R: Support Vector Machines

ksvm {kernlab} | R Documentation

## Support Vector Machines

### Description

Support Vector Machines are an excellent tool for classification, novelty detection, and regression. `ksvm` supports the well known C-svc, nu-svc, (classification) one-class-svc (novelty) eps-svr, nu-svr (regression) formulations along with native multi-class classification formulations and the bound-constraint SVM formulations.
`ksvm` also supports class-probabilities output and confidence intervals for regression.

### Usage

    ## S4 method for signature 'formula'
    ksvm(x, data = NULL, ..., subset, na.action = na.omit, scaled = TRUE)

    ## S4 method for signature 'vector'
    ksvm(x, ...)

    ## S4 method for signature 'matrix'
    ksvm(x, y = NULL, scaled = TRUE, type = NULL,
         kernel ="rbfdot", kpar = "automatic",
         C = 1, nu = 0.2, epsilon = 0.1, prob.model = FALSE,
         class.weights = NULL, cross = 0, fit = TRUE, cache = 40,
         tol = 0.001, shrinking = TRUE, ...,
         subset, na.action = na.omit)

    ## S4 method for signature 'kernelMatrix'
    ksvm(x, y = NULL, type = NULL,
         C = 1, nu = 0.2, epsilon = 0.1, prob.model = FALSE,
         class.weights = NULL, cross = 0, fit = TRUE, cache = 40,
         tol = 0.001, shrinking = TRUE, ...)

    ## S4 method for signature 'list'
    ksvm(x, y = NULL, type = NULL,
         kernel = "stringdot", kpar = list(length = 4, lambda = 0.5),
         C = 1, nu = 0.2, epsilon = 0.1, prob.model = FALSE,
         class.weights = NULL, cross = 0, fit = TRUE, cache = 40,
         tol = 0.001, shrinking = TRUE, ...,
         na.action = na.omit)

### Arguments

`x` | a symbolic description of the model to be fit. When not using a formula x can be a matrix or vector containing the training data or a kernel matrix of class `kernelMatrix` of the training data or a list of character vectors (for use with the string kernel). Note, that the intercept is always excluded, whether given in the formula or not.
`data` | an optional data frame containing the training data, when using a formula. By default the data is taken from the environment which ‘ksvm’ is called from.
`y` | a response vector with one label for each row/component of `x`. Can be either a factor (for classification tasks) or a numeric vector (for regression).
`scaled` | A logical vector indicating the variables to be scaled. If `scaled` is of length 1, the value is recycled as many times as needed and all non-binary variables are scaled. Per default, data are scaled internally (both `x` and `y` variables) to zero mean and unit variance. The center and scale values are returned and used for later predictions.
`type` | `ksvm` can be used for classification , for regression, or for novelty detection. Depending on whether `y` is a factor or not, the default setting for `type` is `C-svc` or `eps-svr`, respectively, but can be overwritten by setting an explicit value. Valid options are: `C-svc` C classification `nu-svc` nu classification `C-bsvc` bound-constraint svm classification `spoc-svc` Crammer, Singer native multi-class `kbb-svc` Weston, Watkins native multi-class `one-svc` novelty detection `eps-svr` epsilon regression `nu-svr` nu regression `eps-bsvr` bound-constraint svm regression
`kernel` | the kernel function used in training and predicting. This parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments (see `kernels`). kernlab provides the most popular kernel functions which can be used by setting the kernel parameter to the following strings: `rbfdot` Radial Basis kernel "Gaussian" `polydot` Polynomial kernel `vanilladot` Linear kernel `tanhdot` Hyperbolic tangent kernel `laplacedot` Laplacian kernel `besseldot` Bessel kernel `anovadot` ANOVA RBF kernel `splinedot` Spline kernel `stringdot` String kernel Setting the kernel parameter to "matrix" treats `x` as a kernel matrix calling the `kernelMatrix` interface. The kernel parameter can also be set to a user defined function of class kernel by passing the function name as an argument.
`kpar` | the list of hyper-parameters (kernel parameters). This is a list which contains the parameters to be used with the kernel function. For valid parameters for existing kernels are : `sigma` inverse kernel width for the Radial Basis kernel function "rbfdot" and the Laplacian kernel "laplacedot". `degree, scale, offset` for the Polynomial kernel "polydot" `scale, offset` for the Hyperbolic tangent kernel function "tanhdot" `sigma, order, degree` for the Bessel kernel "besseldot". `sigma, degree` for the ANOVA kernel "anovadot". `length, lambda, normalized` for the "stringdot" kernel where length is the length of the strings considered, lambda the decay factor and normalized a logical parameter determining if the kernel evaluations should be normalized. Hyper-parameters for user defined kernels can be passed through the kpar parameter as well. In the case of a Radial Basis kernel function (Gaussian) kpar can also be set to the string "automatic" which uses the heuristics in `sigest` to calculate a good `sigma` value for the Gaussian RBF or Laplace kernel, from the data. (default = "automatic").
`C` | cost of constraints violation (default: 1) this is the ‘C’-constant of the regularization term in the Lagrange formulation.
`nu` | parameter needed for `nu-svc`, `one-svc`, and `nu-svr`. The `nu` parameter sets the upper bound on the training error and the lower bound on the fraction of data points to become Support Vectors (default: 0.2).
`epsilon` | epsilon in the insensitive-loss function used for `eps-svr`, `nu-svr` and `eps-bsvm` (default: 0.1)
`prob.model` | if set to `TRUE` builds a model for calculating class probabilities or in case of regression, calculates the scaling parameter of the Laplacian distribution fitted on the residuals. Fitting is done on output data created by performing a 3-fold cross-validation on the training data. For details see references. (default: `FALSE`)
`class.weights` | a named vector of weights for the different classes, used for asymmetric class sizes. Not all factor levels have to be supplied (default weight: 1). All components have to be named.
`cache` | cache memory in MB (default 40)
`tol` | tolerance of termination criterion (default: 0.001)
`shrinking` | option whether to use the shrinking-heuristics (default: `TRUE`)
`cross` | if a integer value k>0 is specified, a k-fold cross validation on the training data is performed to assess the quality of the model: the accuracy rate for classification and the Mean Squared Error for regression
`fit` | indicates whether the fitted values should be computed and included in the model or not (default: `TRUE`)
`...` | additional parameters for the low level fitting function
`subset` | An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)
`na.action` | A function to specify the action to be taken if `NA`s are found. The default action is `na.omit`, which leads to rejection of cases with missing values on any required variable. An alternative is `na.fail`, which causes an error if `NA` cases are found. (NOTE: If given, this argument must be named.)

