1.4. Support Vector Machines

    The advantages of support vector machines are:

    The disadvantages of support vector machines include:

    • If the number of features is much greater than the number ofsamples, avoid over-fitting in choosing Kernel functions and regularizationterm is crucial.

    • SVMs do not directly provide probability estimates, these arecalculated using an expensive five-fold cross-validation(see , below).

    The support vector machines in scikit-learn support both dense( and convertible to that by numpy.asarray) andsparse (any scipy.sparse) sample vectors as input. However, to usean SVM to make predictions for sparse data, it must have been fit on suchdata. For optimal performance, use C-ordered numpy.ndarray (dense) orscipy.sparse.csr_matrix (sparse) with dtype=float64.

    SVC, and LinearSVC are classescapable of performing multi-class classification on a dataset.

    SVC and are similar methods, but acceptslightly different sets of parameters and have different mathematicalformulations (see section Mathematical formulation). On theother hand, is another implementation of SupportVector Classification for the case of a linear kernel. Note thatLinearSVC does not accept keyword kernel, as this isassumed to be linear. It also lacks some of the members of and NuSVC, like support_.

    As other classifiers, , NuSVC and take as input two arrays: an array X of size [n_samples,n_features] holding the training samples, and an array y of class labels(strings or integers), size [n_samples]:

    >>>

    After being fitted, the model can then be used to predict new values:

    >>>

    1. >>> clf.predict([[2., 2.]])
    2. array([1])

    SVMs decision function depends on some subset of the training data,called the support vectors. Some properties of these support vectorscan be found in members supportvectors, support_ andn_support:

    >>>

    1. >>> # get support vectors
    2. >>> clf.support_vectors_
    3. array([[0., 0.],
    4.        [1., 1.]])
    5. >>> # get indices of support vectors
    6. >>> clf.support_
    7. >>> # get number of support vectors for each class
    8. >>> clf.n_support_
    9. array([1, 1]...)

    SVC and implement the “one-against-one”approach (Knerr et al., 1990) for multi- class classification. Ifn_class is the number of classes, then n_class * (n_class - 1) / 2classifiers are constructed and each one trains data from two classes.To provide a consistent interface with other classifiers, thedecision_function_shape option allows to monotically transform the results of the“one-against-one” classifiers to a decision function of shape (n_samples,n_classes).

    >>>

    On the other hand, LinearSVC implements “one-vs-the-rest”multi-class strategy, thus training n_class models. If there are onlytwo classes, only one model is trained:

    >>>

    1. >>> lin_clf = svm.LinearSVC()
    2. >>> lin_clf.fit(X, Y)
    3. LinearSVC()
    4. >>> dec = lin_clf.decision_function([[1]])
    5. >>> dec.shape[1]
    6. 4

    See for a complete description ofthe decision function.

    Note that the LinearSVC also implements an alternative multi-classstrategy, the so-called multi-class SVM formulated by Crammer and Singer, byusing the option multi_class='crammer_singer'. This method is consistent,which is not true for one-vs-rest classification.In practice, one-vs-rest classification is usually preferred, since the resultsare mostly similar, but the runtime is significantly less.

    For “one-vs-rest” the attributes coef and intercepthave the shape [n_class, n_features] and [n_class] respectively.Each row of the coefficients corresponds to one of the n_class many“one-vs-rest” classifiers and similar for the intercepts, in theorder of the “one” class.

    In the case of “one-vs-one” SVC, the layout of the attributesis a little more involved. In the case of having a linear kernel, theattributes coef and intercept have the shape[n_class (n_class - 1) / 2, n_features] and[n_class (n_class - 1) / 2] respectively. This is similar to thelayout for described above, with each row now correspondingto a binary classifier. The order for classes0 to n is “0 vs 1”, “0 vs 2” , … “0 vs n”, “1 vs 2”, “1 vs 3”, “1 vs n”, . .. “n-1 vs n”.

    The shape of dualcoef is [n_class-1, n_SV] witha somewhat hard to grasp layout.The columns correspond to the support vectors involved in anyof the n_class * (n_class - 1) / 2 “one-vs-one” classifiers.Each of the support vectors is used in n_class - 1 classifiers.The n_class - 1 entries in each row correspond to the dual coefficientsfor these classifiers.

    This might be made more clear by an example:

    Consider a three class problem with class 0 having three support vectors

    1.4. Support Vector Machines - 图2 and class 1 and 2 having two support vectors and1.4. Support Vector Machines - 图4 respectively. For eachsupport vector, there are two dual coefficients. Let’s callthe coefficient of support vector1.4. Support Vector Machines - 图6 in the classifier betweenclasses and1.4. Support Vector Machines - 图8

    .Then dualcoef looks like this:

    1.4.1.2. Scores and probabilities

    The decision_function method of and NuSVC givesper-class scores for each sample (or a single score per sample in the binarycase). When the constructor option probability is set to True,class membership probability estimates (from the methods predict_proba andpredict_log_proba) are enabled. In the binary case, the probabilities arecalibrated using Platt scaling: logistic regression on the SVM’s scores,fit by an additional cross-validation on the training data.In the multiclass case, this is extended as per Wu et al. (2004).

