The whitening Package

Description

The "whitening" package implements the whitening methods (ZCA, PCA, Cholesky, ZCA-cor, and PCA-cor) discussed in Kessy, Lewin, and Strimmer (2018) as well as the whitening approach to canonical correlation analysis allowing negative canonical correlations described in Jendoubi and Strimmer (2019).

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io/) with Takoua Jendoubi, Agnan Kessy, and Alex Lewin.

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. <DOI:10.1080/00031305.2016.1277159>

Jendoubi, T., and K. Strimmer 2019. A whitening approach to probabilistic canonical correlation analysis for omics data integration. BMC Bioinformatics 20: 15. <DOI:10.1186/s12859-018-2572-9>

Website: https://strimmerlab.github.io/software/whitening/

See Also

whiteningMatrix, whiten, whiteningLoadings, explainedVariation, cca, and scca.

Plots of Correlations and Loadings

Description

corplot computes the correlation within and between X and Y and displays the three corresponding matrices visusally.

loadplot computes the squared loadings for X and Y and plots the resulting matrices.

Usage

corplot(cca.out, X, Y)
loadplot(cca.out, numScores)

Arguments

Value

A plot.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

Part of the plot code was adapted from the img.matcor function in the CCA package and from the image.plot function in the fields package.

See Also

scca.

Compute Explained Variation from Loadings

Description

explainedVariation computes the explained variation for each whitened variables from the loadings (both covariance loadings and correlation loadings).

Usage

  explainedVariation(Phi)

Arguments

Details

explainedVariation computes for each column of the loading matrix the sum of squares of the elements in that column.

Value

explainedVariation returns a vector with the explained variation contributed by each whitened variable.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. <DOI:10.1080/00031305.2016.1277159>

See Also

whiteningLoadings

Examples

# 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)

# PCA-cor loadings
ldgs = whiteningLoadings(S, method="PCA-cor")

# Explained variation from correlation loadings
explainedVariation( ldgs$Psi )

Forina 1986 Wine Data - Extended UCI Wine Data

Description

The forina1986 dataset describes 27 properties of 178 samples of wine from three grape varieties (59 Barolo, 71 Grignolino, 48 Barbera) as reported in Forina et al. (1986).

Usage

data(forina1986)

Format

A list containing the following components:

attribs collects measurements for 27 attributes of 178 wine samples.

type describes the variety ("Barolo", "Grignolino", or "Barbera").

Details

This data set contains the full set of covariates described in Forina et al. (1986) except for Sulphate (variable 15 in Forina et al. 1986). These are: 1) Alcohol, 2) Sugar-free extract, 3) Fixed acidity, 4) Tartaric acid, 5) Malic acid, 6) Uronic acids, 7) pH, 8) Ash, 9) Alcalinity of ash, 10) Potassium, 11) Calcium, 12) Magnesium, 13) Phosphate, 14) Chloride, 15) Total phenols, 16) Flavanoids, 17) Nonflavanoid phenols, 18) Proanthocyanins, 19) Color intensity, 20) Hue, 21) OD280/OD315 of diluted wines, 22) OD280/OD315 of flavonoids, 23) Glycerol, 24) 2-3-butanediol, 25) Total nitrogen, 26) Proline, and 27) Methanol.

The UCI wine data set (https://archive.ics.uci.edu/ml/datasets/wine) is a subset of the Forina et al. (1986) data set comprising only 13 variables.

Source

The original data matrix is available from https://www.researchgate.net/publication/271908647_Wines_MForina_CArmanino_MCastino_MUbigli_Multivariate_data_analysis_as_discriminating_method_of_the_origin_of_wines_Vitis_25_189-201_1986.

References

Forina, M., Armanino, C., Castino, M., and Ubigli, M. Multivariate data analysis as a discriminating method of the origin of wines. Vitis 25:189-201 (1986). https://ojs.openagrar.de/index.php/VITIS/article/view/5950.

