机器学习|机器学习----支持向量机 (Support Vector Machine,SVM)算法原理及python实现机器学习|算法|支持向量机|算

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。
如下图：H1 没有将这两个类分开。但 H2 有，不过只有很小的边距。而 H3 以最大的边距将它们分开了。

文章图片

python实现代码如下：

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, labelMatdef selectJrand(i, m): j = i while (j == i): j = int(random.uniform(0, m)) return jdef clipAlpha(aj, H, L): if aj > H: aj = H if L > aj: aj = L return ajclass 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 Ekdef 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, Ejdef 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 0def 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.alphasdef 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 wdef 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.clock() main() end = time.clock() print('finish all in %s' % str(end - start))

所用数据集 SVMSet.txt 如下：

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

该代码运行三次结果不一，第一个拟合情况较差，第二次才显示为正确的SVM划分，第三次及之后趋于稳定，如下所示：