Machine_Learning_in_Action06 - SVM
SVM
介绍 SVM
使用SMO优化算法
使用核函数对数据进行空间转化
与其他分类器对比
目录
- SVM 的实现方法有多种,本章介绍最常见的一种:序列最小化优化算法(SMO, sequential minimal optimization algorithm)
通过最大化间隔分离数据
- 优点
- 泛化错误率低,计算量小,结果易解释
- 缺点
- 对参数和核函数的选择敏感,原始分类器不加修改仅适用于二分类问题
- 适用数据类型:数值型和标称型数据
线性可分的数据，分界线叫做超平面。
我们希望分类器能将数据尽可能地远离超平面，我们想要最大化边界距离（margin），目的是为了能在有限的数据集上训练时有尽可能好的健壮性。
与超平面距离最近的点叫做支持向量，我们要做的是最大化支持向量与分类超平面的距离。
寻找最大边界距离
分类超平面往往是 \(\mathbf{w}^T\mathbf{x} + b\) 的形式，A 点到超平面的距离是 \(|\mathbf{w}^T\mathbf{x} + b| / ||\mathbf{w}||\) 。b 是一个常数，类似于逻辑回归中的 \(w_0\) ，w 和 b 描述了分类超平面。
以分类器的形式处理优化问题
理解分类器是如何工作的有助于理解优化问题。对于分类函数sigmoid，我们将数据输入函数，输出一个类别，类别是0或1，这里我们将sigmoid函数替换为阶跃函数，输入小于0时取值为-1，否则为1.
为何要将类别从0和1替换为-1和1，是因为这样在数学上是可控的，-1和1只差了一个符号。这样我们就只用一个公式就能表示点到分类面的距离 \(label*(\mathbf{w}^T\mathbf{x}+b)\) 。如果一个点距离分类面很远，并且是在正数据集一侧，则 \(\mathbf{w}^T\mathbf{x}+b\) 将会是一个很大的正值，\(label*(\mathbf{w}^T\mathbf{x}+b)\) 是一个比较大的正数。而对于负数据集，它也是一个正的值。
现在的目标是找到一组 \(\mathbf{w}\) 和 b 来定义分类器。为此我们需要找到最小的边界距离的点（支持向量），我们必须最大化边界距离：
\[arg\max_{w,b}\{\min_n(label·(\mathbf{w^Tx} + b)) · \frac{1}{||w||}\}
\]
直接求解这个问题是很困难的，所以我们换一种形式。先看花括号内的部分，多重优化是很困难的，所以我们可以保持一部分不变，最大化另一部分。如果我们令支持向量的 \(label·(\mathbf{w^Tx} + b)\) 为1，那么我们需要最大化 \(||w||^{-1}\)。不是所有的 \(label·(\mathbf{w^Tx} + b)\) 都是1，只有最靠近分类面的是1，远离分类面的，内积将会更大。
优化问题变成了一个限定性优化问题，为了得到最好的值，\(label·(\mathbf{w^Tx} + b)\) 被限制大于等于1。这种问题是常见的限定性优化问题，通常使用拉格朗日乘子法。
上述的优化过程有个假设，就是假设数据集是100% 线性可分的，而现实并不如此，所以可以引入一个松弛因子（slack variables），允许有数据点分布在分类面的错误的一边
SVM 的一般框架
SVM 的一般方法：
- 收集数据
- 准备：数值型数据
- 分析：可视化来找到分类超平面
- 训练
- 测试
- 应用
- 目前SVM是二分类，想要应用于多分类，需要手动修改代码
用 SMO 算法进行效率优化
在此之前，人们使用二次规划求解优化问题，它会消耗大量计算资源并且很复杂。
现在我们要使用SMO算法，然后会写一个简化版本来说明它是如何工作。简化版可以处理少量数据。下一节我们会使用完整版本，它的运行速度比简化版快很多。
Platt 的 SMO 算法
1996年 John Platt 发表了SMO算法，用来训练 SVM。SMO代表 Sequential Minimal Optimization。它会将大的优化问题拆分成小的问题。
SMO算法的工作原理是：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slpha，那么就增大其中一个同时减小另一个。这里所谓“合适”就是指两个alpha必须要符合一定的条件，一是这两个alpha必须要在间隔边界之外，二是这两个alpha还没有进行过区间化处理货不在边界上。
用简化的SMO 解决小数据集问题
Platt SMO算法的完整版需要更多的代码，简化版代码量小，但是执行需要的时间却很长。外循环步骤循环决定了了要优化的最佳alpha，而简化版会跳过这一步。首先在数据集上遍历每个alpha，然后再剩下的alpha集合中随机选择另一个alpha，构建alpha对。这里有个地方要注意，就是我们需要同时改变alpha的值，因为他们满足一个约束 \(\sum{\alpha_i·label^{(i)}} = 0\)。改变一个alpha的值可能会破坏这个约束，所以我们一般会同时改变这两个值。
为此，我们创建一个辅助函数，从一个范围中随机选择一个值。再创建另一个辅助函数，对较大的值进行截断。
import numpy as np
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
while (j==i):
j = int(np.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 = np.mat(dataMatIn); labelMat = np.mat(classLabels).transpose()
b = 0; m,n = np.shape(dataMatrix)
alphas = np.mat(np.zeros((m,1)))
iter = 0
while (iter < maxIter):
alphaPairsChanged = 0
for i in range(m):
fXi = float(np.multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + b
Ei = fXi - float(labelMat[i])
if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or \
((labelMat[i]*Ei > toler) and \
(alphas[i] > 0)):
j = selectJrand(i,m)
fXj = float(np.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])
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
if __name__ == '__main__':
dataArr,labelArr = loadDataSet('testSet.txt')
print(dataArr)
b, alphas = smoSimple(dataArr, labelArr, 0.6, 0.01, 40)