### Details

`ksvm` uses John Platt's SMO algorithm for solving the SVM QP problem an most SVM formulations. On the `spoc-svc`, `kbb-svc`, `C-bsvc` and `eps-bsvr` formulations a chunking algorithm based on the TRON QP solver is used.
For multiclass-classification with k classes, k > 2, `ksvm` uses the ‘one-against-one’-approach, in which k(k-1)/2 binary classifiers are trained; the appropriate class is found by a voting scheme, The `spoc-svc` and the `kbb-svc` formulations deal with the multiclass-classification problems by solving a single quadratic problem involving all the classes.
If the predictor variables include factors, the formula interface must be used to get a correct model matrix.
In classification when `prob.model` is `TRUE` a 3-fold cross validation is performed on the data and a sigmoid function is fitted on the resulting decision values f. The data can be passed to the `ksvm` function in a `matrix` or a `data.frame`, in addition `ksvm` also supports input in the form of a kernel matrix of class `kernelMatrix` or as a list of character vectors where a string kernel has to be used.
The `plot` function for binary classification `ksvm` objects displays a contour plot of the decision values with the corresponding support vectors highlighted.
The predict function can return class probabilities for classification problems by setting the `type` parameter to "probabilities".
The problem of model selection is partially addressed by an empirical observation for the RBF kernels (Gaussian , Laplace) where the optimal values of the sigma width parameter are shown to lie in between the 0.1 and 0.9 quantile of the \|x- x'\| statistics. When using an RBF kernel and setting `kpar` to "automatic", `ksvm` uses the `sigest` function to estimate the quantiles and uses the median of the values.

### Value

An S4 object of class `"ksvm"` containing the fitted model, Accessor functions can be used to access the slots of the object (see examples) which include:

`alpha` | The resulting support vectors, (alpha vector) (possibly scaled).
`alphaindex` | The index of the resulting support vectors in the data matrix. Note that this index refers to the pre-processed data (after the possible effect of `na.omit` and `subset`)
`coef` | The corresponding coefficients times the training labels.
`b` | The negative intercept.
`nSV` | The number of Support Vectors
`obj` | The value of the objective function. In case of one-against-one classification this is a vector of values
`error` | Training error
`cross` | Cross validation error, (when cross > 0)
`prob.model` | Contains the width of the Laplacian fitted on the residuals in case of regression, or the parameters of the sigmoid fitted on the decision values in case of classification.

### Note

Data is scaled internally by default, usually yielding better results.

