orientlm: Linear models for orientation data

Description

Regression models for matched pairs of orientations.

Usage

orientlm(observed, leftformula, trueorient = rotmatrix(diag(3)), 
         rightformula, data = list(), subset, weights, na.action, 
         iterations = 5)

Value

Returns a list containing the following components:

leftfit: Result of lm call based on leftformula
rightfit: Result of lm call based on rightformula
A1: Fitted values of $A_{1}$ for each observation
A2: Fitted values of $A_{2}$ for each observation
predict: Fitted values of $A_{1}^{t} U_{i} A_{2}$ for each observation

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 leftformula and rightformula are specified.

Author

Duncan Murdoch

Details

The Prentice (1989) model for matched pairs of orientations was

$E (V_{i}) = k A_{1}^{t} U_{i} A_{2}$

where $V_{i}$ is the observed orientation, $A_{1}$ and $A_{2}$ are orientation matrices, and $U_{i}$ 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 $A_{1}$ and $A_{2}$ 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.

References

Prentice, M.J. (1989). Spherical regression on matched pairs of orientation statistics. JRSS B 51, 241-248.

Shin, H.S.H. (1999). Experimental Design for Orientation Models. PhD thesis, Queen's University.

Examples

Run this code

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)

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