print(b, alphas[alphas>0])
for i in range(100):
if alphas[i]>0.0: print(dataArr[i], labelArr[i])
使用完整 Platt SMO 算法加速优化
import numpy as np
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
class optStruct:
def __init__(self,dataMatIn, classLabels, C, toler):
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = np.shape(dataMatIn)[0]
self.alphas = np.mat(np.zeros((self.m,1)))
self.b = 0
self.eCache = np.mat(np.zeros((self.m,2)))
def calcEk(oS, k):
fXk = float(np.multiply(oS.alphas,oS.labelMat).T*\
(oS.X*oS.X[k,:].T)) + oS.b
Ek = fXk - float(oS.labelMat[k])
return Ek
def selectJ(i, oS, Ei):
maxK = -1; maxDeltaE = 0; Ej = 0
oS.eCache[i] = [1,Ei]
validEcacheList = np.nonzero(oS.eCache[:,0].A)[0]
if (len(validEcacheList)) > 1:
for k in validEcacheList:
if k == i: continue
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k; maxDeltaE = deltaE; Ej = Ek
return maxK, Ej
else:
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
def updateEk(oS, k):
Ek = calcEk(oS, k)
oS.eCache[k] = [1,Ek]
def selectJrand(i,m):
j=i
while (j==i):
j = int(np.random.uniform(0,m))
return j
def clipAlpha(aj,H,L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
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)
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)
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])
updateEk(oS, i)
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 smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):
oS = optStruct(np.mat(dataMatIn),np.mat(classLabels).transpose(),C,toler)
iter = 0
entireSet = True; alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet:
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:
nonBoundIs = np.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
elif (alphaPairsChanged == 0): entireSet = True
print("iteration number: %d" % iter)
return oS.b,oS.alphas
if __name__ == '__main__':
dataArr,labelArr = loadDataSet('testSet.txt')
print(dataArr)
b, alphas = smoP(dataArr, labelArr, 0.6, 0.01, 40)
print(b, alphas[alphas>0])
for i in range(100):
if alphas[i]>0.0: print(dataArr[i], labelArr[i])
对复杂数据集使用核函数
利用核函数可以处理非线性可分数据的情况，径向基函数（radial bisis function）是一种常用的核函数。
使用径向基函数作为核函数
所谓径向基函数，就是基于向量距离进行运算的函数，其高斯版本为：
\[k(x,y) = exp(\frac{-||x-y||^2}{2\sigma^2})
\]
其中，\(\sigma\) 是用户定义的用于确定到达率（reach）或者函数值跌落到0的速度参数
import numpy as np
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 selectJ(i, oS, Ei):
maxK = -1; maxDeltaE = 0; Ej = 0
oS.eCache[i] = [1,Ei]
validEcacheList = np.nonzero(oS.eCache[:,0].A)[0]
if (len(validEcacheList)) > 1:
for k in validEcacheList:
if k == i: continue
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k; maxDeltaE = deltaE; Ej = Ek
return maxK, Ej
else:
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
def selectJrand(i,m):
j=i
while (j==i):
j = int(np.random.uniform(0,m))
return j
def clipAlpha(aj,H,L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def updateEk(oS, k):
Ek = calcEk(oS, k)
oS.eCache[k] = [1,Ek]
def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):
oS = optStruct(np.mat(dataMatIn),np.mat(classLabels).transpose(),C,toler, kTup)
