**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) or`scipy.sparse.csr_matrix`

(sparse) with `dtype=float64`

.

## 1.4.1. Classification#

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

SVC and NuSVC are similar methods, but accept slightlydifferent sets of parameters and have different mathematical formulations (seesection Mathematical formulation). On the other hand,LinearSVC is another (faster) implementation of Support VectorClassification for the case of a linear kernel. It alsolacks some of the attributes of SVC and NuSVC, like`support_`

. LinearSVC uses `squared_hinge`

loss and due to itsimplementation in `liblinear`

it also regularizes the intercept, if considered.This effect can however be reduced by carefully fine tuning its`intercept_scaling`

parameter, which allows the intercept term to have adifferent regularization behavior compared to the other features. Theclassification results and score can therefore differ from the other twoclassifiers.

As other classifiers, SVC, NuSVC andLinearSVC take as input two arrays: an array `X`

of shape`(n_samples, n_features)`

holding the training samples, and an array `y`

ofclass labels (strings or integers), of shape `(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 (detailed in the Mathematical formulation)depends on some subset of the training data, called the support vectors. Someproperties of these support vectors can be found in attributes`support_vectors_`

, `support_`

and `n_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]...)

Examples

SVM: Maximum margin separating hyperplane

SVM-Anova: SVM with univariate feature selection

### 1.4.1.1. Multi-class classification#

SVC and NuSVC implement the “one-versus-one”approach for multi-class classification. In total,`n_classes * (n_classes - 1) / 2`

classifiers are constructed and each one trains data from two classes.To provide a consistent interface with other classifiers, the`decision_function_shape`

option allows to monotonically transform theresults of the “one-versus-one” classifiers to a “one-vs-rest” decisionfunction 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] # 6 classes: 4*3/2 = 66>>> clf.decision_function_shape = "ovr">>> dec = clf.decision_function([[1]])>>> dec.shape[1] # 4 classes4

On the other hand, LinearSVC implements “one-vs-the-rest”multi-class strategy, thus training `n_classes`

models.

>>> 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.

## Details on multi-class strategies#

Note that the LinearSVC also implements an alternative multi-classstrategy, the so-called multi-class SVM formulated by Crammer and Singer[16], by using the option `multi_class='crammer_singer'`

. In practice,one-vs-rest classification is usually preferred, since the results are mostlysimilar, but the runtime is significantly less.

For “one-vs-rest” LinearSVC the attributes `coef_`

and `intercept_`

have the shape `(n_classes, n_features)`

and `(n_classes,)`

respectively.Each row of the coefficients corresponds to one of the `n_classes`

“one-vs-rest” classifiers and similar for the intercepts, in theorder of the “one” class.

In the case of “one-vs-one” SVC and NuSVC, the layout ofthe attributes is a little more involved. In the case of a linearkernel, the attributes `coef_`

and `intercept_`

have the shape`(n_classes * (n_classes - 1) / 2, n_features)`

and `(n_classes *(n_classes - 1) / 2)`

respectively. This is similar to the layout forLinearSVC 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 `dual_coef_`

is `(n_classes-1, n_SV)`

witha somewhat hard to grasp layout.The columns correspond to the support vectors involved in anyof the `n_classes * (n_classes - 1) / 2`

“one-vs-one” classifiers.Each support vector `v`

has a dual coefficient in each of the`n_classes - 1`

classifiers comparing the class of `v`

against another class.Note that some, but not all, of these dual coefficients, may be zero.The `n_classes - 1`

entries in each column are these dual coefficients,ordered by the opposing class.

This might be clearer with an example: consider a three class problem withclass 0 having three support vectors\(v^{0}_0, v^{1}_0, v^{2}_0\) and class 1 and 2 having two support vectors\(v^{0}_1, v^{1}_1\) and \(v^{0}_2, v^{1}_2\) respectively. For eachsupport vector \(v^{j}_i\), there are two dual coefficients. Let’s callthe coefficient of support vector \(v^{j}_i\) in the classifier betweenclasses \(i\) and \(k\) \(\alpha^{j}_{i,k}\).Then `dual_coef_`

looks like this:

\(\alpha^{0}_{0,1}\) | \(\alpha^{1}_{0,1}\) | \(\alpha^{2}_{0,1}\) | \(\alpha^{0}_{1,0}\) | \(\alpha^{1}_{1,0}\) | \(\alpha^{0}_{2,0}\) | \(\alpha^{1}_{2,0}\) |

