pdSpecEst (version 1.2.4)

pdMean: Weighted Karcher mean of HPD matrices

Description

pdMean calculates an (approximate) weighted Karcher or Frechet mean of a sample of \((d,d)\)-dimensional HPD matrices intrinsic to a user-specified metric. In the case of the affine-invariant Riemannian metric as detailed in e.g., B09pdSpecEst[Chapter 6] or PFA05pdSpecEst, the weighted Karcher mean is either approximated via the fast recursive algorithm in H13pdSpecEst or computed via the slower, but more accurate, gradient descent algorithm in P06pdSpecEst. By default, the unweighted Karcher mean is computed.

Usage

pdMean(M, w, metric = "Riemannian", grad_desc = FALSE, 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'.

grad_desc

if metric = "Riemannian", a logical value indicating if the gradient descent algorithm in P06pdSpecEst should be used, defaults to FALSE.

maxit

maximum number of iterations in gradient descent algorithm, only used if grad_desc = TRUE and metric = "Riemannian". Defaults to 1000

reltol

optional tolerance parameter in gradient descent algorithm, only used if grad_desc = TRUE and metric = "Riemannian". Defaults to 1E-10.

References

See Also

Mid, pdMedian

Examples

# NOT RUN {
## 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())
z <- rnorm(100)
## Generate random weight vector
w <- abs(z)/sum(abs(z))
## Compute weighted (Riemannian) Karcher mean
pdMean(M, w)

# }