rotations (version 1.5)

zhang: M-estimator theory pivotal bootstrap confidence region

Description

Compute the radius of a $100(1-\alpha)$% confidence region for the central orientation based on M-estimation theory.

Usage

zhang(x, estimator, alp, m)
  "zhang" (x, estimator, alp = NULL, m = 300)
  "zhang" (x, estimator, alp = NULL, m = 300)

Arguments

x

$n-by-p$ matrix where each row corresponds to a random rotation in matrix ($p=9$) or quaternion ($p=4$) form.

estimator

character string either "mean" or "median."

alp

alpha level desired, e.g. 0.05 or 0.10.

m

number of replicates to use to estimate the critical value.

Value

Radius of the confidence region centered at the specified estimator.

Details

Compute the radius of a $100(1-\alpha)$% confidence region for the central orientation based on the projected mean estimator using the method due to Zhang & Nordman (2009) (unpublished MS thesis). By construction each axis will have the same radius so the radius reported is for all three axis. A normal theory version of this procedure uses the theoretical chi-square limiting distribution and is given by the chang option. This method is called "direct" because it used M-estimation theory for SO(3) directly instead of relying on transforming a result from directional statistics as prentice and fisheretal do.

See Also

bayesCR, prentice, fisheretal, chang

Examples

Rs <- ruars(20, rcayley, kappa = 100)

#The zhang method can be accesed from the "region" function or the "zhang" function
#They will be different because it is a bootstrap.
region(Rs, method = "direct", type = "bootstrap", alp = 0.1, estimator = "mean")
zhang(Rs, estimator = "mean", alp = 0.1)