whitening (version 1.2.0)

whiten: Whiten Data Matrix

Description

whiten whitens a data matrix \(X\) using the empirical covariance matrix \(cov(X)\) as basis for computing the whitening transformation.

Usage

whiten(X, center=FALSE, method=c("ZCA", "ZCA-cor", "PCA", "PCA-cor",
    "Chol-prec", "Chol-cov", "Cholesky"))

Arguments

X

Data matrix, with samples in rows and variables in columns.

center

Center columns to mean zero.

method

Determines the type of whitening transformation.

Value

whiten returns the whitened data matrix \(Z = X W'\).

Details

The following six different whitening approaches can be selected:

method="ZCA": ZCA whitening, also known as Mahalanobis whitening, ensures that the average covariance between whitened and orginal variables is maximal.

method="ZCA-cor": Likewise, ZCA-cor whitening leads to whitened variables that are maximally correlated (on average) with the original variables.

method="PCA": In contrast, PCA whitening lead to maximally compressed whitened variables, as measured by squared covariance.

method="PCA-cor": PCA-cor whitening is similar to PCA whitening but uses squared correlations.

method="Chol-prec" and method="Cholesky": computes a whitening matrix by applying Cholesky decomposition on the precision matrix. This yields an upper triangular positive diagonal whitening matrix and lower triangular positive diagonal cross-covariance and cross-correlation matrices.

method="Chol-cov": computes a whitening matrix by applying Cholesky decomposition on the precision matrix. This yields a lower triangular positive diagonal whitening matrix and upper triangular positive diagonal cross-covariance and cross-correlation matrices.

ZCA-cor whitening is implicitely employed in computing CAT and CAR scores used for variable selection in classification and regression, see the functions catscore in the sda package and carscore in the care package.

In both PCA and PCA-cor whitening there is a sign-ambiguity in the eigenvector matrices. In order to resolve the sign-ambiguity we use eigenvector matrices with a positive diagonal. This has the effect to make cross-correlations and cross-correlations positive diagonal for PCA and PCA-cor.

For details see Kessy, Lewin, and Strimmer (2018).

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. https://doi.org/10.1080/00031305.2016.1277159

See Also

whiteningMatrix.

Examples

# NOT RUN {
# load whitening library
library("whitening")

######

# example data set
# E. Anderson. 1935.  The irises of the Gaspe Peninsula.
# Bull. Am. Iris Soc. 59: 2--5
data("iris")
X = as.matrix(iris[,1:4])
d = ncol(X) # 4
n = nrow(X) # 150
colnames(X) # "Sepal.Length" "Sepal.Width"  "Petal.Length" "Petal.Width"

# whitened data
Z.ZCAcor = whiten(X, method="ZCA-cor")

# check covariance matrix
zapsmall( cov(Z.ZCAcor) )
# }