Directional (version 4.8)

MLE of (hyper-)spherical distributions: MLE of (hyper-)spherical distributions

Description

MLE of (hyper-)spherical distributions.

Usage

vmf.mle(x, fast = FALSE, 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.

fast

IF you want a faster version, but with fewer information returned, set this equal to TRUE.

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.

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.

Details

The vm estimates the mean direction and concentration of a fitted von Mises-Fisher distribution.

The von Mises-Fisher distribution for groups of data is also implemented.

The acg.mle fits 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.

The iag.mle implements MLE of the projected normal distribution, on the sphere.

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. (2018). An Elliptically Symmetric Angular Gaussian Distribution. Statistics and Computing, 28: 689-697.

See Also

racg, vm.mle, rvmf

Examples

# NOT RUN {
m <- c(0, 0, 0, 0)
s <- cov(iris[, 1:4])
x <- racg(100, s)
mod <- acg.mle(x)
mod
cov2cor(mod$cova)  ## estimated covariance matrix turned into a correlation matrix
cov2cor(s)  ## true covariance matrix turned into a correlation matrix
vmf.mle(x)
x <- rbind( rvmf(100,rnorm(4), 10), rvmf(100,rnorm(4), 20) )
a <- multivmf.mle(x, rep(1:2, each = 100) )
# }