Mstep.hh.MSAR.VM: 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
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 $\kappa$ parameter (A preciser)
Value
List containing
muintercepts
kappavon Mises AR coefficients
priorprior probabilities
transmattransition matrix
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 $$p_2(y_t|x_t,y_{t-s}^{t-1}) = \frac {1}{b(x_t,y_{t-s}^{t-1})} \exp\left(\kappa_0^{(x_t)} \cos(y_t-\phi_0^{(x_t)})+
\sum_{\ell=1}^s\kappa_\ell^{(x_t)} \cos(y_t-y_{t-\ell}-\phi_\ell^{(x)})\right)$$
which is equivalent to
$$p_2(y_t|x_t,y_{t-s}^{t-1}) = \frac {1}{b(x_t,y_{t-s}^{t-1})} \left|\exp\left([\gamma_0^{(x_t)} +
\sum_{\ell=1}^s\gamma_\ell^{(x_t)} e^{iy_{t-\ell}}]e^{-iy_t}\right)\right|$$
$b(x_t,y_{t-s}^{t-1})$ is a normalisation constant.
Both the real and the complex formulation are implemented. In practice, the complex version is used if the initial $\kappa$ is complex.
References
Ailliot P., Bessac J., Monbet V., Pene F., (2014) Non-homogeneous hidden Markov-switching models for wind time series. JSPI.