支持向量机（Support Vector Machine）
前置知识：拉格朗日数乘法、对偶问题、核技巧
拉格朗日数乘法
针对的是约束优化问题：
例题：
已知x>0,y>0,x+2y+2xy=8,则x+2y的最小值__。
解：
引入参数 λ \lambda λ 构造新函数L： x + 2 y + λ ( x + 2 y + 2 x y − 8 ) x+2y+\lambda(x+2y+2xy-8) x+2y+λ(x+2y+2xy−8)
分别对x,y,
λ
\lambda
λ求偏导：
L
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0
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λ
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λ
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8
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0
L_x = 1+\lambda(1+2y)=0\\ L_y = 2+\lambda(2+2x)=0\\ \ \ \ \ \ \ L_\lambda = x+2y+2xy-8=0\\
Lx=1+λ(1+2y)=0Ly=2+λ(2+2x)=0 Lλ=x+2y+2xy−8=0
三个方程三个未知数，可以求得x=2,y=1。
即当x=2,y=1时x+2y的 最小值为4。
对偶问题
用于对优化问题的转换
例如：maxmin -> minmax
默认约束优化问题是弱对偶关系，当满足KNN条件时，具有强对偶关系，即二者等价。
核技巧
用于对高维特征的扩展，当样本数或超平面维数过大时，可以利用核技巧优化，将问题转化为有限维问题。
Machine Learning action
本章学习的是支持向量机中最流行的一种实现–序列最小优化（Sequential Minimal Optimization）算法。
优点：泛化错误率低，计算开销不大，结果容易理解
缺点：对参数调节和和函数的选择敏感，原始分类器不加修改仅适用于处理二分类问题。
适用数据类型：数值型和标称型数据
原理性部分
分割超平面集将不同类别的数据点分割的平面，支持向量机就是由这些分割超平面组成的分类器。
**支持向量(support vector)是指离分割超平面(separating byperplane)**最近的那些点，令支持向量的间隔最大化，就是构造支持向量机的优化目标
图中a,b,c都可以认为是一个分割超平面，显然，超分割平面b的鲁棒性即泛化能力要优于a,c。
如何建立数理模型来寻找到优质超分割平面？
那么就需要定义一个优化目标，即距离超分割面最近的那些的间隔最大化。
a
r
g
m
a
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w
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{
m
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arg\ max_{w,b}\ \{ min(w^T*x+b) \}
arg maxw,b {min(wT∗x+b)}
其中w,b就是待优化的参数，假如直接求解该模型是十分困难的一件事，于是开始了一系列的模型转换，感兴趣的可以结合文章开头的视频以及西瓜书进一步学习，这里只将流程梳理一遍。
- 利用拉格朗朗日乘子法，得到拉格朗日方程组
- 将拉格朗日函数进行转换，转化为 a r g m i n m a x arg\ min max arg minmax问题
- 利用对偶问题的强对偶条件(KKT条件)将 a r g m i n m a x arg\ minmax arg minmax转化为 a r g m a x m i n arg\ maxmin arg maxmin问题
- 由于样本空间的非必线性可分，因此增加松弛变量
同时问题的参数也变为 α \alpha α,接下来便是对该问题进行求解
机器学习实战中使用的方法时SMO算法。
SMO算法伪代码：
创建一个alpha向量并将其初始化为0向量
当迭代次数小于最大迭代次数时(外循环)：
对数据集中的每个数据向量(内循环)：
如果该数据向量可以被优化：
随机选择另外一个数据向量
同时优化这两个向量
如果两个向量都不能被优化，退出内循环
如果所有向量都没有被优化，增加迭代数目，继续下一次循环
实际上SMO算法是一个较为有效的贪心算法。
code
'''
Created on Nov 4, 2010
Chapter 5 source file for Machine Learing in Action
@author: Peter
'''
from numpy import *
from time import sleep
def loadDataSet(fileName):
dataMat = []
labelMat = []
fr = open(fileName)
for line in fr.readlines():
lineArr = line.strip().split('\t')
dataMat.append([float(lineArr[0]), float(lineArr[1])])
labelMat.append(float(lineArr[2]))
return dataMat, labelMat
def selectJrand(i, m):
j = i # we want to select any J not equal to i
while (j == i):
j = int(random.uniform(0, m))
return j
def clipAlpha(aj, H, L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
dataMatrix = mat(dataMatIn)
labelMat = mat(classLabels).transpose()
b = 0
m, n = shape(dataMatrix)
alphas = mat(zeros((m, 1)))
iter = 0
while (iter < maxIter):
alphaPairsChanged = 0
for i in range(m):
fXi = float(multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[i, :].T)) + b
Ei = fXi - float(labelMat[i]) # if checks if an example violates KKT conditions
if ((labelMat[i] * Ei < -toler) and (alphas[i] < C)) or ((labelMat[i] * Ei > toler) and (alphas[i] > 0)):
j = selectJrand(i, m)
fXj = float(multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[j, :].T)) + b
Ej = fXj - float(labelMat[j])
alphaIold = alphas[i].copy()
alphaJold = alphas[j].copy()
if (labelMat[i] != labelMat[j]):
L = max(0, alphas[j] - alphas[i])
H = min(C, C + alphas[j] - alphas[i])
else:
L = max(0, alphas[j] + alphas[i] - C)
H = min(C, alphas[j] + alphas[i])
if L == H:
print("L==H")
continue
eta = 2.0 * dataMatrix[i, :] * dataMatrix[j, :].T - dataMatrix[i, :] * dataMatrix[i, :].T - dataMatrix[
j,
:] * dataMatrix[
j, :].T
if eta >= 0 :
print("eta>=0")
continue
alphas[j] -= labelMat[j] * (Ei - Ej) / eta
alphas[j] = clipAlpha(alphas[j], H, L)
if (abs(alphas[j] - alphaJold) < 0.00001):
print("j not moving enough")
continue
alphas[i] += labelMat[j] * labelMat[i] * (alphaJold - alphas[j]) # update i by the same amount as j
# the update is in the oppostie direction
b1 = b - Ei - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[i, :].T - labelMat[