\(\alpha^{0}_{0,2}\) | \(\alpha^{1}_{0,2}\) | \(\alpha^{2}_{0,2}\) | \(\alpha^{0}_{1,2}\) | \(\alpha^{1}_{1,2}\) | \(\alpha^{0}_{2,1}\) | \(\alpha^{1}_{2,1}\) |

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

Examples

Plot different SVM classifiers in the iris dataset

### 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`

and`predict_log_proba`

) are enabled. In the binary case, the probabilities arecalibrated using Platt scaling [9]: 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 [10].

Note

The same probability calibration procedure is available for all estimatorsvia the CalibratedClassifierCV (seeProbability calibration). In the case of SVC and NuSVC, thisprocedure is builtin in libsvm which is used under the hood, so it doesnot rely on scikit-learn’sCalibratedClassifierCV.

The cross-validation involved in Platt scalingis an expensive operation for large datasets.In addition, the probability estimates may be inconsistent with the scores:

the “argmax” of the scores may not be the argmax of the probabilities

in binary classification, a sample may be labeled by

`predict`

asbelonging to the positive class even if the output of`predict_proba`

isless than 0.5; and similarly, it could be labeled as negative even if theoutput of`predict_proba`

is more than 0.5.

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=False`

and 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. SeeSVM Tie Breaking Example for an example ontie breaking.

### 1.4.1.3. Unbalanced problems#

In problems where it is desired to give more importance to certainclasses or certain individual samples, the parameters `class_weight`

and`sample_weight`

can be used.

SVC (but not NuSVC) implements the parameter`class_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`

.The figure below illustrates the decision boundary of an unbalanced problem,with and without weight correction.

SVC, NuSVC, SVR, NuSVR, LinearSVC,LinearSVR and OneClassSVM implement also weights forindividual samples in the `fit`

method through the `sample_weight`

parameter.Similar to `class_weight`

, this sets the parameter `C`

for the i-thexample to `C * sample_weight[i]`

, which will encourage the classifier toget these samples right. The figure below illustrates the effect of sampleweighting on the decision boundary. The size of the circles is proportionalto the sample weights:

Examples

SVM: Separating hyperplane for unbalanced classes

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 ignores samples whose prediction is close to theirtarget.

There are three different implementations of Support Vector Regression:SVR, NuSVR and LinearSVR. LinearSVRprovides a faster implementation than SVR but only considers thelinear kernel, while NuSVR implements a slightly different formulationthan SVR and LinearSVR. Due to its implementation in`liblinear`

LinearSVR also regularizes the intercept, if considered.This effect can however be reduced by carefully fine tuning its`intercept_scaling`

parameter, which allows the intercept term to have adifferent regularization behavior compared to the other features. Theclassification results and score can therefore differ from the other twoclassifiers. See Implementation 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]>>> regr = svm.SVR()>>> regr.fit(X, y)SVR()>>> regr.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 the libsvm-based implementation scales between\(O(n_{features} \times n_{samples}^2)\) and\(O(n_{features} \times n_{samples}^3)\) depending on how efficientlythe libsvm cache is used in practice (dataset dependent). If the datais very sparse \(n_{features}\) should be replaced by the average numberof non-zero features in a sample vector.

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:decreasing C corresponds to more regularization.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 in [11].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. This can be doneeasily by using a Pipeline:>>> from sklearn.pipeline import make_pipeline>>> from sklearn.preprocessing import StandardScaler>>> from sklearn.svm import SVC>>> clf = make_pipeline(StandardScaler(), SVC())

See section Preprocessing data for more details on scaling andnormalization.

Regarding the

`shrinking`

parameter, quoting [12]:*We found that if thenumber of iterations is large, then shrinking can shorten the trainingtime. However, if we loosely solve the optimization problem (e.g., byusing a large stopping tolerance), the code without using shrinking maybe much faster*Parameter

`nu`

in NuSVC/OneClassSVM/NuSVRapproximates the fraction of training errors and support vectors.In SVC, if the data is 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 (when`probability`

is set to`True`

). This randomness can be controlledwith the`random_state`

parameter. If`probability`

is set to`False`

these 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 uncommonto 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(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 features are selected).The`C`

value that yields a “null” model (all weights equal to zero) canbe calculated using l1_min_c.

