# init.theta.MSAR.VM: Initialisation function for von Mises MSAR model fitting In NHMSAR: Non-Homogeneous Markov Switching Autoregressive Models

## Description

Initialization before fitting von Mises (non) homogeneous Markov switching autoregressive models by EM algorithm. Non homogeneity may be introduce in the probability transitions. The link function is defined here.

## Usage

 1 2 3 4 init.theta.MSAR.VM(data, ..., M, order, regime_names = NULL, nh.emissions = NULL, nh.transitions = NULL, label = NULL, ncov.emis = 0, ncov.trans = 0) 

## Arguments

 data array of univariate or multivariate series with dimension T*N.samples*d with T: number of time steps of each sample, N.samples: number of realisations of the same stationary process, d: dimension M number of regimes order order of AR processes label "HH" (default) for homogeneous MS AR model "NH" for non homogeneous transitions regime_names (optional) regime's names may be chosen nh.emissions not available - under development. nh.transitions link function for non homogeneous transitions. Default: von Mises (see details). ncov.emis not available - under development. ncov.trans number of covariates in NH model ...

## Details

The model with non homogeneous transitions is labeled "NH" and it is written

P(X_t|X_{t-1}=x_{t-1}) = q(z_t,θ_{z_t})

with X_t the hidden process and q von Mises link function such that

p_1(x_t|x_{t-1},z_{t}) =\frac{ q_{x_{t-1},x_t}≤ft|\exp ≤ft(\tildeλ_{x_{t-1},x_t} e^{-iz_{t}} \right)\right|} {∑_{x'=1}^M q_{x_{t-1},x'}≤ft|\exp ≤ft(\tildeλ_{x_{t-1},x'} e^{-iz_{t}} \right)\right|},

with \tildeλ_{x,x'} a complex parameter (by taking \tildeλ_{x,x'}=λ_{x,x'} e^{iψ_{x,x'}}).

## Value

return a list of class MSAR including

 theta parameter ..$transmat transition matrix ..$prior prior probabilities ..$mu vector of intercepts ..$kappa matrix of 'AR' coefficients (not complex by default) ..$par.emis parameters of non homogeneous emissions (not used) ..$par.trans parameters of non homogeneous transitions label model's label

## Author(s)

Val\'erie Monbet, [email protected]

## References

Ailliot P., Bessac J., Monbet V., Pene F., (2014) Non-homogeneous hidden Markov-switching models for wind time series. JSPI.