Kernel Principal Component Analysis

Kernel principal component analysis (KPCA/Kernel PCA) is a nonlinear extension of classical PCA using techniques called kernel trick, a common method of introducing nonlinearity by transforming, usually, covariance structure or other gram-type estimate to make it flexible in Reproducing Kernel Hilbert Space.

do.kpca(
  X,
  ndim = 2,
  preprocess = c("null", "center", "scale", "cscale", "whiten", "decorrelate"),
  kernel = c("gaussian", 1)
)

Arguments

X

an \((n\times p)\) matrix or data frame whose rows are observations and columns represent independent variables.

ndim

an integer-valued target dimension.

preprocess

an additional option for preprocessing the data. Default is "null". See also aux.preprocess for more details.

kernel

a vector containing name of a kernel and corresponding parameters. See also aux.kernelcov for complete description of Kernel Trick.

Value

a named list containing

Y

an \((n\times ndim)\) matrix whose rows are embedded observations.

trfinfo

a list containing information for out-of-sample prediction.

vars

variances of projected data / eigenvalues from kernelized covariance matrix.

References

Schölkopf B, Smola A, Müller K (1997). “Kernel Principal Component Analysis.” In Goos G, Hartmanis J, van Leeuwen J, Gerstner W, Germond A, Hasler M, Nicoud J (eds.), Artificial Neural Networks — ICANN'97, volume 1327, 583--588. Springer Berlin Heidelberg, Berlin, Heidelberg. ISBN 978-3-540-63631-1 978-3-540-69620-9.

See also

aux.kernelcov

Author

Kisung You

Examples

# \donttest{
## load iris data
data(iris)
set.seed(100)
subid = sample(1:150,50)
X     = as.matrix(iris[subid,1:4])
label = as.factor(iris[subid,5])

## try out different settings
output1 <- do.kpca(X)                         # default setting
output2 <- do.kpca(X,kernel=c("gaussian",5))  # gaussian kernel with large bandwidth
output3 <- do.kpca(X,kernel=c("laplacian",1)) # laplacian kernel

## visualize three different projections
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(output1$Y, col=label, pch=19, main="Gaussian kernel")
plot(output2$Y, col=label, pch=19, main="Gaussian kernel with sigma=5")
plot(output3$Y, col=label, pch=19, main="Laplacian kernel")

par(opar)
# }