reform_parameters function

Reform any parameter vector into standard form.

reform_parameters takes a parameter vector of any (non-constrained) GMAR, StMAR, or G-StMAR model and returns a list with the parameter vector in the standard form, parameter matrix containing AR coefficients and component variances, mixing weights alphas, and in case of StMAR or G-StMAR model also degrees of freedom parameters.

reform_parameters(
  p,
  M,
  params,
  model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE
)

Arguments

• p: a positive integer specifying the autoregressive order of the model.

• M: - For GMAR and StMAR models:: a positive integer specifying the number of mixture components.

  • For G-StMAR models:: a size (2x1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

• params: a real valued parameter vector specifying the model.

  • For non-restricted models:: Size $(M(p+3)+M-M1-1x1)$ vector theta $=$(upsilon_{1} $,...,$upsilon_{M} , $\alpha_{1},...,\alpha_{M-1},$nu ) where

      * upsilon_{m} $=(\phi_{m,0},$phi_{m} $,$$\sigma_{m}^2)$
      * phi_{m} $=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M$
      * nu $=(\nu_{M1+1},...,\nu_{M})$
      * $M1$ is the number of GMAR type regimes.
     
     In the GMAR model, $M1=M$ and the parameter nu dropped. In the StMAR model, $M1=0$.
     
     If the model imposes linear constraints on the autoregressive parameters: Replace the vectors phi_{m} with the vectors psi_{m} that satisfy phi_{m} $=$C_{m}psi_{m} (see the argument `constraints`).
    

  • For restricted models:: Size $(3M+M-M1+p-1x1)$ vector theta $=(\phi_{1,0},...,\phi_{M,0},$phi $,$

      $\sigma_{1}^2,...,\sigma_{M}^2,$$\alpha_{1},...,\alpha_{M-1},$nu ), where phi =$(\phi_{1},...,\phi_{p})$
     
     contains the AR coefficients, which are common for all regimes.
     
     If the model imposes linear constraints on the autoregressive parameters: Replace the vector phi with the vector psi that satisfies phi $=$Cpsi (see the argument `constraints`).
    

  Symbol $\phi$ denotes an AR coefficient, $\sigma^2$ a variance, $\alpha$ a mixing weight, and $\nu$ a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term $\phi_{m,0}$ with the regimewise mean $\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})$. In the G-StMAR model, the first M1 components are GMAR type

  and the rest M2 components are StMAR type. Note that in the case M=1 , the mixing weight parameters $\alpha$ are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters $\nu$ have to be larger than $2$.

• model: is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type.

• restricted: a logical argument stating whether the AR coefficients $\phi_{m,1},...,\phi_{m,p}$ are restricted to be the same for all regimes.

Returns

Returns a list with...

• $params: parameter vector in the standard form.
• $pars: corresponding parameter matrix containing AR coefficients and component variances. First row for phi0 or means depending on the parametrization. Column for each component.
• $alphas: numeric vector containing mixing weight parameters for all of the components (also for the last one).
• $dfs: numeric vector containing degrees of freedom parameters for all of components. Returned only if model == "StMAR" or model == "G-StMAR".

@keywords internal

Details

This function does not support models imposing linear constraints. No argument checks in this function.