Examples

# load whitening library
library("whitening")

# load Forina 1986 wine data set
data(forina1986)

table(forina1986$type)
#    Barolo Grignolino    Barbera 
#        59         71         48

dim(forina1986$attrib)
# 178  27

colnames(forina1986$attrib)
# [1] "Alcohol"                      "Sugar-free extract"          
# [3] "Fixed acidity"                "Tartaric acid"               
# [5] "Malic acid"                   "Uronic acids"                
# [7] "pH"                           "Ash"                         
# [9] "Alkalinity of ash"            "Potassium"                   
#[11] "Calcium"                      "Magnesium"                   
#[13] "Phosphate"                    "Chloride"                    
#[15] "Total phenols"                "Flavanoids"                  
#[17] "Nonflavanoid phenols"         "Proanthocyanins"             
#[19] "Color intensity"              "Hue"                         
#[21] "OD280/OD315 of diluted wines" "OD280/OD315 of flavonoids"   
#[23] "Glycerol"                     "2-3-butanediol"              
#[25] "Total nitrogen"               "Proline"                     
#[27] "Methanol"  

# PCA-cor whitened data
Z = whiten(forina1986$attrib, method="PCA-cor") 

wt = as.integer(forina1986$type)
plot(Z[,1], Z[,2], xlab=expression(paste(Z[1])), ylab=expression(paste(Z[2])), 
  main="Forina 1986 Wine Data", sub="PCA-cor Whitening", col=wt, pch=wt+14)
legend("topright", levels(forina1986$type)[1:3], col=1:3, pch=(1:3)+14 )



## relationship to UCI wine data

# UCI wine data is a subset
uciwine.attrib = forina1986$attrib[, c("Alcohol", "Malic acid", "Ash", 
  "Alcalinity of ash", "Magnesium", "Total phenols", "Flavanoids", 
  "Nonflavanoid phenols", "Proanthocyanins", "Color intensity", "Hue", 
  "OD280/OD315 of diluted wines", "Proline")]

# two small differences compared to UCI wine data matrix
uciwine.attrib[172,"Color intensity"]  # 9.9 but 9.899999 in UCI matrix
uciwine.attrib[71,"Hue"] # 0.91 but 0.906 in UCI matrix

TCGA LUSC Data

Description

A preprocessed sample of gene expression and methylation data as well as selected clinical covariates for 130 patients with lung squamous cell carcinoma (LUSC) as available from The Cancer Genome Atlas (TCGA) database (Kandoth et al. 2013).

Usage

data(lusc)

Format

lusc$rnaseq2 is a 130 x 206 matrix containing the calibrated gene expression levels of 206 genes for 130 patients.

lusc$methyl is a 130 x 234 matrix containing the methylation levels of 234 probes for 130 patients.

sex is a vector recording the sex (male vs. female) of the 130 patients.

packs is the number of cigarette packs per year smoked by each patient.

survivalTime is number of days to last follow-up or the days to death.

censoringStatus is the vital status (0=alive, 1=dead).

Details

This data set is used to illustrate CCA-based data integration in Jendoubi and Strimmer (2019) and also described in Wan et al. (2016).

Source

The data were retrieved from TCGA (Kandoth et al. 2014) using the TCGA2STAT tool following the guidelines and the preprocessing steps detailed in Wan et al. (2016).

References

Jendoubi, T., Strimmer, K.: A whitening approach to probabilistic canonical correlation analysis for omics data integration. BMC Bioinformatics 20:15 <DOI:10.1186/s12859-018-2572-9>

Kandoth, C., McLellan, M.D., Vandin, F., Ye, K., Niu, B., Lu, C., Xie, M., andJ. F. McMichael, Q.Z., Wyczalkowski, M.A., Leiserson, M.D.M., Miller, C.A., Welch, J.S., Walter, M.J., Wendl, M.C., Ley, T.J., Wilson, R.K., Raphael, B.J., Ding, L.: Mutational landscape and significance across 12 major cancer types. Nature 502, 333–339 (2013). <DOI:10.1038/nature12634>

Wan, Y.-W., Allen, G.I., Liu, Z.: TCGA2STAT: simple TCGA data access for integrated statistical analysis in R. Bioinformatics 32, 952–954 (2016). <DOI:10.1093/bioinformatics/btv677>

Examples

# load whitening library
library("whitening")

# load TGCA LUSC data set
data(lusc)

names(lusc)
#"rnaseq2"         "methyl"          "sex"             "packs"          
#"survivalTime"    "censoringStatus" 

dim(lusc$rnaseq2) # 130 206 gene expression
dim(lusc$methyl)  # 130 234 methylation level

## Not run: 
library("survival")
s = Surv(lusc$survivalTime, lusc$censoringStatus)
plot(survfit(s ~ lusc$sex), xlab = "Years", ylab = "Probability of survival", lty=c(2,1), lwd=2)
legend("topright", legend = c("male", "female"), lty =c(1,2), lwd=2)

## End(Not run)

Nutrimouse Data

Description

The nutrimouse dataset is a collection of gene expression and lipid measurements collected in a nutrigenomic study in the mouse studying 40 animals by Martin et al. (2007).

