BKPC (version 1.0.1)

kPCA: Kernel Principal Components Analysis

Description

Kernel PCA is a nonlinear generalization of principal component analysis.

Usage

# S3 method for default
kPCA(x, theta = NULL, ...)
# S3 method for kern
kPCA(x, ...)
# S3 method for kernelMatrix
kPCA(x, ...)

Arguments

x

either: a data matrix, a kernel matrix of class "kernelMatrix" or a kernel matrix of class "kern".

theta

the inverse kernel bandwidth parameter for the Gaussian kernel.

…

Currently not used.

Value

An object of class "kPCA" including:

KPCs

The original data projected on the principal components.

Es

the corresponding eigenvalues.

Vecs

a matrix containing principal component vectors (column wise).

K

a kernel matrix of class "kernelMatrix" or of class "kern".

theta

if Gaussian kernel was calculated, this is the bandwidth parameter used in its calculation.

x

if supplied, the original data matrix.

Details

The data can be passed to the kPCA function in a matrix and the Gaussian kernel (via the gaussKern function) is used to map the data to the high-dimensional feature space where the principal components are computed. The bandwidth parameter theta can be supplied to the gaussKern function, else a default value is used. Alternatively, the Gaussian kernel matrix of class "kern" can be supplied to the kPCA function directly. In addition, kPCA also supports input in the form of a kernel matrix of class "kernelMatrix" (in package kernlab) thus allowing for other kernel functions.

References

Schoelkopf B., A. Smola, K.-R. Mueller : Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299-1319.

See Also

gaussKern kpca (in package kernlab) kernelMatrix (in package kernlab)

Examples

# NOT RUN {
data(iris)
testset <- sample(1:150,20)

train <- as.matrix(iris[-testset,-5])
test <- as.matrix(iris[testset,-5])


# make training set kernel

gk <- gaussKern(train)
Ktrain <- gk$K
image(Ktrain)

# make testing set kernel

gk2 <- gaussKern(train, test, gk$theta) 
Ktest <- gk2$K


#  make training set kernel using kernelMatrix from library kernlab.

library(kernlab)

kfunc <- laplacedot(sigma = 1)
Ktrain2 <- kernelMatrix(kfunc, train)
image(Ktrain2)

# make testing set kernel using kernelMatrix {kernlab}

Ktest2 <- kernelMatrix(kfunc, test, train)



# Do KPCA:

kpcData <- kPCA(train)
kpcKern <- kPCA(Ktrain)
kpcKern2 <- kPCA(Ktrain2)


# plot the data projection on the principal components

pairs(kpcData$KPCs[ , 1 : 3], col = iris[-testset, 5])

# proportion of variance explained by each PC

plot(kpcData$Es/sum(kpcData$Es), xlab = "PC", ylab = "Proportion of variance")


# embed data from the testing set on the new space:


KPCpred1 <- predict(kpcData, test)

KPCpred2 <- predict(kpcKern, Ktest)

KPCpred3 <- predict(kpcKern2, Ktest2)

#plot the test data projection on the principal components

pairs(KPCpred3[ , 1 : 3], col = iris[testset, 5])




# }