sscor (version 0.2)

SSCM2Shape: Calculation of the shape matrix

Description

SSCM2Shape transforms the spatial sign covariance matrix of an elliptical distribution into its standardized shape matrix.

Usage

SSCM2Shape(V,itermax=100,tol=10^(-10))

Arguments

V

(required) p x p matrix representing the theoretical SSCM.

tol

(optional) numeric, defines the stopping rule of the approximation procedure, see the help of evSSCM2evShape for details.

itermax

(optional) numeric, defines the maximal number of iterations, see the help of evSSCM2evShape for details.

Value

p x p symmetric numerical matrix, representing the shape matrix with trace 1, which corresponds to the spatial sign covariance matrix.

Details

The calculation consists of three steps. First one calculates eigenvectors and eigenvalues of the SSCM matrix by the function eigen. Then one determines the eigenvalues of the related Shape matrix using the function evSSCM2evShape. Finally one expands the eigendecomposition consisting of the eigenvectors of the SSCM and the eigenvalues of the shape matrix. The resulting shape matrix is standardized to have a trace of 1. Note that this procedure only works for elliptical distributions.

References

Dürre, A., Vogel, D., Fried, R. (2015): Spatial sign correlation, Journal of Multivariate Analyis, vol. 135, 89--105. arvix 1403.7635

Dürre, A., Tyler, D. E., Vogel, D. (2016): On the eigenvalues of the spatial sign covariance matrix in more than two dimensions, to appear in: Statistics and Probability Letters. arvix 1512.02863

See Also

Calculating the theoretical shape from the theoretical SSCM SSCM2Shape

Calculating the eigenvalues of the SSCM evShape2evSSCM

Examples

# defining a shape matrix with trace 1
V <- matrix(c(1,0.8,-0.2,0.8,1,0,-0.2,0,1),ncol=3)/3
V

# calculating the related SSCM
SSCM <- Shape2SSCM(V)

# recalculate the shape based on the SSCM
V2 <- SSCM2Shape(SSCM)
V2

# error is negligible
sum(abs(V-V2))