### Author(s)

Alexandros Karatzoglou (SMO optimizers in C++ by Chih-Chung Chang & Chih-Jen Lin)
alexandros.karatzoglou@ci.tuwien.ac.at

### References

- Chang Chih-Chung, Lin Chih-Jen
  LIBSVM: a library for Support Vector Machines
  http://www.csie.ntu.edu.tw/~cjlin/libsvm

- Chih-Wei Hsu, Chih-Jen Lin
  BSVM http://www.csie.ntu.edu.tw/~cjlin/bsvm/

- J. Platt
  Probabilistic outputs for support vector machines and comparison to regularized likelihood methods
  Advances in Large Margin Classifiers, A. Smola, P. Bartlett, B. Schoelkopf and D. Schuurmans, Eds. Cambridge, MA: MIT Press, 2000.
  http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1639

- H.-T. Lin, C.-J. Lin and R. C. Weng
  A note on Platt's probabilistic outputs for support vector machines
  http://www.csie.ntu.edu.tw/~htlin/paper/doc/plattprob.pdf

- C.-W. Hsu and C.-J. Lin
  A comparison on methods for multi-class support vector machines
  IEEE Transactions on Neural Networks, 13(2002) 415-425.
  http://www.csie.ntu.edu.tw/~cjlin/papers/multisvm.ps.gz

- K. Crammer, Y. Singer
  On the learnability and design of output codes for multiclass prolems
  Computational Learning Theory, 35-46, 2000.
  http://webee.technion.ac.il/people/koby/publications/ecoc-mlj02.pdf

- J. Weston, C. Watkins
  Multi-class support vector machines
  In M. Verleysen, Proceedings of ESANN99 Brussels, 1999
  http://citeseer.ist.psu.edu/8884.html

### See Also

`predict.ksvm`, `ksvm-class`, `couple`

### Examples

    ## simple example using the spam data set
    data(spam)

    ## create test and training set
    index <- sample(1:dim(spam)[1])
    spamtrain <- spam[index[1:floor(dim(spam)[1]/2)], ]
    spamtest <- spam[index[((ceiling(dim(spam)[1]/2)) + 1):dim(spam)[1]], ]

    ## train a support vector machine
    filter <- ksvm(type~.,data=spamtrain,kernel="rbfdot",
                   kpar=list(sigma=0.05),C=5,cross=3)
    filter

    ## predict mail type on the test set
    mailtype <- predict(filter,spamtest[,-58])

    ## Check results
    table(mailtype,spamtest[,58])

    ## Another example with the famous iris data
    data(iris)

    ## Create a kernel function using the build in rbfdot function
    rbf <- rbfdot(sigma=0.1)
    rbf

    ## train a bound constraint support vector machine
    irismodel <- ksvm(Species~.,data=iris,type="C-bsvc",
                      kernel=rbf,C=10,prob.model=TRUE)

    irismodel

    ## get fitted values
    fitted(irismodel)

    ## Test on the training set with probabilities as output
    predict(irismodel, iris[,-5], type="probabilities")

    ## Demo of the plot function
    x <- rbind(matrix(rnorm(120),,2),matrix(rnorm(120,mean=3),,2))
    y <- matrix(c(rep(1,60),rep(-1,60)))

    svp <- ksvm(x,y,type="C-svc")
    plot(svp,data=x)

    ### Use kernelMatrix
    K <- as.kernelMatrix(crossprod(t(x)))

    svp2 <- ksvm(K, y, type="C-svc")

    svp2

    # test data
    xtest <- rbind(matrix(rnorm(20),,2),matrix(rnorm(20,mean=3),,2))
    # test kernel matrix i.e. inner/kernel product of test data with
    # Support Vectors

    Ktest <- as.kernelMatrix(crossprod(t(xtest),t(x[SVindex(svp2), ])))

    predict(svp2, Ktest)

    #### Use custom kernel

    k <- function(x,y) {(sum(x*y) +1)*exp(-0.001*sum((x-y)^2))}
    class(k) <- "kernel"

    data(promotergene)

    ## train svm using custom kernel
    gene <- ksvm(Class~.,data=promotergene[c(1:20, 80:100),],kernel=k,
                 C=5,cross=5)

    gene

    #### Use text with string kernels
    data(reuters)
    is(reuters)
    tsv <- ksvm(reuters,rlabels,kernel="stringdot",
                kpar=list(length=5),cross=3,C=10)
    tsv

    ## regression
    # create data
    x <- seq(-20,20,0.1)
    y <- sin(x)/x + rnorm(401,sd=0.03)

    # train support vector machine
    regm <- ksvm(x,y,epsilon=0.01,kpar=list(sigma=16),cross=3)
    plot(x,y,type="l")
    lines(x,predict(regm,x),col="red")

[Package kernlab version 0.9-25 Index]