## 1.4.6. Kernel functions#

The *kernel function* can be any of the following:

linear: \(\langle x, x'\rangle\).

polynomial: \((\gamma \langle x, x'\rangle + r)^d\), where\(d\) is specified by parameter

`degree`

, \(r\) by`coef0`

.rbf: \(\exp(-\gamma \|x-x'\|^2)\), where \(\gamma\) isspecified by parameter

`gamma`

, must be greater than 0.sigmoid \(\tanh(\gamma \langle x,x'\rangle + r)\),where \(r\) is specified by

`coef0`

.

Different kernels are specified by the `kernel`

parameter:

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

See also Kernel Approximation for a solution to use RBF kernels that is much faster and more scalable.

### 1.4.6.1. 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 GridSearchCV with`C`

and `gamma`

spaced exponentially far apart to choose good values.

Examples

RBF SVM parameters

Scaling the regularization parameter for SVCs

### 1.4.6.2. 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

`support_vectors_`

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.

## Using Python functions as kernels#

You can use your own defined kernels by passing a function to the`kernel`

parameter.

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)

## Using the Gram matrix#

You can pass pre-computed kernels by using the `kernel='precomputed'`

option. You should then pass Gram matrix instead of X to the `fit`

and`predict`

methods. The kernel values between *all* training vectors and thetest vectors must be provided:

>>> import numpy as np>>> from sklearn.datasets import make_classification>>> from sklearn.model_selection import train_test_split>>> from sklearn import svm>>> X, y = make_classification(n_samples=10, random_state=0)>>> X_train , X_test , y_train, y_test = train_test_split(X, y, random_state=0)>>> clf = svm.SVC(kernel='precomputed')>>> # linear kernel computation>>> gram_train = np.dot(X_train, X_train.T)>>> clf.fit(gram_train, y_train)SVC(kernel='precomputed')>>> # predict on training examples>>> gram_test = np.dot(X_test, X_train.T)>>> clf.predict(gram_test)array([0, 1, 0])

Examples

SVM with custom kernel

## 1.4.7. Mathematical formulation#

A support vector machine constructs a hyper-plane or set of hyper-planes in ahigh 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. The figure below shows the decisionfunction for a linearly separable problem, with three samples on themargin boundaries, called “support vectors”:

In general, when the problem isn’t linearly separable, the support vectorsare the samples *within* the margin boundaries.

We recommend [13] and [14] as good references for the theory andpracticalities of SVMs.

### 1.4.7.1. SVC#

Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, in two classes, and avector \(y \in \{1, -1\}^n\), our goal is to find \(w \in\mathbb{R}^p\) and \(b \in \mathbb{R}\) such that the prediction given by\(\text{sign} (w^T\phi(x) + b)\) is correct for most samples.

SVC solves the following primal problem:

\[ \begin{align}\begin{aligned}\min_ {w, b, \zeta} \frac{1}{2} w^T w + C \sum_{i=1}^{n} \zeta_i\\\begin{split}\textrm {subject to } & y_i (w^T \phi (x_i) + b) \geq 1 - \zeta_i,\\& \zeta_i \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \]

Intuitively, we’re trying to maximize the margin (by minimizing\(||w||^2 = w^Tw\)), while incurring a penalty when a sample ismisclassified or within the margin boundary. Ideally, the value \(y_i(w^T \phi (x_i) + b)\) would be \(\geq 1\) for all samples, whichindicates a perfect prediction. But problems are usually not always perfectlyseparable with a hyperplane, so we allow some samples to be at a distance \(\zeta_i\) fromtheir correct margin boundary. The penalty term `C`

controls the strength ofthis penalty, and as a result, acts as an inverse regularization parameter(see note below).

The dual problem to the primal is

\[ \begin{align}\begin{aligned}\min_{\alpha} \frac{1}{2} \alpha^T Q \alpha - e^T \alpha\\\begin{split}\textrm {subject to } & y^T \alpha = 0\\& 0 \leq \alpha_i \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \]

where \(e\) is the vector of all ones,and \(Q\) is an \(n\) by \(n\) positive semidefinite matrix,\(Q_{ij} \equiv y_i y_j K(x_i, x_j)\), where \(K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\)is the kernel. The terms \(\alpha_i\) are called the dual coefficients,and they are upper-bounded by \(C\).This dual representation highlights the fact that training vectors areimplicitly mapped into a higher (maybe infinite)dimensional space by the function \(\phi\): see kernel trick.

