# Mstep.nh.MSAR: M step of the EM algorithm. In NHMSAR: Non-Homogeneous Markov Switching Autoregressive Models

## Description

M step of the EM algorithm for fitting Markov switching auto-regressive models with non homogeneous transitions.

## Usage

 ```1 2 3``` ```Mstep.nh.MSAR(data,theta,FB,covar=NULL,method=method, ARfix=FALSE,reduct=FALSE,penalty=FALSE,sigma.diag=FALSE, lambda1=lambda1,lambda2=lambda2,par = NULL) ```

## Arguments

 `data` array of univariate or multivariate series with dimension T*N.samples*d. T: number of time steps of each sample, N.samples: number of realisations of the same stationary process, d: dimension. `theta` model's parameter; object of class MSAR. See also init.theta.MSAR. `FB` Forward-Backward results, obtained by calling Estep.MSAR function `covar` transitions covariates `method` permits to choice the optimization algorithm. default is "ucminf", other possible choices are "BFGS" or "L-BFGS-B" `sigma.diag` if TRUE the innovation covariance matrices are diagonal. `reduct` if TRUE, autoregressive matrices and innovation covariance matrices are constrained to have the same pattern (zero and non zero coefficients) as the one of initial matrices. `ARfix` if TRUE the AR parameters are not estimated, they stay fixed at their initial value. `lambda1` penalization constant for the precision matrices. It may be a scalar or a vector of length M (with M the number of regimes). If it is equal to0 no penalization is introduced for the precision matrices. `lambda2` penalization constant for the autoregressive matrices. It may be a scalar or a vector of length M (with M the number of regimes). If it is equal to0 no penalization is introduced for the atoregression matrices. `penalty` choice of the penalty for the autoregressive matrices. Possible values are ridge, lasso or SCAD (default). `par` allows to give an initial value to the precision matrices.

## Value

List containing

 `..\$A0` intercepts `..\$A` AR coefficients `..\$sigma` variance of innovation `..\$prior` prior probabilities `..\$transmat` transition matrix `..\$par.trans` transitions parameters

## Author(s)

Valerie Monbet, [email protected]

## References

Ailliot P., Monbet V., (2012), Markov switching autoregressive models for wind time series. Environmental Modelling & Software, 30, pp 92-101.