.. _svm:
=======================
Support Vector Machines
=======================
.. currentmodule:: sklearn.svm
**Support vector machines (SVMs)** are a set of supervised learning
methods used for :ref:`classification `,
:ref:`regression ` and :ref:`outliers detection
`.
The advantages of support vector machines are:
- Effective in high dimensional spaces.
- Still effective in cases where number of dimensions is greater
than the number of samples.
- Uses a subset of training points in the decision function (called
support vectors), so it is also memory efficient.
- Versatile: different :ref:`svm_kernels` can be
specified for the decision function. Common kernels are
provided, 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 of
samples, avoid over-fitting in choosing :ref:`svm_kernels` and regularization
term is crucial.
- SVMs do not directly provide probability estimates, these are
calculated using an expensive five-fold cross-validation
(see :ref:`Scores and probabilities `, below).
The support vector machines in scikit-learn support both dense
(``numpy.ndarray`` and convertible to that by ``numpy.asarray``) and
sparse (any ``scipy.sparse``) sample vectors as input. However, to use
an SVM to make predictions for sparse data, it must have been fit on such
data. For optimal performance, use C-ordered ``numpy.ndarray`` (dense) or
``scipy.sparse.csr_matrix`` (sparse) with ``dtype=float64``.
.. _svm_classification:
Classification
==============
:class:`SVC`, :class:`NuSVC` and :class:`LinearSVC` are classes
capable of performing multi-class classification on a dataset.
.. figure:: ../auto_examples/svm/images/sphx_glr_plot_iris_svc_001.png
:target: ../auto_examples/svm/plot_iris_svc.html
:align: center
:class:`SVC` and :class:`NuSVC` are similar methods, but accept
slightly different sets of parameters and have different mathematical
formulations (see section :ref:`svm_mathematical_formulation`). On the
other hand, :class:`LinearSVC` is another implementation of Support
Vector Classification for the case of a linear kernel. Note that
:class:`LinearSVC` does not accept keyword ``kernel``, as this is
assumed to be linear. It also lacks some of the members of
:class:`SVC` and :class:`NuSVC`, like ``support_``.
As other classifiers, :class:`SVC`, :class:`NuSVC` and
:class:`LinearSVC` 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 vectors
can be found in members ``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]...)
.. _svm_multi_class:
Multi-class classification
--------------------------
:class:`SVC` and :class:`NuSVC` implement the "one-against-one"
approach (Knerr et al., 1990) for multi- class classification. If
``n_class`` is the number of classes, then ``n_class * (n_class - 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 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, :class:`LinearSVC` implements "one-vs-the-rest"
multi-class strategy, thus training n_class models. If there are only
two 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 :ref:`svm_mathematical_formulation` for a complete description of
the decision function.
Note that the :class:`LinearSVC` also implements an alternative multi-class
strategy, the so-called multi-class SVM formulated by Crammer and Singer, by
using 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 results
are mostly similar, but the runtime is significantly less.
For "one-vs-rest" :class:`LinearSVC` the attributes ``coef_`` and ``intercept_``
have 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 the
order of the "one" class.
In the case of "one-vs-one" :class:`SVC`, the layout of the attributes
is a little more involved. In the case of having a linear kernel, the
attributes ``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 the
layout for :class:`LinearSVC` described above, with each row now corresponding
to a binary classifier. The order for classes
0 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_class-1, n_SV]`` with
a somewhat hard to grasp layout.
The columns correspond to the support vectors involved in any
of 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 coefficients
for these classifiers.
This might be made more clear by an example:
Consider a three class problem with class 0 having three support vectors
:math:`v^{0}_0, v^{1}_0, v^{2}_0` and class 1 and 2 having two support vectors
:math:`v^{0}_1, v^{1}_1` and :math:`v^{0}_2, v^{1}_2` respectively. For each
support vector :math:`v^{j}_i`, there are two dual coefficients. Let's call
the coefficient of support vector :math:`v^{j}_i` in the classifier between
classes :math:`i` and :math:`k` :math:`\alpha^{j}_{i,k}`.
