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# Support Vector Machines

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).

SVM implementation in OpenCV is based on [LibSVM].

[Burges98]

3. 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] | (1, 2) 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)

## CvParamGrid

struct `CvParamGrid`

  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.

  double `CvParamGrid::``min_val`

    Minimum value of the statmodel parameter.

  double `CvParamGrid::``max_val`

    Maximum value of the statmodel parameter.

  double `CvParamGrid::``step`

    Logarithmic step for iterating the statmodel parameter.

The grid determines the following iteration sequence of the statmodel parameter values:

(min\_val, min\_val*step, min\_val*{step}^2, \dots, min\_val*{step}^n),

where n is the maximal index satisfying

\texttt{min\_val} * \texttt{step} ^n < \texttt{max\_val}

The grid is logarithmic, so `step` must always be greater then 1.

## CvParamGrid::CvParamGrid

The constructors.

C++:`` `CvParamGrid::``CvParamGrid`()

C++:`` `CvParamGrid::``CvParamGrid`(double min_val, double max_val, double log_step)

The full constructor initializes corresponding members. The default constructor creates a dummy grid:

    CvParamGrid::CvParamGrid()
    {
        min_val = max_val = step = 0;
    }

## CvParamGrid::check

Checks validness of the grid.

C++:`` bool `CvParamGrid::``check`()

Returns `true` if the grid is valid and `false` otherwise. The grid is valid if and only if:

- Lower bound of the grid is less then the upper one.
- Lower bound of the grid is positive.
- Grid step is greater then 1.

## CvSVMParams

struct `CvSVMParams`

SVM training parameters.

The structure must be initialized and passed to the training method of `CvSVM`.

## CvSVMParams::CvSVMParams

The constructors.

C++:`` `CvSVMParams::``CvSVMParams`()

C++:`` `CvSVMParams::``CvSVMParams`(int svm_type, int kernel_type, double degree, double gamma, double coef0, double Cvalue, double nu, double p, CvMat* class_weights, CvTermCriteria term_crit)

  Parameters:

  - svm_type –

    Type of a SVM formulation. Possible values are:

    - CvSVM::C_SVC C-Support Vector Classification. `n`-class classification (`n` \geq 2), allows imperfect separation of classes with penalty multiplier `C` for outliers.
    - CvSVM::NU_SVC \nu-Support Vector Classification. `n`-class classification with possible imperfect separation. Parameter \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 \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 \nu-Support Vector Regression. \nu is used instead of `p`.

    See [LibSVM] for details.

  - 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. K(x_i, x_j) = x_i^T x_j.
    - CvSVM::POLY Polynomial kernel: K(x_i, x_j) = (\gamma x_i^T x_j + coef0)^{degree}, \gamma > 0.
    - CvSVM::RBF Radial basis function (RBF), a good choice in most cases. K(x_i, x_j) = e^{-\gamma ||x_i - x_j||^2}, \gamma > 0.
    - CvSVM::SIGMOID Sigmoid kernel: K(x_i, x_j) = \tanh(\gamma x_i^T x_j + coef0).
  - degree – Parameter `degree` of a kernel function (POLY).
  - gamma – Parameter \gamma of a kernel function (POLY / RBF / SIGMOID).
  - coef0 – Parameter `coef0` of a kernel function (POLY / SIGMOID).
  - Cvalue – Parameter `C` of a SVM optimization problem (C_SVC / EPS_SVR / NU_SVR).
  - nu – Parameter \nu of a SVM optimization problem (NU_SVC / ONE_CLASS / NU_SVR).
  - p – Parameter \epsilon of a SVM optimization problem (EPS_SVR).
  - 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 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.
  - 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.

The default constructor initialize the structure with following values:

    CvSVMParams::CvSVMParams() :
        svm_type(CvSVM::C_SVC), kernel_type(CvSVM::RBF), degree(0),
        gamma(1), coef0(0), C(1), nu(0), p(0), class_weights(0)
    {
        term_crit = cvTermCriteria( CV_TERMCRIT_ITER+CV_TERMCRIT_EPS, 1000, FLT_EPSILON );
    }

## CvSVM

class `CvSVM` : public `CvStatModel`

Support Vector Machines.

## CvSVM::CvSVM

Default and training constructors.

C++:`` `CvSVM::``CvSVM`()

C++:`` `CvSVM::``CvSVM`(const Mat& trainData, const Mat& responses, const Mat& varIdx=Mat(), const Mat& sampleIdx=Mat(), CvSVMParams params=CvSVMParams() )

C++:`` `CvSVM::``CvSVM`(const CvMat* trainData, const CvMat* responses, const CvMat* varIdx=0, const CvMat* sampleIdx=0, CvSVMParams params=CvSVMParams() )

Python:`` `cv2.``SVM`([trainData, responses[, varIdx[, sampleIdx[, params]]]]) → <SVM object>

The constructors follow conventions of `CvStatModel::CvStatModel()`. See `CvStatModel::train()` for parameters descriptions.

## CvSVM::train

Trains an SVM.