iter = 0
entireSet = True; alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet:
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:
nonBoundIs = np.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
elif (alphaPairsChanged == 0): entireSet = True
print("iteration number: %d" % iter)
return oS.b,oS.alphas
def kernelTrans(X, A, kTup):
m,n = np.shape(X)
K = np.mat(np.zeros((m,1)))
if kTup[0]=='lin': K = X * A.T
elif kTup[0]=='rbf':
for j in range(m):
deltaRow = X[j,:] - A
K[j] = deltaRow*deltaRow.T
K = np.exp(K /(-1*kTup[1]**2))
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):
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = np.shape(dataMatIn)[0]
self.alphas = np.mat(np.zeros((self.m,1)))
self.b = 0
self.eCache = np.mat(np.zeros((self.m,2)))
self.K = np.mat(np.zeros((self.m,self.m)))
for i in range(self.m):
self.K[:,i] = kernelTrans(self.X, self.X[i,:], kTup)
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)
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]
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)
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])
updateEk(oS, i)
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 calcEk(oS, k):
fXk = float(np.multiply(oS.alphas,oS.labelMat).T*oS.K[:,k] + oS.b)
Ek = fXk - float(oS.labelMat[k])
return Ek
def testRbf(k1=1.3):
dataArr,labelArr = loadDataSet('testSetRBF.txt')
b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1))
datMat=np.mat(dataArr); labelMat = np.mat(labelArr).transpose()
svInd=np.nonzero(alphas.A>0)[0]
sVs=datMat[svInd]
labelSV = labelMat[svInd];
print("there are %d Support Vectors" % np.shape(sVs)[0])
m,n = np.shape(datMat)
errorCount = 0
for i in range(m):
kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
predict=kernelEval.T * np.multiply(labelSV,alphas[svInd]) + b
if np.sign(predict)!=np.sign(labelArr[i]): errorCount += 1
print("the training error rate is: %f" % (float(errorCount)/m))
dataArr,labelArr = loadDataSet('testSetRBF2.txt')
errorCount = 0
datMat=np.mat(dataArr); labelMat = np.mat(labelArr).transpose()
m,n = np.shape(datMat)
for i in range(m):
kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
predict=kernelEval.T * np.multiply(labelSV,alphas[svInd]) + b
if np.sign(predict)!=np.sign(labelArr[i]): errorCount += 1
print("the test error rate is: %f" % (float(errorCount)/m))
if __name__ == '__main__':
testRbf()
例：重新回顾手写数字分类问题
import numpy as np
import os
def img2vector(filename):
returnVect = np.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 smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):
oS = optStruct(np.mat(dataMatIn),np.mat(classLabels).transpose(),C,toler, kTup)
iter = 0
entireSet = True; alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet:
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:
nonBoundIs = np.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
elif (alphaPairsChanged == 0): entireSet = True
print("iteration number: %d" % iter)
return oS.b,oS.alphas
def kernelTrans(X, A, kTup):
m,n = np.shape(X)
K = np.mat(np.zeros((m,1)))
if kTup[0]=='lin': K = X * A.T
elif kTup[0]=='rbf':
