OCSVM 学习笔记

前言

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

OCSVM 简介

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:

min⁡w∈F,ξ∈Rn,ρ∈R12∥w∥2+1vn∑i=1nξi−ρs.t.(w⋅Φ(xi))≥ρ−ξi,ξi≥0\begin{aligned} &\min_{w \in F, \xi\in R^n, \rho \in R} \frac{1}{2}\Vert w\Vert^2 + \frac{1}{vn}\sum_{i=1}^n\xi_i - \rho\\ &s.t.\quad (w \cdot \Phi(x_i)) \geq \rho - \xi_i , \quad \xi_i\geq 0 \end{aligned}

其中 ξi\xi_i 为松弛变量，v∈(0,1)v\in (0,1) 用于调节松弛程度 ，F 为特征空间：

k(x,y)=(Φ(x)⋅Φ(y))k(x, y) = (\Phi (x) \cdot \Phi (y))

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:

min⁡α12∑ijαiαjk(xi,xj)s.t.0≤αi≤1vn,∑iαi=1\begin{aligned} &\min_{\alpha} \frac {1}{2} \sum_{ij} \alpha_i \alpha_j k (x_i , x_j) \\ &s.t.\quad 0 \leq \alpha_i \leq \frac{1}{vn} , \sum_{i}\alpha_i = 1 \end{aligned}

另外一方面，有分类函数 f(x)

f(x)=sgn⁡((w⋅Φ(x))−ρ)=sgn⁡(∑i=1nαik(xi,x)−ρ)f(x) = \operatorname {sgn} ((w \cdot \Phi (x) ) - \rho ) = \operatorname { s g n } ( \sum _ { i = 1 } ^ { n } \alpha _ { i } k ( x_i , x) - \rho )

算法示例

我们可以通过 sklearn 简单的实现 OCSVM。

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>>> 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:

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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()

参考文献

  • B. Schölkopf, R. Williamson, A. Smola, J. Shawe-Taylor, and J. Platt, “Support vector method for novelty detection,” in Advances in Neural Information Processing Systems, vol. 12. Denver, CO, USA: MIT Press, Aug./Sep. 1999, pp. 582–588.