Then ``dual_coef_`` looks like this:
+------------------------+------------------------+------------------+
|:math:`\alpha^{0}_{0,1}`|:math:`\alpha^{0}_{0,2}`|Coefficients |
+------------------------+------------------------+for SVs of class 0|
|:math:`\alpha^{1}_{0,1}`|:math:`\alpha^{1}_{0,2}`| |
+------------------------+------------------------+ |
|:math:`\alpha^{2}_{0,1}`|:math:`\alpha^{2}_{0,2}`| |
+------------------------+------------------------+------------------+
|:math:`\alpha^{0}_{1,0}`|:math:`\alpha^{0}_{1,2}`|Coefficients |
+------------------------+------------------------+for SVs of class 1|
|:math:`\alpha^{1}_{1,0}`|:math:`\alpha^{1}_{1,2}`| |
+------------------------+------------------------+------------------+
|:math:`\alpha^{0}_{2,0}`|:math:`\alpha^{0}_{2,1}`|Coefficients |
+------------------------+------------------------+for SVs of class 2|
|:math:`\alpha^{1}_{2,0}`|:math:`\alpha^{1}_{2,1}`| |
+------------------------+------------------------+------------------+
.. _scores_probabilities:
Scores and probabilities
------------------------
The ``decision_function`` method of :class:`SVC` and :class:`NuSVC` gives
per-class scores for each sample (or a single score per sample in the binary
case). 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 are
calibrated 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 scaling
is 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 scores
may not be the argmax of the probabilities.
(E.g., in binary classification,
a sample may be labeled by ``predict`` as belonging to a class
that 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=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 ties
by default. You can set ``break_ties=True`` for the output of ``predict`` to be
the same as ``np.argmax(clf.decision_function(...), axis=1)``, otherwise the
first class among the tied classes will always be returned; but have in mind
that it comes with a computational cost.
.. figure:: ../auto_examples/svm/images/sphx_glr_plot_svm_tie_breaking_001.png
:target: ../auto_examples/svm/plot_svm_tie_breaking.html
:align: center
.. topic:: 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"
`_.
Unbalanced problems
--------------------
In problems where it is desired to give more importance to certain
classes or certain individual samples keywords ``class_weight`` and
``sample_weight`` can be used.
:class:`SVC` (but not :class:`NuSVC`) implement a keyword
``class_weight`` in the ``fit`` method. It's a dictionary of the form
``{class_label : value}``, where value is a floating point number > 0
that sets the parameter ``C`` of class ``class_label`` to ``C * value``.
.. figure:: ../auto_examples/svm/images/sphx_glr_plot_separating_hyperplane_unbalanced_001.png
:target: ../auto_examples/svm/plot_separating_hyperplane_unbalanced.html
:align: center
:scale: 75
:class:`SVC`, :class:`NuSVC`, :class:`SVR`, :class:`NuSVR` and
:class:`OneClassSVM` implement also weights for individual samples in method
``fit`` through keyword ``sample_weight``. Similar to ``class_weight``, these
set the parameter ``C`` for the i-th example to ``C * sample_weight[i]``.
.. figure:: ../auto_examples/svm/images/sphx_glr_plot_weighted_samples_001.png
:target: ../auto_examples/svm/plot_weighted_samples.html
:align: center
:scale: 75
.. topic:: Examples:
* :ref:`sphx_glr_auto_examples_svm_plot_iris_svc.py`,
* :ref:`sphx_glr_auto_examples_svm_plot_separating_hyperplane.py`,
* :ref:`sphx_glr_auto_examples_svm_plot_separating_hyperplane_unbalanced.py`
* :ref:`sphx_glr_auto_examples_svm_plot_svm_anova.py`,
* :ref:`sphx_glr_auto_examples_svm_plot_svm_nonlinear.py`
* :ref:`sphx_glr_auto_examples_svm_plot_weighted_samples.py`,
.. _svm_regression:
Regression
==========
The method of Support Vector Classification can be extended to solve
regression problems. This method is called Support Vector Regression.
The model produced by support vector classification (as described
above) depends only on a subset of the training data, because the cost
function for building the model does not care about training points
that lie beyond the margin. Analogously, the model produced by Support
Vector Regression depends only on a subset of the training data,
because the cost function for building the model ignores any training
data close to the model prediction.