    Needless to say, the cross-validation involved in Platt scalingis an expensive operation for large datasets.In addition, the probability estimates may be inconsistent with the scores,in the sense that the “argmax” of the scoresmay not be the argmax of the probabilities.(E.g., in binary classification,a sample may be labeled by predict as belonging to a classthat has probability <½ according to predict_proba.)Platt’s method is also known to have theoretical issues.If confidence scores are required, but these do not have to be probabilities,then it is advisable to set probability=Falseand use instead of predict_proba.

    Please note that when decision_function_shape='ovr' and n_classes > 2,unlike decision_function, the predict method does not try to break tiesby default. You can set break_ties=True for the output of predict to bethe same as np.argmax(clf.decision_function(…), axis=1), otherwise thefirst class among the tied classes will always be returned; but have in mindthat it comes with a computational cost.

    References:

    In problems where it is desired to give more importance to certainclasses or certain individual samples keywords class_weight andsample_weight can be used.

    SVC (but not ) implement a keywordclass_weight in the fit method. It’s a dictionary of the form{class_label : value}, where value is a floating point number > 0that sets the parameter C of class class_label to C * value.

    , NuSVC, , NuSVR, ,LinearSVR and implement also weights forindividual samples in method fit through keyword sample_weight. Similarto class_weight, these set the parameter C for the i-th example toC * sample_weight[i].

    ../_images/sphx_glr_plot_weighted_samples_0011.png

    Examples:

    1.4.2. Regression

    The method of Support Vector Classification can be extended to solveregression problems. This method is called Support Vector Regression.

    The model produced by support vector classification (as describedabove) depends only on a subset of the training data, because the costfunction for building the model does not care about training pointsthat lie beyond the margin. Analogously, the model produced by SupportVector Regression depends only on a subset of the training data,because the cost function for building the model ignores any trainingdata close to the model prediction.

    There are three different implementations of Support Vector Regression:, NuSVR and . LinearSVRprovides a faster implementation than but only considerslinear kernels, while NuSVR implements a slightly differentformulation than and LinearSVR. See for further details.

    As with classification classes, the fit method will take asargument vectors X, y, only that in this case y is expected to havefloating point values instead of integer values:

    >>>

    1. >>> from sklearn import svm
    2. >>> X = [[0, 0], [2, 2]]
    3. >>> y = [0.5, 2.5]
    4. >>> clf = svm.SVR()
    5. >>> clf.fit(X, y)
    6. SVR()
    7. >>> clf.predict([[1, 1]])
    8. array([1.5])

    Examples:

    The class implements a One-Class SVM which is used inoutlier detection.

    See Novelty and Outlier Detection for the description and usage of OneClassSVM.

    1.4.4. Complexity

    Support Vector Machines are powerful tools, but their compute andstorage requirements increase rapidly with the number of trainingvectors. The core of an SVM is a quadratic programming problem (QP),separating support vectors from the rest of the training data. The QPsolver used by this libsvm-based implementation scales between

    and1.4. Support Vector Machines - 图28 depending on how efficientlythe cache is used in practice (dataset dependent). If the datais very sparse should be replaced by the average numberof non-zero features in a sample vector.

    Also note that for the linear case, the algorithm used inLinearSVC by the implementation is much moreefficient than its libsvm-based counterpart and canscale almost linearly to millions of samples and/or features.

    References:

    1.4.6. Kernel functions

    The kernel function can be any of the following:

    • linear:

      1.4. Support Vector Machines - 图30
      .

    • polynomial:

      .
      1.4. Support Vector Machines - 图32
      is specified by keyword degree,
      by coef0.

    • rbf:

      1.4. Support Vector Machines - 图34
      .
      isspecified by keyword gamma, must be greater than 0.

    • sigmoid (

      1.4. Support Vector Machines - 图36
      ),where
      is specified by coef0.

    Different kernels are specified by keyword kernel at initialization:

    >>>

    1.4.6.1. Custom Kernels

    You can define your own kernels by either giving the kernel as apython function or by precomputing the Gram matrix.

    Classifiers with custom kernels behave the same way as any otherclassifiers, except that:

    1.4.6.1.1. Using Python functions as kernels

    You can also use your own defined kernels by passing a function to thekeyword kernel in the constructor.

    Your kernel must take as arguments two matrices of shape(n_samples_1, n_features), (n_samples_2, n_features)and return a kernel matrix of shape (n_samples_1, n_samples_2).

