whitening (version 1.2.0)

whiteningCrossCov: Compute Whitening Cross-Covariance Matrix

Description

whiteningCrossCov computes the cross-covariance \(Phi=Cov(z, x)\) between the whitened and the original variables.

Usage

whiteningCrossCov(Sigma, method=c("ZCA", "ZCA-cor", "PCA", "PCA-cor", 
    "Chol-prec", "Chol-cov", "Cholesky"))

Arguments

Sigma

Covariance matrix.

method

Determines the type of whitening transformation.

Value

whiteningCrossCov returns a square cross-covariance matrix \(Phi\).

Details

\(Phi=Cov(z, x)\) is the cross-covariance matrix between the whitened and the original variables. It satisfies \(Phi' Phi = Sigma\). The transpose of the cross-covariance matrix is the inverse of the whitening matrix so that \(Phi' = W^{-1}\). The cross-covariance matrix is therefore relevant in inverse whitening transformations (=coloring transformations) \(x = Phi' z\).

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"

# estimate covariance
S = cov(X)

# ZCA-cor whitening matrix
W.ZCAcor = whiteningMatrix(S, method="ZCA-cor")

# ZCA-cor cross-covariance matrix
Phi.ZCAcor = whiteningCrossCov(S, method="ZCA-cor")

# check constraint of cross-covariance matrix
crossprod(Phi.ZCAcor)
S

# transpose of cross-covariance matrix is equal to the inverse whitening matrix
t(Phi.ZCAcor)
solve(W.ZCAcor)
# }