The unit hypersphere (sphere, for short) is one of the most fundamental curved
space in studying geometry. Precisely, we denote \((p-1)\) sphere in \(\mathbf{R}^p\) by
$$\mathcal{S}^{p-1} = \lbrace x \in \mathbf{R}^p ~ \vert ~ x^\top x = \|x\|^2 = 1 \rbrace$$
where vectors are of unit norm. In wrap.sphere, normalization is applied when
each data point is not on the unit sphere.
Arguments
- input
data vectors to be wrapped as
riemdataclass. Following inputs are considered,- matrix
an \((n \times p)\) matrix of row observations of unit norm.
- list
a length-\(n\) list whose elements are length-\(p\) vectors of unit norm.
Value
a named riemdata S3 object containing
- data
a list of \((p\times 1)\) matrices in \(\mathcal{S}^{p-1}\).
- size
dimension of the ambient space.
- name
name of the manifold of interests, "sphere"
Examples
#-------------------------------------------------------------------
# Checker for Two Types of Inputs
#
# Generate 5 observations in S^2 embedded in R^3.
#-------------------------------------------------------------------
## DATA GENERATION
d1 = array(0,c(5,3))
d2 = list()
for (i in 1:5){
single = stats::rnorm(3)
d1[i,] = single
d2[[i]] = single
}
## RUN
test1 = wrap.sphere(d1)
test2 = wrap.sphere(d2)