kpca {rchemo}

KPCA

Description

Kernel PCA (Scholkopf et al. 1997, Scholkopf & Smola 2002, Tipping 2001) by SVD factorization of the weighted Gram matrix D^(1/2) * Phi(X) * Phi(X)' * D^(1/2). D is a (n, n) diagonal matrix of weights for the observations (rows of X).

Usage

kpca(X, weights = NULL, nlv, kern = "krbf", ...)

## S3 method for class 'Kpca'
transform(object, X, ..., nlv = NULL)  

## S3 method for class 'Kpca'
summary(object, ...)  

Arguments

X

For the main function: Training X-data (n, p). — For the auxiliary functions: New X-data (m, p) to consider.

weights

Weights (n, 1) to apply to the training observations. Internally, weights are "normalized" to sum to 1. Default to NULL (weights are set to 1 / n).

nlv

The number of PCs to calculate.

kern

Name of the function defining the considered kernel for building the Gram matrix. See krbf for syntax, and other available kernel functions.

...

Optional arguments to pass in the kernel function defined in kern (e.g. gamma for krbf).

object

— For the auxiliary functions: A fitted model, output of a call to the main functions.

Value

For kpca:

X

Training X-data (n, p).

Kt

Gram matrix

T

X-scores matrix.

P

X-loadings matrix.

sv

vector of singular values

eig

vector of eigenvalues.

weights

vector of observation weights.

kern

kern function.

dots

Optional arguments.

For transform.Kpca: X-scores matrix for new X-data.

For summary.Kpca:

explvar

explained variance matrix.

References

Scholkopf, B., Smola, A., Muller, K.-R., 1997. Kernel principal component analysis, in: Gerstner, W., Germond, A., Hasler, M., Nicoud, J.-D. (Eds.), Artificial Neural Networks - ICANN 97, Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, pp. 583-588. https://doi.org/10.1007/BFb0020217

Scholkopf, B., Smola, A.J., 2002. Learning with kernels: support vector machines, regularization, optimization, and beyond, Adaptive computation and machine learning. MIT Press, Cambridge, Mass.

Tipping, M.E., 2001. Sparse kernel principal component analysis. Advances in neural information processing systems, MIT Press. http://papers.nips.cc/paper/1791-sparse-kernel-principal-component-analysis.pdf

Examples

## EXAMPLE 1

n <- 5 ; p <- 4
X <- matrix(rnorm(n * p), ncol = p)

nlv <- 3
kpca(X, nlv = nlv, kern = "krbf")

fm <- kpca(X, nlv = nlv, kern = "krbf", gamma = .6)
fm$T
transform(fm, X[1:2, ])
transform(fm, X[1:2, ], nlv = 1)
summary(fm)

## EXAMPLE 2

n <- 5 ; p <- 4
X <- matrix(rnorm(n * p), ncol = p)
nlv <- 3
pcasvd(X, nlv = nlv)$T
kpca(X, nlv = nlv, kern = "kpol")$T