目录
1.什么是SVM
2.最大间隔
3.SMO高效算法
4.代码
SVM(全称Support Vector Machine)中文名支持向量机。SVM是一种监督机器学习算法,是一种二分类模型,它的目的是寻找一个超平面来对样本进行分割,分割的原则是间隔最大化,最终转化为一个凸二次规划问题来求解。可用于分类或回归挑战。然而,它主要用于分类问题。
如果一个线性函数能够将样本分开,称这些数据样本是线性可分的。那么什么是线性函数呢?其实很简单,在二维空间中就是一条直线,在三维空间中就是一个平面,以此类推,如果不考虑空间维数,这样的线性函数统称为超平面。我们看一个简单的二维空间的例子,+代表正类,-代表负类,样本是线性可分的,但是很显然不只有这一条直线可以将样本分开,而是有无数条,我们所说的线性可分支持向量机就对应着能将数据正确划分并且间隔最大的直线。
在样本空间中寻找一个超平面, 将不同类别的样本分开。
就必须给出点到分隔面的法线或垂线的长度,点X到超平面的距离为:
其中 表示的是,所有元素的平方和的开平方。
如下图,支持向量(support vector)就是离分隔超平面最近的那些点。
超平面方程:
上述将数据集分隔开来的直线称为分隔超平面,即。
通过数学知识可知,求的最大值,就是求
的最小值,求最大值我们利用求导获取极值来解题,为了简化计算,因此问题可以等价于求
的最小值:
到了这里,就得出了求解最大间隔超平面的最终表达式。
现在,让我们来看一下简单的间隔最大化样例计算:
如上,D表示了是三个二维数据,第一列表示x坐标,第二类列表示y坐标,第三列表示样本类型,+1表示正样本,-1表示负样本,画出了这三个点的二维坐标图。
其中,,和
和
,可以看成是,在两直线的约束区间内,取满足原点为中心的圆的最小半径,如下图
因为圆心,在约束区间外,所以当圆与约束区间相切的时候,就可以获得解, ③
把③带入,原①②式子:
SMO算法的目标是求出一系列alpha和b,一旦求出这些alpha,就很容易算出权重向量w并得到分割超平面。
SMO算法工作原理:每次循环中选择两个alpha进行优化处理。一旦找到一对合适的alpha,那么就增大其中一个,减小另外一个
算法流程:每次选取两个a进行更新
import random
from numpy import zeros, mat, nonzero, shape, multiply, sign, exp
def selectJrand(i,m):
j=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 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)
m = len(trainingFileList)
trainingMat = zeros((m,1024))
for i in range(m):
fileNameStr = trainingFileList[i]
fileStr = fileNameStr.split('.')[0]
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 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
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
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 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 Digits(kTup=('rbf',20)):
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))
if __name__ =="__main__":
Digits()
运行结果:
参考:http://t.csdn.cn/UhQwV