M step of the EM algorithm for fitting homogeneous Markov switching auto-regressive models, called in fit.MSAR.VM.
Mstep.hh.MSAR.VM(data, theta, FB, constr = 0)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.
model's parameter; object of class MSAR. See also init.theta.MSAR.
Forward-Backward results, obtained by calling Estep.MSAR function
constraints are added to the \(\kappa\) parameter (A preciser)
List containing
intercepts
von Mises AR coefficients
prior probabilities
transition matrix
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.
Ailliot P., Bessac J., Monbet V., Pene F., (2014) Non-homogeneous hidden Markov-switching models for wind time series. JSPI.
fit.MSAR.VM, Estep.MSAR.VM