ipca: Independent Principal Component Analysis

Description

Performs independent principal component analysis on the given data matrix, a combination of Principal Component Analysis and Independent Component Analysis.

Usage

ipca(X, ncomp = 3, mode = c("deflation","parallel"),
          fun = c("logcosh", "exp"),
          scale = FALSE, max.iter = 200, 
          tol = 1e-04, w.init=NULL)

Arguments

a numeric matrix (or data frame) which provides the data for the principal component analysis.

ncomp

integer, number of independent component to choose. Set by default to 3.

mode

character string. What type of algorithm to use when estimating the unmixing matrix, choose one of "deflation", "parallel". Default set to deflation.

fun

the function used in approximation to neg-entropy in the FastICA algorithm. Default set to logcosh, see details of FastICA.

scale

a logical value indicating whether the variables (columns) of the data matrix X should be standardized beforehand. By default, X is centered.

max.iter

integer, maximum number of iterations to perform.

tol

a positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged, see fastICA package.

w.init

initial un-mixing matrix (unlike FastICA, this matrix is fixed here).

Value

ipca returns a list with class "ipca" containing the following components:
ncompthe number of independent principal components used.
unmixingthe unmixing matrix of size (ncomp x ncomp)
mixingthe mixing matrix of size (ncomp x ncomp)
Xthe centered data matrix
xthe indepenent principal components
loadingsthe independent loading vectors
kurtosisthe kurtosis measure of the independent loading vectors

encoding

latin1

Details

In PCA, the loading vectors indicate the importance of the variables in the principal components. In large biological data sets, the loading vectors should only assign large weights to important variables (genes, metabolites ...). That means the distribution of any loading vector should be super-Gaussian: most of the weights are very close to zero while only a few have large (absolute) values. However, due to the existence of noise, the distribution of any loading vector is distorted and tends toward a Gaussian distribtion according to the Central Limit Theroem. By maximizing the non-Gaussianity of the loading vectors using FastICA, we obtain more noiseless loading vectors. We then project the original data matrix on these noiseless loading vectors, to obtain independent principal components, which should be also more noiseless and be able to better cluster the samples according to the biological treatment (note, IPCA is an unsupervised approach). Algorithm 1. The original data matrix is centered. 2. PCA is used to reduce dimension and generate the loading vectors. 3. ICA (FastICA) is implemented on the loading vectors to generate independent loading vectors. 4. The centered data matrix is projected on the independent loading vectors to obtain the independent principal components.

References

Yao, F., Coquery, J. and L^e Cao, K.-A.(2011) Principal component analysis with independent loadings: a combination of PCA and ICA. (in preparation) A. Hyvarinen and E. Oja (2000) Independent Component Analysis: Algorithms and Applications, Neural Networks, 13(4-5):411-430 J L Marchini, C Heaton and B D Ripley (2010). fastICA: FastICA Algorithms to perform ICA and Projection Pursuit. R package version 1.1-13.

Examples

Run this code

data(liver.toxicity)

# implement IPCA on a microarray dataset
ipca.res <- ipca(liver.toxicity$gene, ncomp = 3, mode="deflation")
ipca.res

# samples representation
plotIndiv(ipca.res, ind.names = liver.toxicity$treatment[, 4], cex = 0.5, 
          col = as.numeric(as.factor(liver.toxicity$treatment[, 4])))
plot3dIndiv(ipca.res, cex = 0.01,
            col = as.numeric(as.factor(liver.toxicity$treatment[, 4])))

# variables representation
plotVar(ipca.res, var.label = TRUE, cex = 0.5)

plot3dVar(ipca.res, rad.in = 0.5, cex = 0.5, 
          col = as.numeric(as.factor(liver.toxicity$treatment[, 4])))

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