MLE of (hyper-)spherical distributions | R Documentation |

MLE of (hyper-)spherical distributions.

```
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)
```

`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. |

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.

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. |

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

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).

```
racg, vm.mle, rvmf
```

```
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) )
```

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