Usage

data(nutrimouse)

Format

A list containing the following components:

gene collects gene expression of 120 genes in liver tissue for 40 mice.

lipid collects concentrations of 21 lipids for 40 mice.

diet describes the diet of each mouse ("coc", "fish", "lin", "ref", or "sun").

genotype describes the genotype of each mouse: wild type ("wt") or PPARalpha deficient ("ppar").

Details

This data set is used to illustrate CCA-based data integration in Jendoubi and Strimmer (2019) and is also described in Gonzalez et al. (2008).

Source

The original data are available in the CCA package by Gonzalez et al. (2008), see their function nutrimouse.

References

Gonzalez, I., Dejean, S., Martin, P.G.P, Baccini, A. CCA: an R package to extend canonical correlation analysis. J. Statist. Software 23:1–13 (2008)

Jendoubi, T., Strimmer, K.: A whitening approach to probabilistic canonical correlation analysis for omics data integration. BMC Bioinformatics 20:15 <DOI:10.1186/s12859-018-2572-9>

Martin, P.G.P., Guillou, H., Lasserre, F., Dejean, S., Lan, A., Pascussi, J.-M., Cristobal, M.S., Legrand, P., Besse, P., Pineau, T.: Novel aspects of PPARalpha-mediated regulation of lipid and xenobiotic metabolism revealed through a multigenomic study. Hepatology 54, 767–777 (2007) <DOI:10.1002/hep.21510>

Examples

# load whitening library
library("whitening")

# load nutrimouse data set
data(nutrimouse)

dim(nutrimouse$gene) # 40 120
dim(nutrimouse$lipid) # 40 21
levels( nutrimouse$diet ) #  "coc"  "fish" "lin"  "ref"  "sun"
levels( nutrimouse$genotype ) # "wt"   "ppar"

Pitprops Correlation Data for 14 Variables

Description

Pit prop timber is used in construction to build mining tunnels. The pitprops14 data is described in Jeffers (1967) and is a correlation matrix that was calculated from measuring 14 physical properties of 180 pit props made from wood from Corsican pines grown in East Anglia, UK.

Usage

data(pitprops14)

Format

A correlation matrix of dimension 14 times 14.

Details

The 14 variables are described in Jeffers (1967) as follows:

topdiam: the top diameter of the prop in inches; length: the length of the prop in inches; moist: the moisture content of the prop, expressed as a percentage of the dry weight; testsg: the specific gravity of the timber at the time of the test; ovensg: the oven-dry specific gravity of the timber; rinotop: the number of annual rings at the top of the prop; ringbut: the number of annual rings at the base of the prop; bowmax: the maximum bow in inches; bowdist: the distance of the point of maximum bow from the top of the prop in inches; whorls: the number of knot whorls; clear: the length of clear prop from the top of the prop in inches; knots: the average number of knots per whorl; diaknot: the average diameter of the knots in inches; maxcs: the maximum compressive strength in lb per square inch.

Source

The data set is printed in Jeffers (1967) in Table 2 and Table 5.

References

Jeffers, J. N. R. 1967. Two case studies in the application of principal component analysis. JRSS C (Applied Statistics) 16: 225-236. <DOI:10.2307/2985919>

Examples

# load whitening library
library("whitening")

# load pitprops14 data set
data(pitprops14)
colnames(pitprops14)

# correlation matrix for the first 13 variables
pitprops13 = pitprops14[1:13, 1:13]

# correlation loadings for PCA whitening
Psi = whiteningLoadings(pitprops13, "PCA")$Psi

# corresponding explained variation
Psi.explained = explainedVariation(Psi)

# the first six whitened variables account for 87% of the variation
cumsum(Psi.explained)/13*100

Perform Canonical Correlation Analysis

Description

scca computes canonical correlations and directions using a shrinkage estimate of the joint correlation matrix of X and Y.

cca computes canonical correlations and directions based on empirical correlations.

Usage

scca(X, Y, lambda.cor, scale=TRUE, verbose=TRUE)
cca(X, Y, scale=TRUE)

Arguments

Details

The canonical directions in this function are scaled in such a way that they correspond to whitening matrices - see Jendoubi and Strimmer (2019) for details. Note that the sign convention for the canonical directions employed here allows purposely for both positive and negative canonical correlations.

