init.theta.MSAR.VM: Initialisation function for von Mises MSAR model fitting

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

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

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

…

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

Details

The model with non homogeneous transitions is labeled "NH" and it is written $$P(X_t|X_{t-1}=x_{t-1}) = q(z_t,\theta_{z_t})$$ with $X_t$ the hidden process and $q$ von Mises link function such that $$p_1(x_t|x_{t-1},z_{t}) =\frac{ q_{x_{t-1},x_t}\left|\exp \left(\tilde\lambda_{x_{t-1},x_t} e^{-iz_{t}} \right)\right|} {\sum_{x'=1}^M q_{x_{t-1},x'}\left|\exp \left(\tilde\lambda_{x_{t-1},x'} e^{-iz_{t}} \right)\right|}, $$ with $\tilde\lambda_{x,x'}$ a complex parameter (by taking $\tilde\lambda_{x,x'}=\lambda_{x,x'} e^{i\psi_{x,x'}}$).

References