支持向量机

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支持向量机是一种有监督学习的方法，它广泛的应用于统计分类以及回归分析中. 支持向量机属于一般化线性分类器.他们也可以认为是提克洛夫规则化（Tikhonov Regularization）方法的一个特例.这族分类器的特点是他们能够同时最小化经验误差与最大化几何边缘区.因此支持向量机也被称为最大边缘区分类器.

[编辑] 介绍

[编辑] 动机

有很多个分类器(超平面）可以把数据分开,但是只有一个能够达到最大分割.

我们通常希望分类的过程是一个机器学习的过程。这些数据点并不需要是 $\mathbb{R}^2$ 中的点，而可以是任意 $\mathbb{R}^p$ (统计学符号)中或者 $\mathbb{R}^n$ (计算机科学符号) 的点.我们希望能够把这些点通过一个n-1维的超平面分开.通常这个被称为线性分类器.有很多分类器都符合这个要求.但是我们还希望找到分类最佳的平面，即使得属于两个不同类的数据点间隔最大的那个面，该面亦称为最大间隔超平面.如果我们能够找到这个面，那么这个分类器就称为最大间隔分类器.

[编辑] 问题定义

设样本属于两个类，用该样本训练svm得到的最大间隔超平面。在超平面上的样本点也称为支持向量.

我们考虑以下形式的样本点

$\{ (\mathbf{x}_1, c_1), (\mathbf{x}_2, c_2), \ldots, (\mathbf{x}_n, c_n)\}$

其中c_i为1或−1 --用以表示数据点属于哪个类. $\mathbf{x}_i$ 是一个 $p -$ (统计学符号), 或 $n -$ (计算机科学符号) 维向量，其每个元素都被缩放到[0,1]或[-1,1].缩放的目的是防止方差大的随机变量主导分类过程.我们可以把这些数据称为训练数据,希望我们的支持向量机能够通过一个超平面正确的把他们分开.超平面的数学形式可以写作

$\mathbf{w}\cdot\mathbf{x} - b=0.$

根据几何知识,我们知道 $\mathbf{w}$ 向量垂直于分类超平面.加入位移b的目的是增加间隔.如果没有b的话,那超平面将不得不通过原点,限制了这个方法的灵活性.

由于我们要求最大间隔,因此我们需要知道支持向量以及(与最佳超平面)平行的并且离支持向量最近的超平面.我们可以看到这些平行超平面可以由方程族

$\mathbf{w}\cdot\mathbf{x} - b=1,$

$\mathbf{w}\cdot\mathbf{x} - b=-1.$

来表示.

如果这些训练数据是线性可分的,那就可以找到这样两个超平面,在它们之间没有任何样本点并且这两个超平面之间的距离也最大.通过几何不难得到这两个超平面之间的距离是 2/|w|,因此我们需要最小化 |w|. 同时为了使得样本数据点都在超平面的间隔区以外,我们需要保证对于所有的 $i$ 满足其中的一个条件

$\mathbf{w}\cdot\mathbf{x_i} - b \ge 1\qquad\mathrm{or}$

$\mathbf{w}\cdot\mathbf{x_i} - b \le -1\qquad\mathrm{}$

这两个式子可以写作:

$c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1, \quad 1 \le i \le n.\qquad\qquad(1)$

[编辑] 原型

现在寻找最佳超平面这个问题就变成了在(1)这个约束条件下最小化|w|.这是一个二次项规划QP(quadratic programming)最优化中的问题.

更清楚的，它可以表示如下

最小化 $(1/2)||\mathbf{w}||^2$ , 满足 $c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1, \quad 1 \le i \le n.$ .

1/2 这个因子是为了数学上表达的方便加上的.

[编辑] 对偶型(Dual Form)

把原型的分类规则写作对偶型，可以看到分类器其实是一个关于支持向量（即那些在间隔区边缘的训练样本点）的函数。

支持向量机的对偶型如下： $\max \sum_{i=1}^n\alpha_i - \sum_{i,j} \alpha_i \alpha_j c_i c_j x_i^Tx_j$ 并满足 $α i > = 0$

[编辑] 软间隔

1995年, Corinna Cortes 与Vapnik 提出了一种改进的最大间隔区方法,这种方法可以处理标记错误的样本. If there exists no hyperplane that can split the "yes" and "no" examples, the Soft Margin method will choose a hyperplane that splits the examples as cleanly as possible, while still maximizing the distance to the nearest cleanly split examples. This work popularized the expression Support Vector Machine or SVM. The method introduces slack variables, $ξ i$ , which measure the degree of misclassification of the datum $x i$

$c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i \quad 1 \le i \le n \quad\quad(2)$ .

