R: Least Squares Support Vector Machine

lssvm {kernlab} | R Documentation

## Least Squares Support Vector Machine

### Description

The `lssvm` function is an implementation of the Least Squares SVM. `lssvm` includes a reduced version of Least Squares SVM using a decomposition of the kernel matrix which is calculated by the `csi` function.

### Usage

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

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

    ## S4 method for signature 'matrix'
    lssvm(x, y, scaled = TRUE, kernel = "rbfdot", kpar = "automatic",
          type = NULL, tau = 0.01, reduced = TRUE, tol = 0.0001,
          rank = floor(dim(x)[1]/3), delta = 40, cross = 0, fit = TRUE,
          ..., subset, na.action = na.omit)

    ## S4 method for signature 'kernelMatrix'
    lssvm(x, y, type = NULL, tau = 0.01,
          tol = 0.0001, rank = floor(dim(x)[1]/3), delta = 40, cross = 0,
          fit = TRUE, ...)

    ## S4 method for signature 'list'
    lssvm(x, y, scaled = TRUE,
          kernel = "stringdot", kpar = list(length=4, lambda = 0.5),
          type = NULL, tau = 0.01, reduced = TRUE, tol = 0.0001,
          rank = floor(dim(x)[1]/3), delta = 40, cross = 0, fit = TRUE,
          ..., subset)

### Arguments

`x` | a symbolic description of the model to be fit, a matrix or vector containing the training data when a formula interface is not used or a `kernelMatrix` or a list of character vectors.
`data` | an optional data frame containing the variables in the model. By default the variables are taken from the environment which ‘lssvm’ 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 classification or regression - currently nor supported -).
`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 to zero mean and unit variance. The center and scale values are returned and used for later predictions.
`type` | Type of problem. Either "classification" or "regression". Depending on whether `y` is a factor or not, the default setting for `type` is "classification" or "regression" respectively, but can be overwritten by setting an explicit value. (regression is currently not supported)
`kernel` | the kernel function used in training and predicting. This parameter can be set to any function, of class kernel, which computes a dot product between two vector arguments. 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. `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").
`tau` | the regularization parameter (default 0.01)
`reduced` | if set to `FALSE` the full linear problem of the lssvm is solved, when `TRUE` a reduced method using `csi` is used.
`rank` | the maximal rank of the decomposed kernel matrix, see `csi`
`delta` | number of columns of cholesky performed in advance, see `csi` (default 40)
`tol` | tolerance of termination criterion for the `csi` function, lower tolerance leads to more precise approximation but may increase the training time and the decomposed matrix size (default: 0.0001)
`fit` | indicates whether the fitted values should be computed and included in the model or not (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 Mean Squared Error for regression
`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.)
`...` | additional parameters

### Details

Least Squares Support Vector Machines are reformulation to the standard SVMs that lead to solving linear KKT systems. The algorithm is based on the minimization of a classical penalized least-squares cost function. The current implementation approximates the kernel matrix by an incomplete Cholesky factorization obtained by the `csi` function, thus the solution is an approximation to the exact solution of the lssvm optimization problem. The quality of the solution depends on the approximation and can be influenced by the "rank" , "delta", and "tol" parameters.

### Value

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

`alpha` | the parameters of the `"lssvm"`
`coef` | the model coefficients (identical to alpha)
`b` | the model offset.
`xmatrix` | the training data used by the model

### Author(s)

Alexandros Karatzoglou
alexandros.karatzoglou@ci.tuwien.ac.at

### References

J. A. K. Suykens and J. Vandewalle
Least Squares Support Vector Machine Classifiers
Neural Processing Letters vol. 9, issue 3, June 1999

### See Also

`ksvm`, `gausspr`, `csi`

### Examples

    ## simple example
    data(iris)

    lir <- lssvm(Species~.,data=iris)

    lir

    lirr <- lssvm(Species~.,data= iris, reduced = FALSE)

    lirr

    ## Using the kernelMatrix interface

    iris <- unique(iris)

    rbf <- rbfdot(0.5)

    k <- kernelMatrix(rbf, as.matrix(iris[,-5]))

    klir <- lssvm(k, iris[, 5])

    klir

    pre <- predict(klir, k)

[Package kernlab version 0.9-25 Index]