rbase.mean: Fréchet Mean of Manifold-valued Data
In RiemBase: Functions and C++ Header Files for Computation on Manifolds

Description Usage Arguments Value Author(s) References Examples

View source: R/rbase.mean.R

For manifold-valued data, Fréchet mean is the solution of following cost function,

\textrm{min}_x ∑_{i=1}^n ρ^2 (x, x_i),\quad x\in\mathcal{M}

for a given data \{x_i\}_{i=1}^n and ρ(x,y) is the geodesic distance between two points on manifold \mathcal{M}. It uses a gradient descent method with a backtracking search rule for updating.

1	rbase.mean(input, maxiter = 496, eps = 1e-06, parallel = FALSE)

`input`	a S3 object of `riemdata` class. See `riemfactory` for more details.
`maxiter`	maximum number of iterations for gradient descent algorithm.
`eps`	stopping criterion for the norm of gradient.
`parallel`	a flag for enabling parallel computation.

a named list containing

x: an estimate Fréchet mean.
iteration: number of iterations until convergence.

Kisung You

\insertRef

karcher_riemannian_1977RiemBase

\insertRef

kendall_probability_1990RiemBase

\insertRef

afsari_convergence_2013RiemBase

### Generate 100 data points on Sphere S^2 near (0,0,1).
ndata = 100
theta = seq(from=-0.99,to=0.99,length.out=ndata)*pi
tmpx  = cos(theta) + rnorm(ndata,sd=0.1)
tmpy  = sin(theta) + rnorm(ndata,sd=0.1)

### Wrap it as 'riemdata' class
data  = list()
for (i in 1:ndata){
  tgt = c(tmpx[i],tmpy[i],1)
  data[[i]] = tgt/sqrt(sum(tgt^2)) # project onto Sphere
}
data = riemfactory(data, name="sphere")

### Compute Fréchet Mean
out1 = rbase.mean(data)
out2 = rbase.mean(data,parallel=TRUE) # test parallel implementation