The function scca uses some clever matrix algebra to avoid computation of full correlation matrices, and hence can be applied to high-dimensional data sets - see Jendoubi and Strimmer (2019) for details.

cca it is a shortcut for running scca with lambda.cor=0 and verbose=FALSE.

If scale=FALSE the standard deviations needed for the canonical directions are estimated by apply(X, 2, sd) and apply(X, 2, sd).

If X or Y contains only a single variable the correlation-adjusted cross-correlations K reduce to the CAR score (see function carscore in the care package) described in Strimmer and Zuber (2011).

Value

scca and cca return a list with the following components:

K - the correlation-adjusted cross-correlations.

lambda - the canonical correlations.

WX and WY - the whitening matrices for X and Y, with canonical directions in the rows. If scale=FALSE then canonical directions include scale of the data, if scale=TRUE then only correlations are needed to compute the canonical directions.

PhiX and PhiY - the loadings for X and Y. If scale=TRUE then these are the correlation loadings, i.e. the correlations between the whitened variables and the original variables.

scale - whether data was standardized (if scale=FALSE then canonical directions include scale of the data).

lambda.cor - shrinkage intensity used for estimating the correlations (0 for empirical estimator)

lambda.cor.estimated - indicates whether shrinkage intenstiy was specified or estimated.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io) with Takoua Jendoubi.

References

Jendoubi, T., and K. Strimmer 2019. A whitening approach to probabilistic canonical correlation analysis for omics data integration. BMC Bioinformatics 20: 15. <DOI:10.1186/s12859-018-2572-9>

Zuber, V., and K. Strimmer. 2011. High-dimensional regression and variable selection using CAR scores. Statist. Appl. Genet. Mol. Biol. 10: 34. <DOI:10.2202/1544-6115.1730>

See Also

cancor and whiteningMatrix.

Examples

# load whitening library
library("whitening")

# example data set
data(LifeCycleSavings)
X = as.matrix( LifeCycleSavings[, 2:3] )
Y = as.matrix( LifeCycleSavings[, -(2:3)] )
n = nrow(X)
colnames(X) # "pop15" "pop75"
colnames(Y) # "sr"   "dpi"  "ddpi" 

# CCA

cca.out = cca(X, Y, scale=TRUE)
cca.out$lambda  # canonical correlations
cca.out$WX      # whitening matrix / canonical directions X
cca.out$WY      # whitening matrix / canonical directions Y
cca.out$K       # correlation-adjusted cross-correlations
cca.out$PhiX    # correlation loadings X
cca.out$PhiX    # correlation loadings Y

corplot(cca.out, X, Y)
loadplot(cca.out, 2)
# column sums of squared correlation loadings add to 1
colSums(cca.out$PhiX^2) 

# CCA whitened data
CCAX = tcrossprod( scale(X), cca.out$WX )
CCAY = tcrossprod( scale(Y), cca.out$WY )
zapsmall(cov(CCAX))
zapsmall(cov(CCAY))
zapsmall(cov(CCAX,CCAY)) # canonical correlations


# compare with built-in function cancor 
# note different signs in correlations and directions!
cancor.out = cancor(scale(X), scale(Y))
cancor.out$cor                  # canonical correlations
t(cancor.out$xcoef)*sqrt(n-1)   # canonical directions X
t(cancor.out$ycoef)*sqrt(n-1)   # canonical directions Y


## see "User guides, package vignettes and other documentation"
## for examples with high-dimensional data using the scca function

Simulate Random Orthogonal Matrix

Description

simOrtho generates a random orthogonal matrix.

Usage

simOrtho(d, nonNegDiag = FALSE)

Arguments

Details

The algorithm is based on QR decomposition of a random matrix, see Section 3 page 404 in Stewart (1980).

Value

simOrtho returns a real matrix of size d times d.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

References

G.W. Stewart. 1980. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17:403-409. <DOI:10.1137/0717034>

See Also

whiteningMatrix

Examples

# load whitening library
library("whitening")

# simulate random orthogonal matrix
Q = simOrtho(4) # matrix of dimension 4x4
Q

zapsmall( crossprod(Q) )
zapsmall( tcrossprod(Q) )

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", "Cholesky"))

Arguments

Details

The following whitening approaches can be selected:

method="ZCA" and method="ZCA-cov": 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" and method="PCA-cov": 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="Cholesky": computes a whitening matrix by applying Cholesky decomposition. This yields both a lower triangular positive diagonal whitening matrix and lower triangular positive diagonal loadings (cross-covariance and cross-correlation).

