# M step of the EM algorithm for fitting von Mises Markov switching auto-regressive models.

### Description

M step of the EM algorithm for fitting homogeneous Markov switching auto-regressive models, called in fit.MSAR.VM.

### Usage

1 | ```
Mstep.hh.MSAR.VM(data, theta, FB, constr = 0)
``` |

### 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 |

`constr` |
constraints are added to the |

### Details

The homogeneous MSAR model is labeled "HH" and it is written

*
P(X_t|X_{t-1}=x_{t-1}) = Q_{x_{t-1},x_t}*

with *X_t* the hidden univariate process defined on *\{1,\cdots,M \}*

*
Y_t|X_t=x_t,y_{t-1},...,y_{t-p}*

has a von Mises distribution with density

* p_2(y_t|x_t,y_{t-s}^{t-1}) = \frac {1}{b(x_t,y_{t-s}^{t-1})} \exp≤ft(κ_0^{(x_t)} \cos(y_t-φ_0^{(x_t)})+
∑_{\ell=1}^sκ_\ell^{(x_t)} \cos(y_t-y_{t-\ell}-φ_\ell^{(x)})\right)*

which is equivalent to

* p_2(y_t|x_t,y_{t-s}^{t-1}) = \frac {1}{b(x_t,y_{t-s}^{t-1})} ≤ft|\exp≤ft([γ_0^{(x_t)} +
∑_{\ell=1}^sγ_\ell^{(x_t)} e^{iy_{t-\ell}}]e^{-iy_t}\right)\right|*

*b(x_t,y_{t-s}^{t-1})* is a normalisation constant.

Both the real and the complex formulation are implemented. In practice, the complex version is used if the initial *κ* is complex.

### Value

List containing

`mu` |
intercepts |

`kappa` |
von Mises AR coefficients |

`prior` |
prior probabilities |

`transmat` |
transition matrix |

### Author(s)

Valerie Monbet, valerie.monbet@univ-rennes1.fr

### References

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

### See Also

fit.MSAR.VM, Estep.MSAR.VM

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