mean {rotations}

Mean rotation

Description

Compute the sample geometric or projected mean.

Usage

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

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

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

Value

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

See Also

median.SO3, bayes.mean, weighted.mean.SO3

Examples

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"))