Description Usage Arguments Details Value References See Also Examples
evShape2evSSCM
transforms the eigenvalues of the shape matrix of an elliptical distribution into that of the spatial sign covariance matrix.
1 | evShape2evSSCM(evShape)
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evShape |
(required) p-dimensional numeric, representing the eigenvalues of the shape matrix. |
The eigenvalues of the SSCM can be calculated from the eigenvalues of the shape matrix by numerical evaluation of onedimensional integrals, see Proposition 3 of Dürre, Tyler, Vogel (2016). We use the substitution
x=(1+t)/(1-t)
and Gaussian quadrature with Jacobi polynomials up to order 500 and beta=0 as well as alpha=p/2-1, see chapter 2.4 (iv) of Gautschi (1997) for details.
The nodes and weights of the Gauss-Jacobi-quadrature are originally computed by the gaussquad
package and saved in the file jacobiquad
for faster computation.
p-dimensional numeric, representing the eigenvalues of the corresponding spatial sign covariance matrix.
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
Gautschi, W. (1997): Numerical Analysis - An Introduction, Birkhäuser, Basel.
Novomestky, F. (2013): gaussquad: Collection of functions for Gaussian quadrature. R package version 1.0-2.
Calculating the theoretical SSCM from the theoretical shape matrix Shape2SSCM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | # defining eigenvalues of the shape matrix
evShape <- seq(from=0,to=1,by=0.1)
# standardized to have sum 1
evShape <- evShape/sum(evShape)
# calculating the related eigenvalues of the SSCM
evSSCM <- evShape2evSSCM(evShape)
plot(evShape,evSSCM)
# recalculate the eigenvalues of the shape matrix
evShape2 <- evSSCM2evShape(evSSCM)
# error is negligible
sum(abs(evShape-evShape2))
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