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Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. They belong to a family of generalized linear classifiers. They can also be considered a special case of Tikhonov regularization. A special property of SVMs is that they simultaneously minimize the empirical classification error and maximize the geometric margin; hence they are also known as maximum margin classifiers. Image File history File links Circle-question-red. ... Supervised learning is a machine learning technique for creating a function from training data. ... Statistical classification is a type of supervised learning problem in which labeled training data is used to create a function that will correctly predict the label of future data. ... In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Tikhonov regularization is the most commonly used method of regularization of ill-posed problems. ...

Introduction

Support vector machines map input vectors to a higher dimensional space where a maximal separating hyperplane is constructed. Two parallel hyperplanes are constructed on each side of the hyperplane that separates the data. The separating hyperplane is the hyperplane that maximizes the distance between the two parallel hyperplanes. An assumption is made that the larger the margin or distance between these parallel hyperplanes the better the generalisation error of the classifier will be. An excellent tutorial is "A tutorial on Support Vector Machines for pattern recognition" by C.J.C Burges. A comparison of the SVM to other classifiers are performed by van der Walt and Barnard (see reference section). ^{[original research?]} A hyperplane is a concept in geometry. ...

Motivation

Many linear classifiers (hyperplanes) separate the data. However, only one achieves maximum separation.

Often we are interested in classifying data as a part of a machine-learning process. Each data point will be represented by a p-dimensional vector (a list of p numbers). Each of these data points belongs to only one of two classes. We are interested in whether we can separate them with an "p minus 1" dimensional hyperplane. This is a typical form of linear classifier. There are many linear classifiers that might satisfy this property. However, we are additionally interested in finding out if we can achieve maximum separation (margin) between the two classes. By this we mean that we pick the hyperplane so that the distance from the hyperplane to the nearest data point is maximized. That is to say that the nearest distance between a point in one separated hyperplane and a point in the other separated hyperplane is maximized. Now, if such a hyperplane exists, it is clearly of interest and is known as the maximum-margin hyperplane and such a linear classifier is known as a maximum margin classifier. Image File history File links Classifier. ... Image File history File links Classifier. ... A hyperplane is a concept in geometry. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The term margin has many meanings: In telecommunication, margin has the following meanings: In communications systems, the maximum degree of signal distortion that can be tolerated without affecting the restitution, without its being interpreted incorrectly by the decision circuit. ... In geometry, a maximum-margin hyperplane is a hyperplane which separates two clouds of points and is at equal distance from the two. ...

Formalization

We consider data points of the form:

where the c_i is either 1 or −1, a constant denoting the class to which the point belongs. Each is a p-dimensional real vector, usually of normalised (Normalizing constant) [0,1] or [-1,1] values. The scaling is important to guard against variables (attributes) with larger variance that might otherwise dominate the classification. We can view this as training data, which denotes the correct classification which we would like the SVM to eventually distinguish, by means of the dividing (or separating) hyperplane, which takes the form In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...

Maximum-margin hyperplanes for a SVM trained with samples from two classes. Samples along the hyperplanes are called the support vectors.

The vector points perpendicular to the separating hyperplane. Adding the offset parameter b allows us to increase the margin. In its absence, the hyperplane is forced to pass through the origin, restricting the solution. Image File history File links Download high resolution version (740x903, 6 KB) Summary Description: Support vector machine optimal hyperplanes and training samples Source: self-made Date: created 15. ... Image File history File links Download high resolution version (740x903, 6 KB) Summary Description: Support vector machine optimal hyperplanes and training samples Source: self-made Date: created 15. ...

As we are interested in the maximum margin, we are interested in the support vectors and the parallel hyperplanes (to the optimal hyperplane) closest to these support vectors in either class. It can be shown that these parallel hyperplanes can be described by equations

If the training data are linearly separable, we can select these hyperplanes so that there are no points between them and then try to maximize their distance. By using geometry, we find the distance between the hyperplanes is 2/|w|, so we want to minimize |w|. To exclude data points, we need to ensure that for all i either In geometry, when two sets of points in a two-dimensional graph can be completely separated by a single line, they are said to be linearly separable. ...

This can be rewritten as:

Primal Form

The problem now is to minimize |w| subject to the constraint (1). This is a quadratic programming (QP) optimization problem. More clearly, Quadratic programming (QP) is a special type of mathematical optimization problem. ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ...

minimize , subject to .

The factor of 1/2 is used for mathematical convenience.

Dual Form

Writing the classification rule in its dual form reveals that classification is only a function of the support vectors, i.e., the training data that lie on the margin. The dual of the SVM can be shown to be:

subject to ,

where the α terms constitute a dual representation for the weight vector in terms of the training set:

Soft margin

In 1995, Corinna Cortes and Vladimir Vapnik suggested a modified maximum margin idea that allows for mislabeled examples. 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 Year 1995 (MCMXCV) was a common year starting on Sunday (link will display full 1995 Gregorian calendar). ... Vladimir Naumovich Vapnik is one of the main developers of Vapnik Chervonenkis theory. ...

.

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

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. Fig. ...