The objective function is then increased by a function which penalises non-zero $ξ i$ , and the optimisation becomes a trade off between a large margin, and a small error penalty. If the penalty function is linear, the equation (3) now transforms to

$\min ||w||^2 + C \sum_i \xi_i \quad \mathbf{such \; that \;}c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i \quad 1 \le i \le n$

This constraint in (2) along with the objective of minimizing |w| can be solved using Lagrange multipliers. The key advantage of a linear penalty function is that the slack variables vanish from the dual problem, with the constant C appearing only as an additional constraint on the Lagrange multipliers. Non-linear penalty functions have been used, particularly to reduce the effect of outliers on the classifier, but unless care is taken, the problem becomes non-convex, and thus it is considerably more difficult to find a global solution.

[编辑] 非线性分类

The original optimal hyperplane algorithm proposed by Vladimir Vapnik in 1963 was a linear classifier. However, in 1992, Bernhard Boser, Isabelle Guyon and Vapnik suggested a way to create non-linear classifiers by applying the kernel trick (originally proposed by Aizerman) to maximum-margin hyperplanes. The resulting algorithm is formally similar, except that every dot product is replaced by a non-linear kernel function. This allows the algorithm to fit the maximum-margin hyperplane in the transformed feature space. The transformation may be non-linear and the transformed space high dimensional; thus though the classifier is a hyperplane in the high-dimensional feature space it may be non-linear in the original input space.

If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimension. Maximum margin classifiers are well regularized, so the infinite dimension does not spoil the results. Some common kernels include,

Polynomial (homogeneous): $k(\mathbf{x},\mathbf{x}')=(\mathbf{x} \cdot \mathbf{x'})^d$
Polynomial (inhomogeneous): $k(\mathbf{x},\mathbf{x}')=(\mathbf{x} \cdot \mathbf{x'} + 1)^d$
Radial Basis Function: $k(\mathbf{x},\mathbf{x}')=\exp(-\gamma \|\mathbf{x} - \mathbf{x'}\|^2)$ , for $γ > 0$
Gaussian Radial basis function: $k(\mathbf{x},\mathbf{x}')=\exp(- \frac{\|\mathbf{x} - \mathbf{x'}\|^2}{2 \sigma^2})$
Sigmoid: $k(\mathbf{x},\mathbf{x}')=\tanh(\kappa \mathbf{x} \cdot \mathbf{x'}+c)$ , for some (not every) $κ > 0$ and $c < 0$

[编辑] 回归

A version of a SVM for regression was proposed in 1996 by Vapnik, Harris Drucker, Chris Burges, Linda Kaufman and Alex Smola. This method is called support vector regression (SVR). The model produced by support vector classification (as described above) only depends 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 SVR only depends on a subset of the training data, because the cost function for building the model ignores any training data that is close (within a threshold $ε$ ) to the model prediction.

[编辑] 实现

The parameters of the maximum-margin hyperplane are derived by solving the optimization. There exist several specialized algorithms for quickly solving the QP problem that arises from SVMs, mostly reliant on heuristics for breaking the problem down into smaller, more-manageble chunks. A common method for solving the QP problem is Platt's SMO algorithm, breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm such as conjugate gradient methods.

