SVM kernel approximation with Python

Introduction

A high-performance SVM classifier will likely need thousands of support vectors, and the resulting high complexity of classification prevents their use in many practical applications, with large numbers of training samples or large numbers of features in the input space. To handle this, several approximations to the RBF kernel (and similar kernels) have been devised. Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel.

where Phi is the implicit mapping embedded in the RBF kernel.

Implementation

Scikit-learn has already implemented Fourier transform and Nystroem approximation techniques using RBFSampler and Nystroem classes accordingly.

import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets, svm, pipeline
from sklearn.kernel_approximation import (RBFSampler, Nystroem)
# The digits dataset
digits = datasets.load_digits(n_class=9)

# To apply an classifier on this data, we need to flatten the image, to
# turn the data in a (samples, feature) matrix:
n_samples = len(digits.data)
data = digits.data / 16.
data -= data.mean(axis=0)

# We learn the digits on the first half of the digits
data_train, targets_train = data[:n_samples / 2], digits.target[:n_samples / 2]

# Now predict the value of the digit on the second half:
data_test, targets_test = data[n_samples / 2:], digits.target[n_samples / 2:]

# Create a classifier: a support vector classifier
kernel_svm = svm.SVC(gamma=.2)
linear_svm = svm.LinearSVC()

# create pipeline from kernel approximation
# and linear svm
feature_map_fourier = RBFSampler(gamma=.2, random_state=1)
feature_map_nystroem = Nystroem(gamma=.2, random_state=1)
fourier_approx_svm = pipeline.Pipeline([("feature_map", feature_map_fourier),
                                        ("svm", svm.LinearSVC())])

nystroem_approx_svm = pipeline.Pipeline([("feature_map", feature_map_nystroem),
                                        ("svm", svm.LinearSVC())])

# fit and predict using linear and kernel svm:
kernel_svm.fit(data_train, targets_train)
kernel_svm_score = kernel_svm.score(data_test, targets_test)

linear_svm.fit(data_train, targets_train)
linear_svm_score = linear_svm.score(data_test, targets_test)

sample_sizes = 30 * np.arange(1, 10)
fourier_scores = []
nystroem_scores = []
fourier_times = []
nystroem_times = []

for D in sample_sizes:
    fourier_approx_svm.set_params(feature_map__n_components=D)
    nystroem_approx_svm.set_params(feature_map__n_components=D)
    
    nystroem_approx_svm.fit(data_train, targets_train)
    fourier_approx_svm.fit(data_train, targets_train)

    fourier_score = fourier_approx_svm.score(data_test, targets_test)
    nystroem_score = nystroem_approx_svm.score(data_test, targets_test)
    nystroem_scores.append(nystroem_score)
    fourier_scores.append(fourier_score)

# plot the results:
accuracy = plt.figure(figsize=(8, 8))
accuracy = plt.subplot()
accuracy.plot(sample_sizes, nystroem_scores, label="Nystroem approx. kernel")
accuracy.plot(sample_sizes, fourier_scores, label="Fourier approx. kernel")
accuracy.plot([sample_sizes[0], sample_sizes[-1]],
              [linear_svm_score, linear_svm_score], label="linear svm")
accuracy.plot([sample_sizes[0], sample_sizes[-1]],
              [kernel_svm_score, kernel_svm_score], label="rbf svm")

# legends and labels
accuracy.set_title("Classification accuracy")
accuracy.set_xlim(sample_sizes[0], sample_sizes[-1])
accuracy.set_ylim(np.min(fourier_scores), 1)
accuracy.set_ylabel("Classification accuracy")
accuracy.legend(loc='best')

Take a look at score plot of different techniques, the more features we use the closer the approximation to original kernel. And since the whole point of approximation was to use it on large number of features, this is exactly what we want to achieve.

Conclusion

SVM kernel approximation can dramatically speed up the prediction process with minimal accuracy loss, enabling the usage of SVM in production resource limited systems.

Anomaly detection with Python

Introduction

The goal of anomaly detection is to identify cases that are unusual within data that is seemingly homogeneous. Anomaly detection is an important tool for detecting fraud, network intrusion, and other rare events that may have great significance but are hard to find.

Outliers are cases that are unusual because they fall outside the distribution that is considered normal for the data. The distance from the center of a normal distribution indicates how typical a given point is with respect to the distribution of the data. Each case can be ranked according to the probability that it is either typical or atypical.

Anomaly detection is a form of classification and is implemented as one-class classification, because only one class is represented in the training data. An anomaly detection model predicts whether a data point is typical for a given distribution or not. An atypical data point can be either an outlier or an example of a previously unseen class. Normally, a classification model must be trained on data that includes both examples and counter-examples for each class so that the model can learn to distinguish between them.

Implementation

Scikit-learn implements One-class SVM algorithm, which detects the soft boundary of that set so as to classify new points as belonging to that set or not. The class that implements this is called OneClassSVM.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.font_manager
from sklearn import svm

# Generate train data
X = 0.3 * np.random.randn(100, 2)
X_train = np.r_[X + 2, X - 2]
# Generate some regular novel observations
X = 0.3 * np.random.randn(20, 2)
X_test = np.r_[X + 2, X - 2]
# Generate some abnormal novel observations
X_outliers = np.random.uniform(low=-4, 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_outliers = y_pred_outliers[y_pred_outliers == 1].size

# plot the line, the points, and the nearest vectors to the plane
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.title("Novelty Detection")
plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), 0, 7), cmap=plt.cm.Blues_r)
a = plt.contour(xx, yy, Z, levels=[0], linewidths=2, colors='red')
plt.contourf(xx, yy, Z, levels=[0, Z.max()], colors='orange')

b1 = plt.scatter(X_train[:, 0], X_train[:, 1], c='white')
b2 = plt.scatter(X_test[:, 0], X_test[:, 1], c='green')
c = plt.scatter(X_outliers[:, 0], X_outliers[:, 1], c='red')
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_outliers))
plt.show()

Since we only have one category, normal points are marked by 1 and outliners are marked by -1.

Conclusion

Anomaly detection is applicable in a variety of domains, such as intrusion detection, fraud detection, fault detection, system health monitoring, event detection in sensor networks, and detecting Eco-system disturbances. It is often used in preprocessing to remove anomalous data from the dataset. In supervised learning, removing the anomalous data from the dataset often results in a statistically significant increase in accuracy.