支持向量机
1 支持向量
1.1 线性可分
如上图所示，在图A中的两组数据已经分的足够开，因此很容易就可以画一条直线将两组数据点分开。
这种情况下，这组数据就被称为线性可分数据。
将数据集分开的直线称为分割超平面。当数据点都在二维平面上时，分割超平面只是一条直线。当数据集是三维时就是一个平面。如果数据集是N维的，那么就需要N-1维的某某对象对数据进行分割。该对象就被称为超平面也是分类的决策边界。
根据这种方法我们构造一个分类器，如果数据点离决策边界越远，那么其最后的预测结果也就越可信
如上图所示，我们希望找到离分割超平面最近的点，确保他们离分割面的距离尽可能远。此时，点到分割线的距离称为间隔
而支持向量就是离分隔超平面最近的那些点。
2. SMO优化算法
2.1 Platt的SMO算法
SMO，表示序列最小优化。Platt的SMO算法是将大优化问题分解为多个小优化问题来求解的。
SMO算法的目标是求出一系列alpha和b，一旦求出来这些alpha，就很容易计算出权重向量W并得到分隔超平面。
2.1.1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同时减小另一个。
alpha合适的条件：两个alpha必须在间隔边界之外；两个alpha没有进行过区间化处理或者不在边界上
2.2 处理小规模数据集
2.2.1 辅助函数
构建一个辅助函数用于在某个区间范围内随机选择一个整数，同时构造另一个辅助函数，用于在数值太大时对其进行调整。
def loadDataSet(fileName):
dataMat = []
labelMat = []
fr = open('testSet.txt')
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
34
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
上面的代码块中，loadDataSet()函数作用是打开文件并对其进行逐行解析，从而得到每行的类标签和整个数据矩阵。
selectJrand()的两个参数，i是第一个alpha的下标，m是所有alpha的数目。只要函数值不等于输入值，函数就会进行随机选择。
clipAlpha()是辅助函数，用于调整大于H或小于L事务alpha值。
执行以上代码可以得到的结果如下所示：
[-1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, -1.0, 1.0, -1.0, -1.0, -1.0, 1.0, -1.0, -1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, 1.0, 1.0, 1.0, 1.0, -1.0, 1.0, -1.0, -1.0, 1.0, -1.0, -1.0, -1.0, -1.0, 1.0, 1.0, 1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0, -1.0, 1.0, 1.0, -1.0, 1.0, -1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, -1.0, -1.0, 1.0, 1.0, 1.0, -1.0, 1.0, 1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0]
2.2.2 简化版SMO函数
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
以上代码中有五个参数输入，分别表示：数据集、类别标签、常数C、容错率和取消前最大的循环次数。在函数中将多个列表和输入参数转换成Numpy矩阵,达到简化数学处理操作的目的。由于转置了类别标签，因此我们得到的是一个列向量。此时类别标签向量的每行4元素和数据矩阵中的行一一对应。我们也可以通过矩阵dataMatIn的shape属性得到常数m和n。最后构建一个alpha列矩阵，矩阵中元素初始化为0，并建立一个Iter变量。该变量存储的是在没有任何alpha改变的情况下遍历数据集的次数。当该变量达到输入值maxIter时，函数结束运行并推出。
部分运行结果为：
b的值为：
alpha矩阵
alphas[alphas>0] 命令是数组过滤的一个实例，并且只对numpy类型有用
3 完整SMO算法加速优化
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
# 外循环代码
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
运行结果（部分）：
4.手写体识别–课内
代码：
# from math import exp
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()
结果：
5. 人脸检测–课外
代码：
from time import time
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.model_selection import RandomizedSearchCV
from sklearn.datasets import fetch_lfw_people
from sklearn.metrics import classification_report
from sklearn.metrics import ConfusionMatrixDisplay
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA
from sklearn.svm import SVC
from sklearn.utils.fixes import loguniform
lfw_people = fetch_lfw_people(min_faces_per_person=70, resize=0.4)
n_samples, h, w = lfw_people.images.shape
X = lfw_people.data
n_features = X.shape[1]
y = lfw_people.target
target_names = lfw_people.target_names
n_classes = target_names.shape[0]
print("Total dataset size:")
print("n_samples: %d" % n_samples)
print("n_features: %d" % n_features)
print("n_classes: %d" % n_classes)
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.25, random_state=42
)
scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)
n_components = 150
print(
"Extracting the top %d eigenfaces from %d faces" % (n_components, X_train.shape[0])
)
t0 = time()
pca = PCA(n_components=n_components, svd_solver="randomized", whiten=True).fit(X_train)
print("done in %0.3fs" % (time() - t0))
eigenfaces = pca.components_.reshape((n_components, h, w))
print("Projecting the input data on the eigenfaces orthonormal basis")
t0 = time()
X_train_pca = pca.transform(X_train)
X_test_pca = pca.transform(X_test)
print("done in %0.3fs" % (time() - t0))
print("Fitting the classifier to the training set")
t0 = time()
param_grid = {
"C": loguniform(1e3, 1e5),
"gamma": loguniform(1e-4, 1e-1),
}
clf = RandomizedSearchCV(
SVC(kernel="rbf", class_weight="balanced"), param_grid, n_iter=10
)
clf = clf.fit(X_train_pca, y_train)
print("done in %0.3fs" % (time() - t0))
print("Best estimator found by grid search:")
print(clf.best_estimator_)
print("Predicting people's names on the test set")
t0 = time()
y_pred = clf.predict(X_test_pca)
print("done in %0.3fs" % (time() - t0))
print(classification_report(y_test, y_pred, target_names=target_names))
ConfusionMatrixDisplay.from_estimator(
clf, X_test_pca, y_test, display_labels=target_names, xticks_rotation="vertical"
)
plt.tight_layout()
plt.show()
def plot_gallery(images, titles, h, w, n_row=3, n_col=4):
"""Helper function to plot a gallery of portraits"""
plt.figure(figsize=(1.8 * n_col, 2.4 * n_row))
plt.subplots_adjust(bottom=0, left=0.01, right=0.99, top=0.90, hspace=0.35)
for i in range(n_row * n_col):
plt.subplot(n_row, n_col, i + 1)
plt.imshow(images[i].reshape((h, w)), cmap=plt.cm.gray)
plt.title(titles[i], size=12)
plt.xticks(())
plt.yticks(())
def title(y_pred, y_test, target_names, i):
pred_name = target_names[y_pred[i]].rsplit(" ", 1)[-1]
true_name = target_names[y_test[i]].rsplit(" ", 1)[-1]
return "predicted: %s\ntrue: %s" % (pred_name, true_name)
prediction_titles = [
title(y_pred, y_test, target_names, i) for i in range(y_pred.shape[0])
]
plot_gallery(X_test, prediction_titles, h, w)
eigenface_titles = ["eigenface %d" % i for i in range(eigenfaces.shape[0])]
plot_gallery(eigenfaces, eigenface_titles, h, w)
plt.show()
结果：
6.总结
支持向量机是一种分类器，会产生一个二值决策结果。支持向量机的泛化错误率较低，具有良好的学习能力，且学习到的结果具有很好的推广性。
支持向量机试图通过求解一个二次优化问题来解决最大化分类间隔。核方法会将微信从一个低维空间映射到一个高维空间，可以将一个在低维空间中的非线性问题转化为高维空间下的线性问题来解决。
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