rotations (version 1.5)

UARS: Generic UARS Distribution

Description

Density, distribution function and random generation for the the generic uniform axis-random spin (UARS) class of distributions.

Usage

duars(R, dangle, S = id.SO3, kappa = 1, ...)
  puars(R, pangle = NULL, S = id.SO3, kappa = 1, ...)
  ruars(n, rangle, S = NULL, kappa = 1, space = "SO3", ...)

Arguments

R

Value at which to evaluate the UARS density.

n

number of observations. If length(n)>1, the length is taken to be the number required.

dangle

The function to evaluate the angles from, e.g. dcayley, dvmises, dfisher, dhaar.

pangle

The form of the angular density, e.g. pcayley, pvmises, pfisher, phaar.

rangle

The function from which to simulate angles, e.g. rcayley, rvmises, rhaar, rfisher.

S

central orientation of the distribution.

kappa

concentration parameter.

space

indicates the desired representation: matrix ("SO3") or quaternion ("Q4").

...

additional arguments.

Value

duars

gives the density

puars

gives the distribution function. If pangle is left empty, the empirical CDF is returned.

ruars

generates random deviates

Details

For the rotation R with central orientation S and concentration $\kappa$ the UARS density is given by $$f(R|S,\kappa)=\frac{4\pi}{3-tr(S^\top R)}C(\cos^{-1}[tr(S^\top R)-1]/2|\kappa)$$ where $C(r|\kappa)$ is one of the Angular-distributions.

References

Bingham M, Nordman D and Vardeman S (2009). "Modeling and Inference for Measured Crystal Orientations and a Tractable Class of Symmetric Distributions for Rotations in three Dimensions." Journal of the American Statistical Association, 104(488), pp. 1385-1397.

See Also

For more on the angular distribution options see Angular-distributions.

Examples

#Generate random rotations from the Cayley-UARS distribution with central orientation
#rotated about the y-axis through pi/2 radians
S <- as.SO3(c(0, 1, 0), pi/2)
Rs <- ruars(20, rangle = rcayley, kappa = 1, S = S)

rs <- mis.angle(Rs-S)                          #Find the associated misorientation angles
frs <- duars(Rs, dcayley, kappa = 10, S = S)   #Compute UARS density evaluated at each rotations
plot(rs, frs)

cdf <- puars(Rs, pcayley, S = S)               #By supplying 'pcayley', it is used to compute the
plot(rs, cdf)                                  #the CDF

ecdf <- puars(Rs, S = S)                       #No 'puars' arguement is supplied so the empirical
plot(rs, ecdf)                                 #cdf is returned