Intuition: Separating Normal Data in Feature Space
To understand how One-Class SVM separates normal from novel data, imagine mapping your input data into a high-dimensional feature space using a kernel function, such as the radial basis function (RBF). In this transformed space, the algorithm constructs a flexible boundary that tightly encloses the majority of normal data points, treating anything outside as a potential anomaly. This boundary is not a simple geometric shape like a circle or rectangle, but adapts to the underlying distribution and spread of your data. The SVM learns this boundary by maximizing the margin between the bulk of the data and the origin, effectively distinguishing between what is considered "normal" and what lies outside as "novel".
1234567891011121314151617181920212223242526272829303132333435363738import numpy as np import matplotlib.pyplot as plt from sklearn.svm import OneClassSVM # Generate normal data rng = np.random.RandomState(42) X = 0.3 * rng.randn(100, 2) X_train = np.r_[X + 2, X - 2] # Generate some novel (outlier) points X_outliers = rng.uniform(low=-4, high=4, size=(20, 2)) # Fit the One-Class SVM clf = OneClassSVM(kernel="rbf", gamma=0.5, nu=0.1) clf.fit(X_train) # Create a grid for plotting xx, yy = np.meshgrid(np.linspace(-5, 5, 500), np.linspace(-5, 5, 500)) Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) plt.figure(figsize=(8, 8)) # Plot decision boundary and margin plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), 0, 7), cmap=plt.cm.PuBu) a = plt.contour(xx, yy, Z, levels=[0], linewidths=2, colors='darkred') # Plot training points plt.scatter(X_train[:, 0], X_train[:, 1], c='white', s=20, edgecolor='k', label="Normal") # Plot outliers plt.scatter(X_outliers[:, 0], X_outliers[:, 1], c='gold', s=20, edgecolor='k', label="Novel") # Highlight support vectors plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1], s=100, facecolors='none', edgecolors='navy', label="Support Vectors") plt.legend() plt.title("One-Class SVM with RBF Kernel: Decision Boundary and Support Vectors") plt.xlim((-5, 5)) plt.ylim((-5, 5)) plt.xlabel("Feature 1") plt.ylabel("Feature 2") plt.show()
The boundary produced by One-Class SVM is highly flexible due to the RBF kernel. It can adapt to different data shapes, wrapping tightly around clusters or stretching to enclose elongated distributions. This adaptability means the SVM is sensitive to the true structure of your data, but also to parameter choices like gamma and nu that control how closely the boundary follows the data.
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To understand how One-Class SVM separates normal from novel data, imagine mapping your input data into a high-dimensional feature space using a kernel function, such as the radial basis function (RBF). In this transformed space, the algorithm constructs a flexible boundary that tightly encloses the majority of normal data points, treating anything outside as a potential anomaly. This boundary is not a simple geometric shape like a circle or rectangle, but adapts to the underlying distribution and spread of your data. The SVM learns this boundary by maximizing the margin between the bulk of the data and the origin, effectively distinguishing between what is considered "normal" and what lies outside as "novel".
1234567891011121314151617181920212223242526272829303132333435363738import numpy as np import matplotlib.pyplot as plt from sklearn.svm import OneClassSVM # Generate normal data rng = np.random.RandomState(42) X = 0.3 * rng.randn(100, 2) X_train = np.r_[X + 2, X - 2] # Generate some novel (outlier) points X_outliers = rng.uniform(low=-4, high=4, size=(20, 2)) # Fit the One-Class SVM clf = OneClassSVM(kernel="rbf", gamma=0.5, nu=0.1) clf.fit(X_train) # Create a grid for plotting xx, yy = np.meshgrid(np.linspace(-5, 5, 500), np.linspace(-5, 5, 500)) Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) plt.figure(figsize=(8, 8)) # Plot decision boundary and margin plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), 0, 7), cmap=plt.cm.PuBu) a = plt.contour(xx, yy, Z, levels=[0], linewidths=2, colors='darkred') # Plot training points plt.scatter(X_train[:, 0], X_train[:, 1], c='white', s=20, edgecolor='k', label="Normal") # Plot outliers plt.scatter(X_outliers[:, 0], X_outliers[:, 1], c='gold', s=20, edgecolor='k', label="Novel") # Highlight support vectors plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1], s=100, facecolors='none', edgecolors='navy', label="Support Vectors") plt.legend() plt.title("One-Class SVM with RBF Kernel: Decision Boundary and Support Vectors") plt.xlim((-5, 5)) plt.ylim((-5, 5)) plt.xlabel("Feature 1") plt.ylabel("Feature 2") plt.show()
The boundary produced by One-Class SVM is highly flexible due to the RBF kernel. It can adapt to different data shapes, wrapping tightly around clusters or stretching to enclose elongated distributions. This adaptability means the SVM is sensitive to the true structure of your data, but also to parameter choices like gamma and nu that control how closely the boundary follows the data.
Merci pour vos commentaires !
