R: MLE of (hyper-)spherical distributions

MLE of (hyper-)spherical distributions {Rfast} | R Documentation

## MLE of (hyper-)spherical distributions

### Description

MLE of (hyper-)spherical distributions.

### Usage

    vmf.mle(x, tol = 1e-07)
    multivmf.mle(x, ina, tol = 1e-07, ell = FALSE)
    acg.mle(x, tol = 1e-07)
    iag.mle(x, tol = 1e-07)

### Arguments

`x` | A matrix with directional data, i.e. unit vectors.
`ina` | A numerical vector with discrete numbers starting from 1, i.e. 1, 2, 3, 4,... or a factor variable. Each number denotes a sample or group. If you supply a continuous valued vector the function will obviously provide wrong results.
`ell` | This is for the multivmf.mle only. Do you want the log-likelihood returned? The default value is TRUE.
`tol` | The tolerance value at which to terminate the iterations.

### Details

For the von Mises-Fisher, the normalised mean is the mean direction. For the concentration parameter, a Newton-Raphson is implemented. For the angular central Gaussian distribution there is a constraint on the estimated covariance matrix; its trace is equal to the number of variables. An iterative algorithm takes place and convergence is guaranteed. Newton-Raphson for the projected normal distribution, on the sphere, is implemented as well. Finally, the von Mises-Fisher distribution for groups of data is also implemented.

### Value

For the von Mises-Fisher a list including:

`loglik` | The maximum log-likelihood value.
`mu` | The mean direction.
`kappa` | The concentration parameter.

For the multi von Mises-Fisher a list including:

`loglik` | A vector with the maximum log-likelihood values if ell is set to TRUE. Otherwise NULL is returned.
`mi` | A matrix with the group mean directions.
`ki` | A vector with the group concentration parameters.

For the angular central Gaussian a list including:

`iter` | The number if iterations required by the algorithm to converge to the solution.
`cova` | The estimated covariance matrix.

For the spherical projected normal a list including:

`iters` | The number of iteration required by the Newton-Raphson.
`mesi` | A matrix with two rows. The first row is the mean direction and the second is the mean vector. The first comes from the second by normalising to have unit length.
`param` | A vector with the elements, the norm of mean vector, the log-likelihood and the log-likelihood of the spherical uniform distribution. The third value helps in case you want to do a log-likleihood ratio test for uniformity.

### Author(s)

Michail Tsagris R implementation and documentation: Michail Tsagris <mtsagris@uoc.gr>

### References

Mardia, K. V. and Jupp, P. E. (2000). Directional statistics. Chicester: John Wiley & Sons.

Sra, S. (2012). A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of Is(x). Computational Statistics, 27(1): 177–190.

Tyler D. E. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika 74(3): 579-589.

Paine P.J., Preston S.P., Tsagris M and Wood A.T.A. (2017). An Elliptically Symmetric Angular Gaussian Distribution. Statistics and Computing (To appear).

### See Also

`racg, vm.mle, rvmf`

### Examples

    m <- c(0, 0, 0, 0)
    s <- cov(iris[, 1:4])
    x <- racg(100, s)
    mod <- acg.mle(x)
    mod
    res<-cov2cor(mod$cova)  ## estimated covariance matrix turned into a correlation matrix
    res<-cov2cor(s)  ## true covariance matrix turned into a correlation matrix
    res<-vmf.mle(x)
    x <- rbind( rvmf(100,rnorm(4), 10), rvmf(100,rnorm(4), 20) )
    a <- multivmf.mle(x, rep(1:2, each = 100) )

[Package Rfast version 2.0.4 Index]