1(array(trainingSet), trainingLabels, 1000) #计算回归系数
errorCount = 0; numTestVec = 0.0
for line in frTest.readlines():
numTestVec += 1.0
currLine = line.strip().split('\t')
lineArr =[]
for i in range(21):
lineArr.append(float(currLine[i]))
if int(classifyVector(array(lineArr), trainWeights))!= int(currLine[21]):
errorCount += 1
errorRate = (float(errorCount)/numTestVec)
print "the error rate of this test is: %f" % errorRate
return errorRate
def multiTest(): #调用colicTest()10次并求平均值
numTests = 10; errorSum=0.0
for k in range(numTests):
errorSum += colicTest()
print "after %d iterations the average error rate is: %f" % (numTests, errorSum/float(numTests))
在python提示符下，敲下如下代码：
logRegres.multiTest()
输出：
the error rate of this test is: 0.313433
the error rate of this test is: 0.402985
the error rate of this test is: 0.298507
the error rate of this test is: 0.402985
the error rate of this test is: 0.388060
the error rate of this test is: 0.313433
the error rate of this test is: 0.328358
the error rate of this test is: 0.343284
the error rate of this test is: 0.283582
the error rate of this test is: 0.388060
after 10 iterations the average error rate is: 0.346269