Linear models for orientation data

Regression models for matched pairs of orientations.

Usage

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

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.

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.

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

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.

Author

Duncan Murdoch

Examples

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)