mean: Mean rotation

Description

Compute the sample geometric or projected mean.

Usage

# S3 method for SO3
mean(x, type = "projected", epsilon = 1e-05, maxIter = 2000, ...)
# S3 method for Q4
mean(x, type = "projected", epsilon = 1e-05, maxIter = 2000, ...)

Value

Estimate of the projected or geometric mean of the sample in the same parametrization.

Arguments

x: $n\times p$ matrix where each row corresponds to a random rotation in matrix form ($p=9$) or quaternion ($p=4$) form.
type: string indicating "projected" or "geometric" type mean estimator.
epsilon: stopping rule for the geometric-mean.
maxIter: maximum number of iterations allowed for geometric-mean.
...: additional arguments.

Details

This function takes a sample of 3D rotations (in matrix or quaternion form) and returns the projected arithmetic mean denoted $\widehat{\bm S}_P$ or geometric mean $\widehat{\bm S}_G$ according to the type option. For a sample of $n$ rotations in matrix form $\bm{R}_i\in SO(3), i=1,2,\dots,n$, the mean-type estimator is defined as $$\widehat{\bm{S}}=argmin_{\bm{S}\in SO(3)}\sum_{i=1}^nd^2(\bm{R}_i,\bm{S})$$ where $d$ is the Riemannian or Euclidean distance. For more on the projected mean see moakher02 and for the geometric mean see manton04. For the projected mean from a quaternion point of view see tyler1981.

tyler1981, moakher02, manton04

Examples

Run this code

Rs <- ruars(20, rvmises, kappa = 0.01)

# Projected mean
mean(Rs)

# Same as mean(Rs)
project.SO3(colMeans(Rs))

# Geometric mean
mean(Rs, type = "geometric")

# Bias of the projected mean
rot.dist(mean(Rs))

# Bias of the geometric mean
rot.dist(mean(Rs, type = "geometric"))

# Same thing with quaternion form
Qs <- as.Q4(Rs)
mean(Qs)
mean(Qs, type = "geometric")
rot.dist(mean(Qs))
rot.dist(mean(Qs, type = "geometric"))