Von Mises mixtures: mu and kappa?

Hello

I am trying to calculate the values of the concentration parameters (kappa) and preferred direction (mu) for a Von Mises mixture model. I currently have some R code that gives me optimised values for the product of kappa and mu, but I'm not sure how to calculate them when both are unknown? How could I calculate mu and kappa from y2 if I didn't know either in the 1st place? I what to use movMF to give me values of kappa from some directional data where I don't know either kappa or mu.


## Generate and fit a "small-mix" data set a la Banerjee et al.
mu <- rbind(c(-0.251, -0.968),
            c(0.399, 0.917))
kappa <- c(4, 4)

theta <- kappa * mu
theta
alpha <- c(0.48, 0.52)

## Generate a sample of size n = 50 from the von Mises-Fisher mixture
## with the above parameters.
set.seed(123)
x <- rmovMF(50, theta, alpha)
## Fit a von Mises-Fisher mixture with the "right" number of components,
## using 10 EM runs.
y2 <- movMF(x, 2, nruns = 10)

Y2 gives

y2

theta:
       [,1]      [,2]
1  2.443225  5.259337
2 -1.851384 -4.291278
alpha:
[1] 0.4823648 0.5176352
L:
[1] 24.98124

How could I calculate kappa and mu if I didn't know either in the 1st place?

Thanks
Peter


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