Non-linear classification

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. Vladimir Naumovich Vapnik is one of the main developers of Vapnik Chervonenkis theory. ... Year 1963 (MCMLXIII) was a common year starting on Tuesday (link will display full calendar) of the Gregorian calendar. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Year 1992 (MCMXCII) was a leap year starting on Wednesday (link will display full 1992 Gregorian calendar). ... In machine learning, the kernel trick is a method for converting a linear classifier algorithm into a non-linear one by using a non-linear function to map the original observations into a higher-dimensional space; this makes a linear classification in the new space equivalent to non-linear classification... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In analysis, consider an integral transform T which transforms a function f into a function Tf given by the integral formula The function k(x,y) that appears in this formula is the kernel of the operator T. See also: Dirichlet kernel convolution kernel trick Categories: Stub | Mathematical analysis ... Space has been an interest for philosophers and scientists for much of human history. ...

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, Generally, the word gaussian pertains to Carl Friedrich Gauss and his ideas. ... Radial basis functions are a means for interpolation in a stream of data. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, inverse problems are often ill-posed. ...

• Polynomial (homogeneous):
• Polynomial (inhomogeneous):
• Radial Basis Function: , for γ > 0
• Gaussian Radial basis function:
• Sigmoid: , for some (not every) κ > 0 and c < 0

The logistic curve A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an S shape. ...

Regression

A version of a SVM for regression was proposed in 1996 by Vladimir 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 are close (within a threshold ε) to the model prediction. Year 1996 (MCMXCVI) was a leap year starting on Monday (link will display full 1996 Gregorian calendar). ... Vladimir Naumovich Vapnik is one of the main developers of Vapnik Chervonenkis theory. ...

Implementation

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-manageable chunks. A common method for solving the QP problem is Platt's SMO algorithm, which 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. In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive definite. ...

See also

• Predictive analytics
• Relevance Vector Machine, a probabilistic sparse kernel model identical in functional form to SVM.

Predictive analytics encompasses a variety of techniques from statistics and data mining that process current and historical data in order to make “predictions” about future events. ... Relevance Vector Machine (RVMs) is a machine learning technique that uses Bayesian theory to obtain sparse solutions for regression and classification. ...

References

• C.M. van der Walt and E. Barnard,“Data characteristics that determine classifier performance”, in Proceedings of the Sixteenth Annual Symposium of the Pattern Recognition Association of South Africa, pp.160-165, 2006.
• 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 [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]
• 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. (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
• Vladimir Vapnik, S.Kotz "Estimation of Dependences Based on Empirical Data" Springer, 2006. ISBN: 0387308652, 510 pages [this is a reprint of Vapnik's early book describing philosophy behind SVM approach. The 2006 Appendix describes recent development].
• Dmitriy Fradkin and Ilya Muchnik "Support Vector Machines for Classification" in J. Abello and G. Carmode (Eds) "Discrete Methods in Epidemiology", DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 70, pp. 13-20, 2006. [9]. Succinctly describes theoretical ideas behind SVM.
• Kristin P. Bennett and Colin Campbell, "Support Vector Machines: Hype or Hallelujah?", SIGKDD Explorations, 2,2, 2000, 1-13. [10]. Excellent introduction to SVMs with helpful figures.

Minimum message length (MML) is a formal information theory restatement of Occams Razor: even when models are not equal in goodness of fit accuracy to the observed data, the one generating the shortest overall message is more likely to be correct (where the message consists of a statement of... Minimum message length (MML) is a formal information theory restatement of Occams Razor: even when models are not equal in goodness of fit accuracy to the observed data, the one generating the shortest overall message is more likely to be correct (where the message consists of a statement of... In operations research, specifically in decision analysis, a decision tree is a decision support tool that uses a graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. ...

External links

General

• 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)

Software

• The Kernel-Machine Library (GNU) C++ template library for Support Vector Machines
• Lush -- a 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.
• SVMProt -- Protein Functional Family Prediction.
• 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. (The SVM implementation is called "SMO". It can be found in the Weka Explorer GUI, under the "functions" category.)
• OSU SVM - Matlab implementation based on LIBSVM
• Torch - C++ machine learning library with SVM
• Shogun - Large Scale Machine Learning Toolbox that provides several SVM implementations (like libSVM, SVMlight) under a common framework and interfaces to Octave, Matlab, Python, R
• Spider - Machine learning library for Matlab
• kernlab - Kernel-based Machine Learning library for R
• 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++
• pcSVM is an object oriented SVM framework written in C++ and provides wrapping to Python classes. The site provides a stand alone demo tool for experimenting with SVMs.
• PyML -- a Python machine learning package. Includes: SVM, nearest neighbor classifiers, ridge regression, Multi-class methods (one-against-one and one-against-rest), Feature selection (filter methods, RFE, multiplicative update, Model selection, Classifier testing (cross-validation, error rates, ROC curves, statistical test for comparing classifiers).
• Algorithm::SVM -- Perl bindings for the libsvm Support Vector Machine library

YALE (Yet Another Learning Environment) is an environment for machine learning experiments and data mining. ... Weka is a suite of machine learning software written in Java at the University of Waikato which implements several machine learning algorithms from various learning paradigms. ...

Interactive SVM applications

• 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