Fréchet Mean and Variation

Given \(N\) observations \(X_1, X_2, \ldots, X_N \in \mathcal{M}\), compute Fréchet mean and variation with respect to the geometry by minimizing $$\textrm{min}_x \sum_{n=1}^N w_n \rho^2 (x, x_n),\quad x\in\mathcal{M}$$ where \(\rho (x, y)\) is a distance for two points \(x,y\in\mathcal{M}\). If non-uniform weights are given, normalized version of the mean is computed and if weight=NULL, it automatically sets equal weights (\(w_i = 1/n\)) for all observations.

Usage

riem.mean(riemobj, weight = NULL, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

riemobj

a S3 "riemdata" class for \(N\) manifold-valued data.

weight

weight of observations; if NULL it assumes equal weights, or a nonnegative length-\(N\) vector that sums to 1 should be given.

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

mean

a mean matrix on \(\mathcal{M}\).

variation

sum of (weighted) squared distances.

Examples

#-------------------------------------------------------------------
#        Example on Sphere : points near (0,1) on S^1 in R^2
#-------------------------------------------------------------------
## GENERATE DATA
ndata = 50
mydat = array(0,c(ndata,2))
for (i in 1:ndata){
  tgt = c(stats::rnorm(1, sd=2), 1)
  mydat[i,] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydat)

## COMPUTE TWO MEANS
mean.int = as.vector(riem.mean(myriem, geometry="intrinsic")$mean)
mean.ext = as.vector(riem.mean(myriem, geometry="extrinsic")$mean)

## VISUALIZE
opar <- par(no.readonly=TRUE)
plot(mydat[,1], mydat[,2], pch=19, xlim=c(-1.1,1.1), ylim=c(0,1.1),
     main="BLUE-extrinsic vs RED-intrinsic")
arrows(x0=0,y0=0,x1=mean.int[1],y1=mean.int[2],col="red")
arrows(x0=0,y0=0,x1=mean.ext[1],y1=mean.ext[2],col="blue")

par(opar)