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更新完 6.SVM支持向量机
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jiangzhonglian
@@ -101,7 +101,7 @@ def calcEk(oS, k):
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k 具体的某一行
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Returns:
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Ek 预测结果与真实结果比对，计算误差Ek
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"""
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fXk = float(multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b)
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Ek = fXk - float(oS.labelMat[k])
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@@ -142,7 +142,7 @@ def selectJ(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej
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maxK = -1
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maxDeltaE = 0
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Ej = 0
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# # 首先将输入值Ei在缓存中设置成为有效的。这里的有效意味着它已经计算好了。
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# 首先将输入值Ei在缓存中设置成为有效的。这里的有效意味着它已经计算好了。
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oS.eCache[i] = [1, Ei]
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# print 'oS.eCache[%s]=%s' % (i, oS.eCache[i])
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@@ -515,8 +515,8 @@ if __name__ == "__main__":
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# plotfig_SVM(dataArr, labelArr, ws, b, alphas)
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# # 有核函数的测试
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# testRbf(1)
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testRbf(0.8)
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# 项目实战
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# 示例：手写识别问题回顾
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testDigits(('rbf', 20))
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# testDigits(('rbf', 20))
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@@ -11,6 +11,18 @@ from numpy import *
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import matplotlib.pyplot as plt
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class optStruct:
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def __init__(self, dataMatIn, classLabels, C, toler): # Initialize the structure with the parameters
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self.X = dataMatIn
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self.labelMat = classLabels
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self.C = C
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self.tol = toler
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self.m = shape(dataMatIn)[0]
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self.alphas = mat(zeros((self.m, 1)))
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self.b = 0
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self.eCache = mat(zeros((self.m, 2))) # first column is valid flag
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def loadDataSet(fileName):
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"""loadDataSet（对文件进行逐行解析，从而得到第行的类标签和整个数据矩阵）
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@@ -61,6 +73,222 @@ def clipAlpha(aj, H, L):
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return aj
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def calcEk(oS, k):
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"""calcEk（求 Ek误差：预测值-真实值的差）
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该过程在完整版的SMO算法中陪出现次数较多，因此将其单独作为一个方法
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Args:
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oS optStruct对象
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k 具体的某一行
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Returns:
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Ek 预测结果与真实结果比对，计算误差Ek
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"""
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fXk = float(multiply(oS.alphas, oS.labelMat).T * (oS.X * oS.X[k, :].T)) + oS.b
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Ek = fXk - float(oS.labelMat[k])
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return Ek
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def selectJ(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej
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"""selectJ（返回最优的j和Ej）
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内循环的启发式方法。
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选择第二个(内循环)alpha的alpha值
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这里的目标是选择合适的第二个alpha值以保证每次优化中采用最大步长。
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该函数的误差与第一个alpha值Ei和下标i有关。
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Args:
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i 具体的第i一行
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oS optStruct对象
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Ei 预测结果与真实结果比对，计算误差Ei
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Returns:
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j 随机选出的第j一行
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Ej 预测结果与真实结果比对，计算误差Ej
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"""
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maxK = -1
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maxDeltaE = 0
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Ej = 0
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# 首先将输入值Ei在缓存中设置成为有效的。这里的有效意味着它已经计算好了。
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oS.eCache[i] = [1, Ei]
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# print 'oS.eCache[%s]=%s' % (i, oS.eCache[i])
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# print 'oS.eCache[:, 0].A=%s' % oS.eCache[:, 0].A.T
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# """
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# # 返回非0的：行列值
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# nonzero(oS.eCache[:, 0].A)= (
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# 行： array([ 0, 2, 4, 5, 8, 10, 17, 18, 20, 21, 23, 25, 26, 29, 30, 39, 46,52, 54, 55, 62, 69, 70, 76, 79, 82, 94, 97]),
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# 列： array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0])
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# )
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# """
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# print 'nonzero(oS.eCache[:, 0].A)=', nonzero(oS.eCache[:, 0].A)
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# # 取行的list
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# print 'nonzero(oS.eCache[:, 0].A)[0]=', nonzero(oS.eCache[:, 0].A)[0]
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# 非零E值的行的list列表，所对应的alpha值
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validEcacheList = nonzero(oS.eCache[:, 0].A)[0]
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if (len(validEcacheList)) > 1:
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for k in validEcacheList: # 在所有的值上进行循环，并选择其中使得改变最大的那个值
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if k == i:
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continue # don't calc for i, waste of time
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# 求 Ek误差：预测值-真实值的差
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Ek = calcEk(oS, k)
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deltaE = abs(Ei - Ek)
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if (deltaE > maxDeltaE):
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maxK = k
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maxDeltaE = deltaE
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Ej = Ek
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return maxK, Ej
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else: # 如果是第一次循环，则随机选择一个alpha值
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j = selectJrand(i, oS.m)
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# 求 Ek误差：预测值-真实值的差
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Ej = calcEk(oS, j)
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return j, Ej
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def updateEk(oS, k): # after any alpha has changed update the new value in the cache
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"""updateEk（计算误差值并存入缓存中。）
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在对alpha值进行优化之后会用到这个值。
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Args:
