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

## 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

 1 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 #------------------------------------------------------------------- # 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 opar = par(no.readonly=TRUE) par(mfrow=c(1,2), pty="s") plot(embed2int, main="intrinsic MDS", ylim=c(-2,2), col=mylabs, pch=19) plot(embed2ext, main="extrinsic MDS", ylim=c(-2,2), col=mylabs, pch=19) par(opar) 

Riemann documentation built on June 20, 2021, 5:07 p.m.