riem.sammon {Riemann}

Sammon Mapping

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, apply Sammon mapping, a non-linear dimensionality reduction method. Since the method depends only on the pairwise distances of the data, it can be adapted to the manifold-valued data.

Usage

riem.sammon(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.

...

extra parameters including

maxiter

maximum number of iterations to be run (default:50).

eps

tolerance level for stopping criterion (default: 1e-5).

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

Sammon JW (1969). “A Nonlinear Mapping for Data Structure Analysis.” IEEE Transactions on Computers, C-18(5), 401–409. ISSN 0018-9340.

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)

## COMPARE SAMMON WITH MDS
embed2mds = riem.mds(myriem, ndim=2)$embed
embed2sam = riem.sammon(myriem, ndim=2)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2mds, col=mylabs, pch=19, main="MDS")
plot(embed2sam, col=mylabs, pch=19, main="Sammon mapping")
par(opar)