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oS optStruct对象
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k 某一列的行号
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"""
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# 求 误差：预测值-真实值的差
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Ek = calcEk(oS, k)
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oS.eCache[k] = [1, Ek]
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def innerL(i, oS):
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"""innerL
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内循环代码
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Args:
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i 具体的某一行
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oS optStruct对象
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Returns:
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0 找不到最优的值
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1 找到了最优的值，并且oS.Cache到缓存中
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"""
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# 求 Ek误差：预测值-真实值的差
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Ei = calcEk(oS, i)
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# 约束条件 (KKT条件是解决最优化问题的时用到的一种方法。我们这里提到的最优化问题通常是指对于给定的某一函数，求其在指定作用域上的全局最小值。)
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# 0<=alphas[i]<=C，但由于0和C是边界值，我们无法进行优化，因为需要增加一个alphas和降低一个alphas。
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# 表示发生错误的概率：labelMat[i]*Ei 如果超出了 toler， 才需要优化。至于正负号，我们考虑绝对值就对了。
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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)):
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# 选择最大的误差对应的j进行优化。效果更明显
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j, Ej = selectJ(i, oS, Ei)
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alphaIold = oS.alphas[i].copy()
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alphaJold = oS.alphas[j].copy()
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# L和H用于将alphas[j]调整到0-C之间。如果L==H，就不做任何改变，直接return 0
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if (oS.labelMat[i] != oS.labelMat[j]):
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L = max(0, oS.alphas[j] - oS.alphas[i])
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H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
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else:
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L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
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H = min(oS.C, oS.alphas[j] + oS.alphas[i])
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if L == H:
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print("L==H")
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return 0
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# eta是alphas[j]的最优修改量，如果eta==0，需要退出for循环的当前迭代过程
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# 如果ETA为0，那么计算新的alphas[j]就比较麻烦了, 为什么呢？ 因为2个值一样。
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# 2ab <= a^2 + b^2
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eta = 2.0 * oS.X[i, :] * oS.X[j, :].T - oS.X[i, :] * oS.X[i, :].T - oS.X[j, :] * oS.X[j, :].T
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if eta >= 0:
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print("eta>=0")
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return 0
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# 计算出一个新的alphas[j]值
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oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
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# 并使用辅助函数，以及L和H对其进行调整
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oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
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# 更新误差缓存
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updateEk(oS, j)
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# 检查alpha[j]是否只是轻微的改变，如果是的话，就退出for循环。
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if (abs(oS.alphas[j] - alphaJold) < 0.00001):
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print("j not moving enough")
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return 0
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# 然后alphas[i]和alphas[j]同样进行改变，虽然改变的大小一样，但是改变的方向正好相反
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oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j])
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# 更新误差缓存
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updateEk(oS, i)
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# 在对alpha[i], alpha[j] 进行优化之后，给这两个alpha值设置一个常数b。
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# w= Σ[1~n] ai*yi*xi => b = yj Σ[1~n] ai*yi(xi*xj)
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# 所以： b1 - b = (y1-y) - Σ[1~n] yi*(a1-a)*(xi*x1)
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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
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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
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if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]):
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oS.b = b1
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elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]):
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oS.b = b2
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else:
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oS.b = (b1 + b2) / 2.0
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return 1
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else:
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return 0
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def smoP(dataMatIn, classLabels, C, toler, maxIter):
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"""
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完整SMO算法外循环，与smoSimple有些类似，但这里的循环退出条件更多一些
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Args:
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dataMatIn 数据集
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classLabels 类别标签
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C 松弛变量(常量值)，允许有些数据点可以处于分隔面的错误一侧。
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控制最大化间隔和保证大部分的函数间隔小于1.0这两个目标的权重。
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可以通过调节该参数达到不同的结果。
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toler 容错率
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maxIter 退出前最大的循环次数
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Returns:
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b 模型的常量值
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alphas 拉格朗日乘子
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"""
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# 创建一个 optStruct 对象
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oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)
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iter = 0
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entireSet = True
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alphaPairsChanged = 0
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# 循环遍历：循环maxIter次 并且 （alphaPairsChanged存在可以改变 or 所有行遍历一遍）
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while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
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alphaPairsChanged = 0
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# 当entireSet=true or 非边界alpha对没有了；就开始寻找 alpha对，然后决定是否要进行else。
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if entireSet:
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# 在数据集上遍历所有可能的alpha
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for i in range(oS.m):
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# 是否存在alpha对，存在就+1
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alphaPairsChanged += innerL(i, oS)
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print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
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iter += 1
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# 对已存在 alpha对，选出非边界的alpha值，进行优化。
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else:
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# 遍历所有的非边界alpha值，也就是不在边界0或C上的值。
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nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
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for i in nonBoundIs:
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alphaPairsChanged += innerL(i, oS)
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print("non-bound, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
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iter += 1
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# 如果找到alpha对，就优化非边界alpha值，否则，就重新进行寻找，如果寻找一遍 遍历所有的行还是没找到，就退出循环。
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if entireSet:
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entireSet = False # toggle entire set loop
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elif (alphaPairsChanged == 0):
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entireSet = True
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print("iteration number: %d" % iter)
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return oS.b, oS.alphas
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def calcWs(alphas, dataArr, classLabels):
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"""
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基于alpha计算w值
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@@ -81,126 +309,6 @@ def calcWs(alphas, dataArr, classLabels):
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return w
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'''
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#######********************************
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Non-Kernel VErsions below
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#######********************************
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'''
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class optStruct:
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def __init__(self, dataMatIn, classLabels, C, toler): # Initialize the structure with the parameters
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self.X = dataMatIn
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self.labelMat = classLabels
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self.C = C
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self.tol = toler
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self.m = shape(dataMatIn)[0]
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self.alphas = mat(zeros((self.m, 1)))
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self.b = 0
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self.eCache = mat(zeros((self.m, 2))) # first column is valid flag
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def calcEk(oS, k):
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fXk = float(multiply(oS.alphas, oS.labelMat).T * (oS.X * oS.X[k, :].T)) + oS.b
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Ek = fXk - float(oS.labelMat[k])
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return Ek
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def selectJ(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej
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maxK = -1
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maxDeltaE = 0
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Ej = 0
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oS.eCache[i] = [1, Ei] # set valid #choose the alpha that gives the maximum delta E
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validEcacheList = nonzero(oS.eCache[:, 0].A)[0]
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if (len(validEcacheList)) > 1:
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for k in validEcacheList: # loop through valid Ecache values and find the one that maximizes delta E
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if k == i: continue # don't calc for i, waste of time
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Ek = calcEk(oS, k)
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deltaE = abs(Ei - Ek)
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if (deltaE > maxDeltaE):
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maxK = k
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maxDeltaE = deltaE
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Ej = Ek
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return maxK, Ej
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else: # in this case (first time around) we don't have any valid eCache values
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j = selectJrand(i, oS.m)
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Ej = calcEk(oS, j)
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return j, Ej
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def updateEk(oS, k): # after any alpha has changed update the new value in the cache
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Ek = calcEk(oS, k)
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oS.eCache[k] = [1, Ek]
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def innerL(i, oS):
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Ei = calcEk(oS, i)
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if ((oS.labelMat[i] * Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or (
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(oS.labelMat[i] * Ei > oS.tol) and (oS.alphas[i] > 0)):
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j, Ej = selectJ(i, oS, Ei) # this has been changed from selectJrand
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alphaIold = oS.alphas[i].copy()
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alphaJold = oS.alphas[j].copy()
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if (oS.labelMat[i] != oS.labelMat[j]):
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L = max(0, oS.alphas[j] - oS.alphas[i])
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H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
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else:
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L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
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H = min(oS.C, oS.alphas[j] + oS.alphas[i])
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if L == H:
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print("L==H")
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return 0
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eta = 2.0 * oS.X[i, :] * oS.X[j, :].T - oS.X[i, :] * oS.X[i, :].T - oS.X[j, :] * oS.X[j, :].T
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if eta >= 0:
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print("eta>=0")
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return 0
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oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
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oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
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updateEk(oS, j) # added this for the Ecache
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if (abs(oS.alphas[j] - alphaJold) < 0.00001):
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print("j not moving enough")
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return 0
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oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j]) # update i by the same amount as j
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updateEk(oS, i) # added this for the Ecache #the update is in the oppostie direction
|
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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): # full Platt SMO
|
||||
oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)
|
||||
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 plotfig_SVM(xArr, yArr, ws, b, alphas):
|
||||
"""
|
||||
参考地址：
|
||||
|
||||
@@ -145,7 +145,7 @@ def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
|
||||
# 然后alphas[i]和alphas[j]同样进行改变，虽然改变的大小一样，但是改变的方向正好相反
|
||||
alphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j])
|
||||
# 在对alpha[i], alpha[j] 进行优化之后，给这两个alpha值设置一个常数b。
|
||||
# w= Σ[1~n] ai*yi*xi => b = yi- Σ[1~n] ai*yi(xi*xj)
|
||||
# w= Σ[1~n] ai*yi*xi => b = yj- Σ[1~n] ai*yi(xi*xj)
|
||||
# 所以： b1 - b = (y1-y) - Σ[1~n] yi*(a1-a)*(xi*x1)
|
||||
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
|
||||
@@ -158,6 +158,7 @@ def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
|
||||
alphaPairsChanged += 1
|
||||
print("iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
|
||||
# 在for循环外，检查alpha值是否做了更新，如果在更新则将iter设为0后继续运行程序
|
||||
# 知道更新完毕后，iter次循环无变化，才推出循环。
|
||||
if (alphaPairsChanged == 0):
|
||||
iter += 1
|
||||
else:
|
||||
|
||||
Reference in New Issue
Block a user