1.4. Support Vector Machines

Support vector machines (SVMs) are a set of supervised learningmethods used for classification,regression and outliers detection.

The advantages of support vector machines are:

• Effective in high dimensional spaces.

• Still effective in cases where number of dimensions is greaterthan the number of samples.

• Uses a subset of training points in the decision function (calledsupport vectors), so it is also memory efficient.

• Versatile: different Kernel functions can bespecified for the decision function. Common kernels areprovided, but it is also possible to specify custom kernels.

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 Scores and probabilities, below).

The support vector machines in scikit-learn support both dense(numpy.ndarray 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.

1.4.1. Classification

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

SVC and NuSVC are similar methods, but acceptslightly different sets of parameters and have different mathematicalformulations (see section Mathematical formulation). On theother hand, LinearSVC 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 ofSVC and NuSVC, like support_.

As other classifiers, SVC, NuSVC andLinearSVC 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]:

>>>

>>> from sklearn import svm
>>> X = [[0, 0], [1, 1]]
>>> y = [0, 1]
>>> clf = svm.SVC()
>>> clf.fit(X, y)
SVC()

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

>>>

>>> clf.predict([[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:

>>>

>>> # get support vectors
>>> clf.support_vectors_
array([[0., 0.],
       [1., 1.]])
>>> # get indices of support vectors
>>> clf.support_
array([0, 1]...)
>>> # get number of support vectors for each class
>>> clf.n_support_
array([1, 1]...)

1.4.1.1. Multi-class classification

SVC and NuSVC 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).

>>>

>>> X = [[0], [1], [2], [3]]
>>> Y = [0, 1, 2, 3]
>>> clf = svm.SVC(decision_function_shape='ovo')
>>> clf.fit(X, Y)
SVC(decision_function_shape='ovo')
>>> dec = clf.decision_function([[1]])
>>> dec.shape[1] # 4 classes: 4*3/2 = 6
6
>>> clf.decision_function_shape = "ovr"
>>> dec = clf.decision_function([[1]])
>>> dec.shape[1] # 4 classes
4

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:

>>>

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

See Mathematical formulation 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” LinearSVC 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 LinearSVC 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

and class 1 and 2 having two support vectors and respectively. For eachsupport vector, there are two dual coefficients. Let’s callthe coefficient of support vector in the classifier betweenclasses and

.Then dualcoef looks like this:

		Coefficientsfor SVs of class 0
		Coefficientsfor SVs of class 1
		Coefficientsfor SVs of class 2

1.4.1.2. Scores and probabilities

The decision_function method of SVC 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 decision_function 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:

• Wu, Lin and Weng,“Probability estimates for multi-class classification by pairwise coupling”,JMLR 5:975-1005, 2004.

• Platt“Probabilistic outputs for SVMs and comparisons to regularized likelihood methods”.

1.4.1.3. Unbalanced problems

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 NuSVC) 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.

SVC, NuSVC, SVR, NuSVR, LinearSVC,LinearSVR and OneClassSVM 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].

Examples:

• Plot different SVM classifiers in the iris dataset,

• SVM: Maximum margin separating hyperplane,

• SVM: Separating hyperplane for unbalanced classes

• SVM-Anova: SVM with univariate feature selection,

• Non-linear SVM

• SVM: Weighted samples,

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:SVR, NuSVR and LinearSVR. LinearSVRprovides a faster implementation than SVR but only considerslinear kernels, while NuSVR implements a slightly differentformulation than SVR and LinearSVR. SeeImplementation details 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:

>>>

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

Examples:

• Support Vector Regression (SVR) using linear and non-linear kernels

1.4.3. Density estimation, novelty detection

The class OneClassSVM 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

and depending on how efficientlythe libsvm 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 liblinear implementation is much moreefficient than its libsvm-based SVC counterpart and canscale almost linearly to millions of samples and/or features.

1.4.5. Tips on Practical Use

• Avoiding data copy: For SVC, SVR, NuSVC andNuSVR, if the data passed to certain methods is not C-orderedcontiguous, and double precision, it will be copied before calling theunderlying C implementation. You can check whether a given numpy array isC-contiguous by inspecting its flags attribute.

  For LinearSVC (and LogisticRegression) any input passed as a numpyarray will be copied and converted to the liblinear internal sparse datarepresentation (double precision floats and int32 indices of non-zerocomponents). If you want to fit a large-scale linear classifier withoutcopying a dense numpy C-contiguous double precision array as input wesuggest to use the SGDClassifier class instead. The objectivefunction can be configured to be almost the same as the LinearSVCmodel.

• Kernel cache size: For SVC, SVR, NuSVC andNuSVR, the size of the kernel cache has a strong impact on runtimes for larger problems. If you have enough RAM available, it isrecommended to set cache_size to a higher value than the default of200(MB), such as 500(MB) or 1000(MB).

• Setting C: C is 1 by default and it’s a reasonable defaultchoice. If you have a lot of noisy observations you should decrease it.It corresponds to regularize more the estimation.

