OCSVM(OneClass SVM) 算法是一种经典的异常检测算法,基本原理与 SVM 类似,与 SVM 关注的二分类问题不同的是,就像它的名字 OneClass SVM 那样,OCSVM 只有一个分类,这也正是异常检测所需要的,我们不关注那些异常的数据点。
如果你不了解 SVM 可以点击这里: 支持向量机(SVM)学习笔记
Our strategy is to map the data into the feature space corresponding to the kernel, and to separate them from the origin with maximum margin.
OCSVM 的基本思想是: 将数据映射到与内核相对应的特征空间,在数据与原点间构建超平面,并且最大化超平面到零点的距离
类似的,我们可以得到最优化问题的目标函数和优化条件:
其中
为松弛变量,
用于调节松弛程度 ,F 为特征空间:
目标函数和优化条件之后,类似的,写出拉格朗日方程,得到对偶问题求解:
另外一方面,有分类函数 f(x)
我们可以通过 sklearn
简单的实现 OCSVM。
>>> from sklearn.svm import OneClassSVM
>>> X = [[0], [0.44], [0.45], [0.46], [1]]
>>> clf = OneClassSVM(gamma='auto').fit(X)
>>> clf.predict(X)
array([-1, 1, 1, 1, -1])
>>> clf.score_samples(X)
array([1.7798..., 2.0547..., 2.0556..., 2.0561..., 1.7332...])
下面是 sklearn
上的一个简单案例:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.font_manager
from sklearn import svm
xx, yy = np.meshgrid(np.linspace(-5, 5, 500), np.linspace(-5, 5, 500))
# Generate train data
X = 0.3 * np.random.randn(100, 2)
X_train = np.r_[X + 2, X - 2]
X_test = np.r_[X + 2, X-2]
# Generate some abnormal novel observations
X_outliers = np.random.uniform(low=0.1, high=4, size=(20, 2))
# fit the model
clf = svm.OneClassSVM(nu=0.1, kernel='rbf', gamma=0.1)
clf.fit(X_train)
y_pred_train = clf.predict(X_train)
y_pred_test = clf.predict(X_test)
y_pred_outliers = clf.predict(X_outliers)
n_error_train = y_pred_train[y_pred_train == -1].size
n_error_test = y_pred_test[y_pred_test == -1].size
n_error_outlier = y_pred_outliers[y_pred_outliers == 1].size
# plot the line , the points, and the nearest vectors to the plane
Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
plt.title("Novelty Detection")
plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), 0, 7), cmap=plt.cm.PuBu)
a = plt.contour(xx, yy, Z, levels=[0, Z.max()], colors='palevioletred')
s =40
b1 = plt.scatter(X_train[:, 0], X_train[:, 1], c='white', s=s, edgecolors='k')
b2 = plt.scatter(X_test[:, 0], X_test[:, 1], c='blueviolet', s=s, edgecolors='k')
c = plt.scatter(X_outliers[:, 0], X_outliers[:, 1], c='gold', s=s, edgecolors='k')
plt.axis('tight')
plt.xlim((-5, 5))
plt.ylim((-5, 5))
plt.legend([a.collections[0], b1, b2, c],
["learned frontier", 'training observations',
"new regular observations", "new abnormal observations"],
loc="upper left",
prop=matplotlib.font_manager.FontProperties(size=11))
plt.xlabel(
"error train: %d/200; errors novel regular: %d/40; errors novel abnormal:%d/40"%(
n_error_train, n_error_test, n_error_outlier) )
plt.show()