# Support Vector Machines (SVMs)

## Support Vector Machines from [LIBSVM]

class mlpy.LibSvm(svm_type='c_svc', kernel_type='linear', degree=3, gamma=0.001, coef0=0, C=1, nu=0.5, eps=0.001, p=0.1, cache_size=100, shrinking=True, probability=False, weight={})

  LibSvm.

  Parameters:
    svm_typestring

      SVM type, can be one of: ‘c_svc’, ‘nu_svc’, ‘one_class’, ‘epsilon_svr’, ‘nu_svr’

    kernel_typestring

      kernel type, can be one of: ‘linear’ (uT*v), ‘poly’ ((gamma*uT*v + coef0)^degree), ‘rbf’ (exp(-gamma*|u-v|^2)), ‘sigmoid’ (tanh(gamma*uT*v + coef0))

    degreeint (for ‘poly’ kernel_type)

      degree in kernel

    gammafloat (for ‘poly’, ‘rbf’, ‘sigmoid’ kernel_type)

      gamma in kernel (e.g. 1 / number of features)

    coef0float (for ‘poly’, ‘sigmoid’ kernel_type)

      coef0 in kernel

    Cfloat (for ‘c_svc’, ‘epsilon_svr’, ‘nu_svr’)

      cost of constraints violation

    nufloat (for ‘nu_svc’, ‘one_class’, ‘nu_svr’)

      nu parameter

    epsfloat

      stopping criterion, usually 0.00001 in nu-SVC, 0.001 in others

    pfloat (for ‘epsilon_svr’)

      p is the epsilon in epsilon-insensitive loss function of epsilon-SVM regression

    cache_sizefloat [MB]

      size of the kernel cache, specified in megabytes

    shrinkingbool

      use the shrinking heuristics

    probabilitybool

      predict probability estimates

    weightdict

      changes the penalty for some classes (if the weight for a class is not changed, it is set to 1). For example, to change penalty for classes 1 and 2 to 0.5 and 0.8 respectively set weight={1:0.5, 2:0.8}

Example on `spiral dataset`:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> import mlpy
    >>> f = np.loadtxt("spiral.data")
    >>> x, y = f[:, :2], f[:, 2]
    >>> svm = mlpy.LibSvm(svm_type='c_svc', kernel_type='rbf', gamma=100)
    >>> svm.learn(x, y)
    >>> xmin, xmax = x[:,0].min()-0.1, x[:,0].max()+0.1
    >>> ymin, ymax = x[:,1].min()-0.1, x[:,1].max()+0.1
    >>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.01), np.arange(ymin, ymax, 0.01))
    >>> xnew = np.c_[xx.ravel(), yy.ravel()]
    >>> ynew = svm.pred(xnew).reshape(xx.shape)
    >>> fig = plt.figure(1)
    >>> plt.set_cmap(plt.cm.Paired)
    >>> plt.pcolormesh(xx, yy, ynew)
    >>> plt.scatter(x[:,0], x[:,1], c=y)
    >>> plt.show()

[LIBSVM]

Chih-Chung Chang and Chih-Jen Lin. LIBSVM: a library for support vector machines. 2001. Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm

[Cristianini]

N Cristianini and J Shawe-Taylor. An introduction to support vector machines. Cambridge University Press.

[Vapnik95]

V Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, 1995.

## Kernel Adatron

class mlpy.KernelAdatron(C=1000, maxsteps=1000, eps=0.01)

  Kernel Adatron algorithm without-bias-term (binary classifier).

  The algoritm handles a version of the 1-norm soft margin support vector machine. If C is very high the algoritm handles a version of the hard margin SVM.

  Use positive definite kernels (such as Gaussian and Polynomial kernels)

  Parameters:
    Cfloat

      upper bound on the value of alpha

    maxstepsinteger (> 0)

      maximum number of steps

    epsfloat (>=0)

      the algoritm stops when abs(1 - margin) < eps

Example:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> import mlpy
    >>> np.random.seed(0)
    >>> mean1, cov1, n1 = [1, 4.5], [[1,1],[1,2]], 20  # 20 samples of class 1
    >>> x1 = np.random.multivariate_normal(mean1, cov1, n1)
    >>> y1 = np.ones(n1, dtype=int)
    >>> mean2, cov2, n2 = [2.5, 2.5], [[1,1],[1,2]], 30 # 30 samples of class 2
    >>> x2 = np.random.multivariate_normal(mean2, cov2, n2)
    >>> y2 = 2 * np.ones(n2, dtype=int)
    >>> x = np.concatenate((x1, x2), axis=0) # concatenate the samples
    >>> y = np.concatenate((y1, y2))
    >>> K = mlpy.kernel_gaussian(x, x, sigma=2) # kernel matrix
    >>> xmin, xmax = x[:,0].min()-1, x[:,0].max()+1
    >>> ymin, ymax = x[:,1].min()-1, x[:,1].max()+1
    >>> xx, yy = np.meshgrid(np.arange(xmin, xmax, 0.02), np.arange(ymin, ymax, 0.02))
    >>> xt = np.c_[xx.ravel(), yy.ravel()] # test points
    >>> Kt = mlpy.kernel_gaussian(xt, x, sigma=2) # test kernel matrix
    >>> fig = plt.figure(1)
    >>> cmap = plt.set_cmap(plt.cm.Paired)
    >>> for i, c in enumerate([1, 10, 100, 1000]):
    ...     ka = mlpy.KernelAdatron(C=c)
    ...     ax = plt.subplot(2, 2, i+1)
    ...     ka.learn(K, y)
    ...     ytest = ka.pred(Kt).reshape(xx.shape)
    ...     title = ax.set_title('C: %s; margin: %.3f; steps: %s;' % (c, ka.margin(), ka.steps()))
    ...     plot1 = plt.pcolormesh(xx, yy, ytest)
    ...     plot2 = plt.scatter(x[:,0], x[:,1], c=y)
    >>> plt.show()

[Friess]

Friess, Cristianini, Campbell. The Kernel-Adatron Algorithm: a Fast and Simple Learning Procedure for Support Vector Machines.

[Kecman03]

Kecman, Vogt, Huang. On the Equality of Kernel AdaTron and Sequential Minimal Optimization in Classification and Regression Tasks and Alike Algorithms for Kernel Machines. ESANN’2003 proceedings - European Symposium on Artificial Neural Networks, ISBN 2-930307-03-X, pp. 215-222.