# riem.pga: Principal Geodesic Analysis In Riemann: Learning with Data on Riemannian Manifolds

 riem.pga R Documentation

## 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