# riem.mds: Multidimensional Scaling In Riemann: Learning with Data on Riemannian Manifolds

 riem.mds R Documentation

## Multidimensional Scaling

### Description

Given N observations X_1, X_2, …, X_N \in \mathcal{M}, apply multidimensional scaling to get low-dimensional embedding in Euclidean space. Usually, ndim=2,3 are chosen for visualization.

### Usage

riem.mds(riemobj, ndim = 2, geometry = c("intrinsic", "extrinsic"))

### Arguments

 riemobj a S3 "riemdata" class for N manifold-valued data. ndim an integer-valued target dimension (default: 2). geometry (case-insensitive) name of geometry; either geodesic ("intrinsic") or embedded ("extrinsic") geometry.

### Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

stress

discrepancy between embedded and original distances as a measure of error.

### References

\insertRef

torgerson_multidimensional_1952aRiemann

### Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
tgt = c(1, stats::rnorm(2, sd=0.1))
mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
tgt = c(stats::rnorm(2, sd=0.1), 1)
mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## MDS EMBEDDING WITH TWO GEOMETRIES
embed2int = riem.mds(myriem, geometry="intrinsic")$embed embed2ext = riem.mds(myriem, geometry="extrinsic")$embed

## VISUALIZE