for j in range(m):
deltaRow = X[j,:] - A
K[j] = deltaRow*deltaRow.T
K = np.exp(K /(-1*kTup[1]**2))
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):
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = np.shape(dataMatIn)[0]
self.alphas = np.mat(np.zeros((self.m,1)))
self.b = 0
self.eCache = np.mat(np.zeros((self.m,2)))
self.K = np.mat(np.zeros((self.m,self.m)))
for i in range(self.m):
self.K[:,i] = kernelTrans(self.X, self.X[i,:], kTup)
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)
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]
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)
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])
updateEk(oS, i)
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 calcEk(oS, k):
fXk = float(np.multiply(oS.alphas,oS.labelMat).T*oS.K[:,k] + oS.b)
Ek = fXk - float(oS.labelMat[k])
return Ek
def selectJ(i, oS, Ei):
maxK = -1; maxDeltaE = 0; Ej = 0
oS.eCache[i] = [1,Ei]
validEcacheList = np.nonzero(oS.eCache[:,0].A)[0]
if (len(validEcacheList)) > 1:
for k in validEcacheList:
if k == i: continue
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k; maxDeltaE = deltaE; Ej = Ek
return maxK, Ej
else:
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
def selectJrand(i,m):
j=i
while (j==i):
j = int(np.random.uniform(0,m))
return j
def clipAlpha(aj,H,L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def updateEk(oS, k):
Ek = calcEk(oS, k)
oS.eCache[k] = [1,Ek]
def loadImages(dirName):
from os import listdir
hwLabels = []
trainingFileList = os.listdir(dirName)
m = len(trainingFileList)
trainingMat = np.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):
m,n = np.shape(X)
K = np.mat(np.zeros((m,1)))
if kTup[0]=='lin': K = X * A.T
elif kTup[0]=='rbf':
for j in range(m):
deltaRow = X[j,:] - A
K[j] = deltaRow*deltaRow.T
K = np.exp(K /(-1*kTup[1]**2))
else: raise NameError('Houston We Have a Problem -- That Kernel is not recognized')
return K
def testDigits(kTup=('rbf', 10)):
dataArr,labelArr = loadImages('digits/trainingDigits')
b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 300, kTup)
datMat=np.mat(dataArr); labelMat = np.mat(labelArr).transpose()
svInd=np.nonzero(alphas.A>0)[0]
sVs=datMat[svInd]
labelSV = labelMat[svInd];
print("there are %d Support Vectors" % np.shape(sVs)[0])
m,n = np.shape(datMat)
errorCount = 0
for i in range(m):
kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
predict=kernelEval.T * np.multiply(labelSV,alphas[svInd]) + b
if np.sign(predict)!=np.sign(labelArr[i]): errorCount += 1
print("the training error rate is: %f" % (float(errorCount)/m))
dataArr,labelArr = loadImages('digits/testDigits')
errorCount = 0
datMat=np.mat(dataArr); labelMat = np.mat(labelArr).transpose()
m,n = np.shape(datMat)
for i in range(m):
kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
predict=kernelEval.T * np.multiply(labelSV,alphas[svInd]) + b
if np.sign(predict)!=np.sign(labelArr[i]): errorCount += 1
print("the test error rate is: %f" % (float(errorCount)/m))
if __name__ == '__main__':
testDigits()
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