riem.pga: Principal Geodesic Analysis

In Riemann: Learning with Data on Riemannian Manifolds

Principal Geodesic Analysis

Description

Given N observations X_1, X_2, …, X_N \in \mathcal{M}, Principal Geodesic Analysis (PGA) finds a low-dimensional embedding by decomposing 2nd-order information in tangent space at an intrinsic mean of the data.

Usage

riem.pga(riemobj, ndim = 2)

Arguments

riemobj	a S3 "riemdata" class for N manifold-valued data.
ndim	an integer-valued target dimension.

Value

a named list containing

center

an intrinsic mean in a matrix representation form.

embed

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

References

\insertRef

fletcher_principal_2004Riemann

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)

## EMBEDDING WITH MDS AND PGA
embed2mds = riem.mds(myriem, ndim=2, geometry="intrinsic")$embed
embed2pga = riem.pga(myriem, ndim=2)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2mds, main="Multidimensional Scaling",    col=mylabs, pch=19)
plot(embed2pga, main="Principal Geodesic Analysis", col=mylabs, pch=19)
par(opar)

Riemann documentation built on March 18, 2022, 7:55 p.m.