C++:`` bool `CvSVM::``train`(const Mat& trainData, const Mat& responses, const Mat& varIdx=Mat(), const Mat& sampleIdx=Mat(), CvSVMParams params=CvSVMParams() )

C++:`` bool `CvSVM::``train`(const CvMat* trainData, const CvMat* responses, const CvMat* varIdx=0, const CvMat* sampleIdx=0, CvSVMParams params=CvSVMParams() )

Python:`` `cv2.SVM.``train`(trainData, responses[, varIdx[, sampleIdx[, params]]]) → retval

The method trains the SVM model. It follows the conventions of the generic `CvStatModel::train()` approach with the following limitations:

- Only the `CV_ROW_SAMPLE` data layout is supported.
- Input variables are all ordered.
- 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`).
- Missing measurements are not supported.

All the other parameters are gathered in the `CvSVMParams` structure.

## CvSVM::train_auto

Trains an SVM with optimal parameters.

C++:`` bool `CvSVM::``train_auto`(const Mat& trainData, const Mat& responses, const Mat& varIdx, const Mat& sampleIdx, CvSVMParams params, int k_fold=10, CvParamGrid Cgrid=CvSVM::get_default_grid(CvSVM::C), CvParamGrid gammaGrid=CvSVM::get_default_grid(CvSVM::GAMMA), CvParamGrid pGrid=CvSVM::get_default_grid(CvSVM::P), CvParamGrid nuGrid=CvSVM::get_default_grid(CvSVM::NU), CvParamGrid coeffGrid=CvSVM::get_default_grid(CvSVM::COEF), CvParamGrid degreeGrid=CvSVM::get_default_grid(CvSVM::DEGREE), bool balanced=false)

C++:`` bool `CvSVM::``train_auto`(const CvMat* trainData, const CvMat* responses, const CvMat* varIdx, const CvMat* sampleIdx, CvSVMParams params, int kfold=10, CvParamGrid Cgrid=get_default_grid(CvSVM::C), CvParamGrid gammaGrid=get_default_grid(CvSVM::GAMMA), CvParamGrid pGrid=get_default_grid(CvSVM::P), CvParamGrid nuGrid=get_default_grid(CvSVM::NU), CvParamGrid coeffGrid=get_default_grid(CvSVM::COEF), CvParamGrid degreeGrid=get_default_grid(CvSVM::DEGREE), bool balanced=false )

Python:`` `cv2.SVM.``train_auto`(trainData, responses, varIdx, sampleIdx, params[, k_fold[, Cgrid[, gammaGrid[, pGrid[, nuGrid[, coeffGrid[, degreeGrid[, balanced]]]]]]]]) → retval

  Parameters:

  - k_fold – Cross-validation parameter. The training set is divided into `k_fold` subsets. One subset is used to test the model, the others form the train set. So, the SVM algorithm is executed `k_fold` times.
  - *Grid – Iteration grid for the corresponding SVM parameter.
  - 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.

The method trains the SVM model automatically by choosing the optimal parameters `C`, `gamma`, `p`, `nu`, `coef0`, `degree` from `CvSVMParams`. Parameters are considered optimal when the cross-validation estimate of the test set error is minimal.

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`.

And, finally, if the optimization in a parameter is required but the corresponding grid is unknown, you may call the function `CvSVM::get_default_grid()`. To generate a grid, for example, for `gamma`, call `CvSVM::get_default_grid(CvSVM::GAMMA)`.

This function works for the classification (`params.svm_type=CvSVM::C_SVC` or `params.svm_type=CvSVM::NU_SVC`) as well as for the regression (`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.

## CvSVM::predict

Predicts the response for input sample(s).

C++:`` float `CvSVM::``predict`(const Mat& sample, bool returnDFVal=false )`const`

C++:`` float `CvSVM::``predict`(const CvMat* sample, bool returnDFVal=false )`const`

C++:`` float `CvSVM::``predict`(const CvMat* samples, CvMat* results)`const`

Python:`` `cv2.SVM.``predict`(sample[, returnDFVal]) → retval

Python:`` `cv2.SVM.``predict_all`(samples[, results]) → results

  Parameters:

  - sample – Input sample for prediction.
  - samples – Input samples for prediction.
  - 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).
  - 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.

The function is parallelized with the TBB library.

## CvSVM::get_default_grid

Generates a grid for SVM parameters.

C++:`` CvParamGrid `CvSVM::``get_default_grid`(int param_id)

  Parameters:

  - param_id –

    SVM parameters IDs that must be one of the following:

    > - CvSVM::C
    > - CvSVM::GAMMA
    > - CvSVM::P
    > - CvSVM::NU
    > - CvSVM::COEF
    > - CvSVM::DEGREE

    The grid is generated for the parameter with this ID.

The function generates a grid for the specified parameter of the SVM algorithm. The grid may be passed to the function `CvSVM::train_auto()`.

## CvSVM::get_params

Returns the current SVM parameters.

C++:`` CvSVMParams `CvSVM::``get_params`()`const`

This function may be used to get the optimal parameters obtained while automatically training `CvSVM::train_auto()`.

## CvSVM::get_support_vector

Retrieves a number of support vectors and the particular vector.

C++:`` int `CvSVM::``get_support_vector_count`()`const`

C++:`` const float* `CvSVM::``get_support_vector`(int i)`const`

Python:`` `cv2.SVM.``get_support_vector_count`() → retval

  Parameters: | i – Index of the particular support vector.

The methods can be used to retrieve a set of support vectors.

## CvSVM::get_var_count

Returns the number of used features (variables count).

C++:`` int `CvSVM::``get_var_count`()`const`

Python:`` `cv2.SVM.``get_var_count`() → retval

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