evShape2evSSCM transforms the eigenvalues of the shape matrix of an elliptical distribution into that of the spatial sign covariance matrix.
(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
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
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# 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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