Note that Cholesky whitening depends on the ordering of input variables. In the convention used here the first input variable is linked with the first latent variable only, the second input variable is linked to the first and second latent variable only, and so on, and the last variable is linked to all latent variables.

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 so that PCA and PCA-cor cross-correlations and cross-covariances have a positive diagonal for the given ordering of the original variables.

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

Canonical correlation analysis (CCA) can also be understood as a special form of whitening (also implemented in this package).

Value

whiten returns the whitened data matrix Z = X W^T.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io) with Agnan Kessy and Alex Lewin.

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. <DOI:10.1080/00031305.2016.1277159>

See Also

whiteningMatrix, whiteningLoadings, scca.

Examples

# 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) )

Internal whitening Functions

Description

Internal whitening functions.

Note

These are not to be called by the user (or in some cases are just waiting for proper documentation to be written).

Compute Whitening Loadings

Description

whiteningLoading computes the loadings (= cross-covariance matrix \Phi=Cov(x, z)) between the original and the whitened variables as well as the correlation loadings (= cross-correlation matrix \Psi=Cor(x, z)). The original variables are in the rows and the whitened variables in the columns.

Usage

  whiteningLoadings(Sigma, method=c("ZCA", "ZCA-cor", "PCA", "PCA-cor", "Cholesky"))

Arguments

Details

\Phi=Cov(x, z) is the cross-covariance matrix between the original and the whitened variables. It satisfies \Phi \Phi^{T} = \Sigma = Var(x). This 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.

\Psi=Cor(x, z) is the cross-correlation matrix between the original and the whitened variables.

The following 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="Cholesky": computes a whitening matrix by applying Cholesky decomposition. This yields both a lower triangular positive diagonal whitening matrix and lower triangular positive diagonal loadings (cross-covariance and cross-correlation).

Note that Cholesky whitening depends on the ordering of input variables. In the convention used here the first input variable is linked with the first latent variable only, the second input variable is linked to the first and second latent variable only, and so on, and the last variable is linked to all latent variables.

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 so that PCA and PCA-cor cross-correlations and cross-covariances have a positive diagonal for the given ordering of the original variables.

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

Value

whiteningLoadings returns a list with the following items:

Phi - cross-covariance matrix \Phi - the loadings.

Psi - cross-correlation matrix \Psi - the correlation loadings.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. <DOI:10.1080/00031305.2016.1277159>

See Also

explainedVariation, whiten, whiteningMatrix.

Examples

# 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 loadings
ldgs = whiteningLoadings(S, method="ZCA-cor")
ldgs

# cross-covariance matrix
Phi.ZCAcor = ldgs$Phi

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

# cross-covariance matrix aka loadings is equal to the inverse whitening matrix
Phi.ZCAcor
solve(W.ZCAcor)

Compute Whitening Matrix

Description

whiteningMatrix computes the whitening matrix W.

Usage

  whiteningMatrix(Sigma, method=c("ZCA", "ZCA-cor", "PCA", "PCA-cor", "Cholesky"))

Arguments

Details

Whitening is a linear transformation z = W x where the whitening matrix satisfies the constraint W^T W = \Sigma^{-1} where \Sigma = Cov(x).

This function implements various natural whitening transformations discussed in Kessy, Lewin, and Strimmer (2018).

The following 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="Cholesky": computes a whitening matrix by applying Cholesky decomposition. This yields both a lower triangular positive diagonal whitening matrix and lower triangular positive diagonal loadings (cross-covariance and cross-correlation).

Note that Cholesky whitening depends on the ordering of input variables. In the convention used here the first input variable is linked with the first latent variable only, the second input variable is linked to the first and second latent variable only, and so on, and the last variable is linked to all latent variables.

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 so that PCA and PCA-cor cross-correlations and cross-covariances have a positive diagonal for the given ordering of the original variables.

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

Canonical correlation analysis (CCA) can also be understood as a special form of whitening (also implemented in this package).

Value

whiteningMatrix returns a square whitening matrix W.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io) with Agnan Kessy and Alex Lewin.

References

Kessy, A., A. Lewin, and K. Strimmer. 2018. Optimal whitening and decorrelation. The American Statistician. 72: 309-314. <DOI:10.1080/00031305.2016.1277159>

See Also

whiten, whiteningLoadings, scca.

Examples

# 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")

# check constraint on the whitening matrix
crossprod(W.ZCAcor)
solve(S)

# whitened data
Z = tcrossprod(X, W.ZCAcor) 
Z
zapsmall( cov(Z) )