j] * (alphas[j] - alphaJold) * dataMatrix[i, :] * dataMatrix[j, :].T
b2 = b - Ej - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[j, :].T - labelMat[
j] * (alphas[j] - alphaJold) * dataMatrix[j, :] * dataMatrix[j, :].T
if (0 < alphas[i]) and (C > alphas[i]):
b = b1
elif (0 < alphas[j]) and (C > alphas[j]):
b = b2
else:
b = (b1 + b2) / 2.0
alphaPairsChanged += 1
print("iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
if (alphaPairsChanged == 0):
iter += 1
else:
iter = 0
print("iteration number: %d" % iter)
return b, alphas
def kernelTrans(X, A, kTup): # calc the kernel or transform data to a higher dimensional space
m, n = shape(X)
K = mat(zeros((m, 1)))
if kTup[0] == 'lin':
K = X * A.T # linear kernel
elif kTup[0] == 'rbf':
for j in range(m):
deltaRow = X[j, :] - A
K[j] = deltaRow * deltaRow.T
K = exp(K / (-1 * kTup[1] ** 2)) # divide in NumPy is element-wise not matrix like Matlab
else:
raise NameError('Houston We Have a Problem -- \
That Kernel is not recognized')
return K
class optStruct:
def __init__(self, dataMatIn, classLabels, C, toler, kTup): # Initialize the structure with the parameters
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = shape(dataMatIn)[0]
self.alphas = mat(zeros((self.m, 1)))
self.b = 0
self.eCache = mat(zeros((self.m, 2))) # first column is valid flag
self.K = mat(zeros((self.m, self.m)))
for i in range(self.m):
self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup)
def calcEk(oS, k):
fXk = float(multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b)
Ek = fXk - float(oS.labelMat[k])
return Ek
def selectJ(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej
maxK = -1
maxDeltaE = 0
Ej = 0
oS.eCache[i] = [1, Ei] # set valid #choose the alpha that gives the maximum delta E
validEcacheList = nonzero(oS.eCache[:, 0].A)[0]
if (len(validEcacheList)) > 1:
for k in validEcacheList: # loop through valid Ecache values and find the one that maximizes delta E
if k == i: continue # don't calc for i, waste of time
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k
maxDeltaE = deltaE
Ej = Ek
return maxK, Ej
else: # in this case (first time around) we don't have any valid eCache values
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
def updateEk(oS, k): # after any alpha has changed update the new value in the cache
Ek = calcEk(oS, k)
oS.eCache[k] = [1, Ek]
def innerL(i, oS):
Ei = calcEk(oS, i)
if ((oS.labelMat[i] * Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or (
(oS.labelMat[i] * Ei > oS.tol) and (oS.alphas[i] > 0)):
j, Ej = selectJ(i, oS, Ei) # this has been changed from selectJrand
alphaIold = oS.alphas[i].copy()
alphaJold = oS.alphas[j].copy()
if (oS.labelMat[i] != oS.labelMat[j]):
L = max(0, oS.alphas[j] - oS.alphas[i])
H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
else:
L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
H = min(oS.C, oS.alphas[j] + oS.alphas[i])
if L == H:
print("L==H")
return 0
eta = 2.0 * oS.K[i, j] - oS.K[i, i] - oS.K[j, j] # changed for kernel
if eta >= 0:
print("eta>=0")
return 0
oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
updateEk(oS, j) # added this for the Ecache
if (abs(oS.alphas[j] - alphaJold) < 0.00001):
print("j not moving enough")
return 0
oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j]) # update i by the same amount as j
updateEk(oS, i) # added this for the Ecache #the update is in the oppostie direction
b1 = oS.b - Ei - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, i] - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.K[i, j]
b2 = oS.b - Ej - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, j] - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.K[j, j]
if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]):
oS.b = b1
elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]):
oS.b = b2
else:
oS.b = (b1 + b2) / 2.0
return 1
else:
return 0
def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)): # full Platt SMO
oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler, kTup)
iter = 0
entireSet = True
alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet: # go over all