There are three different implementations of Support Vector Regression:
:class:`SVR`, :class:`NuSVR` and :class:`LinearSVR`. :class:`LinearSVR`
provides a faster implementation than :class:`SVR` but only considers
linear kernels, while :class:`NuSVR` implements a slightly different
formulation than :class:`SVR` and :class:`LinearSVR`. See
:ref:`svm_implementation_details` for further details.
As with classification classes, the fit method will take as
argument vectors X, y, only that in this case y is expected to have
floating 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])
.. topic:: Examples:
* :ref:`sphx_glr_auto_examples_svm_plot_svm_regression.py`
.. _svm_outlier_detection:
Density estimation, novelty detection
=======================================
The class :class:`OneClassSVM` implements a One-Class SVM which is used in
outlier detection.
See :ref:`outlier_detection` for the description and usage of OneClassSVM.
Complexity
==========
Support Vector Machines are powerful tools, but their compute and
storage requirements increase rapidly with the number of training
vectors. The core of an SVM is a quadratic programming problem (QP),
separating support vectors from the rest of the training data. The QP
solver used by this `libsvm`_-based implementation scales between
:math:`O(n_{features} \times n_{samples}^2)` and
:math:`O(n_{features} \times n_{samples}^3)` depending on how efficiently
the `libsvm`_ cache is used in practice (dataset dependent). If the data
is very sparse :math:`n_{features}` should be replaced by the average number
of non-zero features in a sample vector.
Also note that for the linear case, the algorithm used in
:class:`LinearSVC` by the `liblinear`_ implementation is much more
efficient than its `libsvm`_-based :class:`SVC` counterpart and can
scale almost linearly to millions of samples and/or features.
Tips on Practical Use
=====================
* **Avoiding data copy**: For :class:`SVC`, :class:`SVR`, :class:`NuSVC` and
:class:`NuSVR`, if the data passed to certain methods is not C-ordered
contiguous, and double precision, it will be copied before calling the
underlying C implementation. You can check whether a given numpy array is
C-contiguous by inspecting its ``flags`` attribute.
For :class:`LinearSVC` (and :class:`LogisticRegression
`) any input passed as a numpy
array will be copied and converted to the liblinear internal sparse data
representation (double precision floats and int32 indices of non-zero
components). If you want to fit a large-scale linear classifier without
copying a dense numpy C-contiguous double precision array as input we
suggest to use the :class:`SGDClassifier
` class instead. The objective
function can be configured to be almost the same as the :class:`LinearSVC`
model.
* **Kernel cache size**: For :class:`SVC`, :class:`SVR`, :class:`NuSVC` and
:class:`NuSVR`, the size of the kernel cache has a strong impact on run
times for larger problems. If you have enough RAM available, it is
recommended to set ``cache_size`` to a higher value than the default of
200(MB), such as 500(MB) or 1000(MB).
* **Setting C**: ``C`` is ``1`` by default and it's a reasonable default
choice. If you have a lot of noisy observations you should decrease it.
It corresponds to regularize more the estimation.
:class:`LinearSVC` and :class:`LinearSVR` are less sensitive to ``C`` when
it becomes large, and prediction results stop improving after a certain
threshold. 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 **it
is highly recommended to scale your data**. For example, scale each
attribute on the input vector X to [0,1] or [-1,+1], or standardize it
to have mean 0 and variance 1. Note that the *same* scaling must be
applied to the test vector to obtain meaningful results. See section
:ref:`preprocessing` for more details on scaling and normalization.
* Parameter ``nu`` in :class:`NuSVC`/:class:`OneClassSVM`/:class:`NuSVR`
approximates the fraction of training errors and support vectors.
* In :class:`SVC`, if data for classification are unbalanced (e.g. many
positive and few negative), set ``class_weight='balanced'`` and/or try
different penalty parameters ``C``.
* **Randomness of the underlying implementations**: The underlying
implementations of :class:`SVC` and :class:`NuSVC` use a random number
generator only to shuffle the data for probability estimation (when
``probability`` is set to ``True``). This randomness can be controlled
with the ``random_state`` parameter. If ``probability`` is set to ``False``
these estimators are not random and ``random_state`` has no effect on the
results. The underlying :class:`OneClassSVM` implementation is similar to
the ones of :class:`SVC` and :class:`NuSVC`. As no probability estimation
is provided for :class:`OneClassSVM`, it is not random.