Once the optimization problem is solved, the output ofdecision_function for a given sample \(x\) becomes:

\[\sum_{i\in SV} y_i \alpha_i K(x_i, x) + b,\]

and the predicted class correspond to its sign. We only need to sum over thesupport vectors (i.e. the samples that lie within the margin) because thedual coefficients \(\alpha_i\) are zero for the other samples.

These parameters can be accessed through the attributes `dual_coef_`

which holds the product \(y_i \alpha_i\), `support_vectors_`

whichholds the support vectors, and `intercept_`

which holds the independentterm \(b\)

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 Ridge regression,the relation between them is given as \(C = \frac{1}{alpha}\).

## LinearSVC#

The primal problem can be equivalently formulated as

\[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}^{n}\max(0, 1 - y_i (w^T \phi(x_i) + b)),\]

where we make use of the hinge loss. This is the form that isdirectly optimized by LinearSVC, but unlike the dual form, this onedoes not involve inner products between samples, so the famous kernel trickcannot be applied. This is why only the linear kernel is supported byLinearSVC (\(\phi\) is the identity function).

## NuSVC#

The \(\nu\)-SVC formulation [15] is a reparameterization of the\(C\)-SVC and therefore mathematically equivalent.

We introduce a new parameter \(\nu\) (instead of \(C\)) whichcontrols the number of support vectors and *margin errors*:\(\nu \in (0, 1]\) is an upper bound on the fraction of margin errors anda lower bound of the fraction of support vectors. A margin error correspondsto a sample that lies on the wrong side of its margin boundary: it is eithermisclassified, or it is correctly classified but does not lie beyond themargin.

### 1.4.7.2. SVR#

Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, and avector \(y \in \mathbb{R}^n\) \(\varepsilon\)-SVR solves the following primal problem:

\[ \begin{align}\begin{aligned}\min_ {w, b, \zeta, \zeta^*} \frac{1}{2} w^T w + C \sum_{i=1}^{n} (\zeta_i + \zeta_i^*)\\\begin{split}\textrm {subject to } & y_i - w^T \phi (x_i) - b \leq \varepsilon + \zeta_i,\\ & w^T \phi (x_i) + b - y_i \leq \varepsilon + \zeta_i^*,\\ & \zeta_i, \zeta_i^* \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \]

Here, we are penalizing samples whose prediction is at least \(\varepsilon\)away from their true target. These samples penalize the objective by\(\zeta_i\) or \(\zeta_i^*\), depending on whether their predictionslie above or below the \(\varepsilon\) tube.

The dual problem is

\[ \begin{align}\begin{aligned}\min_{\alpha, \alpha^*} \frac{1}{2} (\alpha - \alpha^*)^T Q (\alpha - \alpha^*) + \varepsilon e^T (\alpha + \alpha^*) - y^T (\alpha - \alpha^*)\\\begin{split}\textrm {subject to } & e^T (\alpha - \alpha^*) = 0\\& 0 \leq \alpha_i, \alpha_i^* \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \]

where \(e\) is the vector of all ones,\(Q\) is an \(n\) by \(n\) positive semidefinite matrix,\(Q_{ij} \equiv K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\)is the kernel. Here training vectors are implicitly mapped into a higher(maybe infinite) dimensional space by the function \(\phi\).

The prediction is:

\[\sum_{i \in SV}(\alpha_i - \alpha_i^*) K(x_i, x) + b\]

These parameters can be accessed through the attributes `dual_coef_`

which holds the difference \(\alpha_i - \alpha_i^*\), `support_vectors_`

whichholds the support vectors, and `intercept_`

which holds the independentterm \(b\)

## LinearSVR#

The primal problem can be equivalently formulated as

\[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}^{n}\max(0, |y_i - (w^T \phi(x_i) + b)| - \varepsilon),\]

where we make use of the epsilon-insensitive loss, i.e. errors of less than\(\varepsilon\) are ignored. This is the form that is directly optimizedby LinearSVR.

## 1.4.8. Implementation details#

Internally, we use libsvm [12] and liblinear [11] to handle allcomputations. These libraries are wrapped using C and Cython.For a description of the implementation and details of the algorithmsused, please refer to their respective papers.

References