Originally, support vector machines (SVM) was a technique for building an optimal binary (2-class) classifier. Later the technique was extended to regression and clustering problems. SVM is a partial case of kernel-based methods. It maps feature vectors into a higher-dimensional space using a kernel function and builds an optimal linear discriminating function in this space or an optimal hyper-plane that fits into the training data. In case of SVM, the kernel is not defined explicitly. Instead, a distance between any 2 points in the hyper-space needs to be defined.
The solution is optimal, which means that the margin between the separating hyper-plane and the nearest feature vectors from both classes (in case of 2-class classifier) is maximal. The feature vectors that are the closest to the hyper-plane are called *support vectors*, which means that the position of other vectors does not affect the hyper-plane (the decision function).
..[Burges98] C. Burges. *A tutorial on support vector machines for pattern recognition*, Knowledge Discovery and Data Mining 2(2), 1998 (available online at http://citeseer.ist.psu.edu/burges98tutorial.html)
..[LibSVM] C.-C. Chang and C.-J. Lin. *LIBSVM: a library for support vector machines*, ACM Transactions on Intelligent Systems and Technology, 2:27:1--27:27, 2011. (http://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf)
The structure represents the logarithmic grid range of statmodel parameters. It is used for optimizing statmodel accuracy by varying model parameters, the accuracy estimate being computed by cross-validation.
:param svm_type:Type of a SVM formulation. Possible values are:
***CvSVM::C_SVC** C-Support Vector Classification. ``n``-class classification (``n``:math:`\geq` 2), allows imperfect separation of classes with penalty multiplier ``C`` for outliers.
***CvSVM::NU_SVC**:math:`\nu`-Support Vector Classification. ``n``-class classification with possible imperfect separation. Parameter :math:`\nu` (in the range 0..1, the larger the value, the smoother the decision boundary) is used instead of ``C``.
***CvSVM::ONE_CLASS** Distribution Estimation (One-class SVM). All the training data are from the same class, SVM builds a boundary that separates the class from the rest of the feature space.
***CvSVM::EPS_SVR**:math:`\epsilon`-Support Vector Regression. The distance between feature vectors from the training set and the fitting hyper-plane must be less than ``p``. For outliers the penalty multiplier ``C`` is used.
***CvSVM::NU_SVR**:math:`\nu`-Support Vector Regression. :math:`\nu` is used instead of ``p``.
See [LibSVM]_ for details.
:param kernel_type:Type of a SVM kernel. Possible values are:
***CvSVM::LINEAR** Linear kernel. No mapping is done, linear discrimination (or regression) is done in the original feature space. It is the fastest option. :math:`K(x_i, x_j) = x_i^T x_j`.
:param degree:Parameter ``degree`` of a kernel function (POLY).
:param gamma:Parameter :math:`\gamma` of a kernel function (POLY / RBF / SIGMOID).
:param coef0:Parameter ``coef0`` of a kernel function (POLY / SIGMOID).
:param Cvalue:Parameter ``C`` of a SVM optimiazation problem (C_SVC / EPS_SVR / NU_SVR).
:param nu:Parameter :math:`\nu` of a SVM optimization problem (NU_SVC / ONE_CLASS / NU_SVR).
:param p:Parameter :math:`\epsilon` of a SVM optimization problem (EPS_SVR).
:param class_weights:Optional weights in the C_SVC problem , assigned to particular classes. They are multiplied by ``C`` so the parameter ``C`` of class ``#i`` becomes :math:`class\_weights_i * C`. Thus these weights affect the misclassification penalty for different classes. The larger weight, the larger penalty on misclassification of data from the corresponding class.
:param term_crit:Termination criteria of the iterative SVM training procedure which solves a partial case of constrained quadratic optimization problem. You can specify tolerance and/or the maximum number of iterations.
* Output variables can be either categorical (``params.svm_type=CvSVM::C_SVC`` or ``params.svm_type=CvSVM::NU_SVC``), or ordered (``params.svm_type=CvSVM::EPS_SVR`` or ``params.svm_type=CvSVM::NU_SVR``), or not required at all (``params.svm_type=CvSVM::ONE_CLASS``).
:param k_fold:Cross-validation parameter. The training set is divided into ``k_fold`` subsets. One subset is used to train the model, the others form the test set. So, the SVM algorithm is executed ``k_fold`` times.
:param \*Grid:Iteration grid for the corresponding SVM parameter.
:param balanced:If ``true`` and the problem is 2-class classification then the method creates more balanced cross-validation subsets that is proportions between classes in subsets are close to such proportion in the whole train dataset.
If there is no need to optimize a parameter, the corresponding grid step should be set to any value less than or equal to 1. For example, to avoid optimization in ``gamma``, set ``gamma_grid.step = 0``, ``gamma_grid.min_val``, ``gamma_grid.max_val`` as arbitrary numbers. In this case, the value ``params.gamma`` is taken for ``gamma``.
the corresponding grid is unknown, you may call the function :ocv:func:`CvSVM::get_default_grid`. To generate a grid, for example, for ``gamma``, call ``CvSVM::get_default_grid(CvSVM::GAMMA)``.
(``params.svm_type=CvSVM::EPS_SVR`` or ``params.svm_type=CvSVM::NU_SVR``). If ``params.svm_type=CvSVM::ONE_CLASS``, no optimization is made and the usual SVM with parameters specified in ``params`` is executed.
:param returnDFVal:Specifies a type of the return value. If ``true`` and the problem is 2-class classification then the method returns the decision function value that is signed distance to the margin, else the function returns a class label (classification) or estimated function value (regression).
:param results:Output prediction responses for corresponding samples.
If you pass one sample then prediction result is returned. If you want to get responses for several samples then you should pass the ``results`` matrix where prediction results will be stored.