  LinearSVC and LinearSVR are less sensitive to C whenit becomes large, and prediction results stop improving after a certainthreshold. Meanwhile, larger C values will take more time to train,sometimes up to 10 times longer, as shown by Fan et al. (2008)

• Support Vector Machine algorithms are not scale invariant, so itis highly recommended to scale your data. For example, scale eachattribute on the input vector X to [0,1] or [-1,+1], or standardize itto have mean 0 and variance 1. Note that the same scaling must beapplied to the test vector to obtain meaningful results. See sectionPreprocessing data for more details on scaling and normalization.

• Parameter nu in NuSVC/OneClassSVM/NuSVRapproximates the fraction of training errors and support vectors.

• In SVC, if data for classification are unbalanced (e.g. manypositive and few negative), set class_weight='balanced' and/or trydifferent penalty parameters C.

• Randomness of the underlying implementations: The underlyingimplementations of SVC and NuSVC use a random numbergenerator only to shuffle the data for probability estimation (whenprobability is set to True). This randomness can be controlledwith the random_state parameter. If probability is set to Falsethese estimators are not random and random_state has no effect on theresults. The underlying OneClassSVM implementation is similar tothe ones of SVC and NuSVC. As no probability estimationis provided for OneClassSVM, it is not random.

  The underlying LinearSVC implementation uses a random numbergenerator to select features when fitting the model with a dual coordinatedescent (i.e when dual is set to True). It is thus not uncommon,to have slightly different results for the same input data. If thathappens, try with a smaller tol parameter. This randomness can also becontrolled with the random_state parameter. When dual isset to False the underlying implementation of LinearSVC isnot random and random_state has no effect on the results.

• Using L1 penalization as provided by LinearSVC(loss='l2', penalty='l1',dual=False) yields a sparse solution, i.e. only a subset of featureweights is different from zero and contribute to the decision function.Increasing C yields a more complex model (more feature are selected).The C value that yields a “null” model (all weights equal to zero) canbe calculated using l1_min_c.

References:

• Fan, Rong-En, et al.,“LIBLINEAR: A library for large linear classification.”,Journal of machine learning research 9.Aug (2008): 1871-1874.

1.4.6. Kernel functions

The kernel function can be any of the following:

• linear:

  
  .

• polynomial:

  
  .
  
  is specified by keyword degree,
  
  by coef0.

• rbf:

  
  .
  
  isspecified by keyword gamma, must be greater than 0.

• sigmoid (

  
  ),where
  
  is specified by coef0.

Different kernels are specified by keyword kernel at initialization:

>>>

>>> linear_svc = svm.SVC(kernel='linear')
>>> linear_svc.kernel
'linear'
>>> rbf_svc = svm.SVC(kernel='rbf')
>>> rbf_svc.kernel
'rbf'

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:

• Field supportvectors is now empty, only indices of supportvectors are stored in support_

• A reference (and not a copy) of the first argument in the fit()method is stored for future reference. If that array changes between theuse of fit() and predict() you will have unexpected results.

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:

>>>

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

Examples:

• SVM with custom kernel.

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.

>>>

>>> import numpy as np
>>> from sklearn import svm
>>> X = np.array([[0, 0], [1, 1]])
>>> y = [0, 1]
>>> clf = svm.SVC(kernel='precomputed')
>>> # linear kernel computation
>>> gram = np.dot(X, X.T)
>>> clf.fit(gram, y)
SVC(kernel='precomputed')
>>> # predict on training examples
>>> clf.predict(gram)
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 sklearn.model_selection.GridSearchCV withC and gamma spaced exponentially far apart to choose good values.

Examples:

• RBF SVM parameters

1.4.7. Mathematical formulation

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.

1.4.7.1. SVC

Given training vectors

, i=1,…, n, in two classes, and avector, SVC solves the following primal problem:

Its dual is

where

is the vector of all ones, is the upper bound, is an by positive semidefinite matrix,, whereis the kernel. Here training vectors are implicitly mapped into a higher(maybe infinite) dimensional space by the function.

The decision function is:

Note

While SVM models derived from libsvm and liblinear 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

.

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

, supportvectors whichholds the support vectors, and intercept_ which holds the independentterm :

References:

• “Automatic Capacity Tuning of Very Large VC-dimension Classifiers”,I. Guyon, B. Boser, V. Vapnik - Advances in neural informationprocessing 1993.

• “Support-vector networks”,C. Cortes, V. Vapnik - Machine Learning, 20, 273-297 (1995).

1.4.7.2. NuSVC

We introduce a new parameter

which controls the number ofsupport vectors and training errors. The parameter 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 the-SVC and therefore mathematically equivalent.

1.4.7.3. SVR

Given training vectors

, i=1,…, n, and avector

-SVR solves the following primal problem:

Its dual is

where

is the vector of all ones, is the upper bound, is an by positive semidefinite matrix,is the kernel. Here training vectors are implicitly mapped into a higher(maybe infinite) dimensional space by the function.

The decision function is:

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

, supportvectors whichholds the support vectors, and intercept_ which holds the independentterm

References:

• “A Tutorial on Support Vector Regression”,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 libsvm 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

• LIBSVM: A Library for Support Vector Machines.

• LIBLINEAR – A Library for Large Linear Classification.