# covAxis: One step Tyler Shape Matrix In ICS: Tools for Exploring Multivariate Data via ICS/ICA

## Description

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

## Usage

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

## Arguments

 `X` numeric data matrix or dataframe. `na.action` a function which indicates what should happen when the data contain 'NA's. Default is to fail.

## 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.

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. <doi:10.1111/j.1467-9868.2009.00706.x>.

`ics`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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) ```