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.
***SVM::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``.
***SVM::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.
***SVM::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.
***SVM::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 classWeights:Optional weights in the C_SVC problem , assigned to particular classes. They are multiplied by ``C`` so the parameter ``C`` of class ``#i`` becomes ``classWeights(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 termCrit: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.
A comparison of different kernels on the following 2D test case with four classes. Four C_SVC SVMs have been trained (one against rest) with auto_train. Evaluation on three different kernels (CHI2, INTER, RBF). The color depicts the class with max score. Bright means max-score > 0, dark means max-score < 0.
Use ``StatModel::train`` to train the model, ``StatModel::train<RTrees>(traindata, params)`` to create and train the model, ``StatModel::load<RTrees>(filename)`` to load the pre-trained model. Since SVM has several parameters, you may want to find the best parameters for your problem. It can be done with ``SVM::trainAuto``.
:param kFold:Cross-validation parameter. The training set is divided into ``kFold`` subsets. One subset is used to test the model, the others form the train set. So, the SVM algorithm is executed ``kFold`` times.
: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 ``gammaGrid.step = 0``, ``gammaGrid.minVal``, ``gamma_grid.maxVal`` 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:`SVM::getDefaulltGrid`. To generate a grid, for example, for ``gamma``, call ``SVM::getDefaulltGrid(SVM::GAMMA)``.
(``params.svmType=SVM::EPS_SVR`` or ``params.svmType=SVM::NU_SVR``). If ``params.svmType=SVM::ONE_CLASS``, no optimization is made and the usual SVM with parameters specified in ``params`` is executed.
..ocv:function:: double SVM::getDecisionFunction(int i, OutputArray alpha, OutputArray svidx) const
:param i:the index of the decision function. If the problem solved is regression, 1-class or 2-class classification, then there will be just one decision function and the index should always be 0. Otherwise, in the case of N-class classification, there will be N*(N-1)/2 decision functions.
:param alpha:the optional output vector for weights, corresponding to different support vectors. In the case of linear SVM all the alpha's will be 1's.
:param svidx:the optional output vector of indices of support vectors within the matrix of support vectors (which can be retrieved by ``SVM::getSupportVectors``). In the case of linear SVM each decision function consists of a single "compressed" support vector.
StatModel::predict(samples, results, flags) should be used. Pass ``flags=StatModel::RAW_OUTPUT`` to get the raw response from SVM (in the case of regression, 1-class or 2-class classification problem).
