Linear models for orientation data
Description
Regression models for matched pairs of orientations.
Usage
1 2 3 
Arguments
observed 
Observed orientations 
leftformula 
Formula for “left” model (see below) 
trueorient 
“True” orientation (see below) 
rightformula 
Formula for “right” model (see below) 
data 
Optional data frame for predictors in linear model 
subset 
Optional logical vector indicating subset of data 
weights 
Optional weights 
na.action 
Optional NA function for predictors 
iterations 
How many iterations to use. Ignored unless both

Details
The Prentice (1989) model for matched pairs of orientations was
E(Vi) = k t(A1) %*% Ui %*% A2
where Vi is the observed orientation, A1 and A2 are orientation matrices, and Ui is the “true” orientation, and k is a constant. It was assumed that errors were symmetrically distributed about the identity matrix.
This function generalizes this model, allowing A1 and A2 to depend on
regressor variables through leftformula
and rightformula
respectively.
These formulas should include the predictor variables (right hand side) only, e.g. use
~ x + y + z
rather than response ~ x + y + z
. Specify the response using
the observed
argument. If
both formulas are ~ 1
, i.e. intercepts only, then Prentice's original model is
recovered. More general models are fit by coordinatewise linear regression in the rotmatrix
representation of the orientation, with fitted values projected onto SO(3) using the
nearest.SO3
function.
When both left and right models are given, Prentice's iterative approach is used with a fixed number of iterations. Note that Shin (1999) found that Prentice's scheme sometimes fails to find the global minimum; this function presumably suffers from the same failing.
Value
Returns a list containing the following components:
leftfit 
Result of 
rightfit 
Result of 
A1 
Fitted values of A1 for each observation 
A2 
Fitted values of A2 for each observation 
predict 
Fitted values of t(A1) %*% Ui %*% A2 for each observation 
Author(s)
Duncan Murdoch
References
Prentice, M.J. (1989). Spherical regression on matched pairs of orientation statistics. JRSS B 51, 241248.
Shin, H.S.H. (1999). Experimental Design for Orientation Models. PhD thesis, Queen's University.
Examples
1 2 3 4 5 6 7 8 9 10 11 12  x < rep(1:10,10)
y < rep(1:10,each=10)
A1 < skewvector(cbind(x/10,y/10,rep(0,100)))
A2 < skewvector(c(1,1,1))
trueorientation < skewvector(matrix(rnorm(300),100))
noise < skewvector(matrix(rnorm(300)/10,100))
obs < t(A1) %*% trueorientation %*% A2 %*% noise
fit < orientlm(obs, ~ x + y, trueorientation, ~ 1)
context < boat3d(A1, x, z=y, col = 'green', graphics='scatterplot3d')
boat3d(fit$A1, x, z=y, add=context)
