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

## 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

 1 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

  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) ## 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 June 20, 2021, 5:07 p.m.