pdMedian: Weighted intrinsic median of HPD matrices

In JorisChau/pdSpecEst: An Analysis Toolbox for Hermitian Positive Definite Matrices

Description

pdMedian calculates a weighted intrinsic median of a sample of (d,d)-dimensional HPD matrices based on a Weiszfeld algorithm intrinsic to the chosen metric.In the case of the affine-invariant Riemannian metric as detailed in e.g., \insertCiteB09pdSpecEst[Chapter 6] or \insertCitePFA05pdSpecEst, the intrinsic Weiszfeld algorithm in \insertCiteF09pdSpecEst is used. By default, the unweighted intrinsic median is computed.

Usage

pdMedian(M, w, metric = "Riemannian", maxit = 1000, reltol)

Arguments

M	a (d,d,S)-dimensional array corresponding to a sample of (d,d)-dimensional HPD matrices of size S.
w	an S-dimensional nonnegative weight vector, such that sum(w) = 1.
metric	the distance measure, one of 'Riemannian', 'logEuclidean', 'Cholesky', 'Euclidean' or 'rootEuclidean'. Defaults to 'Riemannian'.
maxit	maximum number of iterations in gradient descent algorithm. Defaults to 1000
reltol	optional tolerance parameter in gradient descent algorithm. Defaults to 1E-10.

Note

The function does not check for positive definiteness of the input matrices, and (depending on the specified metric) may fail if matrices are close to being singular.

References

\insertAllCited

See Also

pdMean

Examples

## Generate random sample of HPD matrices
m <- function(){
 X <- matrix(complex(real=rnorm(9), imaginary=rnorm(9)), nrow=3)
 t(Conj(X)) %*% X
}
M <- replicate(100, m())
## Generate random weight vector
z <- rnorm(100)
w <- abs(z)/sum(abs(z))
## Compute weighted intrinsic (Riemannian) median
pdMedian(M, w)

JorisChau/pdSpecEst documentation built on Jan. 13, 2020, 6:08 p.m.