机器学习：支持向量机 (Support Vector Machine)算法原理及python实现
文章目录
- 机器学习：支持向量机 (Support Vector Machine)算法原理及python实现
- SVM算法概述
- SVM算法python实现
- 1.创建样本，例中使用二维平面中的两类点来表示两种不同样本
- 2.处理数据集方法，每条数据的前两个数据为坐标，最后为类别
- 3.主方法代码
- 4.完整代码
- 运行结果
- SVM 算法特性
SVM算法概述
支持向量机（Support Vector Machine，SVM） 是一种用于分类问题的监督算法。SVM模型将实例表示为空间中的点，将使用一条直线（超平面）分隔数据点，且是两类数据间隔（边距：超平面与最近的类点之间的距离）最大。只通过几个支持向量就确定了超平面，说明它不在乎细枝末节，所以不容易过拟合，但不能确保一定不会过拟合。可以处理复杂的非线性问题。
如图所示的几个将训练样本分开的超平面可能有很多，应选择”正中间”, 容忍性好, 鲁棒性高, 泛化能力最强。且此个超平面以最大边距将样本分开。
SVM算法python实现
1.创建样本，例中使用二维平面中的两类点来表示两种不同样本
3.542485 1.977398 -1
3.018896 2.556416 -1
7.551510 -1.580030 1
2.114999 -0.004466 -1
8.127113 1.274372 1
7.108772 -0.986906 1
8.610639 2.046708 1
2.326297 0.265213 -1
3.634009 1.730537 -1
0.341367 -0.894998 -1
3.125951 0.293251 -1
2.123252 -0.783563 -1
0.887835 -2.797792 -1
7.139979 -2.329896 1
1.696414 -1.212496 -1
8.117032 0.623493 1
8.497162 -0.266649 1
4.658191 3.507396 -1
8.197181 1.545132 1
1.208047 0.213100 -1
1.928486 -0.321870 -1
2.175808 -0.014527 -1
7.886608 0.461755 1
3.223038 -0.552392 -1
3.628502 2.190585 -1
7.407860 -0.121961 1
7.286357 0.251077 1
2.301095 -0.533988 -1
-0.232542 -0.547690 -1
3.457096 -0.082216 -1
3.023938 -0.057392 -1
8.015003 0.885325 1
8.991748 0.923154 1
7.916831 -1.781735 1
7.616862 -0.217958 1
2.450939 0.744967 -1
7.270337 -2.507834 1
1.749721 -0.961902 -1
1.803111 -0.176349 -1
8.804461 3.044301 1
1.231257 -0.568573 -1
2.074915 1.410550 -1
-0.743036 -1.736103 -1
3.536555 3.964960 -1
8.410143 0.025606 1
7.382988 -0.478764 1
6.960661 -0.245353 1
8.234460 0.701868 1
8.168618 -0.903835 1
1.534187 -0.622492 -1
9.229518 2.066088 1
7.886242 0.191813 1
2.893743 -1.643468 -1
1.870457 -1.040420 -1
5.286862 -2.358286 1
6.080573 0.418886 1
2.544314 1.714165 -1
6.016004 -3.753712 1
0.926310 -0.564359 -1
0.870296 -0.109952 -1
2.369345 1.375695 -1
1.363782 -0.254082 -1
7.279460 -0.189572 1
1.896005 0.515080 -1
8.102154 -0.603875 1
2.529893 0.662657 -1
1.963874 -0.365233 -1
8.132048 0.785914 1
8.245938 0.372366 1
6.543888 0.433164 1
-0.236713 -5.766721 -1
8.112593 0.295839 1
9.803425 1.495167 1
1.497407 -0.552916 -1
1.336267 -1.632889 -1
9.205805 -0.586480 1
1.966279 -1.840439 -1
8.398012 1.584918 1
7.239953 -1.764292 1
7.556201 0.241185 1
9.015509 0.345019 1
8.266085 -0.230977 1
8.545620 2.788799 1
9.295969 1.346332 1
2.404234 0.570278 -1
2.037772 0.021919 -1
1.727631 -0.453143 -1
1.979395 -0.050773 -1
8.092288 -1.372433 1
1.667645 0.239204 -1
9.854303 1.365116 1
7.921057 -1.327587 1
8.500757 1.492372 1
1.339746 -0.291183 -1
3.107511 0.758367 -1
2.609525 0.902979 -1
3.263585 1.367898 -1
2.912122 -0.202359 -1
1.731786 0.589096 -1
2.387003 1.573131 -1
2.处理数据集方法，每条数据的前两个数据为坐标，最后为类别
def loadDataSet(fileName):
dataMat = []
labelMat = []
with open(fileName) as fr:
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
3.主方法代码
def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):
"""
输入：数据集, 类别标签, 常数C, 容错率, 最大循环次数
输出：目标b, 参数alphas
"""
oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)
iterr = 0
entireSet = True
alphaPairsChanged = 0
