1 # Kernal Princlple Component Analysis之实现
2
3 import numpy
4 from matplotlib import pyplot as plt
5
6 numpy.random.seed(0)
7
8
9 def getOriData(type1Size=100, type2Size=100):
10 '''
11 生成特征提取数据集
12 '''
13 theta1 = numpy.random.uniform(0, 2 * numpy.pi, type1Size)
14 theta2 = numpy.random.uniform(0, 2 * numpy.pi, type2Size)
15
16 R1 = numpy.random.uniform(0, 0.3, type1Size).reshape((-1, 1))
17 R2 = numpy.random.uniform(0.7, 1, type2Size).reshape((-1, 1))
18 X1 = R1 * numpy.hstack((numpy.cos(theta1).reshape((-1, 1)), numpy.sin(theta1).reshape((-1, 1))))
19 X2 = R2 * numpy.hstack((numpy.cos(theta1).reshape((-1, 1)), numpy.sin(theta1).reshape((-1, 1))))
20 return X1, X2
21
22
23 class KPCA(object):
24
25 def __init__(self, X, mu=1):
26 '''
27 X: 特征提取数据集, 1行代表1个样本
28 '''
29 self.__X = X # 特征提取数据集
30 self.__mu = mu # gaussian kernel之超参数
31
32 self.__mean = numpy.mean(X, axis=0)
33 self.__A = (X - self.__mean).T
34
35
36 def get_XTra(self, featureCnt=1):
37 '''
38 获取所有样本特征提取结果
39 featureCnt: 待提取特征数
40 '''
41 XTra = list()
42 for xOri in self.__X:
43 xTra = self.get_xTra(xOri, featureCnt)
44 XTra.append(xTra)
45 XTra = numpy.array(XTra)
46 return XTra
47
48
49 def get_xTra(self, xOri, featureCnt=1):
50 '''
51 获取指定样本特征提取结果
52 '''
53 A = self.__A
54 mu = self.__mu
55 mean = self.__mean.reshape((-1, 1))
56 if not hasattr(self, "eigVals"):
57 KAA = self.__get_KAA(A, mu)
58 self.__calc_eigs(KAA)
59 eigVecs = numpy.real(self.eigVecs)
60
61 x = numpy.array(xOri).reshape((-1, 1))
62 xTra = list()
63 for i in range(featureCnt):
64 alpha = eigVecs[:, i:i+1]
65 hVal = self.__get_hVal(x, mean, alpha, A, mu)
66 xTra.append(hVal)
67 xTra = numpy.array(xTra)
68 return xTra
69
70
71 def __get_hVal(self, x, mean, alpha, A, mu):
72 KAx = self.__get_KAx(A, x - mean, mu)
73 hVal = numpy.matmul(alpha.T, KAx)[0, 0]
74 return hVal
75
76
77 def __calc_eigs(self, KAA):
78 oriEigVals, oriEigVecs = numpy.linalg.eig(KAA)
79 seqArr = numpy.argsort(-oriEigVals)
80 self.eigVals = oriEigVals[seqArr]
81 self.eigVecs = oriEigVecs[:, seqArr]
82
83
84 def __get_KAx(self, A, x, mu):
85 KAx = numpy.zeros((A.shape[1], 1))
86 for rowIdx in range(KAx.shape[0]):
87 x1 = A[:, rowIdx:rowIdx+1]
88 val = self.__calc_gaussian(x1, x, mu)
89 KAx[rowIdx, 0] = val
90 return KAx
91
92
93 def __get_KAA(self, A, mu):
94 KAA = numpy.zeros((A.shape[1], A.shape[1]))
95 for rowIdx in range(KAA.shape[0]):
96 for colIdx in range(rowIdx + 1):
97 x1 = A[:, rowIdx:rowIdx+1]
98 x2 = A[:, colIdx:colIdx+1]
99 val = self.__calc_gaussian(x1, x2, mu)
100 KAA[rowIdx, colIdx] = KAA[colIdx, rowIdx] = val
101 return KAA
102
103
104 def __calc_gaussian(self, x1, x2, mu):
105 val = numpy.math.exp(-mu * numpy.linalg.norm(x1 - x2) ** 2)
106 # val = (numpy.sum(x1 * x2) + 1)
107 return val
108
109
110 class KPCAPlot(object):
111
112 @classmethod
113 def show_XTra(cls, X1Ori, X2Ori, mu=1):
114 XOri = numpy.vstack((X1Ori, X2Ori))
115 kpcaObj = KPCA(XOri, mu)
116 XTra = kpcaObj.get_XTra(2)
117
118 X1Tra = XTra[:X1Ori.shape[0], :]
119 X2Tra = XTra[X1Ori.shape[0]:, :]
120
121 fig = plt.figure(figsize=(10, 3))
122 ax1 = plt.subplot(1, 2, 1)
123 ax2 = plt.subplot(1, 2, 2)
124
125 ax1.scatter(X1Ori[:, 0], X1Ori[:, 1], c="red", s=5, label="cluster 0")
126 ax1.scatter(X2Ori[:, 0], X2Ori[:, 1], c="green", s=5, label="cluster 1")
127 ax1.set(title="Ori Data", xlabel="$x_1$", ylabel="$x_2$")
128
129 ax2.scatter(X1Tra[:, 0], X1Tra[:, 1], c="red", s=5, label="cluster 0")
130 ax2.scatter(X2Tra[:, 0], X2Tra[:, 1], c="green", s=5, label="cluster 1")
131 ax2.set(title="Tra Data", xlabel="$c_1$", ylabel="$c_2$")
132
133 ax1.legend()
134 ax2.legend()
135 fig.tight_layout()
136 fig.savefig("./show_XTra.png", dpi=100)
137
138
139
140 if __name__ == "__main__":
141 X1, X2 = getOriData(100, 100)
142 KPCAPlot.show_XTra(X1, X2, 1)
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左侧为特征提取数据集分布情况, 右侧为该数据集经过KPCA映射后取前2个主成分的分布情况. 很显然, 在合适的超参数选取下, 经过KPCA映射后, 原始非线性可分的数据集转变为线性可分.