    The following code defines a linear kernel and creates a classifierinstance that will use that kernel:

    1. >>> import numpy as np
    2. >>> from sklearn import svm
    3. >>> def my_kernel(X, Y):
    4. ...     return np.dot(X, Y.T)
    5. ...
    6. >>> clf = svm.SVC(kernel=my_kernel)

    Examples:

    1.4.6.1.2. Using the Gram matrix

    Set kernel='precomputed' and pass the Gram matrix instead of X in the fitmethod. At the moment, the kernel values between all training vectors and thetest vectors must be provided.

    >>>

    1. >>> import numpy as np
    2. >>> from sklearn import svm
    3. >>> X = np.array([[0, 0], [1, 1]])
    4. >>> y = [0, 1]
    5. >>> clf = svm.SVC(kernel='precomputed')
    6. >>> # linear kernel computation
    7. >>> gram = np.dot(X, X.T)
    8. >>> clf.fit(gram, y)
    9. SVC(kernel='precomputed')
    10. >>> # predict on training examples
    11. >>> clf.predict(gram)
    12. array([0, 1])

    1.4.6.1.3. Parameters of the RBF Kernel

    When training an SVM with the Radial Basis Function (RBF) kernel, twoparameters must be considered: C and gamma. The parameter C,common to all SVM kernels, trades off misclassification of training examplesagainst simplicity of the decision surface. A low C makes the decisionsurface smooth, while a high C aims at classifying all training examplescorrectly. gamma defines how much influence a single training example has.The larger gamma is, the closer other examples must be to be affected.

    Proper choice of C and gamma is critical to the SVM’s performance. Oneis advised to use withC and gamma spaced exponentially far apart to choose good values.

    Examples:

    A support vector machine constructs a hyper-plane or set of hyper-planesin a high or infinite dimensional space, which can be used forclassification, regression or other tasks. Intuitively, a goodseparation is achieved by the hyper-plane that has the largest distanceto the nearest training data points of any class (so-called functionalmargin), since in general the larger the margin the lower thegeneralization error of the classifier.

    Given training vectors

    , i=1,…, n, in two classes, and avector1.4. Support Vector Machines - 图40, SVC solves the following primal problem:

    Its dual is

    1.4. Support Vector Machines - 图42

    where

    is the vector of all ones,1.4. Support Vector Machines - 图44 is the upper bound, is an1.4. Support Vector Machines - 图46 by positive semidefinite matrix,1.4. Support Vector Machines - 图48, whereis the kernel. Here training vectors are implicitly mapped into a higher(maybe infinite) dimensional space by the function1.4. Support Vector Machines - 图50.

    The decision function is:

    Note

    While SVM models derived from libsvm and use C asregularization parameter, most other estimators use alpha. The exactequivalence between the amount of regularization of two models depends onthe exact objective function optimized by the model. For example, when theestimator used is sklearn.linear_model.Ridge regression,the relation between them is given as

    1.4. Support Vector Machines - 图52.

    This parameters can be accessed through the members dualcoefwhich holds the product

    , supportvectors whichholds the support vectors, and intercept_ which holds the independentterm1.4. Support Vector Machines - 图54 :

    References:

    1.4.7.2. NuSVC

    We introduce a new parameter

    which controls the number ofsupport vectors and training errors. The parameter1.4. Support Vector Machines - 图56 is an upper bound on the fraction of training errors and a lowerbound of the fraction of support vectors.

    It can be shown that the

    -SVC formulation is a reparameterizationof the1.4. Support Vector Machines - 图58-SVC and therefore mathematically equivalent.

    Given training vectors

    , i=1,…, n, and avector1.4. Support Vector Machines - 图60

    -SVR solves the following primal problem:

    1.4. Support Vector Machines - 图62

    Its dual is

    where

    1.4. Support Vector Machines - 图64 is the vector of all ones, is the upper bound,1.4. Support Vector Machines - 图66 is an by1.4. Support Vector Machines - 图68 positive semidefinite matrix,is the kernel. Here training vectors are implicitly mapped into a higher(maybe infinite) dimensional space by the function1.4. Support Vector Machines - 图70.

    The decision function is:

    These parameters can be accessed through the members dualcoefwhich holds the difference

    1.4. Support Vector Machines - 图72, whichholds the support vectors, and intercept_ which holds the independentterm

    References:

    • ,Alex J. Smola, Bernhard Schölkopf - Statistics and Computing archiveVolume 14 Issue 3, August 2004, p. 199-222.

    1.4.8. Implementation details

    Internally, we use and liblinear to handle allcomputations. These libraries are wrapped using C and Cython.

    References:

    For a description of the implementation and details of the algorithmsused, please refer to