Python实现支持向量机(SVM)

之前学SVM做的笔记,这里整理下。

Theory

这里先空下,等之后整理补充上来。

Code

代码大部分来自Github,当中有自己写的一些注释。

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import numpy as np
from numpy import linalg
import cvxopt
import cvxopt.solvers

def linear_kernel(x1, x2):
return np.dot(x1, x2)

def polynomial_kernel(x, y, p=3):
return (1 + np.dot(x, y)) ** p

def gaussian_kernel(x, y, sigma=5.0):
return np.exp(-linalg.norm(x-y)**2 / (2 * (sigma ** 2)))

class SVM(object):

def __init__(self, kernel=linear_kernel, C=None):
self.kernel = kernel
self.C = C
if self.C is not None: self.C = float(self.C)

def fit(self, X, y):
n_samples, n_features = X.shape

# Gram matrix
K = np.zeros((n_samples, n_samples))
for i in range(n_samples):
for j in range(n_samples):
K[i,j] = self.kernel(X[i], X[j])

P = cvxopt.matrix(np.outer(y,y) * K)
q = cvxopt.matrix(np.ones(n_samples) * -1)
A = cvxopt.matrix(y, (1,n_samples))
b = cvxopt.matrix(0.0)

if self.C is None:
G = cvxopt.matrix(np.diag(np.ones(n_samples) * -1))
h = cvxopt.matrix(np.zeros(n_samples))
else:
tmp1 = np.diag(np.ones(n_samples) * -1)
tmp2 = np.identity(n_samples)
G = cvxopt.matrix(np.vstack((tmp1, tmp2)))
tmp1 = np.zeros(n_samples)
tmp2 = np.ones(n_samples) * self.C
h = cvxopt.matrix(np.hstack((tmp1, tmp2)))

# solve QP problem
solution = cvxopt.solvers.qp(P, q, G, h, A, b)

# Lagrange multipliers
a = np.ravel(solution['x'])

# Support vectors have non zero lagrange multipliers
'''
这里a>1e-5就将其视为非零
'''
sv = a > 1e-5
ind = np.arange(len(a))[sv]
self.a = a[sv]
self.sv = X[sv]
self.sv_y = y[sv]
print("%d support vectors out of %d points" % (len(self.a), n_samples))

# Intercept
'''
这里相当于对所有的支持向量求得的b取平均值
'''
self.b = 0
for n in range(len(self.a)):
self.b += self.sv_y[n]
self.b -= np.sum(self.a * self.sv_y * K[ind[n],sv])
self.b /= len(self.a)

# Weight vector
if self.kernel == linear_kernel:
self.w = np.zeros(n_features)
for n in range(len(self.a)):
# linear_kernel相当于在原空间,故计算w不用映射到feature space
self.w += self.a[n] * self.sv_y[n] * self.sv[n]
else:
self.w = None

def project(self, X):
# w有值,即kernel function 是 linear_kernel,直接计算即可
if self.w is not None:
return np.dot(X, self.w) + self.b
# w is None --> 不是linear_kernel,w要重新计算
# 这里没有去计算新的w(非线性情况不用计算w),直接用kernel matrix计算预测结果
else:
y_predict = np.zeros(len(X))
for i in range(len(X)):
s = 0
for a, sv_y, sv in zip(self.a, self.sv_y, self.sv):
s += a * sv_y * self.kernel(X[i], sv)
y_predict[i] = s
return y_predict + self.b

def predict(self, X):
return np.sign(self.project(X))

if __name__ == "__main__":
import pylab as pl

def gen_lin_separable_data():
# generate training data in the 2-d case
mean1 = np.array([0, 2])
mean2 = np.array([2, 0])
cov = np.array([[0.8, 0.6], [0.6, 0.8]])
X1 = np.random.multivariate_normal(mean1, cov, 100)
y1 = np.ones(len(X1))
X2 = np.random.multivariate_normal(mean2, cov, 100)
y2 = np.ones(len(X2)) * -1
return X1, y1, X2, y2

def gen_non_lin_separable_data():
mean1 = [-1, 2]
mean2 = [1, -1]
mean3 = [4, -4]
mean4 = [-4, 4]
cov = [[1.0,0.8], [0.8, 1.0]]
X1 = np.random.multivariate_normal(mean1, cov, 50)
X1 = np.vstack((X1, np.random.multivariate_normal(mean3, cov, 50)))
y1 = np.ones(len(X1))
X2 = np.random.multivariate_normal(mean2, cov, 50)
X2 = np.vstack((X2, np.random.multivariate_normal(mean4, cov, 50)))
y2 = np.ones(len(X2)) * -1
return X1, y1, X2, y2

