机器学习6—SVM学习笔记

机器学习实战之SVM

三种SVM的对偶问题

拉格朗日乘子法和KKT条件

解密SVM系列（一）：关于拉格朗日乘子法和KKT条件

【机器学习详解】SMO算法剖析

回顾初高中三角函数转换知识：

1、诱导公式：

sin(-α) = -sinα； cos(-α) = cosα； sin(π/2-α) = cosα； cos(π/2-α) = sinα；　　 sin(π/2+α) = cosα； cos(π/2+α) = -sinα；

sin(π-α) = sinα； cos(π-α) = -cosα；　　 sin(π+α) = -sinα； cos(π+α) = -cosα；

tanA= sinA/cosA；

tan(π/2+α)＝－cotα； tan(π/2-α)＝ cotα； tan(π-α)＝－tanα； tan(π+α)＝tanα

2、两角和差公式：

sin(AB) = sinAcosBcosAsinB cos(AB) = cosAcosBsinAsinB tan(AB) = (tanAtanB)/(1tanAtanB) cot(AB) = (cotAcotB1)/(cotBcotA)

3、倍角公式　　

sin2A=2sinA•cosA cos2A=cosA2-sinA2=1-2sinA2=2cosA2-1 tan2A=2tanA/（1-tanA2）=2cotA/(cotA2-1)

4、半角公式　　

tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA); cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA.

sin^2(a/2)=(1-cos(a))/2 cos^2(a/2)=(1+cos(a))/2

tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
5、和差化积　

sinθ+sinφ = 2 sin[(θ+φ)/2] cos[(θ-φ)/2] sinθ-sinφ = 2 cos[(θ+φ)/2]sin[(θ-φ)/2]

cosθ+cosφ = 2 cos[(θ+φ)/2]cos[(θ-φ)/2] cosθ-cosφ = -2 sin[(θ+φ)/2]sin[(θ-φ)/2]

tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB) tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)

6、积化和差　　

sinαsinβ = -1/2*[cos(α-β)-cos(α+β)] cosαcosβ = 1/2*[cos(α+β)+cos(α-β)]

sinαcosβ = 1/2*[sin(α+β)+sin(α-β)] cosαsinβ = 1/2*[sin(α+β)-sin(α-β)]万能公式

两条直线相互垂直的条件:

两条直线在同一平面内
1、如果斜率为k1和k2，那么这两条直线垂直的充要条件是k1·k2=－1 ， tanαtan(α+π/2) = tanα(－cotα) = -1

2、如果一直线不存在斜率，则两直线垂直时，一直线的斜率必然为零.

3、两直线(A1x+B1y+C1=0, A2x+B2y+C2=0)垂直的充要条件是：A1A2+B1B2=0.

向量1 (x1,y1),长度 L1 =√(x1²+y1²)
向量2 (x2,y2),长度 L2 =√(x2²+y2²)
（x1,y1）到(x2,y2)的距离：D=√[(x1 - x2)² + (y1 - y2)²]
两个向量垂直,根据勾股定理：L1² + L2² = D²
∴ (x1²+y1²) + (x2²+y2²) = (x1 - x2)² + (y1 - y2)²
∴ x1² + y1² + x2² + y2² = x1² -2x1x2 + x2² + y1² - 2y1y2 + y2²
∴ 0 = -2x1x2 - 2y1y2
∴ x1x2 + y1y2 = 0
该定理还可以扩展到三维向量：x1x2 + y1y2 + z1z2 = 0,那么向量(x1,y1,z1)和（x2,y2,z2）垂直
甚至扩展到更高维度的向量,两个向量L1,L2垂直的充分必要条件是:L1×L2=0

test6.py

#-*- coding：utf-8

import sys
sys.path.append("svmMLiA.py")

import svmMLiA
from numpy import *

dataArr, labelArr = svmMLiA.loadDataSet("testSet.txt")
# b, alphas = svmMLiA.smoSimple(dataArr, labelArr, 0.6, 0.001, 40)

b, alphas = svmMLiA.smoP(dataArr, labelArr, 0.6, 0.001, 40)

ws = svmMLiA.calcWs(alphas, dataArr, labelArr)

num, n = shape(alphas[alphas > 0])

print("num = %d, n = %d" % (num, n))

for i in list(range(100)):
if alphas[i] > 0.0:
print(dataArr[i], labelArr[i])

print("ws : ")
print(ws)

datMat = mat(dataArr)
caculLabel = datMat[0]*mat(ws) + b
print("caculLabel : ")
print(caculLabel)

print(labelArr[0])

svmMLiA.testRbf()

svmMLiA.testDigits(("rbf", 20))

print("over!")

svmMLiA.py

'''
Created on Nov 4, 2010
Chapter 5 source file for Machine Learing in Action
@author: Peter
'''
import matplotlib.pyplot as plt
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)

# 添加绘图部分
xcord0 = []
ycord0 = []
xcord1 = []
ycord1 = []

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
if sign(predict) == 1:
xcord0.extend(datMat[i, :].tolist())
else:
xcord1.extend(datMat[i, :].tolist())
print("the test error rate is: %f" % (float(errorCount) / m))

fig = plt.figure()
ax = fig.add_subplot(111)
s0 = shape(xcord0)[0]
for i in range(s0):
ax.scatter(xcord0[i][0], xcord0[i][1], marker='o', s=20, c='blue')
s1 = shape(xcord1)[0]
for i in range(s1):
ax.scatter(xcord1[i][0], xcord1[i][1], marker='v', s=35, c='red')
plt.show()
# 上绘图部分

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