Orthogonalized Gnanadesikan-Kettenring robust scatter

The Gnanadesikan-Kettenring identity turns a covariance into a pair of scale estimates, cov(u, v) = (sigma(u + v)^2 - sigma(u - v)^2) / 4, so a robust scale gives a robust covariance entry by entry. The catch is that a matrix built this way need not be positive semi-definite, which makes it useless as a scatter. Maronna & Zamar's fix is to build it once, take its eigenvectors as a new basis, and rebuild the scatter DIAGONALLY in that basis: (1) y_j = x_j / sigma_j with sigma_j the MAD; (2) U_jk by the GK identity on the standardized columns, U_jj = 1; (3) \(U = E Lambda E'\), \(z = E' y\); (4) Gamma = diag(sigma(z_l)^2), \(Sigma = A Gamma A'\) with A = diag(sigma) E, positive semi-definite by construction; (5) location mu = A m(z), m the coordinatewise median.

Usage

Ogkcv(y, X = NULL)

Arguments

y

Either the full data matrix (n by p) or, when X is given, its first column.

X

Remaining columns; when supplied the data matrix is \([y, X]\) and the robust regression coefficient Sigma_xx^-1 Sigma_xy is returned as well.

Value

List with sigma, location, scales, estimate, eigenvalues, det, beta (only when X is given), n, p.

Details

Determinism: MAD and medians only, and the shared sign-fixed Jacobi eigensolver, so the basis is identical in both language arms.

References

Maronna, R. A. & Zamar, R. H. (2002). Robust estimates of location and dispersion for high-dimensional datasets. Technometrics, 44(4), 307-317. doi:10.1198/004017002188618509 Gnanadesikan, R. & Kettenring, J. R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 28(1), 81-124.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ogkcv(V)
#> $sigma
#>          [,1]
#> [1,] 8.792411
#> 
#> $location
#> [1] 4.5
#> 
#> $scales
#> [1] 2.9652
#> 
#> $estimate
#> [1] 8.792411
#> 
#> $eigenvalues
#> [1] 1
#> 
#> $det
#> [1] 8.792411
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 1
#> 
#> $method
#> [1] "Orthogonalized Gnanadesikan-Kettenring scatter"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"