# tM: Joint M-estimation of Location and Scatter for a Multivariate... In ICS: Tools for Exploring Multivariate Data via ICS/ICA

## Description

The functions implements three EM algorithms to M-estimate the location vector and scatter matrix of a multivariate t-distribution.

## Usage

 ```1 2 3``` ```tM(X, df = 1, alg = "alg3", mu.init = NULL, V.init = NULL, gamma.init = NULL, eps = 1e-06, maxiter = 100, na.action = na.fail) ```

## Arguments

 `X` numeric data matrix or dataframe. `df` assumed degrees of freedom of the t-distribution. Default is `1` which corresponds to the Cauchy distribution. `alg` specifies which algorithm to use. Options are `alg1`, `alg2` or `alg3`. `alg3` is the default. `mu.init` initial value for the location vector if available. `V.init` initial value for the scatter matrix if available. `gamma.init` initial value for gamma if available. Only needed for `alg2`. `eps` convergence tolerance. `maxiter` maximum number of iterations. `na.action` a function which indicates what should happen when the data contain 'NA's. Default is to fail.

## Details

This implements the EM algorithms described in Kent et al. (1994). The norm used to define convergence is as in Arslan et al. (1995).

Algorithm 1 is valid for all degrees of freedom `df` > 0. Algorithm 2 is well defined only for degrees of freedom `df` > 1. Algorithm 3 is the limiting case of Algorithm 2 with degrees of freedom `df` = 1.

The performance of the algorithms are compared in Arslan et al. (1995).

Note that `cov.trob` in the MASS package implements also a covariance estimate for a multivariate t-distribution. That function provides for example also the possibility to fix the location. It requires however that the degrees of freedom exceeds 2.

## Value

A list containing:

 `mu ` vector with the estimated loaction. `V ` matrix of the estimated scatter. `gam ` estimated value of gamma. Only present when `alg2` is used. `iter ` number of iterations.

Klaus Nordhausen

## References

Kent, J.T., Tyler, D.E. and Vardi, Y. (1994), A curious likelihood identity for the multivariate t-distribution, Communications in Statistics, Simulation and Computation, 23, 441–453. <doi:10.1080/03610919408813180>.

Arslan, O., Constable, P.D.L. and Kent, J.T. (1995), Convergence behaviour of the EM algorithm for the multivariate t-distribution, Communications in Statistics, Theory and Methods, 24, 2981–3000. <doi:10.1080/03610929508831664>.

`cov.trob`
 ```1 2 3 4 5``` ```set.seed(654321) cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3) X <- rmvt(100, cov.matrix, 1) tM(X) rm(.Random.seed) ```