The underlying :class:`LinearSVC` implementation uses a random number
generator to select features when fitting the model with a dual coordinate
descent (i.e when ``dual`` is set to ``True``). It is thus not uncommon,
to have slightly different results for the same input data. If that
happens, try with a smaller tol parameter. This randomness can also be
controlled with the ``random_state`` parameter. When ``dual`` is
set to ``False`` the underlying implementation of :class:`LinearSVC` is
not 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 feature
weights 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) can
be calculated using :func:`l1_min_c`.
.. topic:: References:
* Fan, Rong-En, et al.,
`"LIBLINEAR: A library for large linear classification."
`_,
Journal of machine learning research 9.Aug (2008): 1871-1874.
.. _svm_kernels:
Kernel functions
================
The *kernel function* can be any of the following:
* linear: :math:`\langle x, x'\rangle`.
* polynomial: :math:`(\gamma \langle x, x'\rangle + r)^d`.
:math:`d` is specified by keyword ``degree``, :math:`r` by ``coef0``.
* rbf: :math:`\exp(-\gamma \|x-x'\|^2)`. :math:`\gamma` is
specified by keyword ``gamma``, must be greater than 0.
* sigmoid (:math:`\tanh(\gamma \langle x,x'\rangle + r)`),
where :math:`r` 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'
Custom Kernels
--------------
You can define your own kernels by either giving the kernel as a
python function or by precomputing the Gram matrix.
Classifiers with custom kernels behave the same way as any other
classifiers, except that:
* Field ``support_vectors_`` is now empty, only indices of support
vectors 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 the
use of ``fit()`` and ``predict()`` you will have unexpected results.
Using Python functions as kernels
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
You can also use your own defined kernels by passing a function to the
keyword ``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 classifier
instance 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)
.. topic:: Examples:
* :ref:`sphx_glr_auto_examples_svm_plot_custom_kernel.py`.
Using the Gram matrix
~~~~~~~~~~~~~~~~~~~~~
Set ``kernel='precomputed'`` and pass the Gram matrix instead of X in the fit
method. At the moment, the kernel values between *all* training vectors and the
test 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])
Parameters of the RBF Kernel
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When training an SVM with the *Radial Basis Function* (RBF) kernel, two
parameters must be considered: ``C`` and ``gamma``. The parameter ``C``,
common to all SVM kernels, trades off misclassification of training examples
against simplicity of the decision surface. A low ``C`` makes the decision
surface smooth, while a high ``C`` aims at classifying all training examples
correctly. ``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. One
is advised to use :class:`sklearn.model_selection.GridSearchCV` with
``C`` and ``gamma`` spaced exponentially far apart to choose good values.
.. topic:: Examples:
* :ref:`sphx_glr_auto_examples_svm_plot_rbf_parameters.py`
.. _svm_mathematical_formulation:
Mathematical formulation
========================
A support vector machine constructs a hyper-plane or set of hyper-planes
in a high or infinite dimensional space, which can be used for
classification, regression or other tasks. Intuitively, a good
separation is achieved by the hyper-plane that has the largest distance
to the nearest training data points of any class (so-called functional
margin), since in general the larger the margin the lower the
generalization error of the classifier.
.. figure:: ../auto_examples/svm/images/sphx_glr_plot_separating_hyperplane_001.png
:align: center
:scale: 75
SVC
---
Given training vectors :math:`x_i \in \mathbb{R}^p`, i=1,..., n, in two classes, and a
vector :math:`y \in \{1, -1\}^n`, SVC solves the following primal problem:
.. math::
\min_ {w, b, \zeta} \frac{1}{2} w^T w + C \sum_{i=1}^{n} \zeta_i
\textrm {subject to } & y_i (w^T \phi (x_i) + b) \geq 1 - \zeta_i,\\
& \zeta_i \geq 0, i=1, ..., n
Its dual is
.. math::
\min_{\alpha} \frac{1}{2} \alpha^T Q \alpha - e^T \alpha
\textrm {subject to } & y^T \alpha = 0\\
& 0 \leq \alpha_i \leq C, i=1, ..., n
where :math:`e` is the vector of all ones, :math:`C > 0` is the upper bound,
:math:`Q` is an :math:`n` by :math:`n` positive semidefinite matrix,
:math:`Q_{ij} \equiv y_i y_j K(x_i, x_j)`, where :math:`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 :math:`\phi`.
