Prepare Data on Symmetric Positive-Definite (SPD) Manifold

The collection of symmetric positive-definite matrices is a well-known example of matrix manifold. It is defined as $$\mathcal{S}_{++}^p = \lbrace X \in \mathbf{R}^{p\times p} ~\vert~ X^\top = X,~ v^\top Xv>0~\textrm{for all}~v\ne0 \rbrace$$ where positivity is verified by Cholesky factorization after a scale-relative symmetry check. No ridge or positive-definite projection is applied. Tiny physical units alone do not make a matrix ill-conditioned. Please note that the geometry involving semi-definite matrices is considered in wrap.spdk.

Usage

wrap.spd(input)

Arguments

input

SPD data matrices to be wrapped as riemdata class. Following inputs are considered,

array: an $(p\times p\times n)$ array where each slice along 3rd dimension is a SPD matrix.
list: a length-$n$ list whose elements are $(p\times p)$ SPD matrices.

Value

a named riemdata S3 object containing

data: a list of $(p\times p)$ SPD matrices.
size: size of each SPD matrix.
name: name of the manifold of interests, "spd"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations; empirical covariance of normal observations.
#-------------------------------------------------------------------
#  Data Generation
d1 = array(0,c(3,3,5))
d2 = list()
for (i in 1:5){
  dat = matrix(rnorm(10*3),ncol=3)
  d1[,,i] = stats::cov(dat)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.spd(d1)
test2 = wrap.spd(d2)