[编辑] 参见

Predictive analytics

[编辑] 参考书目

B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, 5th Annual ACM Workshop on COLT, pages 144-152, Pittsburgh, PA, 1992. ACM Press.
Corinna Cortes and V. Vapnik, "Support-Vector Networks, Machine Learning, 20, 1995. [1]
Christopher J. C. Burges. "A Tutorial on Support Vector Machines for Pattern Recognition". Data Mining and Knowledge Discovery 2:121 - 167, 1998 (Also available at CiteSeer: [2])
Nello Cristianini and John Shawe-Taylor. An Introduction to Support Vector Machines and other kernel-based learning methods. Cambridge University Press, 2000. ISBN 0-521-78019-5 ([3] SVM Book)
Harris Drucker, Chris J.C. Burges, Linda Kaufman, Alex Smola and Vladimir Vapnik (1997). "Support Vector Regression Machines". Advances in Neural Information Processing Systems 9, NIPS 1996, 155-161, MIT Press.
Huang T.-M., Kecman V., Kopriva I. (2006), Kernel Based Algorithms for Mining Huge Data Sets, Supervised, Semi-supervised, and Unsupervised Learning, Springer-Verlag, Berlin, Heidelberg, 260 pp. 96 illus., Hardcover, ISBN 3-540-31681-7 [4]
Vojislav Kecman: "Learning and Soft Computing - Support Vector Machines, Neural Networks, Fuzzy Logic Systems", The MIT Press, Cambridge, MA, 2001.[5]
Tapio Pahikkala, Sampo Pyysalo, Jorma Boberg, Aleksandr Mylläri and Tapio Salakoski. Improving the Performance of Bayesian and Support Vector Classifiers in Word Sense Disambiguation using Positional Information. In Proceedings of the International and Interdisciplinary Conference on Adaptive Knowledge Representation and Reasoning (AKRR'05), Jun 2005.
Bernhard Schölkopf and A. J. Smola: Learning with Kernels. MIT Press, Cambridge, MA, 2002. (Partly available on line: [6].) ISBN 0-262-19475-9
Bernhard Schölkopf, Christopher J.C. Burges, and Alexander J. Smola (editors). "Advances in Kernel Methods: Support Vector Learning". MIT Press, Cambridge, MA, 1999. ISBN 0-262-19416-3. [7]
John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. ISBN 0-521-81397-2 ([8] Kernel Methods Book)
P.J. Tan and D.L. Dowe (2004), MML Inference of Oblique Decision Trees, Lecture Notes in Artificial Intelligence (LNAI) 3339, Springer-Verlag, pp1082-1088. Links require password. (This paper uses minimum message length (MML) and actually incorporates probabilistic support vector machines in the leaves of decision trees.)
Vladimir Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, 1999. ISBN 0-387-98780-0

[编辑] 外部链接

[编辑] 概念

www.pascal-network.org (EU Funded Network on Pattern Analysis, Statistical Modelling and Computational Learning)
www.kernel-machines.org (general information and collection of research papers)
www.kernel-methods.net (News, Links, Code related to Kernel methods - Academic Site)
www.support-vector.net (News, Links, Code related to Support Vector Machines - Academic Site)
www.support-vector-machines.org (Literature, Review, Software, Links related to Support Vector Machines - Academic Site)
www.support-vector.ws (Free educational MATLAB based software for SVMs, NN and FL , Links, Publications downloads, Semisupervised learning software SemiL, Links)

[编辑] 软件

Lush -- an Lisp-like interpreted/compiled language with C/C++/Fortran interfaces that has packages to interface to a number of different SVM implementations. Interfaces to LASVM, LIBSVM, mySVM, SVQP, SVQP2 (SVQP3 in future) are available. Leverage these against Lush's other interfaces to machine learning, hidden markov models, numerical libraries (LAPACK, BLAS, GSL), and builtin vector/matrix/tensor engine.
SVMlight -- a popular implementation of the SVM algorithm by Thorsten Joachims; it can be used to solve classification, regression and ranking problems.
LIBSVM -- A Library for Support Vector Machines, Chih-Chung Chang and Chih-Jen Lin
YALE -- a powerful machine learning toolbox containing wrappers for SVMLight, LibSVM, and MySVM in addition to many evaluation and preprocessing methods.
LS-SVMLab - Matlab/C SVM toolbox - well-documented, many features
Gist -- implementation of the SVM algorithm with feature selection.
Weka -- a machine learning toolkit that includes an implementation of an SVM classifier; Weka can be used both interactively though a graphical interface or as a software library. (One of them is called "SMO". In the GUI Weka explorer, it is under the "classify" tab if you "Choose" an algorithm.)
OSU SVM - Matlab implementation based on LIBSVM
Torch - C++ machine learning library with SVM
Shogun - Large Scale Machine Learning Toolbox with interfaces to Octave, Matlab, Python, R
Spider - Machine learning library for Matlab
e1071 - Machine learning library for R
SimpleSVM - SimpleSVM toolbox for Matlab
SVM and Kernel Methods Matlab Toolbox
PCP -- C program for supervised pattern classification. Includes LIBSVM wrapper.
TinySVM -- a small SVM implementation, written in C++

[编辑] 互动 SVM 应用

ECLAT classification of Expressed Sequence Tag (EST) from mixed EST pools using codon usage
EST3 classification of Expressed Sequence Tag (EST) from mixed EST pools using nucleotide triples

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