One step Tyler Shape Matrix

Description

This matrix can be used to get from ics the principal axes which is then known as principal axis analysis.

Usage

covAxis(X, na.action = na.fail)

Arguments

Details

The covAxis matrix V is a given for a sample of size n as

p ave{[(x_i-x_bar)S^{-1}(x_i-x_bar)']^(-1) (x_i-x_bar)'(x_i-x_bar)},

where x_bar is the mean vector and S the regular covariance matrix.

covAxis can be used to perform a Prinzipal Axis Analysis (Critchley et al. 2006) using the function ics. In that case for a centered data matrix X covAxis can be used as S2 in ics, where S1 should be in that case the regular covariance matrix.

Value

Matrix of the estimated scatter.

Author(s)

Klaus Nordhausen

References

Critchley , F., Pires, A. and Amado, C. (2006), Principal axis analysis, Technical Report, 06/14, The Open University Milton Keynes.

Tyler, D.E., Critchley, F., Dümbgen, L. and Oja, H. (2009), Invariant co-ordinate selecetion, Journal of the Royal Statistical Society,Series B, 71, 549–592.

See Also

ics

Examples

data(iris)
iris.centered <- sweep(iris[,1:4], 2, colMeans(iris[,1:4]), "-")
iris.paa <- ics(iris.centered, cov, covAxis, stdKurt = FALSE)
summary(iris.paa)
plot(iris.paa, col=as.numeric(iris[,5]))    
mean(iris.paa@gKurt)
emp.align <- iris.paa@gKurt
emp.align

screeplot(iris.paa)
abline(h = 1)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
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> library(ICS)
Loading required package: mvtnorm
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/ICS/covAxis.Rd_%03d_medium.png", width=480, height=480)
> ### Name: covAxis
> ### Title: One step Tyler Shape Matrix
> ### Aliases: covAxis
> ### Keywords: multivariate
> 
> ### ** Examples
> 
> 
> data(iris)
> iris.centered <- sweep(iris[,1:4], 2, colMeans(iris[,1:4]), "-")
> iris.paa <- ics(iris.centered, cov, covAxis, stdKurt = FALSE)
> summary(iris.paa)

ICS based on two scatter matrices 
S1:  cov
S2:  covAxis

The generalized kurtosis measures of the components are:
[1] 1.2336 1.0168 0.9312 0.8184


The Unmixing matrix is:
        [,1]   [,2]    [,3]    [,4]
[1,]  0.4038 0.1019 -0.5451 -0.3912
[2,] -3.1804 1.9819  2.1251 -1.6616
[3,] -0.0739 2.0609 -0.6686  2.2675
[4,] -0.1762 1.6949  2.1843 -4.4315
> plot(iris.paa, col=as.numeric(iris[,5]))    
> mean(iris.paa@gKurt)
[1] 1
> emp.align <- iris.paa@gKurt
> emp.align
[1] 1.2336055 1.0168092 0.9311902 0.8183951
> 
> screeplot(iris.paa)
> abline(h = 1)
> 
> 
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>