The decision function is:
.. math:: \operatorname{sgn}(\sum_{i=1}^n y_i \alpha_i K(x_i, x) + \rho)
.. note::
While SVM models derived from `libsvm`_ and `liblinear`_ use ``C`` as
regularization parameter, most other estimators use ``alpha``. The exact
equivalence between the amount of regularization of two models depends on
the exact objective function optimized by the model. For example, when the
estimator used is :class:`sklearn.linear_model.Ridge ` regression,
the relation between them is given as :math:`C = \frac{1}{alpha}`.
.. TODO multiclass case ?/
This parameters can be accessed through the members ``dual_coef_``
which holds the product :math:`y_i \alpha_i`, ``support_vectors_`` which
holds the support vectors, and ``intercept_`` which holds the independent
term :math:`\rho` :
.. topic:: References:
* `"Automatic Capacity Tuning of Very Large VC-dimension Classifiers"
`_,
I. Guyon, B. Boser, V. Vapnik - Advances in neural information
processing 1993.
* `"Support-vector networks"
`_,
C. Cortes, V. Vapnik - Machine Learning, 20, 273-297 (1995).
NuSVC
-----
We introduce a new parameter :math:`\nu` which controls the number of
support vectors and training errors. The parameter :math:`\nu \in (0,
1]` is an upper bound on the fraction of training errors and a lower
bound of the fraction of support vectors.
It can be shown that the :math:`\nu`-SVC formulation is a reparameterization
of the :math:`C`-SVC and therefore mathematically equivalent.
SVR
---
Given training vectors :math:`x_i \in \mathbb{R}^p`, i=1,..., n, and a
vector :math:`y \in \mathbb{R}^n` :math:`\varepsilon`-SVR solves the following primal problem:
.. math::
\min_ {w, b, \zeta, \zeta^*} \frac{1}{2} w^T w + C \sum_{i=1}^{n} (\zeta_i + \zeta_i^*)
\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
Its dual is
.. math::
\min_{\alpha, \alpha^*} \frac{1}{2} (\alpha - \alpha^*)^T Q (\alpha - \alpha^*) + \varepsilon e^T (\alpha + \alpha^*) - y^T (\alpha - \alpha^*)
\textrm {subject to } & e^T (\alpha - \alpha^*) = 0\\
& 0 \leq \alpha_i, \alpha_i^* \leq C, i=1, ..., n
where :math:`e` is the vector of all ones, :math:`C > 0` is the upper bound,
:math:`Q` is an :math:`n` by :math:`n` positive semidefinite matrix,
:math:`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 :math:`\phi`.
The decision function is:
.. math:: \sum_{i=1}^n (\alpha_i - \alpha_i^*) K(x_i, x) + \rho
These parameters can be accessed through the members ``dual_coef_``
which holds the difference :math:`\alpha_i - \alpha_i^*`, ``support_vectors_`` which
holds the support vectors, and ``intercept_`` which holds the independent
term :math:`\rho`
.. topic:: References:
* `"A Tutorial on Support Vector Regression"
`_,
Alex J. Smola, Bernhard Schölkopf - Statistics and Computing archive
Volume 14 Issue 3, August 2004, p. 199-222.
.. _svm_implementation_details:
Implementation details
======================
Internally, we use `libsvm`_ and `liblinear`_ to handle all
computations. These libraries are wrapped using C and Cython.
.. _`libsvm`: https://www.csie.ntu.edu.tw/~cjlin/libsvm/
.. _`liblinear`: https://www.csie.ntu.edu.tw/~cjlin/liblinear/
.. topic:: References:
For a description of the implementation and details of the algorithms
used, please refer to
- `LIBSVM: A Library for Support Vector Machines
`_.
- `LIBLINEAR -- A Library for Large Linear Classification
`_.