Logistic回归
- 假设有一些数据点，我们用一条直线对这些点进行拟合（该线称为最佳拟合直线），这个拟合过程称作回归。
- 训练分类器时的做法就是寻找最佳拟合参数，使用的是最优化算法。
- 二值型输出分类器
Sigmoid函数
image.png
- 为了实现Logistic回归分类器，在每个特征值上乘以一个回归系数，然后把所有值相加，将这个总和代入上述函数中，进而得到一个范围在0~1之间的数值。任何大于0.5的数据分为1类，小于0.5的归为0类。
- Logistic回归也可以看成一种概率估计。
- 输入记作z，z=wTx，w为系数向量。
梯度上升，随机梯度上升
- 用来确定最佳回归系数
- 实际就是计算偏导数，根据偏导数和步长来迭代。根据迭代次数或某个允许的误差范围来停止迭代。
Logistic回归梯度上升优化算法
from numpy import *
# testSet.txt的一行有三列用空格分割，前两个是特征，最后一个是分类值
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
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
return weights
绘制logistic回归线
def plotBestFit(weights):
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]
ax.plot(x, y)
plt.xlabel('X1'); plt.ylabel('X2');
plt.show()
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
