kernlab (version 0.1-4)

kpca: Kernel Principal Components Analysis

Description

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

Usage

## S3 method for class 'formula':
kpca(x, data = NULL, na.action, ...)
## S3 method for class 'matrix':
kpca(x, kernel = "rbfdot", kpar = list(sigma = 0.1), features = 0, th = 1e-4, ...)

Arguments

x

The data matrix indexed by row or a formula descibing 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 a dot product between two vector arguments. kernlab provides the most popular kernel functions which can be used by

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

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 NAs 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<

...

additional parameters

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.

notes

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

Details

By the use of kernel functions one can efficiently compute principal components in high-dimensional feature spaces, related to input space by some non-linear map.

References

Schoelkopf B., A. Smola, K.-R. Mueller : Nonlinear component analysis as a kernel eigenvalue problem Neural Computation 10, 1299-1319 http://mlg.anu.edu.au/~smola/papers/SchSmoMul98.pdf

See Also

kcca, pca

Examples

# another example using the iris
data(iris)
test <- sample(1:50,20)

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,as.matrix(iris[test,-5]))
points(emb,col=iris[test,5])