for i in range(oS.m):
alphaPairsChanged += innerL(i, oS)
print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
iter += 1
else: # go over non-bound (railed) alphas
nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
for i in nonBoundIs:
alphaPairsChanged += innerL(i, oS)
print("non-bound, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
iter += 1
if entireSet:
entireSet = False # toggle entire set loop
elif (alphaPairsChanged == 0):
entireSet = True
print("iteration number: %d" % iter)
return oS.b, oS.alphas
def calcWs(alphas, dataArr, classLabels):
X = mat(dataArr)
labelMat = mat(classLabels).transpose()
m, n = shape(X)
w = zeros((n, 1))
for i in range(m):
w += multiply(alphas[i] * labelMat[i], X[i, :].T)
return w
def testRbf(k1=1.3):
dataArr, labelArr = loadDataSet('testSetRBF.txt')
b, alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) # C=200 important
datMat = mat(dataArr)
labelMat = mat(labelArr).transpose()
svInd = nonzero(alphas.A > 0)[0]
sVs = datMat[svInd] # get matrix of only support vectors
labelSV = labelMat[svInd]
print("there are %d Support Vectors" % shape(sVs)[0])
m, n = shape(datMat)
errorCount = 0
for i in range(m):
kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', k1))
predict = kernelEval.T * multiply(labelSV, alphas[svInd]) + b
if sign(predict) != sign(labelArr[i]): errorCount += 1
print("the training error rate is: %f" % (float(errorCount) / m))
dataArr, labelArr = loadDataSet('testSetRBF2.txt')
errorCount = 0
datMat = mat(dataArr)
labelMat = mat(labelArr).transpose()
m, n = shape(datMat)
for i in range(m):
kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', k1))
predict = kernelEval.T * multiply(labelSV, alphas[svInd]) + b
if sign(predict) != sign(labelArr[i]): errorCount += 1
print("the test error rate is: %f" % (float(errorCount) / m))
def img2vector(filename):
returnVect = zeros((1, 1024))
fr = open(filename)
for i in range(32):
lineStr = fr.readline()
for j in range(32):
returnVect[0, 32 * i + j] = int(lineStr[j])
return returnVect
def loadImages(dirName):
from os import listdir
hwLabels = []
trainingFileList = listdir(dirName) # load the training set
m = len(trainingFileList)
trainingMat = zeros((m, 1024))
for i in range(m):
fileNameStr = trainingFileList[i]
fileStr = fileNameStr.split('.')[0] # take off .txt
classNumStr = int(fileStr.split('_')[0])
if classNumStr == 9:
hwLabels.append(-1)
else:
hwLabels.append(1)
trainingMat[i, :] = img2vector('%s/%s' % (dirName, fileNameStr))
return trainingMat, hwLabels
def testDigits(kTup=('rbf', 10)):
dataArr, labelArr = loadImages('trainingDigits')
b, alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, kTup)
datMat = mat(dataArr)
labelMat = mat(labelArr).transpose()
svInd = nonzero(alphas.A > 0)[0]
sVs = datMat[svInd]
labelSV = labelMat[svInd]
print("there are %d Support Vectors" % shape(sVs)[0])
m, n = shape(datMat)
errorCount = 0
for i in range(m):
kernelEval = kernelTrans(sVs, datMat[i, :], kTup)
predict = kernelEval.T * multiply(labelSV, alphas[svInd]) + b
if sign(predict) != sign(labelArr[i]): errorCount += 1
print("the training error rate is: %f" % (float(errorCount) / m))
dataArr, labelArr = loadImages('testDigits')
errorCount = 0
datMat = mat(dataArr)
labelMat = mat(labelArr).transpose()
m, n = shape(datMat)
for i in range(m):
kernelEval = kernelTrans(sVs, datMat[i, :], kTup)
predict = kernelEval.T * multiply(labelSV, alphas[svInd]) + b
if sign(predict) != sign(labelArr[i]): errorCount += 1
print("the test error rate is: %f" % (float(errorCount) / m))
'''#######********************************
Non-Kernel VErsions below
''' #######********************************
class optStructK:
def __init__(self, dataMatIn, classLabels, C, toler): # Initialize the structure with the parameters
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = shape(dataMatIn)[0]
self.alphas = mat(zeros((self.m, 1)))
self.b = 0
self.eCache = mat(zeros((self.m, 2))) # first column is valid flag
def calcEkK(oS, k):
fXk = float(multiply(oS.alphas, oS.labelMat).T * (oS.X * oS.X[k, :].T)) + oS.b
Ek = fXk - float(oS.labelMat[k])
return Ek
