一，介绍
在进行分类的时候，大部分数据并不是线性可分的，而是需要通过数据映射，将数据变换到高维度进行分类，这就需要借助核函数来对其进行变换。
我们已经在线性情况下，超平面公式可以写为：
对于线性不可分，我们使用一个非线性映射，将数据映射到特征空间，在特征空间中使用线性学习器，分类函数变形如下：
二，非线性数据高纬度处理
我举一个简单的线性不可分例子，假设有两类，分别满足：
如下图：
像上面这种图形，我们无法用一个线性函数去分类，但是，当我们将其变换到三维后，如下图：
可见红色和蓝色的点被映射到了不同的平面，在更高维空间中是线性可分的（用一个平面去分割）。
三，核函数-径向基函数
常用的核函数有很多，例如，线性核：；多项式核：
，拉普拉斯核：
等等。
我们这里主要介绍径向基函数核（高斯核）：
其中，σ是用户自定义的用于确定到达率(reach)或者说函数值跌落到0的速度参数。如果σ选得很大的话，高次特征上的权重实际上衰减得非常快，所以实际上（数值上近似一下）相当于一个低维的子空间；反过来，如果σ选得很小，则可以将任意的数据映射为线性可分——当然，这并不一定是好事，因为随之而来的可能是非常严重的过拟合问题。不过，总的来说，通过调控参数σ，高斯核实际上具有相当高的灵活性，也是使用最广泛的核函数之一。
四，Python实现
训练集数据：
-0.214824 0.662756 -1.000000 -0.061569 -0.091875 1.000000 0.406933 0.648055 -1.000000 0.223650 0.130142 1.000000 0.231317 0.766906 -1.000000 -0.748800 -0.531637 -1.000000 -0.557789 0.375797 -1.000000 0.207123 -0.019463 1.000000 0.286462 0.719470 -1.000000 0.195300 -0.179039 1.000000 -0.152696 -0.153030 1.000000 0.384471 0.653336 -1.000000 -0.117280 -0.153217 1.000000 -0.238076 0.000583 1.000000 -0.413576 0.145681 1.000000 0.490767 -0.680029 -1.000000 0.199894 -0.199381 1.000000 -0.356048 0.537960 -1.000000 -0.392868 -0.125261 1.000000 0.353588 -0.070617 1.000000 0.020984 0.925720 -1.000000 -0.475167 -0.346247 -1.000000 0.074952 0.042783 1.000000 0.394164 -0.058217 1.000000 0.663418 0.436525 -1.000000 0.402158 0.577744 -1.000000 -0.449349 -0.038074 1.000000 0.619080 -0.088188 -1.000000 0.268066 -0.071621 1.000000 -0.015165 0.359326 1.000000 0.539368 -0.374972 -1.000000 -0.319153 0.629673 -1.000000 0.694424 0.641180 -1.000000 0.079522 0.193198 1.000000 0.253289 -0.285861 1.000000 -0.035558 -0.010086 1.000000 -0.403483 0.474466 -1.000000 -0.034312 0.995685 -1.000000 -0.590657 0.438051 -1.000000 -0.098871 -0.023953 1.000000 -0.250001 0.141621 1.000000 -0.012998 0.525985 -1.000000 0.153738 0.491531 -1.000000 0.388215 -0.656567 -1.000000 0.049008 0.013499 1.000000 0.068286 0.392741 1.000000 0.747800 -0.066630 -1.000000 0.004621 -0.042932 1.000000 -0.701600 0.190983 -1.000000 0.055413 -0.024380 1.000000 0.035398 -0.333682 1.000000 0.211795 0.024689 1.000000 -0.045677 0.172907 1.000000 0.595222 0.209570 -1.000000 0.229465 0.250409 1.000000 -0.089293 0.068198 1.000000 0.384300 -0.176570 1.000000 0.834912 -0.110321 -1.000000 -0.307768 0.503038 -1.000000 -0.777063 -0.348066 -1.000000 0.017390 0.152441 1.000000 -0.293382 -0.139778 1.000000 -0.203272 0.286855 1.000000 0.957812 -0.152444 -1.000000 0.004609 -0.070617 1.000000 -0.755431 0.096711 -1.000000 -0.526487 0.547282 -1.000000 -0.246873 0.833713 -1.000000 0.185639 -0.066162 1.000000 0.851934 0.456603 -1.000000 -0.827912 0.117122 -1.000000 0.233512 -0.106274 1.000000 0.583671 -0.709033 -1.000000 -0.487023 0.625140 -1.000000 -0.448939 0.176725 1.000000 0.155907 -0.166371 1.000000 0.334204 0.381237 -1.000000 0.081536 -0.106212 1.000000 0.227222 0.527437 -1.000000 0.759290 0.330720 -1.000000 0.204177 -0.023516 1.000000 0.577939 0.403784 -1.000000 -0.568534 0.442948 -1.000000 -0.011520 0.021165 1.000000 0.875720 0.422476 -1.000000 0.297885 -0.632874 -1.000000 -0.015821 0.031226 1.000000 0.541359 -0.205969 -1.000000 -0.689946 -0.508674 -1.000000 -0.343049 0.841653 -1.000000 0.523902 -0.436156 -1.000000 0.249281 -0.711840 -1.000000 0.193449 0.574598 -1.000000 -0.257542 -0.753885 -1.000000 -0.021605 0.158080 1.000000 0.601559 -0.727041 -1.000000 -0.791603 0.095651 -1.000000 -0.908298 -0.053376 -1.000000 0.122020 0.850966 -1.000000 -0.725568 -0.292022 -1.000000 测试集数据： 0.676771 -0.486687 -1.000000 0.008473 0.186070 1.000000 -0.727789 0.594062 -1.000000 0.112367 0.287852 1.000000 0.383633 -0.038068 1.000000 -0.927138 -0.032633 -1.000000 -0.842803 -0.423115 -1.000000 -0.003677 -0.367338 1.000000 0.443211 -0.698469 -1.000000 -0.473835 0.005233 1.000000 0.616741 0.590841 -1.000000 0.557463 -0.373461 -1.000000 -0.498535 -0.223231 -1.000000 -0.246744 0.276413 1.000000 -0.761980 -0.244188 -1.000000 0.641594 -0.479861 -1.000000 -0.659140 0.529830 -1.000000 -0.054873 -0.238900 1.000000 -0.089644 -0.244683 1.000000 -0.431576 -0.481538 -1.000000 -0.099535 0.728679 -1.000000 -0.188428 0.156443 1.000000 0.267051 0.318101 1.000000 0.222114 -0.528887 -1.000000 0.030369 0.113317 1.000000 0.392321 0.026089 1.000000 0.298871 -0.915427 -1.000000 -0.034581 -0.133887 1.000000 0.405956 0.206980 1.000000 0.144902 -0.605762 -1.000000 0.274362 -0.401338 1.000000 0.397998 -0.780144 -1.000000 0.037863 0.155137 1.000000 -0.010363 -0.004170 1.000000 0.506519 0.486619 -1.000000 0.000082 -0.020625 1.000000 0.057761 -0.155140 1.000000 0.027748 -0.553763 -1.000000 -0.413363 -0.746830 -1.000000 0.081500 -0.014264 1.000000 0.047137 -0.491271 1.000000 -0.267459 0.024770 1.000000 -0.148288 -0.532471 -1.000000 -0.225559 -0.201622 1.000000 0.772360 -0.518986 -1.000000 -0.440670 0.688739 -1.000000 0.329064 -0.095349 1.000000 0.970170 -0.010671 -1.000000 -0.689447 -0.318722 -1.000000 -0.465493 -0.227468 -1.000000 -0.049370 0.405711 1.000000 -0.166117 0.274807 1.000000 0.054483 0.012643 1.000000 0.021389 0.076125 1.000000 -0.104404 -0.914042 -1.000000 0.294487 0.440886 -1.000000 0.107915 -0.493703 -1.000000 0.076311 0.438860 1.000000 0.370593 -0.728737 -1.000000 0.409890 0.306851 -1.000000 0.285445 0.474399 -1.000000 -0.870134 -0.161685 -1.000000 -0.654144 -0.675129 -1.000000 0.285278 -0.767310 -1.000000 0.049548 -0.000907 1.000000 0.030014 -0.093265 1.000000 -0.128859 0.278865 1.000000 0.307463 0.085667 1.000000 0.023440 0.298638 1.000000 0.053920 0.235344 1.000000 0.059675 0.533339 -1.000000 0.817125 0.016536 -1.000000 -0.108771 0.477254 1.000000 -0.118106 0.017284 1.000000 0.288339 0.195457 1.000000 0.567309 -0.200203 -1.000000 -0.202446 0.409387 1.000000 -0.330769 -0.240797 1.000000 -0.422377 0.480683 -1.000000 -0.295269 0.326017 1.000000 0.261132 0.046478 1.000000 -0.492244 -0.319998 -1.000000 -0.384419 0.099170 1.000000 0.101882 -0.781145 -1.000000 0.234592 -0.383446 1.000000 -0.020478 -0.901833 -1.000000 0.328449 0.186633 1.000000 -0.150059 -0.409158 1.000000 -0.155876 -0.843413 -1.000000 -0.098134 -0.136786 1.000000 0.110575 -0.197205 1.000000 0.219021 0.054347 1.000000 0.030152 0.251682 1.000000 0.033447 -0.122824 1.000000 -0.686225 -0.020779 -1.000000 -0.911211 -0.262011 -1.000000 0.572557 0.377526 -1.000000 -0.073647 -0.519163 -1.000000 -0.281830 -0.797236 -1.000000 -0.555263 0.126232 -1.000000 代码：
import numpy as np
import matplotlib.pyplot as plt
# 读取数据
def loadDataSet(fileName):
dataMat = []; labelMat = []
fr = open(fileName)
for line in fr.readlines(): #逐行读取，滤除空格等
lineArr = line.strip().split('\t')