def gen_lin_separable_overlap_data():
# generate training data in the 2-d case
mean1 = np.array([0, 2])
mean2 = np.array([2, 0])
cov = np.array([[1.5, 1.0], [1.0, 1.5]])
X1 = np.random.multivariate_normal(mean1, cov, 100)
y1 = np.ones(len(X1))
X2 = np.random.multivariate_normal(mean2, cov, 100)
y2 = np.ones(len(X2)) * -1
return X1, y1, X2, y2

def split_train(X1, y1, X2, y2):
X1_train = X1[:90]
y1_train = y1[:90]
X2_train = X2[:90]
y2_train = y2[:90]
X_train = np.vstack((X1_train, X2_train))
y_train = np.hstack((y1_train, y2_train))
return X_train, y_train

def split_test(X1, y1, X2, y2):
X1_test = X1[90:]
y1_test = y1[90:]
X2_test = X2[90:]
y2_test = y2[90:]
X_test = np.vstack((X1_test, X2_test))
y_test = np.hstack((y1_test, y2_test))
return X_test, y_test


# 仅仅在Linears使用此函数作图,即w存在时
def plot_margin(X1_train, X2_train, clf):
def f(x, w, b, c=0):
# given x, return y such that [x,y] in on the line
# w.x + b = c
return (-w[0] * x - b + c) / w[1]

pl.plot(X1_train[:,0], X1_train[:,1], "ro")
pl.plot(X2_train[:,0], X2_train[:,1], "bo")
pl.scatter(clf.sv[:,0], clf.sv[:,1], s=100, c="g")

# w.x + b = 0
a0 = -4; a1 = f(a0, clf.w, clf.b)
b0 = 4; b1 = f(b0, clf.w, clf.b)
pl.plot([a0,b0], [a1,b1], "k")

# w.x + b = 1
a0 = -4; a1 = f(a0, clf.w, clf.b, 1)
b0 = 4; b1 = f(b0, clf.w, clf.b, 1)
pl.plot([a0,b0], [a1,b1], "k--")

# w.x + b = -1
a0 = -4; a1 = f(a0, clf.w, clf.b, -1)
b0 = 4; b1 = f(b0, clf.w, clf.b, -1)
pl.plot([a0,b0], [a1,b1], "k--")

pl.axis("tight")
pl.show()



def plot_contour(X1_train, X2_train, clf):
# 作training sample数据点的图
pl.plot(X1_train[:,0], X1_train[:,1], "ro")
pl.plot(X2_train[:,0], X2_train[:,1], "bo")
# 做support vectors 的图
pl.scatter(clf.sv[:,0], clf.sv[:,1], s=100, c="g")

X1, X2 = np.meshgrid(np.linspace(-6,6,50), np.linspace(-6,6,50))
X = np.array([[x1, x2] for x1, x2 in zip(np.ravel(X1), np.ravel(X2))])
Z = clf.project(X).reshape(X1.shape)
# pl.contour做等值线图
pl.contour(X1, X2, Z, [0.0], colors='k', linewidths=1, origin='lower')
pl.contour(X1, X2, Z + 1, [0.0], colors='grey', linewidths=1, origin='lower')
pl.contour(X1, X2, Z - 1, [0.0], colors='grey', linewidths=1, origin='lower')

pl.axis("tight")
pl.show()

def test_linear():
X1, y1, X2, y2 = gen_lin_separable_data()
X_train, y_train = split_train(X1, y1, X2, y2)
X_test, y_test = split_test(X1, y1, X2, y2)

clf = SVM()
clf.fit(X_train, y_train)

y_predict = clf.predict(X_test)
correct = np.sum(y_predict == y_test)
print("%d out of %d predictions correct" % (correct, len(y_predict)))

plot_margin(X_train[y_train==1], X_train[y_train==-1], clf)

def test_non_linear():
X1, y1, X2, y2 = gen_non_lin_separable_data()
X_train, y_train = split_train(X1, y1, X2, y2)
X_test, y_test = split_test(X1, y1, X2, y2)

clf = SVM(gaussian_kernel)
clf.fit(X_train, y_train)

y_predict = clf.predict(X_test)
correct = np.sum(y_predict == y_test)
print("%d out of %d predictions correct" % (correct, len(y_predict)))

plot_contour(X_train[y_train==1], X_train[y_train==-1], clf)

def test_soft():
X1, y1, X2, y2 = gen_lin_separable_overlap_data()
X_train, y_train = split_train(X1, y1, X2, y2)
X_test, y_test = split_test(X1, y1, X2, y2)

clf = SVM(C=0.1)
clf.fit(X_train, y_train)

y_predict = clf.predict(X_test)
correct = np.sum(y_predict == y_test)
print("%d out of %d predictions correct" % (correct, len(y_predict)))

plot_contour(X_train[y_train==1], X_train[y_train==-1], clf)


test_soft()

输出:

Supplement

关于用cvxopt求解QP问题的使用,可参考优矿

击蒙御寇