def selectJK(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej
maxK = -1
maxDeltaE = 0
Ej = 0
oS.eCache[i] = [1, Ei] # set valid #choose the alpha that gives the maximum delta E
validEcacheList = nonzero(oS.eCache[:, 0].A)[0]
if (len(validEcacheList)) > 1:
for k in validEcacheList: # loop through valid Ecache values and find the one that maximizes delta E
if k == i: continue # don't calc for i, waste of time
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k
maxDeltaE = deltaE
Ej = Ek
return maxK, Ej
else: # in this case (first time around) we don't have any valid eCache values
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
def updateEkK(oS, k): # after any alpha has changed update the new value in the cache
Ek = calcEk(oS, k)
oS.eCache[k] = [1, Ek]
def innerLK(i, oS):
Ei = calcEk(oS, i)
if ((oS.labelMat[i] * Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or (
(oS.labelMat[i] * Ei > oS.tol) and (oS.alphas[i] > 0)):
j, Ej = selectJ(i, oS, Ei) # this has been changed from selectJrand
alphaIold = oS.alphas[i].copy()
alphaJold = oS.alphas[j].copy()
if (oS.labelMat[i] != oS.labelMat[j]):
L = max(0, oS.alphas[j] - oS.alphas[i])
H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
else:
L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
H = min(oS.C, oS.alphas[j] + oS.alphas[i])
if L == H:
print ("L==H")
return 0
eta = 2.0 * oS.X[i, :] * oS.X[j, :].T - oS.X[i, :] * oS.X[i, :].T - oS.X[j, :] * oS.X[j, :].T
if eta >= 0:
print("eta>=0")
return 0
oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
updateEk(oS, j) # added this for the Ecache
if (abs(oS.alphas[j] - alphaJold) < 0.00001):
print("j not moving enough")
return 0
oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j]) # update i by the same amount as j
updateEk(oS, i) # added this for the Ecache #the update is in the oppostie direction
b1 = oS.b - Ei - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.X[i, :] * oS.X[i, :].T - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.X[i, :] * oS.X[j, :].T
b2 = oS.b - Ej - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.X[i, :] * oS.X[j, :].T - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.X[j, :] * oS.X[j, :].T
if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]):
oS.b = b1
elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]):
oS.b = b2
else:
oS.b = (b1 + b2) / 2.0
return 1
else:
return 0
def smoPK(dataMatIn, classLabels, C, toler, maxIter): # full Platt SMO
oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)
iter = 0
entireSet = True
alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet: # go over all
for i in range(oS.m):
alphaPairsChanged += innerL(i, oS)
print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
iter += 1
else: # go over non-bound (railed) alphas
nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
for i in nonBoundIs:
alphaPairsChanged += innerL(i, oS)
print("non-bound, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
iter += 1
if entireSet:
entireSet = False # toggle entire set loop
elif (alphaPairsChanged == 0):
entireSet = True
print("iteration number: %d" % iter)
return oS.b, oS.alphas
import svm as svmMLiA
import numpy as np
import matplotlib.pyplot as plt
dataArr,labelArr =svmMLiA.loadDataSet('testSet.txt')
# 数据集、类别标签、常数C、容错率、退出最大循环数
b,alphas= svmMLiA.smoSimple(dataArr,labelArr,0.6,0.001,40)
plt.rcParams['font.sans-serif']=['SimHei']
plt.rcParams['axes.unicode_minus'] = False
# matplotlib画图中中文显示会有问题，需要这两行设置默认字体
plt.xlabel('X')
plt.ylabel('Y')
plt.xlim(xmax=12,xmin=-2.5)
plt.ylim(ymax=10,ymin=-10)
# 画两条（0-9）的坐标轴并设置轴标签x，y
x1=list();x2=list()
y1=list();y2=list()
for j in range(len(labelArr)):
if labelArr[j]==1:
x1.append(dataArr[j][0])
y1.append(dataArr[j][1])
else :
x2.append(dataArr[j][0])
y2.append(dataArr[j][1])
print(x1)
print(y1)
colors1 = '#00CED1'
colors2 = '#DC143C'
area = np.pi*4**2
plt.scatter(x1,y1,s=area,c=colors1,alpha=0.4,label='类别A')
plt.scatter(x2,y2,s=area,c=colors2,alpha=0.4,label='类别B')
plt.plot()
plt.legend()
plt.savefig('1.png',dpi=300)
plt.show()
结语
- SVM是经过较严格的数理证明的产物。
- 其特征提取方式教为单一，当数据集的分布过于类似时难以划分。