dataMat.append([float(lineArr[0]), float(lineArr[1])]) #添加数据
labelMat.append(float(lineArr[2])) #添加标签
return dataMat,labelMat
# 核函数 ，本例中计算ΣXj*Xi
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 # 线性核函数
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
# 选择一个不同于i的j值，以获得两个乘子
def selectJrand(i,m):
j=i
while (j==i):
j = int(np.random.uniform(0,m))
return j
# 修剪alpha
# aj - alpha值
# H - alpha上限
# L - alpha下限
def clipAlpha(aj, H, L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
class optStruct:
def __init__(self,dataMatIn, classLabels, C, toler, kTup): # Initialize the structure with the parameters
self.X = dataMatIn # 分类数据
self.labelMat = classLabels # 分类标示
self.C = C # 常数
self.tol = toler # 学习目标
self.m = np.shape(dataMatIn)[0]
self.alphas = np.mat(np.zeros((self.m,1))) # 参数α
self.b = 0
self.eCache = np.mat(np.zeros((self.m,2))) # 存储误差
self.K = np.mat(np.zeros((self.m,self.m))) # 存储ΣXj*Xi
for i in range(self.m):
self.K[:,i] = kernelTrans(self.X, self.X[i,:], kTup)
print(self.K[:,i])
# 计算误差
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
# 选取第二个参数并计算其误差，并返回误差最大的j和Ej
def selectJ(i, oS, Ei):
maxK = -1; maxDeltaE = 0; Ej = 0
oS.eCache[i] = [1,Ei] # 保存误差
# 取出保存了误差的行
validEcacheList = np.nonzero(oS.eCache[:,0].A)[0] # oS.eCache[:,0].A 去第一列转化为数组
if (len(validEcacheList)) > 1:
for k in validEcacheList: # 循环计算获取最大步长
if k == i: continue # 参数相同不计算
Ek = calcEk(oS, k)
deltaE = abs(Ei - Ek)
if (deltaE > maxDeltaE):
maxK = k; maxDeltaE = deltaE; Ej = Ek # 选取步长最大的j
return maxK, Ej
else: # 第一次获取
j = selectJrand(i, oS.m)
Ej = calcEk(oS, j)
return j, Ej
#α更新后重新计算误差
def updateEk(oS, k):
Ek = calcEk(oS, k)
oS.eCache[k] = [1,Ek]
def innerL(i, oS):
Ei = calcEk(oS, i) # 第一步，计算误差
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) # 随机选取第二个参数
alphaIold = oS.alphas[i].copy(); alphaJold = oS.alphas[j].copy(); # 保存αold
# 第二步，计算边界
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] # 第三步，计算学习速率η
if eta >= 0: print ("eta>=0"); return 0
oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta # 第四步：更新αj
oS.alphas[j] = clipAlpha(oS.alphas[j],H,L) # 第五步：修剪αj
updateEk(oS, j) # 更新误差率
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]) # 第六步：更新αi
updateEk(oS, i) # 更新误差率
# 第七步：更新b1,b2
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]
# 第八步：更新b
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(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: #go over all
for i in range(oS.m):
alphaPairsChanged += innerL(i,oS)
print ("全样本遍历:第%d次迭代 样本:%d, alpha优化次数:%d" % (iter,i,alphaPairsChanged))
iter += 1
else:#go over non-bound (railed) alphas
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 #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 = np.mat(dataArr); labelMat = np.mat(classLabels).transpose()
m,n = np.shape(X)
w = np.zeros((n,1))
for i in range(m):
w += np.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))
datMat=np.mat(dataArr); labelMat = np.mat(labelArr).transpose()
svInd=np.nonzero(alphas.A>0)[0]
sVs=datMat[svInd]
labelSV = labelMat[svInd]
print ("支持向量个数： %d " % 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 ("训练集错误率: %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 ("测试集错误率: %f" % (float(errorCount)/m))
if __name__ == '__main__':
testRbf()