while (iterr < 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" % (iterr, i, alphaPairsChanged))
iterr += 1
else:
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" % (iterr, i, alphaPairsChanged))
iterr += 1
if entireSet:
entireSet = False
elif (alphaPairsChanged == 0):
entireSet = True
# print("iteration number: %d" % iterr)
return oS.b, oS.alphas
def calcWs(alphas, dataArr, classLabels):
"""
输入：alphas, 数据集, 类别标签
输出：目标w
"""
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
4.完整代码
from numpy import *
import matplotlib.pyplot as plt
import operator
import time
#处理数据集
def loadDataSet(fileName):
dataMat = []
labelMat = []
with open(fileName) as fr:
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(random.uniform(0, m))
return j
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):
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)))
def calcEk(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 selectJ(i, oS, Ei):
maxK = -1
maxDeltaE = 0
Ej = 0
oS.eCache[i] = [1, Ei]
validEcacheList = 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 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 - Ei - 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)):
"""
输入：数据集, 类别标签, 常数C, 容错率, 最大循环次数
输出：目标b, 参数alphas
"""
oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)
iterr = 0
entireSet = True
alphaPairsChanged = 0
while (iterr < 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" % (iterr, i, alphaPairsChanged))
iterr += 1
else:
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" % (iterr, i, alphaPairsChanged))
iterr += 1
if entireSet:
entireSet = False
elif (alphaPairsChanged == 0):
entireSet = True
# print("iteration number: %d" % iterr)
return oS.b, oS.alphas
def calcWs(alphas, dataArr, classLabels):
"""
输入：alphas, 数据集, 类别标签
输出：目标w
"""
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 plotFeature(dataMat, labelMat, weights, b):
dataArr = array(dataMat)
n = shape(dataArr)[0]
xcord1 = []; ycord1 = []
xcord2 = []; ycord2 = []
for i in range(n):
if int(labelMat[i]) == 1:
xcord1.append(dataArr[i, 0])
ycord1.append(dataArr[i, 1])
else:
xcord2.append(dataArr[i, 0])
ycord2.append(dataArr[i, 1])
fig = plt.figure()
ax = fig.add_subplot(111)
ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
ax.scatter(xcord2, ycord2, s=30, c='green')
x = arange(2, 7.0, 0.1)
y = (-b[0, 0] * x) - 10 / linalg.norm(weights)
ax.plot(x, y)
plt.xlabel('X1'); plt.ylabel('X2')
plt.show()
#调用函数
def main():
#数据集初始化
trainDataSet, trainLabel = loadDataSet('SVMSet.txt')
b, alphas = smoP(trainDataSet, trainLabel, 0.6, 0.0001, 40)
ws = calcWs(alphas, trainDataSet, trainLabel)
print("ws = \n", ws)
print("b = \n", b)
plotFeature(trainDataSet, trainLabel, ws, b)
if __name__ == '__main__':
start = time.perf_counter()
main()
end = time.perf_counter()
print('finish all in %s' % str(end - start))
运行结果
可见此次svm分类方法分类效果较好，可以较好地分割样本
SVM 算法特性
1.训练好的模型的算法复杂度是由支持向量的个数决定的，而不是由数据的维度决定的。所以 SVM 不太容易产生 overfitting。
2.SVM 训练出来的模型完全依赖于支持向量，即使训练集里面所有非支持向量的点都被去除，重复训练过程，结果仍然会得到完全一样的模型。
3.一个 SVM 如果训练得出的支持向量个数比较少，那么SVM 训练出的模型比较容易被泛化。
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