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
mu
intercepts
kappa
von Mises AR coefficients
prior
prior probabilities
transmat
transition 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.