R: Kernel Principal Components Analysis

kpca {kernlab} | R Documentation

## Kernel Principal Components Analysis

### Description

Kernel Principal Components Analysis is a nonlinear form of principal component analysis.

### Usage

    ## S4 method for signature 'formula'
    kpca(x, data = NULL, na.action, ...)

    ## S4 method for signature 'matrix'
    kpca(x, kernel = "rbfdot", kpar = list(sigma = 0.1),
        features = 0, th = 1e-4, na.action = na.omit, ...)

    ## S4 method for signature 'kernelMatrix'
    kpca(x, features = 0, th = 1e-4, ...)

    ## S4 method for signature 'list'
    kpca(x, kernel = "stringdot", kpar = list(length = 4, lambda = 0.5),
        features = 0, th = 1e-4, na.action = na.omit, ...)

### Arguments

`x` | the data matrix indexed by row or a formula describing the model, or a kernel Matrix of class `kernelMatrix`, or a list of character vectors
`data` | an optional data frame containing the variables in the model (when using a formula).
`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 function "Gaussian" `polydot` Polynomial kernel function `vanilladot` Linear kernel function `tanhdot` Hyperbolic tangent kernel function `laplacedot` Laplacian kernel function `besseldot` Bessel kernel function `anovadot` ANOVA RBF kernel function `splinedot` Spline kernel 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. 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". Hyper-parameters for user defined kernels can be passed through the kpar parameter as well.
`features` | Number of features (principal components) to return. (default: 0 , all)
`th` | the value of the eigenvalue under which principal components are ignored (only valid when features = 0). (default : 0.0001)
`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

Using kernel functions one can efficiently compute principal components in high-dimensional feature spaces, related to input space by some non-linear map.
The data can be passed to the `kpca` function in a `matrix` or a `data.frame`, in addition `kpca` 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.

### Value

An S4 object containing the principal component vectors along with the corresponding eigenvalues.

`pcv` | a matrix containing the principal component vectors (column wise)
`eig` | The corresponding eigenvalues
`rotated` | The original data projected (rotated) on the principal components
`xmatrix` | The original data matrix

all the slots of the object can be accessed by accessor functions.

### Note

The predict function can be used to embed new data on the new space

### Author(s)

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

### References

Schoelkopf B., A. Smola, K.-R. Mueller :
Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation 10, 1299-1319
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.1366

### See Also

`kcca`, `pca`

### Examples

    # another example using the iris
    data(iris)
    test <- sample(1:150,20)

    kpc <- kpca(~.,data=iris[-test,-5],kernel="rbfdot",
                kpar=list(sigma=0.2),features=2)

    #print the principal component vectors
    pcv(kpc)

    #plot the data projection on the components
    plot(rotated(kpc),col=as.integer(iris[-test,5]),
         xlab="1st Principal Component",ylab="2nd Principal Component")

    #embed remaining points
    emb <- predict(kpc,iris[test,-5])
    points(emb,col=as.integer(iris[test,5]))

[Package kernlab version 0.9-25 Index]