kha {kernlab} | R Documentation

## Kernel Principal Components Analysis

### Description

Kernel Hebbian Algorithm is a nonlinear iterative algorithm for principal component analysis.

### Usage

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

    ## S4 method for signature 'matrix'
    kha(x, kernel = "rbfdot", kpar = list(sigma = 0.1), features = 5,
             eta = 0.005, th = 1e-4, maxiter = 10000, verbose = FALSE,
            na.action = na.omit, ...)

### Arguments

`x` | The data matrix indexed by row or a formula describing the model. Note, that an intercept is always included, whether given in the formula or not.
`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 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 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: 5)
`eta` | The hebbian learning rate (default : 0.005)
`th` | the smallest value of the convergence step (default : 0.0001)
`maxiter` | the maximum number of iterations.
`verbose` | print convergence every 100 iterations. (default : FALSE)
`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

The original form of KPCA can only be used on small data sets since it requires the estimation of the eigenvectors of a full kernel matrix. The Kernel Hebbian Algorithm iteratively estimates the Kernel Principal Components with only linear order memory complexity. (see ref. for more details)

### Value

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

`pcv` | a matrix containing the principal component vectors (column wise)
`eig` | The normalization values
`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

Kwang In Kim, M.O. Franz and B. Schölkopf
Kernel Hebbian Algorithm for Iterative Kernel Principal Component Analysis
Max-Planck-Institut für biologische Kybernetik, Tübingen (109)
http://www.kyb.tuebingen.mpg.de/publications/pdfs/pdf2302.pdf

### See Also

`kpca`, `kfa`, `kcca`, `pca`

### Examples

    # another example using the iris
    data(iris)
    test <- sample(1:150,70)

    kpc <- kha(~.,data=iris[-test,-5],kernel="rbfdot",
               kpar=list(sigma=0.2),features=2, eta=0.001, maxiter=65)

    #print the principal component vectors
    pcv(kpc)

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

[Package kernlab version 0.9-25 Index]